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Dynamic information model of identification performance Mori, Shuj 1991

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DYNAMIC INFORMATION MODEL OF IDENTIFICATION PERFORMANCE BY SHUJI MORI B.A., Kyoto University, 1983 M.A., Kyoto University, 1986 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES (Psychology) We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA MAY 1991 © Shuji Mori, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Psychology The University of British Columbia Vancouver, Canada Date May 21st, 1991 DE-6 (2/88) ii Abstract This dissertation examined analysis methods and models of sequential dependencies in absolute identification responses. It has been reported that observers' absolute identification responses are strongly affected by previous stimuli and responses, although there is no agreed-upon method of analysis of these sequential dependencies. In this thesis, I used for this purpose multivariate information analysis (Garner, 1962; Garner & McGill, 1956; McGill, 1954), which is an extension of one-input one-output contingent uncertainty (information transmission) to the multivariate case. Multivariate information analysis is preferred to other methods because statistically it is a nonmetric analysis of categorical data, such as those from an absolute identification experiment (Krippendorf, 1986). However, there are some difficulties in its application to empirical data. For example, it is known that information measures are likely to be inflated (or overestimated) when there are a small number of observations per stimulus relative to a large number of variables involved in the calculation (e.g., Houtsma, 1983). Since no previous research had dealt with the inflation problem of multivariate information measures, I ran extensive computer simulations of absolute identification and calculated the multivariate information measures as a function of the number of observations and the number of variables used in the calculation. As expected, the multivariate information measures were inflated for a small number of observations, and they reached their theoretical and/or asymptotic values as larger numbers of observations were used to calculate them. To solve the inflation problem, I used the results of the computer simulations and a method of pooling individual data to estimate the amount of inflation of the information measures and correct them accordingly. Previous studies had suggested that there are three important factors affecting sequential dependencies in absolute identification responses: the amount of stimulus information available to the observers (measured by the amount of information transmission), the number of stimulus/response categories, and giving observers trial-by-trial feedback. To investigate these three factors systematically, I conducted seven absolute identification experiments and analyzed the resulting data by multivariate information analysis with the correction method mentioned above. The results confirmed previous results as follows: (1) The amount of sequential dependencies was inversely related to the amount of information transmission 7 (McGill, 1957; Mori, 1989; Ward & Lockhead, 1971). i i i (2) The amount of sequential dependencies increased with an increasing number of stimulus/response categories (Garner, 1953). (3) The dependency on the previous stimulus was larger when feedback was given than when it was not, and the dependency on the previous response was smaller when feedback was given (Ward & Lockhead, 1971). The results (1) and (2) can be interpreted as an increase of the amount of sequential dependencies with the increasing complexity of making judgments in the task (Garner, 1953). Since the present results were obtained across stimulus modalities (e.g., sound frequency, brightness, visual position), they support the idea that sequential dependencies in absolute identification responses arise mostly from the observer's response processes in the absolute identification task (e.g., Garner, 1953; Ward & Lockhead, 1971). Finally, two general models of absolute identification (Braida & Durlach, 1988; Treisman, 1985) were examined to interpret the pattern of sequential dependencies and other results obtained in this thesis. While some aspects of Braida & Durlach's (1988) model were disconfirmed by the present results (although the model does not make explicit predictions about the type of sequential dependencies obtained in the present study), the present results fit quite well with Treisman's (1985) model, with a few exceptions. iv Table of Contents Abstract • • . . . . . . . • ii Table of Contents • • • • • • • • iv List of Tables . . . . . . . . . . vii List of Figures . . . . . . . . . . viii Acknowledgement . . . . . . . . . . x I. Introduction . . . . . . . . . . j II. Information-theoretic approach in sensation and perception research 3 II. 1. Information-theoretic approach . . . . . . . 4 11.2. Statistical properties of information measures . . . . . . n 11.3. Information concepts and psychological research 15 11.3.1. Effects of redundancy on human behavior 16 11.3.2. Information transmission in an absolute identification experiment • • 18 11.3.3. Channel capacity as a limit on a perceptual system 21 11.3.4. Simple sequential dependencies • • • • - 3 1 11.4. Multivariate information analysis . . . . . . . 33 II. 4.1. A partitioning approach to multivariate information analysis 33 H.4.2. Absolute identification of multidimensional stimuli • • • • . 36 n.4.3. Complex sequential dependencies: Predictability as a function of predictor variables 38 IU. Development of multivariate information analysis • • • • • - 4 4 in.l. Experiment 1 • • • • • . 4 4 III. l . l . Experimental method • - 4 5 ffl.1.2. Results • • • • • - 4 6 III.1.3. Discussion • • . • • • . 4 3 III.2. Negative interaction . . . . . . . . 49 m.3. Experiment 2 . . . . . . . . . 51 III.3.1. Experimental method . . . . . . - 5 2 111.3.2. Results . . . . . . 111.3.3. Discussion • 111.4. Correction methods of inflated information measures : Review 111.4.1. Estimation from a large number of observations 111.4.2. Miller's (1954) correction equation • HI.4.3. Computer simulation 111.5. Computer simulation of absolute identification. 111.5.1. Regression-equation method 111.5.2. Method of matrix manipulation 111.6. Evaluation of correction methods 111.7. Discussion • IV. Dynamic aspects of absolute identification performance I V . l . Experiment 3. IV.1.1. Experimental method IV.1.2. Results • IV.1.3. Discussion IV.2. Experiment 4. ' IV.2.1. Experimental method IV.2.2. Results • IV.2.3. Discussion IV.3. Experiment 5. IV.3.1. Experimental method IV.3.2. Results • IV.3.3. Discussion IV.4. Experiment 6. IV.4.1. Experimental method IV.4.2. Results • IV.4.3. Discussion IVJ5. Experiment 7. V 52 53 56 56 59 62 63 63 64 102 106 107 109 111 112 113 113 114 115 116 116 117 118 120 120 121 122 123 124 vi IV.5.1. Experimental method • • • • • • • 125 IV.5.2. Results . . . . . . . . . 127 IV.5.3. Discussion . . . . . . . . 131 IV. Discussion of Experiments 1 to 7 • • • • • • 132 IV.6.1. The effects of U(Rn:Sn) and the number of stimulus/response categories • 132 IV.6.2. The effects of trial-by-trial feedback • • • • • 139 IV. 6.3. The effects of response categories and that of stimuli used • • • 143 IV. Notes . . . . . . . . . . 145 V.General discussion • • • • • • • 146 V. l . Models of absolute identification performance . . . . . 145 V. l . l . Braida & Durlach's (1988) model of intensity resolution • . . 148 V.1.2. Treisman's (1985) model of criterion setting . . . . 150 V.2. The amount of sequential dependencies and the degree of sequential dependencies • 154 V.3. Conclusions and future directions . . . . . . . 155 References . . . . . . . . . . . 160 Appendices . . . . . . . . . . . 165 vii List of Tables Table 1. Mean and standard deviation (S.D.) of U(R n:S n. 1,R n. 1 |S n) in Experiment 1. • 48 Table 2. The number of cells in confusion matrices for the calculation of information measures in Equations 17 and 20. • • • • • • 52 1 Table 3. The degrees of freedom associated with the information measures in Equations 17 and 20. • • • • • • • • 60 Table 4. Degrees of freedom associated with the information measures of Equations 17 and 20 (df) and five times the number of cells in a confusion matrix for the calculation of information measures (5m). • • • • 61 Table 5. Examples of stimulus-response matrix. 66 Table 6. Examples of stimulus-prestimulus matrix. • • • • 68 Table 7. Corrected estimates of information measures in Experiments 1 and 2. • 105 Table 8. Results of Experiment 3. • • • • 112 Table 9. Results of Experiment 4. • • • • • 115 Table 10. Results of Experiment 5. • • • 119 Table 11. Results of Experiment 6. • • • 123 Table 12. Three sets of stimuli used in Experiment 7. • • • 126 Table 13. Correlation coefficients of information measures of sequential dependencies with factors affecting them. (1) Experiments 1 to 4 (n = 16). 138 Table 14. Correlation coefficients of information measures of sequential dependencies with factors affecting them. (2) Experiments 5 to 6 (n = 10). • • • 142 viii List of Figures Figure 1. A two-way matrix (adapted from Attneave, 1959). • Figure 2. Examples of two way matrices • • Figure 3. A schematic diagram of Equation 1. Figure 4. An absolute identification experiment. Figure 5. Sequential dependencies in an absolute identification experiment. Figure 6. The results of Experiment 1. Figure 7. U(R n:R n. 11 S .A.,) Figure 8. The results of Experiment 2. (1). • Figure 9. The results of Experiment 2. (2). • Figure 10. The information measures of pooled data. Figure 11. The results of simulations with S n X R n matrix only (1). Figure 12. The results of simulations with S n X R n matrix only (2). Figure 13. The results of simulations with S n X R n matrix only (3). Figure 14. The results of simulations with S j^ X R n matrix only (1). Figure 15. The results of simulations with Sn.j x R n matrix only (2). Figure 16. The results of simulations with S,,., X R n matrix only (3). Figure 17. The results of simulations with Kn.l X R n matrix only (1). • Figure 18. The results of simulations with Rn.j x R n matrix only (2). • Figure 19. The results of simulations with X R n matrix only (3). • Figure 20. The results of simulations with S n x R n and S^, X R n matrices (1). Figure 21. The results of simulations with S„ X R n and S j^ X R n matrices (2). Figure 22. The results of simulations with S n X R n and Sn., X R n matrices (3). Figure 23. Simulations with three matrices (1). U(R n:S n) Figure 24. Simulations with three matrices (2). U(R n:SD. 11SJ • Figure 25. Simulations with three matrices (3). U(R n :R n . 1 |S n ) • Figure 26. Simulations with three matrices (4). U(R n:R n. 11 S^S^) Figure 27. Simulations with three matrices (5). U(R n:S n,S n. 1,R n. 1) 5 7 9 19 40 47 50 54 55 58 70 71 72 75 76 77 79 80 81 84 85 86 89 90 91 92 93 ix Figure 28. Simulation and empirical results (1). 4 categories • 96 Figure 29. Simulation and empirical results (2). 6 categories • 97 Figure 30. Simulation and empirical results (3). 10 categories • 98 Figure 31. Simulation and empirical results (4). 16 categories • 99 Figure 32. Corrected information measures of Experiment 1. • 108 Figure 33. Corrected information measures of Experiment 2. • 110 Figure 34. The results of Experiment 7 (1). U ( R J . . . . . 128 Figure 35. The results of Experiment 7 (2). • 129 Figure 36. The results of Experiment 7 (3). • 130 Figure 37. The effect of the number of categories. 134 Figure 38. The effect of ^ R ^ S J . • • • • • 135 Figure 39. Sequential dependencies as a function of U(R n |S n ) (1). 137 Figure 40. Sequential dependencies as a function of U ( R n | S J (2). 140 Figure 41. Thurstonian model and signal detection theory. 147 Figure 42. Illustration of Treisman's (1985) model. . . . . 151 Figure 43. Proportion of sequential dependencies. . . . . 156 X Acknowledgments This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada to Lawrence M . Ward. My sincere gratitude is due to Dr. Lawrence M . Ward for his encouragement and helpful advice throughout the course of the reseach. I also wish to thank Dr. Richard Wright, Odie Geiger, William Turkel and other members of the Psychophysics Laboratory in the Department of Psychology, for helping me running the experiments reported in this thesis. I thank Drs. James Enns, Darrin Lehman, Arthur Burgess, John Petkau for thier useful comments on an early version of this thesis. 1 I. Introduction In an absolute identification experiment, an observer is presented with a stimulus set, a response set, and a mapping between them (the identification function), and is asked to respond to each presentation of an individual stimulus with the appropriate response. In sensation and perception research, absolute identification experiments are of great use in measuring sensitivity to more than 2 stimuli (e.g., Braida & Durlach, 1972, 1988). While the measures of sensitivity (e.g., d', information transmission) only reflect the effect of a presented stimulus on judgment, it has been reported that previous stimuli and responses also affect observers' judgments in absolute identification experiments in a manner similar to that observed in psychophysical scaling tasks (e.g., magnitude estimation). There is a growing recognition among psychophysicists that such sequential dependencies on previous stimuli and responses should be studied thoroughly in order to fully understand the observer's performance in a psychophysical experiment (DeCarlo & Cross, 1990; but Braida & Durlach, 1988). Despite the recognition of importance of studying sequential dependencies, there is no agreed upon method of measuring them. In scaling tasks, sequential dependencies are usually measured by multiple regression analysis, in which the current response is a dependent variable and the presented stimulus and previous stimuli and response are independent variables (e.g., DeCarlo & Cross, 1990; Mori & Ward, 1990). From a psychometric point of view, however, multiple regression analysis is not appropriate for absolute identification data, because absolute identification responses are categorical. In this thesis, I use multivariate information analysis as a method of measuring sequential dependencies in absolute identification experiments. Multivariate information analysis is an extension of information transmission from one input to one output (Shannon, 1948) to include more than one input and one output in the equation (McGill, 1954). Multivariate information analysis is preferred to other analysis methods because statistically it is a nonparametric analysis of categorical data, such as those from absolute identification experiments. Although a few applications of multivariate information analysis were reported in the 1950's (Garner, 1953; McGill, 1957; Pollack, 1953), further applications and theoretical developments are needed to realize the full potential of the method and obtain a more complete picture of sequential dependencies measured by the method. I recently extended the earlier versions of multivariate information analysis and the results of its application to absolute identification data seem promising (Mori, 1989). In what follows, I will discuss various developments of multivariate information analysis and how it can be applied to interpret sequential dependencies and other results from absolute identification experiments. 2 In Chapter II, I will review the literature on the application of information theory to sensation and perception research. Since multivariate information analysis was developed from information theory (Shannon, 1948), it is necessary to present some basic concepts of information theory before I discuss multivariate information analysis. I will also explain the way in which information measures are calculated and their properties. As you shall see, many classical studies of absolute identification (e.g., Miller, 1956) used the concepts and measurement provided by information theory, and the results of those studies still stimulate contemporary research on absolute identification (e.g., Braida & Durlach, 1988; Treisman, 1985). In Chapter III, I will discuss the development" of multivariate information analysis as a technique for the analysis of absolute identification responses. Further applications of multivariate information analysis to absolute identification data revealed two problems in the use of multivariate information analysis; negative interaction and inflation (or overestimation) of information measures from using too few observations. The inflation problem is especially severe for multivariate information analysis because information measures are most inflated when there are a few observations relative to the number of variables used in their calculation. I will discuss three possible methods of correcting inflated information measures; pooling individual data, Miller's (1954) correction equation, and computer simulations. As a result of the discussion and the results of my own computer simulations, I will propose a method of correcting inflated information measures. In Chapter IV, I will examine the factors affecting sequential dependencies in absolute identification responses. Previous studies suggest that there are three major factors: stimulus information available to an observer (measured by information transmission), the number of stimulus/response categories, and giving trial-by-trial feedback. In order to investigate those factors systematically, I conducted absolute identification experiments and analyzed the data by using multivariate information analysis with the correction method proposed in Chapter 2. As you shall see, the results are generally consistent with previous studies and are well explained by the idea that sequential dependencies in absolute identification responses reflect observers' response processes in the task (Garner, 1953; Ward & Lockhead, 1971). In Chapter V, I will discuss the results from the present studies in the context of two recent models of absolute identification (Braida & Durlach, 1988; Treisman, 1985). I will also discuss the differences between multivariate information analysis and multiple correlation/regression analysis as an analysis technique for sequential dependencies. Finally, I will summarize the results obtained in this study and discuss future research suggested by them. 3 II. INFORMATION-THEORETIC APPROACH IN SENSATION A N D PERCEPTION RESEARCH Information theory, in the form formalized by Shannon (1948), made a considerable impact on psychology and stimulated a variety of new research in 1950s - 1960s. Although the widespread application of the concepts of information theory to psychological processes ceased very quickly in the 1970s, some of the concepts remain as a research topic (e.g., channel capacity), and the measurement system provided by information theory is widely used in psychology research (e.g., multivariate information analysis). The value of information theory in psychological research is twofold. One aspect is that information theory is a theory of communication (Shannon, 1948) that provided psychologists with such concepts as uncertainty, information transmission, channel capacity, redundancy, and so on. A l l of them were peculiarly similar to research topics that had been studied in psychology before the coming of information theory. Another aspect, which is often overlooked, is that information theory provided a new statistical technique. The information measures enable psychologists to make a quantitative measurement of those new concepts which would have been impossible to measure without them. Furthermore, the information measures are applicable to data obtained in a wide range of psychological research, and they are preferred in some situations to such statistics as correlation coefficients and analysis of variance (Garner, 1962; Krippendorf, 1986). An important point is that the information concepts and measurement are indispensable to each other, and the information concepts would not have been so useful if they were not associated with their measurement. In what follows, I will review studies using the information concepts and measurement in sensation and perception research. For the sake of space, my review will include only those directly relevant to the topic of this thesis. The majority of studies on other topics, for example, pattern perception and language intelligibility, will not be discussed unless they are relevant to my discussion. Excellent reviews of these topics exist elsewhere (e.g., Garner, 1962). In the first section, I will explain the way in which information measures are calculated and their basic properties. Although the information measures discussed in this section are simple ones, the same principles hold for more complex information measures. I will also emphasize the general applicability of information measures in psychological research. In the second section, I will focus on two important statistical properties of information measures, nonmetricity and additivity. They are what give information measures their great applicability and mathematical simplicity. In the third section, 4 I will discuss information-theoretic concepts and review sensation-perception studies on those concepts. I will emphasize information transmission and channel capacity and relevant studies. Although information-theoretic concepts had a big impact on those studies, it is important to keep in mind that information theory is not a theory of psychology or a theory of sensation and perception. In the fourth section, I will explain multivariate information analysis, which is an extension of the information measures discussed in the first section, and I will review sensation and perception research using the analysis, especially that on sequential dependencies in absolute identification judgments. II.l. Information-theoretic approach In this section, I will briefly explain how information measures are calculated and the basic properties of the measures. For the sake of simplicity, the information measures I will discuss here are simple and their calculation involves only two variables. However, keep in mind that the basic properties remain the same even with the more complex information measures I will discuss later. I start with a situation that is frequently encountered in psychological research. Suppose we have some observations that are classified into categories defined by two variables, A and B, and there are X categories (or levels) in the variable A and Y categories in the variable B. The two variables and their categories can be anything. A common situation is that one of them, say A, is an independent variable and the other, B, is a dependent variable in an experiment; there are X experimental conditions, and subjects in the experiment are classified into Y categories by some criteria. When two variables are involved in a set of observations, we are usually interested in, or curious about, the relation between the two variables. In the above example, we may want to know whether the dependent variable B is affected by, or correlated with, the independent variable. The primitive way to examine the relation between the two variables is to construct a two-way matrix in which one axis corresponds to variable A, and another to variable B. Figure 1 shows an example of a two-way matrix. Each cell in the matrix contains the frequency of a particular combination of two variables in the observations. Simply examining the matrix, we may tell whether the two variables in the matrix are related with each other. Figure 2 presents four examples of such matrices, each of which shows a different degree of the relation between the two variables. Figure 2 (a) seems to show that the two variables are related; the diagonal cells have much higher frequencies than do non-diagonal ones. Compared to Figure 2 (a), the matrix of Figure 2 (b) has a relatively B 1 2 ••• • | • • • • • • X 1 n 2 1 n . 2 n X 2 3 n 1 2 n 2 2 n l 2 n X 2 n , • • • • J • • • Y n 1 Y n X Y n 2 n , n x N Figure 1. A two-way matrix (adapted from Attneave,1959) 6 weak relation between the two variables; the frequencies in diagonal cells are somewhat larger than those in i non-diagonal ones, but the differences are not substantial. The matrices of Figure 2 (c) and (d) show degrees of the relation that are between those shown in (a) and (b); the diagonal cells have relatively higher frequencies than do non-diagonal cells, but the differences are not as large as those observed in (a). A close comparison between the two matrices (c) and (d) seems to indicate that the two variables in the matrix (c) are a bit more related to each other than are those in the matrix (d); diagonal cells in the matrix (c) have higher frequencies than those in the matrix (d), and the frequencies in the matrix (d) are more scattered over non-diagonal cells than those in the matrix (c) are. In scientific research, we usually do not report our "impression" of how the two variables look related with each other. We have to measure the amount of relation between the two variables in an objective manner. The information-theoretic approach provides a measurement of the relation in such a situation. In the information-theoretic approach, the measure of the relation between the two variables is the amount of contingent uncertainty U(A:B) (named by Garner and McGill (1956); see also Garner, 1962), and it is calculated by U(A:B) = U(A) + U(B) - U(A,B) (1), where U(A) = - 2 (n. /N) log 2 (nj/N) (2.1), U(B) = - 2 (n ./N) log 2 (nj/N) (2.2), U(A,B) = - 2 2 (ny/N) log 2 (ny/N) (2.3), and nj is the frequency of category i in the variable A , n^ is the frequency of category j in the variable A , is the frequency of combination of category i in the variable A and category j in the variable B, and N is the sum of the frequencies in all the cells in the matrix (see Figure 1). U(A) and U(B) are the amount of information of the variable A and of the variable B, respectively, and U(A,B) is the amount of joint information of A and B ("information" is sometimes replaced with "uncertainty", e.g., Garner, 1962). The unit of the measures is bits. To explain the validity of contingent uncertainty as the measure of the relation between the two variables, I will change slightly the above equations. I will replace the (relative) frequencies in Equation 2.1-2.3 with probabilities; U(A) = - 2 p, log 2 p. (3.1), B A 47 3 i 50 3 45 2 50 2 45 3 50 3 47 50 50 50 50 50 (a) U(A:B) - 1.553 bits B —I 1 50 40 10 10 33 7 7 33 10 10 40 50 50 50 50 50 50 50 (c) U(A:B) - 1.011 bits B 29 " 17 4 17 21 9 3 4 9 21 ,16 3 16 :31 50 50 50 50 50 50 50 50 (b) U(A:B) - 0.499 bits B —I 1 50 37 12 1 12 30 7 1 1 7 30 12 1 12 37 50 50 50 50 50 50 50 (d) U(A:B) - 0.812 bits Figure 2. Examples of two-way matrices 8 U(B) = - 2 P j log 2 P j (3.2), U(A,B) = - 2 2 P i j log 2 py (3.3), where Pj and pj are the probability of category i in the variable A and of category j in the variable B, respectively, and py is the joint probability of category i in the variable A and category j in the variable B. When there is no relation between two events, the joint probability of the two events is equal to the product of the probabilities associated with them. In the present notation, Py = Pj x Pj- This non-relation is sometimes called independence or orthogonality of the two variables. When the two variables A and B are independent, Equation 3.3 can be rewritten as U(A,B) = - 2 2 (p. X P j ) log 2 (p. x p .) = - 2 2 (p. x pj) (log2 p. + log 2 P j ) = -[22 (p. x P j ) log 2 p. + 2 2 (p. x p j ) log 2 p j] = - 2 p. log 2 pj - 2 pj log 2 P j = U(A) + U(B). For given probabilities P j and pj, U(A,B) will take its maximum value [U(A) + U(B)] when the two variables A and B are independent, and I will use MaxfU(A,B)] to denote U(A,B) for independent A and B. On the other hand, when the variables A and B are totally dependent, categories in the variable A and those in the variable B correspond one to one, and p- = p. = p • (for simplicity, we assume that the number of categories in variable A is equal to that in variable B). For given probabilities p. and pj , the minimum value of U(A,B), denoted as Min[U(A,B)], is thus equal to U(A) and to U(B). For most situations, U(A,B) will take a value between Max[U(A,B)] and Min[U(A,B)]; as the variables A and B become more related with, or dependent on, each other, U(A,B) will decrease. The difference between Max[U(A,B)] and the actual U(A,B) thus will give the measure of the amount of relation between the variables A and B, in terms of the amount of information, U(A:B) = Max[U(A,B)J - U(A,B) = U(A) + U(B) - U(A,B), which is Equation 1. Figure 3 shows a diagram of relations among U(A), U(B), U(A,B), and U(A:B). The two circles represent U(A) and U(B), respectively, and the part in which the two circles overlap represents U(A:B). U(A,B) covers the entire region of the diagram enclosed by the two circles. When there is no relation U ( A ) U ( B ) U ( A , B ) Figure 3. A schematic diagram of Equation 1 between A and B, there will be no overlap of the circles in Figure 3, implying that U(A:B) = 0 bits. When there is overlap of the circle, the area enclosed by them will be maximal, implying Max[U(A,B)] = U(A) + U(B). As the two variables become more related with each other, the overlap between the two circles becomes larger. It is also clear from this diagram that U(A:B) is nondirectional. U(A:B) simply measures the amount of relation between the variable A and B, but not causality from one to the other. This property of U(A:B) is called symmetricity and expressed by 0 U(A:B) = U(B:A). Symmetricity also holds for U(A,B). For demonstration purposes, I calculated the amount of contingent uncertainty for the four matrices in Figure 2 and the values are shown with the matrices in the figure [since every category in both variables A and B has the same frequency (50) in the four matrices, U(A) = U(B) = 2 bits for all of the matrices]. The values seem to agree with our "impression" about the relation between A and B in the four matrices. U(A:B) is much larger (by more than 1 bit) for the matrix (a) than for the matrix (b). U(A:B) from the matrix (c) and from the matrix (d) are between those from (a) and (d), and the matrix (c) has a value of U(A:B) only slightly larger (about 0.2 bits) than that of the matrix (d). The amount of contingent uncertainty is a very general measure of the relation between two variables, and it is applicable to a wide variety of situations. The information measures can be calculated from any set of observations, as long as they are classified by two variables. The most common situation is that of a combination of an independent and a dependent variable. For example, the independent variable A is a set of X tones presented to an observer through a pair of earphones, and the dependent variable B is the observer's responses, which the observer chooses among the Y categories corresponding to the stimuli. Or the two variables can be both dependent variables; for example, one variable is observers' proportion of correct responses to each of X stimuli, and another variable is average confidence rating (1 - Y) of then-responses to the stimuli. The relation of the two dependent variables, measured by the amount of contingent uncertainty, often provides important information. Or, both variables can be independent variables; for example, in an experiment of sound identification, one variable is sound frequency and another is sound intensity. In some situations, we may want to deliberately introduce a correlation between the two independent variables. It may be noted that the above explanation of the information measures and the justification of contingent uncertainty as the measure of relation is rather descriptive. However, it should be understood that I did this because I wanted to emphasize the great applicability of information measures in psychology research, partly because it is how information measures and contingent uncertainty are usually used and interpreted in psychological research. More rigorous and quantitative descriptions of the information measures and rationales for the use of the contingent information, along with the definition of information, can be found in Shannon (1948), Attneave (1959), and Garner (1962). Before closing this section, I will add another basic property of information measures. When I discussed Max[U(A:B)], I always mentioned it with "for given probabilities Pj and P j " . The information measures, U(A), U(B), and U(A,B), will take the "marginal" maximum amount when all the probabilities are equal. In such situations, the amount of those information measures is equal to the logarithm (to the base 2) of the number of categories or combinations of categories. For example, when the probabilities associated with categories in the variable A are all equal, Pi = p 2 = — = Pj = — = Px = */^» ^ d Equation 3.1 is rewritten as U(A) = - 2 ( l / X ) l o g 2 ( l / X ) = - log 2 (1/X) = log 2 X . This is also true for U(B) and U(A,B). Thus, when all the probabilities associated with categories in a variable or a combination of variables are equal, the amount of information of the variable(s) are maximum and equal to a logarithm of the number of categories. II.2. Statistical properties of information measures In the last section, I mentioned that information measures are applicable to a wide range of situations. The general applicability of information measures lies in their statistical properties, namely, nonmetricity and additivity. I will discuss the two properties separately in this section. First of all, information measures are nonmetric For information measures to be used, no assumptions are made about the nature bf the frequency or probability distributions underlying the set of observations to which they are applied. Because of the nonmetric property, information measures require data to be only categorical or qualitative (Krippendorf, 1986). In categorical data, the distinctions (categories) among a set 12 of observations are only nominal, and they need not be ordered along any dimensions of degree or magnitude. Thus the amount of information calculated from categorical data will be unaffected when categories within the data are ordered in a different way. Furthermore, orders based on degree or magnitude differences can be transformed into nominal distinctions; in an identification experiment of sound intensity, for example, the intensity of 50 dB will be called category "1", that of 55 dB will be category "2", and so on. Therefore, the information measures are more general in terms of applicability to data than are such metric statistics as the correlation coefficient; metric statistics require a certain type of probability distribution in the data to which they are applied, and the data must be at least ordinal. Since information measures respond to only a set of frequencies or probabilities associated with categories in the data, the nature of the categories does not matter; as you saw in the last section, the information measures can be calculated from data in which categories are of both independent variables, of both dependent variables, or of an independent and a dependent variable. Reviewing the generality of information measures, Garner (1962) concluded that "[an information measure] is, in a sense, a pure number measure which is applicable to any situation." (p. 61). The nonmetric property of information measures also simplifies their calculation and mathematical development. No matter whether an information measure is calculated from a probability distribution in one variable [e.g., U(A)] or in two variables [e.g., U(A,B)], the basic calculation is the same; an information measure is the sum of logarithms of probabilities, weighted each of them by its own probability. In terms of mathematical development, this property helps in extending bivariate information measures to multivariate ones, without making any extra assumptions of the nature of multivariate probability distributions. Another important statistical property of information measures is their additivity. You have seen in Equation 1 that the amount of contingent uncertainty is the sum (and difference) of the other information measures. In general, any information measure is expressed by the sum (or difference) of other information measures. For example, reordering Equation 1, U(A) is expressed as U(A) = U(A:B) + U(A,B) - U(B) (4). This additivity of information measures arises from their nature as logarithmic functions of probabilities. Remember from the discussion about Max[U(A:B)] that the independence of two probabilities (p^ = pL x p ;) implies the sum of two information measures corresponding to the two probabilities [U(A:B) = U(A) + U(B)]. Needless to say, the logarithm of the product of entities is equal to the sum of the logarithm of each 13 entity: log (X X Y) = log X + log Y. When a given probability distribution involves more than two variables, the interaction (or relation) among the variables is represented by the product of probabilities associated with them, which is, in turn, translated into the addition of information measures corresponding to the probabilities. In addition to their nonmetric property, the additivity property of information measures simplifies their calculation and mathematical development. In the right-hand side of Equation 4, for example, we have a difference between two information measures, [U(A,B) - U(B)]. In the diagram of Figure 3, the difference corresponds to the region in the left circle [U(A)] that is not overlapped by the right circle [U(B)], and the region represents U(A |B) , the amount of information of the variable A with the amount of relation between A and B excluded. Of course, it is possible to calculate U(A | B) from a probability or frequency distribution in the data: -U(A |B) = - 2 P . 2 (py/pj) l 0 g 2 (py/pj). However, when you have already calculated U(B) and U(A,B), you can obtain U(A | B) by simply subtracting U(B) from U(A,B); it is not necessary to count the frequencies in relevant cells and use them in the above equation. This additivity is especially helpful in multivariate information analysis. When there are several variables under study, the number of information measures becomes very large, and their calculation is very complicated. Because of the additivity of the information measures, we are able to derive and calculate them y by simply adding and subtracting the other measures. The nonmetric and additive properties are what gives information measures distinctive characteristics from other statistics. Specifically, the amount of contingent uncertainty has been found to be very useful as a measure of relation between two variables. The contingent uncertainty is often likened to a correlation coefficient and interpreted in the same way as is a correlation coefficient. However the contingent uncertainty and the correlation coefficient are different in two ways (Attneave, 1959; Garner, 1962). First, as I have repeatedly emphasized, the contingent uncertainty is a nonmetric measure, and the correlation coefficient is a metric one. Thus the contingent uncertainty is applicable to a broader range of data than the correlation coefficient is. The second aspect, which was pointed out by Garner (1962), is that the contingent uncertainty measures the "amount" of relation between two variables, whereas the correlation coefficient is a measure of the "degree" of relation. To explain the difference between the "amount" and the "degree" of relation, I will rewrite Equation 4 as U(A) = U(A:B) + U(A|B) :(5). It follows from Equation 5 that the amount of information of a variable A is partitioned into two components, U(A:B) corresponding to the amount in U(A) that is explained, or affected, by the variable B, TJ(A | B) the amount in U(A) that is not explained by the variable B. U(A:B) will take a value between 0 (bits) and that equal to U(A). On the other hand, the correlation coefficient can be regarded as a measure of the ratio of variance of the variable A that is explained by the variable B to the total variance of the variable A . The correlation coefficient always takes a value between 0 and 1, no matter how large the variance of the variable A is. An important point is that the amount of contingent uncertainty is not affected by the amount of U(A | B), whereas the correlation coefficient is changed by the size of error variance, because the change in error variance changes the size of total variance. It thus can be said that the contingent uncertainty is a pure measure of the amount of relation of two variables in data, without respect to how much information is not explained by the relation (Garner, 1962). It is also possible to compare the information measures with other statistics. Garner and McGill (1956) showed a close resemblance between information measures, in a form similar to Equation 5, and the analysis of variance. As the amount of information of the variable A is partitioned into two parts in Equation 5, the analysis of variance partitions the variance of a dependent variable into components of which one is predicted by an independent variable and one is not predicted by the independent variable. By analogy with the analysis of variance, the amount of contingent uncertainty measures the main effect of an independent variable (Garner, 1962). The difference between them is, again, that the analysis of variance requires at least its dependent variable to be metric. Because of their nonmetric property, information measures are also related to X 2 values (Krippendorf, 1986; Miller, 1954; Miller and Madow, 1954). Miller and Madow (1954) derived a transformation function from information measures to X 2 values, and the function has been used to correct biases of information measures due to small number of observations in the data (e.g., MacRae, 1971) and to make significance tests on the information measures (e.g., Krippendorf, 1986). However, recent studies of computer simulations (Houtsma, 1983; Mori and Ward, 1991a) showed that the Miller and Madow^s (1954) transformation function is not practical in most psychology research. I will discuss related issues in more 15 detail in Chapter III. Krippendorf (1986) pointed out the two differences between information measures and X2 statistics. First, information measures are additive in ways X2 statistics are not, so that information measures have more flexibility in statistical analyses. Another difference lies in the number of observations required to obtain unbiased estimates of the two measures. When theoretical (predicted) probabilities of categories are small, a minimum of five observations per each category are required to obtain unbiased estimates of X2 statistics. On the other hand, since information measures are calculated by taking the logarithms of the probabilities and weighting each of them by its own probability, the amount of the bias of information measures is generally smaller than that of X2 statistics is, although the information measures are also biased with relatively few observations (Houtsma, 1983; Mori and Ward, 1991a). It thus seems that although both information measures and X2 statistics are nonmetric statistics, the information measures are more flexible and general as statistics than X2-To summarize this section, information measures have two important statistical properties, nonmetricity and additivity. Not only do the two properties give information measures a broad applicability and simplicity in their calculation and mathematical development, but also the two properties make information measures unique and sometimes preferable to other statistics. II.3. Information concepts and psychological research. In this section, I will discuss concepts derived from information theory and review sensation-perception studies on these concepts. I will review studies using the univariate and/or bivariate information measures discussed above (e.g., the amount of contingent uncertainty). Studies that involve more than two variables and multivariate information analysis will be discussed in the next section. Throughout the discussion in both sections, it is important to keep in mind that information theory is not a theory of psychology or a theory of sensation and perception. Shannon (1948) formulated information theory as a theory of communication, which dealt with an ideal communication system, while biological systems, like humans, are not ideal in the sense of communication engineering. Although the information-theoretic concepts are closely related to psychological ones, and it may be worth comparing biological systems with ideal communication systems, information theory does not by itself provide a model of psychological processes or human behavior. 16 II.3.1. Effect of uncertainty on human behavior Uncertainty is a fundamental concept of information theory, because information theory defines the amount of information in a message as the amount of uncertainty reduced by the message (Shannon, 1948). A psychological interpretation of the relation between uncertainty and information is that a message conveys more information as it reduces our uncertainty about matters under consideration (Garner, 1962). The amount of uncertainty is obtained by U = - 2 p x log 2 p x (6), where p x is a probabUity associated with a possible alternative x about matters. It is obvious from the above definition and Equation 6 that the amount of uncertainty is equal to the amount of information (Equations 2.1 and 3.1), and the terms "information" and "uncertainty" are used interchangeably in psychology (e.g., Attneave, 1959). Psychologists were interested in whether uncertainty, defined in the above way, has any relation to psychological processes, and whether the relation, if any, is reflected by behavioral measures. In order to answer the question experimentally, they used a set of stimuli as alternatives in Equation 6 and manipulated the amount of uncertainty of the set of stimuli, or stimulus uncertainty. The amount of stimulus uncertainty is a function of the number of stimuli in a set and/or the presentation probabilities of the stimuli. As I have shown, the amount of uncertainty is maximum when all stimuli are presented equally often, and it is equal to the logarithm of the number of stimuli in the set. The amount of uncertainty is decreased when the presentation probabilities or frequencies of stimuli are not equal to each other. It should be noted, however, that the increase in the number of stimuli in a set had been already known to have some effects on human behavior, and the effect of the number of stimuli might not have to be interpreted as that of stimulus uncertainty. For example, it may take more time to memorize a larger number of materials than to learn a relatively smaller number of materials, or measures of observers' perceptual identification (e.g., percent correct) may be worse with a set of 20 stimuli to be identified than with a set of only 2 stimuli, ceteris paribus. In order to interpret the obtained change in behavioral measures as the effect of stimulus uncertainty, not that of the number of stimuli, psychologists have used as criteria the linearity of the behavioral measures with the stimulus uncertainty. When behavioral measures are linearly related to the stimulus uncertainty, and not to the number of stimuli (the relation between stimulus uncertainty and the 17 number of stimuli is generally logarithmic), it can be said that the amount of stimulus uncertainty has affected the behavioral measures. The general finding is that behavioral measures of tasks were linearly related to the amount of stimulus uncertainty rather than to the number of stimuli used (Garner, 1962). In experiments of auditory identification of speech words presented in noise, the observers' accuracy of correct identification of words decreased linearly as the amount of stimulus uncertainty increased (Pollack, 1959). More demonstrative is Hyman's (1953) work on choice reaction time. It had been known before Hyman (1953) that the time to identify one of several possible stimuli, called choice reaction time, was linearly related to the number of alternatives. Using choice reaction time to the location of a light in the display, Hyman (1953) conducted a series of experiments in which he manipulated the amount of stimulus information by changing the number of light locations as well as the relative presentation frequencies of light locations. Hyman (1953) found that choice reaction time was linearly related to the stimulus information, no matter whether the stimulus uncertainty was manipulated by the number of alternatives or the relative presentation frequencies. Therefore, behavioral measures of psychological processes seem to reflect the amount of stimulus uncertainty in the tasks. However, it is important to note that the linear relation between the behavioral measures and the amount of stimulus uncertainty was only a rough approximation. Garner (1962) noted in Hyman's (1953) study that the linear relation between the choice reaction time and the stimulus uncertainty did not hold for individual stimuli. There were also factors other than stimulus uncertainty that affected the behavioral measures. In perceptual identification tasks, for example, Pollack (1959) found that the accuracy of correct identification was more affected by the number of response categories the subjects were allowed to use than by the stimulus uncertainty. After reviewing those and other results, Garner (1962) concluded that "... human performance is quite successfully predicted in terms of stimulus [information], but only for average performance with respect to average stimulus [information]. When attempts are made to use [information] measures to predict performance to an individual stimulus within a set, the expected relations are not upheld « to any great extent." (p. 43) Some of the problems (e.g., the effect of the number of response categories) will be also discussed in Section 3.3. 18 II.3.2. Information transmission in an absolute identification experiment Information theory is a theory of communication, and one of the most significant contributions of Shannon's (1948) work is that he modelled a communication system and devised a way to evaluate the flow of information within the system. In a communication system, a sender sends a message (input) through a communication channel, and a receiver receives the message (output) at the other end of the communication channel. When a communication channel is perfect, input and output correspond one to one (they need not be the same). As a communication channel breaks down, the input-output correspondence becomes worse. The input-output correspondence of a communication channel is quantitatively evaluated as the amount of contingent uncertainty. Using the notations in the first section of this chapter, the variable A is, say, input and there are X alternatives used as input, and the variable B is output and there are Y possible alternatives in the output. The amount of input uncertainty, output uncertainty, and joint uncertainty are calculated by Equations 2.1 - 2.3 (or 3.1 - 3.3), respectively, and the amount of contingent uncertainty (Equation 1) measures the input-output correspondence of the communication channel. In this usage, the contingent uncertainty is called information transmission from input to output, or simply information transmission. In psychology, the concept of information transmission has been applied to the human perceptual system. From an information-theoretic point of view, the human perceptual processing system can be regarded as a communication system, in which input is a stimulus from an external world, output is a response made by the system to the stimulus, and information transmission from stimulus to response measures the correspondence between them. To use the concept of information transmission experimentally in human perception studies, an absolute identification experiment is a frequent choice. Figure 4 illustrates the general procedure of absolute identification experiments. In an absolute identification experiment, an observer is presented with a set of stimuli (disks of different size in Figure 4), a set of response categories (usually numbers ranging from 1 to the largest category used in the experiment), and a mapping between them (the identification function) at the beginning of the experiment. The number of response alternatives permitted to an observer is usually the same as that of the stimuli used (but cf. Eriksen and Hake, 1955b). Then the stimuli are randomly presented one by one, and the observer's task is to respond to each presentation of an individual stimulus with the appropriate response. The probabilities (frequencies) of stimulus presentation are often the same for all stimuli in a set (but cf. Chase, Bugnacki, Braida, and Durlach, 1983; Cuddy, 1968). In some experiments, observers are provided feedback as to the identity of the presented stimulus after each response. S T I M U L U S S E T • S T I M U L U S • • R E S P O N S E S E T . r 1 1 '2 ' "31 • • • • '10 ' T R I A L 1 2 3 . 1 0 0 1 0 1 R E S P O N S E '5 ' ' 1 ' '9 ' - . . «2' '3* ( F E E D B A C K ) '4' " T '10' • • • '3 1 '4' Figure 4. An absolute identification experiment 20 In the example of Figure 4, one of the stimuli in the set is randomly chosen and presented on trial 1, and the observer's response to the presented stimulus is '5'. Then, feedback as to the correct identity of the stimulus ('4') is given to the observer. After a certain inter-trial interval, trial 2 starts. In order to obtain a stable, reliable measure of observers' performance in the experiment, they are usually given extensive practice before each experiment, and each experiment consists of from several hundred to several thousand trials and many presentations to each stimulus. The set of observations obtained in an absolute identification experiment is summarized in a two-way matrix like Figure 1, from which we can calculate the amount of stimulus uncertainty, response uncertainty, joint uncertainty, information transmission, and so on. The amount of information transmission has been used as a useful index of subjects' performance for a given experiment or condition. For example, the amount of information transmission has been calculated as an index of discriminability among stimuli to be judged (Mori, 1989; Ward and Lockhead, 1971), and the amount of information transmission decreased when the discriminability among stimuli was decreased by shortening stimulus duration, lowering visibility, and so on. In an attempt to manipulate stimulus discriminability in scaling tasks such as magnitude estimation and cross-modality matching, Ward (1979) and Mori and Ward (1990) calculated the amount of information transmission from absolute identification experiments in which the perceptual difficulty of presented stimuli was manipulated in the same way as was done in the experiments of magnitude estimation and cross-modality matching, and they used the amount of information transmission as an index of perceptual difficulty. The amount of information transmission has also been obtained in absolute identification experiments in which a property of stimuli was systematically changed, like detection and discrimination experiments. Luce, Green, and Weber (1976) conducted absolute identification of sound intensity experiments in which the range of stimulus intensity levels was varied, and they found that the amount of information transmission increased with the intensity range and approached an upper limit (cf. Braida and Durlach, 1972). Ward (1991) manipulated stimulus duration in absolute identification experiments of sound and light intensity, and found that, unlike results usually found in detection and discrimination experiments (measured as d' or proportion of correct responses), stimulus duration had only a small effect on the amount of information transmission. Mori and Ward (1991b) investigated the effect of white-noise masking in absolute identification experiments of sound intensity and frequency, by systematically changing the temporal relation between white-noise masker and a tone to be identified. They found a different pattern of masking effects on sound frequency 21 and on sound intensity: for sound frequency the amount of information transmission showed a white noise masking effect only when the tone was simultaneously presented with white noise, while for sound intensity the amount of information transmission showed the same amount of white-noise masking effect for all the temporal relations used. Mori and Ward (1991b) interpreted the results by reference to the difference in neural coding between sound frequency and intensity and the memory operation involved in absolute identification tasks (Durlach and Braida, 1969). Although the above studies used the amount of information transmission mainly as an index of perceptual processing observed in a given experiment, the vast majority of studies have used information transmission to test another information-theoretic concept, channel capacity. In the next part, I will discuss the concept and psychological studies of channel capacity. II.3.3. Channel capacity as a limit on a perceptual system Although the amount of information transmission provides an index of sensory or perceptual processing for a given experiment or condition, it does not indicate the maximum ability of a sensory or perceptual system to transmit stimulus information. Suppose, for example, we have conducted an absolute identification experiment of 10 sound frequencies and obtained 1 bit as the amount of information transmission. While this 1 bit is used as an index of perceptual processing in this experiment, it does not tell us whether 1 bit is a maximum value, or a minimum value, obtainable for sound frequencies. One way to investigate the ability of a perceptual system is to change some properties of stimulus (e.g., duration) systematically in experiments and see the relation between the change in the properties and behavioral measures obtained from the experiments (e.g., the amount of information transmission). In the last section, I reviewed some of the studies that measured the amount of information transmission as a function of stimulus properties (e.g., Mori and Ward, 1991b). Information theory suggests another approach to investigate an ability of a system to transmit information. In the context of information theory, the amount of information transmission is maximum when input and output correspond one to one, and the maximum amount of information transmission is equal to the amount of stimulus information. To make this point clear, I will change the notation in Equation 5 U(I) = U(LO) + U(I |0) (7), [ 22 . . . / where U(I) is the amount of input information, U(I:0) is the amount of information transmission, and U(I | O) is the amount of input information that is not transmitted to output. When input and output correspond one to one, U(I |0) is zero and U(I:0) = U(I). If all alternatives in the input are equiprobable, we can increase the amount of input uncertainty by increasing the number of alternatives in the input. As the amount of input uncertainty increases, the amount of information transmission also increases and is equal to that of input uncertainty, as long as the amount of input uncertainty does not exceed the capacity of the communication channel. When the amount of input uncertainty exceeds the capacity of the channel to transmit it, the amount of information transmission does not increase further, and it stays at the amount corresponding to the channel capacity although the amount of input uncertainty increases. The channel capacity thus indicates the maximum ability of the communication channel to transmit information, and it is measured by the level of input uncertainty at which the amount of information transmission stops increasing with increases in the amount of input uncertainty. The concept of channel capacity was translated by psychologists as the maximum capacity of a perceptual system to transmit stimulus information, and it has been tested in absolute identification experiments in which the amount of stimulus uncertainty is controlled by the number of stimuli used, and the obtained amount of information transmission is plotted against the amount of stimulus uncertainty. Before I review the channel capacities obtained in absolute identification experiments and explanations for them, one other thing is worth commenting. I mentioned earlier that the amount of contingent uncertainty is the "amount" of relation, not the "degree" of relation. This property of contingent uncertainty (or information transmission in the discussion here) is particularly valuable in absolute identification experiments in which the number of stimuli used is manipulated (Garner, 1962). As Garner (1962) pointed out, it is often difficult to compare an observer's performance in absolute identification experiments in which the number of stimuli is different. A traditional measure was the proportion of errors the observer makes. It is known in studies of absolute identification that the proportion of errors increases with the number of stimuli used, and the simple interpretation of this is that the observer's performance is worse as the number of stimuli used is larger (although a more sophisticated analysis of error proportion is available, e.g., McGill , 1957). However, it is sometimes intuitively obvious that the observer's performance is better with the large number of stimuli. Using Garner's (1962) example, an observer's performance seems better with 1 percent of errors in a 10-category absolute identification experiment than with 0 percent of errors in a 5-category experiment. When 23 the amount of information transmission is used to measure the observer's performance, it is usually larger with the large number of stimuli than with the relatively small number of stimuli (note that the subject's performance does not have to be perfect), which indicates that the observer's performance is better with the large number of stimuli. The difference between the two measures will become clear when we examine Equation 7 carefully. In equation 7, the proportion of errors corresponds to the ratio of U(I|0) to U(I), whereas the amount of information transmission is U(I:0). When U(I) is kept constant, an increase in U(I:0) parallels a decrease in U(I | O), causing the error proportion to be smaller. However, as U(I) increases, an increase in U(I:0) does not necessary result in a decrease in U(I|0). It is even possible for both U(I:0) and U(I|0) to increase when U(I) increases. In such cases, U(I | O), or the proportion of errors, does not reflect the observer's performance in absolute identification experiments in a straightforward manner. The same argument holds for the use of correlation coefficients in absolute identification experiments, because the correlation coefficient also measures the "degree" of relation, i.e., the ratio of two variances corresponding to U(I:0) and U(I). A correlation coefficient decreases as the number of errors increases, even if the variance corresponding to the amount of relation [U(I:0)] is kept constant. Because the number of errors usually increases with increasing number of stimuli used, we need the amount, not the degree, of correspondence between stimuli and responses, to compare the observer's performance with different numbers of stimuli. The amount of information transmission is ideal for this purpose. This aspect of information transmission, first pointed out by Garner (1962), has often been overlooked in studies of absolute identification. The channel capacity is one of the well-known concepts of information theory in psychology, mainly because it was found that the channel capacity of a human sensory system was surprisingly small, implying that the ability of a human sensory system to transmit stimulus information was very limited compared to the ability of mechanical communication systems. For example, Pollack (1952) obtained 2.3 bits of channel capacity for sound frequencies ranging from 100 to 8000 Hz. Since one bit corresponds to 2 equally likely alternatives (log2 2 = 1 bits or 21 = 2 alternative), 2.3 bits is equivalent to 5 alternatives (2" - 5), meaning that the ability of the auditory system to perfectly transmit information about sound frequency will break down when more than 5 alternatives are used. The values obtained for other sensory modalities are not very different from 2.3 bits for sound frequency. Garner (1953) obtained about 2.1 bits of a channel capacity for sound intensity. In a series of experiments, Eriksen and Hake (1955a,b) found 2.2 - 2.8 bits for size, 3.1 bits for hue, and 2.3 bits for brightness (Kintz, Parker, and Boynton, 1969, obtained 2.6 bits for spectral wavelength). Beebe-Center, Rogers, and O'Connel (1955) found that the channel capacity was about 1.7 bits in absolute identification experiments of taste of salt concentrations. Hake and Garner (1951) obtained 3.2 bits for absolute identification of points on a line. Reviewing those studies and others, Miller (1956) concluded that the capacities of the sensory systems to transmit information about such unidimensional stimuli were 2.6 bits on average with a range of 0.6 bits, which is roughly equivalent to 7 ± 2 alternatives. Before I discuss models for such small values of the channel capacities, I should point out that those values were obtained in absolute identification of unidimensional stimuli (e.g., sound frequency, brightness) and that they do not reflect our ability to identify hundreds of objects in everyday life. Our ability to identify those objects seems to be due, for a large part, to the fact that they are multidimensional, that is, they differ in many properties. The increased ability of an observer to identify the multidimensional stimuli has been confirmed experimentally, which I will discuss in the next section. Turning back to the channel capacities for unidimensional stimuli, there are two types of approaches in accounting for the surprisingly small values. One approach is that they reflect a genuine, absolute limit for human sensory systems. Miller (1956) contended that such a small channel capacity would be built into a human sensory system by learning or the design of nervous systems. This view was elaborated by Norwich (1981). Norwich (1981) claimed that the channel capacity is set by the response characteristics of sensory receptors. From his entropy theory of perception (Norwich, 1977), Norwich (1981) derived an equation of information transmission that is a function of the action potential "firing" rate of sensory receptors. Using physiological measures of firing rates of sensory neurons (e.g., auditory fibers), Norwich estimated the amount of information transmission from his equation and obtained values comparable to the channel capacities obtained in absolute identification experiments. Another approach emphasizes the fact that channel capacities are much smaller than corresponding measures of an observer's ability to discriminate two stimuli. In Pollack's (1952) study, for example, the channel capacity for sound frequencies ranging from 100 to 8000 Hz was 23 bits, which is equivalent to 5 categories. In other words, the observers could perfectly identify sound frequencies only when any two adjacent frequencies were separated more than 1780 Hz [= (8000-100)/5]. On the other hand, discrimination thresholds for sound frequency are 3 - 30 Hz for the comparable frequency range when observers are asked to discriminate frequencies of two tones successively presented in 2-interval-forced-choice (2IFC) experiments (Scharf and Buus, 1986). The discrepancies between discriminability measures from absolute identification experiments and from discrimination experiments have been reported in a number of studies for a variety of modalities (e.g., sound intensity). Although not all of the studies were concerned with channel capacity per se, they provided models to explain the surprisingly small channel capacities obtained in absolute identification experiments. In one of the early studies, Garner and Creelman (1964) argued that absolute identification is not a task of perceptual discrimination, but one of judgmental discrimination. In a task of perceptual discrimination, such as 2IFC experiments, factors that affect the receipt of stimulus information at the sensory receptor are critical. On the other hand, what is critical in absolute identification experiments are such judgmental factors as the number of response categories, the compatibility of the response to the stimulus, and decision making (e.g., an observer's tendency to repeat or avoid the same responses). In absolute identification experiments of hue and size of squares, Garner and Creelman (1964) found that changes in stimulus duration, which was known to affect discriminability measures (e.g., just noticeable difference), had little effect on the amount of information transmission. As mentioned before, Ward (1991) also found a very small effect of stimulus duration on the amount of information transmission for sound and light intensity. It thus seems that an observers' performance in absolute identification tasks may be affected by factors different from those affecting their performance in discrimination tasks. Although it may be true that absolute identification is not a task of judgmental discrimination, the findings of Garner and Creelman (1964) and Ward (1991) do not demonstrate it unequivocally. The reason is that the differences between adjacent stimuli in the stimulus sets used in the two studies were far larger than those used in discrimination experiments that measured the effect of the stimulus durations on discrimination thresholds. A generally accepted explanation for the effect of stimulus duration on discrimination thresholds is that a sensory mechanism integrates stimulus energy over time, and that discrimination is based on the integrated stimulus energy (Coren and Ward, 1990). In a very simple model, the integrated stimulus energy (ISE) is expressed by the product of stimulus energy at moment (I) and time (?) ISE = I X T, and an observer can discriminate two stimuli when the difference between ISIs for two stimuli is larger than some criterion. It is obvious that when the difference between stimulus energies of two stimuli is already large enough, the stimulus duration (T) will have little effect on the discrimination threshold. This seems to be the case for the above studies of absolute identification experiments. For example, Ward (1991), in his experiments of absolute identification of sound intensity, used 3 - 6 dB differences between the adjacent stimuli in the range of 40 - 100 dB. On the other hand, Henning (1970) measured the just noticeable difference for sound intensity as a function of stimulus duration in 2IFC discrimination experiments, in which the intensity level of one sound was fixed at 85 dB and the level of the other was 0.3 to 2 dB lower. It should be also noted that discrimination thresholds are smaller for low intensity levels (e.g., 40 dB) than for high intensity levels (e.g., 85 dB). Therefore, in order to conclude that the effect of stimulus duration is different in absolute identification and in discrimination experiments, we require experiments that control the differences among stimuli as well as the stimulus duration. Other researchers have been concerned with procedures of absolute identification experiments that limit an observer's performance in an absolute identification task. Siegel (1972) argued that memory demands in an absolute identification experiment are responsible for the low channel capacity (also see Siegel and Siegel, 1972). In an absolute identification experiment, the observer must remember stimuli (or their categories) to perform the task. According to Siegel (1972), memory demands are minimum when the same stimulus is repeated on two successive trials, and memory demands become greater as repetitions of the same stimulus are separated by a larger number of trials in which other stimuli are judged. The greater the memory demands are, the greater is the proportion of incorrect responses. In experiments of absolute identification of sound frequency, Siegel (1972) found that the accuracy of absolute identification was a function of trials and time separating the presentation of the same stimuli. Furthermore, Siegel (1972) calculated the amount of information transmission from subsets of data classified by the number of intervening trials, and found that i when the same stimuli were repeated on two successive trials, the amount of information transmission was almost perfect for all of a set of up to 16 stimuli used, and that the amount of information transmission decreased as the number of intervening trials increased. Siegel (1972) concluded that as the number of stimuli (that is, the amount of stimulus uncertainty) increases, repetitions of the same stimuli are separated by more trials, lowering the amount of information transmission to a level that is usually taken as channel capacity. Pollack (1956) was concerned with the differences between stimulus ranges used in absolute identification experiments and in discrimination experiments. In a usual 2IFC discrimination experiment, the stimulus range is determined by the difference between the pair of stimuli presented throughout the experiment. On the other hand, the stimulus range in an absolute identification experiment is the difference between the largest and smallest stimuli of the set used in the experiment, and it is usually much larger than that of a discrimination experiment. In order to compare the results of absolute identification and discrimination experiments, Pollack (1956) argued that we must use a comparable stimulus range in both experiments. For this purpose, Pollack (1956) devised a roving discrimination experiment (in turn, the usual discrimination experiment was called a fixed discrimination experiment). The procedures of the roving discrimination experiment are identical to those of the fixed discrimination experiment (i.e., an observer is asked to discriminate two stimuli), except that in the roving discrimination experiment, several different pairs of stimuli to be discriminated are presented randomly throughout the experiment. In Pollack's (1956) experiments, the pairs were chosen in such a way that they covered the same range of stimuli used in absolute identification experiments. Using sound intensity levels as stimuli, Pollack (1956) found that the amount of information transmission from absolute identification experiments was comparable to that from the roving discrimination experiments with the same stimulus range. Following Pollack (1956) and other studies, Durlach and Braida (1969) proposed a model that attempted to explain the differences between the discriminability obtained from absolute identification and discrimination experiments of sound intensity (also see Braida and Durlach, 1988). Their model, which I will discuss in detail in Chapter V, suggests that the difference in discriminabilities obtained from the two experiments are due to the difference in inter-stimulus (trial) interval as well as the difference in the stimulus range. It is assumed in their model that a listener tries to use a sensory memory, or trace, of a sound intensity in order to perform both types of tasks. In 2IFC discrimination experiments, the inter-stimulus interval is usually short (0.5 to 1 second) enough for the sensory trace of the first sound to be still active when the second sound is presented. In absolute identification experiments, however, the inter-stimulus (trial) interval is 6 - 8 seconds and is outside the range in which the sensory trace of a previously presented sound is active, so that an observer does not use the sensory trace to judge a current stimulus. Durlach and Braida (1969) argued that, in order to compare the discriminabilities measured in absolute identification and discrimination experiments, both the stimulus range and the inter-stimulus interval must be the same in the two experiments. Berliner and Durlach (1973) conducted an absolute identification and a roving discrimination experiment with the same stimulus range and the same inter-stimulus (trial) interval, and they 28 found that the discriminabilities from the roving discrimination experiment were comparable to those from the absolute identification experiment. Mori and Ward (1991b) applied an extended version of their model (Braida and Durlach, 1988) to the results of experiments on absolute identification and fixed and roving discrimination of sound intensity and frequency under white noise masking. With additional assumptions of differences in neural coding of sound intensity and frequency on the effect of masking, Mori and Ward (1991b) found that Braida and Durlach (1988) model fitted the results reasonably well. Although Durlach and Braida's (1969) model uses d' measures (from signal detection theory) of discriminability in absolute identification, Braida and Durlach (1972) showed that the d' measures are linearly related to the amount of information transmission. Mori and Ward (1991b) also calculated both d' measures and the amount of information transmission from the results mentioned above and found that the two measures showed the same patterns. Thus it can be said that, like Siegel (1972), Durlach and Braida's (1969) model suggests that the decaying memory of previous stimuli over long inter-trial interval lowers the channel capacity in an absolute identification experiment. I will return to those memory effects in Chapter V when I discuss models of sequential dependencies of psychophysical judgments. As you have seen, the contemporary studies seem to agree that the nature of absolute identification tasks is responsible for the channel capacity and the lower measures of performance than those obtained in discrimination tasks. There are other factors, however, that affect the observer's performance in absolute identification experiments. One of the factors is the number of response categories an observer is allowed to use. I mentioned earlier that the accuracy of correct identification of speech words under intense noise was more affected by the number of response categories than the number of speech words used (Pollack, 1959). In an absolute identification experiment, the number of response categories is usually equal to the number of stimuli used, and it is thus difficult to separate the effect of the number of response categories from that of the number of stimuli used. To separate the two, Eriksen and Hake (1955a) first trained their observers for an ordinary absolute identification task with the number of stimuli equal to that of response categories. In a subsequent session in which the observers were told that the task was the same as in the training session, the number of stimuli used was in fact different from the number of response categories. There were 9 combinations of the number of stimuli used (6, 11, 21) and the number of response categories (6, 11, 21) used in this session. Eriksen and Hake (1955a) found that the observers had a strong tendency to use all response categories permitted even if the number of stimuli used was fewer than that of response categories, and that the amount of information transmission was more dependent on the number of response categories than on that of stimuli used. When the number of response categories was equal to or greater than the number of stimuli used, the amount of information transmission was the same irrespective of the number of stimuli. I will discuss the effect of the number of response categories in Chapter IV in connection with sequential dependencies in absolute identification experiments. Another factor is the use of perceptual anchors. Perceptual anchors exist for almost all perceptual continua and sets of stimuli, and they are used as standards by which an observer judges an individual stimulus. In most absolute identification experiments, perceptual anchors are assumed to be the extreme stimuli in the set of stimuli used (e.g., sound intensity, Braida and Durlach, 1988). However, the accessibility of perceptual anchors is different from one type of stimuli to another, and experiments in which perceptual anchors are very accessible seem to yield high channel capacities. For example, position along a line, in which the perceptual anchors are both end points of the line and are always visible, was found to have 3.2 bits of channel capacity (Hake and Garner, 1951), about 1 bit higher than, e.g., sound intensity of sound frequency (2.1 and 2.3 bits) where the anchors must be remembered. Since perceptual anchors are used as standards for judgments of a presented stimulus, it is predictable that stimuli close to the perceptual anchors are judged more accurately than the other stimuli (e.g., Braida and Durlach, 1988; Garner, 1962). Some researchers have attempted to introduce perceptual anchors experimentally (Chase, et al., 1983; Cuddy, 1968). In experiments of absolute identification of pitch, Cuddy (1968) introduced a perceptual anchor by presenting one of the tones more frequently than the others, and she found that the identification accuracy for the frequently presented tones was improved, and that information transmission for the entire set also increased. In absolute identification of sound intensity, Chase et al. (1983) used three different sets of stimulus presentation probabilities in which some of the tones in the set were presented more frequently than others. Unlike Cuddy (1968), Chase et al. (1983) failed to find either increased identification accuracy for the more frequent tones or any improvement in performance for the entire stimulus set (measured in d'). Chase et al. (1983) also failed to replicate Cuddy's (1968) results for sound frequency. Learning also affects the observer's performance in absolute identification experiments. It is always easy to judge stimuli that are well-learned and familiar to us. For example, we judge horizontal spatial locations everyday, for example, measuring a length on a ruler. Along with apparent perceptual anchors, this familiarity with horizontal locations may be, at least partly, responsible for a high channel capacity for 30 position along a line (Hake and Garner, 1951). We can also control learning effects in experiments. Giving practice always helps observers improve their judgments in absolute identification experiments. In experiments of absolute identification of pitch, Hartman (1954) trained listeners for two sessions a week for 8 weeks, and found in one experiment that information transmission increased from 1.32 bits to 2.20 bits. Cuddy (1970), in experiments of absolute identification of pitch, used college students who had some musical training and those who had no musical training at all. After six sessions of training in two weeks, both group of listeners had information transmission increased by 0.18 to 0.6 bits, and the musically-trained listeners showed more improvement than those without musical training (also Cuddy, 1968). However, it seems that the most drastic improvement in an observer's performance generally occurs at the very beginning of training, and that once the observer passes this stage, his or her performance does not improve very much more. Finally, it has been reported that when an observer is given feedback about a "true" identity of a presented stimulus, it increases information transmission. Eriksen (1958) first reported that giving observers trial-by-trial feedback about their performance increased information transmission to a small but noticeable degree. Ward and Lockhead (1970, 1971) found that information transmission was increased by 0.3 bits when feedback was given. Braida and Durlach (1972) reported that observers' accuracy of absolute identification responses (in d' measure) was improved 20 % when feedback was given (also Chase et al., 1983). Interesting is Siegel's (1972) finding that the repetition of the same stimulus increased the proportion of correct responses to the stimulus when feedback was given, whereas the repetition slightly lowered the proportion of correct response when feedback was not given. Siegel (1972) reasoned that in the presence of feedback, the observers repeat the same response to a repeated stimulus only when the previous response is correct. When feedback is not given, on the other hand, the observers tend to repeat the same response to a repeated stimulus, no matter whether the previous response is correct or not. A similar but more elaborate account for the effect of giving trial-by-trial feedback was given by Treisman (1985). In this section, I reviewed channel capacities obtained in experiments on absolute identification of unidimensional stimuli, and explanations for their surprisingly small values. While some researchers consider the channel capacity as a limit for a human sensory system (Miller, 1956; Norwich, 1981), the majority of studies agree that it reflects the nature of absolute identification tasks (e.g., memory demands, Siegel, 1972). Channel capacity is also affected by such factors as the number of response categories, perceptual anchors, and learning. Finally, it is important to separate the channel capacity for unidimensional stimuli from our ability to identify everyday objects, which differ along many dimensions. II.3.4. Simple sequential dependencies I pointed out that information measures are applicable to any set of observations, as long as the observations are classified by two variables. This generality of information measures is particularly helpful in analyzing inter-dependencies of successive events in a sequence, called sequential dependencies. Using two successive events as the variables in Equation 1, we can calculate the amount of contingent uncertainty as a quantitative measure of sequential dependencies. In cases where the nature of the probability distributions underlying sequential dependencies is unknown, or in which the events are only measured nominally, the nonmetric property of information measures makes them preferable over metric statistics. Sequential dependencies are ubiquitous in any aspect of human behavior. Human language behavior (reading, listening) is known to use sequential characteristics of language. As you may have known, successive letters in English words occur in a dependent manner; that is, particular sequences of letters occur more frequently than others. Such sequential dependencies in a sequence of input are called sequential redundancy (or simply redundancy) in information theory, and redundancy is measured by the amount of contingent uncertainty between successive inputs (for a detailed discussion and description of redundancy, see Garner, 1962). The implication of redundancy in language is twofold. On the one hand, the existence of redundancy in language implies that our language is somewhat inefficient. Redundancy limits the full use of symbols available in our language (i.e., alphabet in English). In this sense, the term "redundancy" is appropriate; each symbol is "redundant" to others. Using the amount of contingent uncertainty and other information measures (i.e., multivariate information analysis), researchers typically found that English is 50 % to 70 % redundant (Attneave, 1959; Garner, 1962; Shannon, 1948). On the other hand, redundancy in language can be regarded as "a powerful safeguard against errors and misunderstanding" (Attneave, 1959, p. 35) in language communication. It should be appreciated that if our language were completely non-redundant, that is, occurrences of letters and words were not related to those of others, every error in input (e.g., typographical error) and noise in output (e.g., telephone conversation in a noisy room) would result in communication that may be "meaningful" but is different from what is intended. This usefulness of redundancy has been confirmed experimentally in speech perception and visual perception of letters and words. Replacement of deleted letters in an English word is more accurate when the deleted letters are in 32 the middle of the word than when they are in the beginning or the end of the word, because letters in the middle of a word are more redundant to other letters before and after them (Miller and Friedman, 1957). Accuracy of perception of eight-letter sequences was better when the letters were redundant in the same way as are those in English words (Miller, Bruner, and Postman, 1954). Therefore, it can be said that, at the expense of efficiency, redundancy provides our language with resilience as a communication device. Sequential dependencies are also found in a sequence of outputs, or responses. Human motor responses are always dependent on the preceding positions and movement. Learning psychologists have also been interested in dependencies in a sequence of their subjects' responses. Again, the amount of sequential dependencies is quantitatively measured by the amount of contingent uncertainty, using two successive responses as two variables in Equation 1. However, sequential dependencies of responses are not limited to those between two successive responses. It has been found that subjects' current response is affected by not only an immediately preceding response but also by responses further back in a sequence. One way to deal with the sequential dependencies on a response preceding another by more than one is simply to use the current response and the preceding response under consideration as variables in Equation 1. Although this technique is simple to use, it ignores the effects of intervening response(s) between the two responses chosen. In order to include the intervening responses in the analysis of sequential dependencies, Miller and Frick (1949) proposed to classify a set of observations by a combination of categories of successive responses. As a simple example, suppose there are only two response categories," +" and For the analysis of simple sequential dependencies, we will measure the amount of contingent uncertainty by using a previous response ("+" or "-") as one variable, say, the variable A and a current response as the variable B. Next step is to use a combination of two successive previous responses as the variable A . In this example, there are now four categories in the variable A ; " + + ","+-","- + ", and Then we will classify a set of observations by the four categories in the variable A and, as before, two categories in the variable B, and calculate the amount of contingent uncertainty from the classification. Next, we will use three successive previous responses as the variable A , and so on. Since the categories used in the variable A are themselves a sequence of previous * responses, the amount of contingent uncertainty measures the amount of dependency of a current response on the sequence of previous responses, not on only one previous response. Although this technique improves the analysis of sequential dependencies, however, its major weakness is that it does not allow us to measure the relative contribution of individual successive responses to a current response. Measurement of the relative contributions of individual previous events on a current one is necessary in the analysis of sequential dependencies in psychophysical judgments, in which previous events are from different sequences, such as a previous stimulus and a previous judgment. Such measures are provided by multivariate information analysis, which I will discuss in the next section. To summarize, although sequential aspects of human behavior already had been known long before the coming of information theory, the lack of proper statistical technique prevented psychologists from studying them. This led Garner (1962) to say "... investigation of temporal or sequential aspects of behavior has not received the attention it deserves." (p. 284) Then information theory provided information measures, particularly contingent uncertainty, which are applicable to a wide range of data. The application of information measures to sequential dependencies then flourished for a while in studies of language and learning. However, for further, detailed analyses, more sophisticated techniques were required, and multivariate information analysis was, as you will see, one of the required techniques. II.4. Multivariate information analysis In this section, I will explain multivariate information analysis and some studies using the analysis. There are several ways to explain multivariate information analysis (e.g., Krippendorf, 1986), but I will do it by extending contingent uncertainty between two variables [U(A:B)] to situations involving more than two variables. This approach, called a partitioning approach, was first developed by McGil l (1954) and later generalized by Garner (1962). Compared to other approaches, the partitioning approach directly leads to the information measures that have been used frequently in sensation and perception research and will be used in this thesis. For the sake of simplicity, I will leave details of the calculation method and statistical properties to Chapter III and the Appendix. Extensive discussions of multivariate information analysis are found in the above mentioned works and Attneave (1959). II.4.1. A partitioning approach to multivariate information analysis To use a partitioning approach to explain multivariate information analysis, I start with Equation 5 U(A) = U(A:B) + U(A |B) (5). Equation 5 shows that the amount of information of variable A [U(A)] is partitioned into two components, U(A:B) and U(A |B) . U(A:B) is the amount in U(A) that is explained by the variable B, and U(A|B) is the amount in U(A) that is not explained by B. Now I extend Equation 5 to a situation in which the variable B is defined in terms of two variables, say B, and B 2 . Such a situation will be found in an absolute identification experiment in which a set of stimuli is defined in terms of two stimulus properties, such as sound intensity and frequency. Another example is that, as Miller and Frick (1949) did, the variable B is a combination of a current response (B,) and a preceding response (B 2). At any rate, Equation 5 is then rewritten as U(A) = U(A:B 1 ,B 2) + U(A|B 1 ,B 2 ) (8). In Equation 8, U(A:B 1 ,B 2) is called multiple contingent uncertainty and is the amount in U(A) that is explained by the combination of Bl and B 2 . U(A | Bj.Bj) is called multiple conditional uncertainty or simply error uncertainty and is the amount in U(A) that is explained by neither B, nor B 2 . Since there is no causal relation assumed among the variables A , Bj, and B 2 , Equation 8 can be written as U(A) = U(B„B2:A) + U(B„B 2 |A) In analogy with the analysis of variance, U(A:B 1,B 2) gives us the total effect of B, and B 2 on the variable A . However, it does not yield the relative amount in U(A) that is explained by Bj or by B 2 . One way to obtain those amounts is to partition U(A:B„B2) into two components U(A:B 1 ,B 2) = U(A:B,) + U(A:B 2 |B , ) (9.1) = U(A:B 2) + U(A:B 1 |B 2 ) (9.2). In the right-hand side of Equation 9.1, the first term is the amount of contingent uncertainty between A and B„ which is identical to that in Equation 5 except the variable B is replaced by B,. The second term is called partial contingent uncertainty and it measures the amount in U(A) that is explained by B 2 but not by B,. In other words, U(A:B 2 |B 1 ) measures the effect of B 2 on A with the effect of B, excluded. The same notation and interpretation are applied to Equation 9.2 except that B } and B 2 are interchanged. Thus multiple contingent uncertainty [UfAiBuBj)] consists of (simple) contingent uncertainty and partial contingent uncertainty. There is another way to partition multiple contingent uncertainty. Using the analogy with analysis of variance once again, the total effect of two variables is partitioned into the main effects of two variables and their interaction effect. In terms of information measures, U(A:B 1 ,B 2) is partitioned into three components U(A:B„B2) = U(A:B,) + U(A:B 2) + INTCA-.Bj.B,) (9.3), where only INT(A:B 2,B 1) is a new information measure and is the amount in U(A) that is explained by the interaction (not combination) of B, and B 2 . Now we have three ways to rewrite Equation 8; U(A) = U(A:B,) + U(A:B 2 |B,) + U(A|Bi,B 2) (10.1) = U(A:B 2) + U(A:B 11 + U(A|Bi,B 2) (10.2) = U(A:B,) + U(A:B 2) + INT(A:B2,B,)-+ U(A|B 1 ,B 2 ) (10.3). ' It should be noted at this point that all the basic properties of simple information measures hold for the information measures in multivariate information analysis. Al l of the information measures so far discussed can be calculated from the relative frequencies or probabilities in a set of observations, and they do not require any assumptions about the underlying probability distributions. This is a real advantage of multivariate information analysis over such metric techniques as analysis of variance, in addition to the fact that the structure of information components in multivariate information analysis is analogous to that of variance components in the analysis of variance (Garner and McGill, 1956). Like simple contingent uncertainty, multiple contingent uncertainty is symmetric; U(A:B 1 ,B J ) = U(A:B 2 ,B 1). I mentioned earlier that the additive property of information measures simplifies their calculation and mathematical development, and this is particularly true for multivariate information analysis. For example, from Equations 10.1 to 10.3 it can be shown that INTXABj.B!) = U(A:B 2 |B, ) - U(A:B 2) = U ( A : B 1 | B 2 ) - U ( A : B 1 ) , which shows that the interaction term is actually the difference of two information measures. The interaction term can also be calculated by reordering Equation 10.3: INTYAiBj.B,) = U(A) - U(A:B,) - U(A:Bj) - U(A|B„B 2). In the right-hand side of the equation, only U(A | BUB^) involves three variables and the calculation of the other information measures is rather simple. The additivity thus helps us to see the relation among different information measures and to save effort and time in their calculation. It is important to keep in mind that the above properties hold for all information measures I will discuss in this thesis. The next step is to extend Equation 8 to include one more variable B 3 U(A) = U(A:B 1 ,B 2 ,B 3) + U(A | B^BJ (11). As I have shown for U(A:B 1 ,B 2), we can partition U(A:B 1,B 2,B 3) into components U(A:B 1 ,B 2 ,B 3 ) = UfArB^B,) + U(A:B 3 |B 1 ,B 2 ) (12.1) = U(A:B,) + U(A:B 2 |B 1 ) + U(A:B 3 |B 1 ,B 2 ) (12.2) = U(A:B,) + U(A:B 2) + I N T t A ^ . B , ) + U(A:B 3 |B 1 ,B 2 ) (12.3). In Equations 12.1 - 12.3, only U(A:B 3 |B„B 2) is a new information measure, which measures the amount in U(A) that is explained by B 3 but by neither B! nor B 2 . Furthermore, U(A:B 3 |B„B 2) can also be partitioned into two components U(A:B 3 |B 1 ,B 2 ) = U(A:B 3 |B 1 ) + INT(A:B2,B31 B J (13), where INT(A:B 2,B 31B,) is the amount in U(A) that is explained by the interaction of B 2 and B 3 but not by Bv Using some of the above equations, Equation 11 can be rewritten as U(A) = U(A:B,) + U(A:B 2) + INT(A:B 1 (B 2) + U(A:B 31B^B,) + U(A | B„B 2,B 3) (14.1) = U(A:B,) + U(A:B 2 |B,) + U(A:B 31B l tBJ + U(A | B„B2,B3) (14.2) = U(A:B,) + U(A:B 2 |B!) + U(A:B 3 |B, ) + INT(A:B 2 )B 31 Bj) + U(A | B ^ B j ) (14.3). Since only the information measures so far discussed are necessary for the remainder of this dissertation, I will not discuss any other information measures in multivariate information analysis. However, it is important to keep in mind that, besides extending the number of variables involved, many other variations can be derived as easily as the above equations are, and that the information measures are interpreted in the same way as was done in the above discussion. Now I will turn attention to empirical studies using multivariate information transmission in sensation and perception research. II.4.2. Absolute identification of multidimensional stimuli As I discussed earlier, studies of absolute identification of unidimensional stimuli indicate that, whatever the reasons are, there is an upper limit for judgments of unidimensional stimuli that cannot be exceeded even with further increase in the amount of stimulus uncertainty. The limited channel capacity for unidimensional stimuli contradicts our everyday experience that we can identify thousands of objects (cars, human faces, etc.) with ease. One possible explanation for the contradiction is that the objects we encounter in everyday life differ in many properties, or dimensions, so that we can identify the objects on the basis of the sum of the (limited) information obtained from each of the dimensions. Using information-theoretic terms, the amount of information transmission of inputs varying along many dimensions is some function of (not necessarily equal to) the sum of information transmission obtained for each of the dimensions. Thus we can increase the amount of information transmission obtained in an absolute identification experiment by increasing the number of dimensions that define a set of stimuli as well as by increasing the number of stimuli in the set (that is, the amount of stimulus uncertainty). In order to test the effects of dimensionality of stimulus on the amount of information transmission, a number of absolute identification experiments have been conducted with a set of stimuli varying along two or more dimensions. The procedures were identical to those of absolute identification experiments of unidimensional stimuli, except that the set of stimuli was defined in terms of two or more dimensions (e.g., sound frequency and intensity) and each response category corresponded to a particular combination of values on the dimensions (for a variation of this procedure, see Pollack, 1953; Tan, Rabinowitz, and Durlach, 1989). The data have been analyzed by multivariate information analysis as discussed earlier, with the different stimulus dimensions used as predictor variables. For example, data from an experiment with two dimensional stimuli have been analyzed by the following equations; U(R) = U(R:S 1,S 2) + U(R|S 1,S 2) = U(R:S,) + U(R:S2) + INT(R:S2,S,) + U(R|S 1,S 2), where R denotes response categories, and Sj and S2 denote each of the two stimulus dimensions. Using multivariate information analysis, we can obtain not only the total amount of information transmission for the two dimensions [UCRiS^Sj)] but also the amount of information transmission for each dimension alone [U(R:S!) and U ^ S j ) ] from the same set of data. For comparison purposes, the amount of information transmission for each of the dimensions has also been obtained from experiments of unidimensional stimuli, that is, a set of stimuli varying along each dimension alone. Since an extensive review of studies of absolute identification of multidimensional stimuli can be found in Garner (1962, 1974) and Treisman (1986), I will describe only one study that used the standard procedures and obtained a typical pattern of results. Beebe-Center et al. (1955) conducted a series of experiments in which the taste of concentrations of salt and sucrose was judged. With all the combinations of four different concentrations of salt and four of sucrose used as stimuli (4X4= 16), Beebe-Center et al. obtained 2.25 bits for the total amount of information transmission [UfRiS^Sj)]. They also calculated from the same data the amount of information transmission for each of salt concentrations and sucrose concentrations taken separately [U(R:S t) and UfRiSJ], and obtained 1.14 bits for salt and 1.00 bits for sucrose. Since the sum of the two amounts of information transmission (2.14 bits) was reasonably close to the total amount of 38 information transmission (U(R:S1,S2) = 2.25 bits), the interaction of the two dimensions [IMYRiS^S,)] was negligible in this experiment (see the previously mentioned Equation 9.3). Beebe-Center et al. also conducted absolute identification experiments of salt concentrations alone and sucrose concentrations alone, with a varying number of stimuli. They obtained 1.7 bits as the channel capacity for both salt alone and for sucrose alone. Thus the combination of salt and sucrose concentrations yielded a larger amount of information transmission (2.25 bits) than the channel capacity for salt alone or for sucrose alone (1.7 bits for both). However, the total amount of information transmission for the combination of the two is less than the sum of the two amounts of maximum information transmission (channel capacity) for each dimension alone (1.7 + 1.7 = 3.4 bits). In other words, the amounts of information transmission for each dimension were smaller when they were obtained from the experiments with multidimensional stimuli than when they were obtained separately from the experiments with unidimensional stimuli. The same pattern of results has been obtained for odor, visual position, and audition (for a review, see Garner, 1962), and Garner (1962) summarized them by saying that"... increase in the dimensionality of the stimulus increases total information transmission [e.g., U(R:S„S2)], but decreases information transmission per dimension [e.g., U(R:Si) and U(R:S2)]." (p. 120) Although I will not discuss any other studies on multidimensional stimuli because they are beyond the scope of this dissertation, I want to emphasize that the finding that the information transmission of multidimensional stimuli is less than the sum of unidimensional information transmissions has led to a variety of perceptual research, including the distinction between perceptually integral and separable dimensions (Coren and Ward, 1989; Lockhead, 1979; Treisman, 1986) and separation of a target dimension from background dimensions (Durlach, Tan, MacMillan, Rabinowitz, and Braida, 1989; Tan et al, 1989). II.4.3. Complex sequential dependencies : Predictability as a function of predictor variables Earlier, I reviewed studies that used simple contingent uncertainty to analyze sequential dependencies. Although it is clear that simple contingent uncertainty is useful in studying sequential characteristics of human behavior and is sometimes to be preferred to other statistical measures (e.g., correlation coefficients) because of its nonmetric property, the use of simple contingent uncertainty is limited to measuring sequential dependencies on only one preceding event. Miller and Frick (1949) provided a way to extend the analysis to more than one preceding event, although it did not allow us to measure the relative contributions of individual preceding events to a current one. Multivariate information analysis gives solutions to those problems; it allows us not only to extend the analysis of sequential dependencies to more than one preceding event, but also to measure the relative contributions of individual preceding events. In analogy with multiple correlation analysis, predictability of a current event is increased by adding more predictor variables in multivariate information analysis. Some of the early studies analyzed sequential dependencies of a current response on previous responses by using the following form of multivariate information analysis: U ( R J = U(R n :R n . 1 ,R n 2 ,R n . 3 , . . ) + U(R n |R n . 1 ) R n . 2 ,R n 3 , . . . ) , where R n is the response on trial n (the current response) and R n . k is the response on trial n-k (a previous response). The multiple contingent uncertainty U(Rn:Rn.„Rn.2,Rn.3,...) measures the amount of sequential dependencies (or predictability) as a function of previous responses. In order to obtain the sequential dependencies on previous responses in a "pure" form, the studies did not involve any stimuli. For example, Senders and Sowards (1952) presented a tone and a light to their observers and asked them to judge whether the light and the tone appeared simultaneously. In their experimental sessions, the tone and light were always presented simultaneously, so that there was no stimulus (stimulus difference) and the observer had to guess the presented "stimulus". Using a somewhat different form of the above equation, Senders and Sowards (1952) and Senders (1953) found that the predictability from previous responses increased by adding previous responses up to RB^. To analyze sequential dependencies in a probability learning experiment, Edwards (1961) took an approach similar to Senders (1953), except that Edwards (1961) used preceding responses and preceding outcomes (answers) as predictor variables. Edwards (1961) found that the predictability increased by adding preceding responses and outcomes up to two trials back in the sequence. In sensation and perception research, multivariate information analysis has been used to analyze sequential dependencies in absolute identification experiments, in which previous events are from different sequences, such as the immediately previous stimulus and immediately previous response. Figure 5 illustrates the sequential dependencies of an observer's response on the previous stimulus (oblique dashed arrow) and a previous response (horizontal dotted arrow), in addition to the effect of the presented stimulus on the current response (vertical solid arrow). As mentioned earlier, the amount of information transmission (e.g., Equation 7) has been a useful index of the effect of the presented stimulus on response. Using the presented stimulus and the immediately previous stimulus and response as predictor variables, multivariate information analysis enables us to measure not only the amount of information transmission but also sequential T R I A L 1 2 3 . . . 1 0 0 1 0 1 S T I M U L U S • • V • • - m w SN \ \ \ \ \ \ X \ ^ R E S P O N S E '5 ' ^ ' 1 ' ^'9' • • ^ ' 2 ' ^ ' 3 ' -->• Figure 5. Sequential dependencies in an absolute identification experiment 41 dependencies on the previous stimulus and response from the same data. As you shall see below, however, the early studies did not measure both the sequential dependency on a previous stimulus and that on a previous response (e.g., Garner, 1953; McGill, 1957). McGill (1957) measured the sequential dependency on the previous response in absolute identification experiments of sound frequency. McGill (1957) used four sound frequencies presented under white noise and changed the intensity level of the tones (with the level of white noise fixed) across experiments. One of the experiments was a guessing experiment in which only the white noise without the tones was presented (without listeners' knowledge). In order to analyze the data, McGill (1957) used the following equation U ( R J = UfR^SJ + UCR.rR^lSJ + U(R n | Sn,Rn.,) (15), where S n is the stimulus on trial n (the current stimulus), and U(R n:S n) is the amount of information transmission from S n to R n . U(Rn:R„.1 |Sn) is the amount in U ( R J that is explained by R n., but not by S n and is taken as the amount of sequential dependency on R n., (see Equation 9.2). Thus Equation 15 measures the effects of a current stimulus and a previous response (solid and dotted arrows in Figure 5) but not the effect of a previous stimulus (dashed arrow). The results showed that U(R n:S n) increased as the intensity level of the tones increased, and that U(R n:R n.! | S J decreased as U(Rn:S„) increased. The latter result implied that the listeners' responses were more dependent on the previous response (R^) as the stimuli became more difficult to perceive (cf, Ward and Lockhead, 1971). In experiments of absolute identification of sound intensity, Garner (1953) measured the amount of sequential dependency on the previous stimulus (S^; dashed arrow in Figure 5) by the following equation U ( R J = U(R n :SJ + U(R n:S n. 1 |S n) + U(R n | Sn,S^) (16), where U(R n:S n., | S J measures the amount of sequential dependency on (Garner, 1953, also used the observer as a predictor variable). Garner (1953) systematically changed the number of tones used as stimuli, and found that U(R n:S n) increased with the number of tones, and that U(R n :S D . 1 1SJ decreased as U(R n:S n) increased. The results, like McGill's (1957), indicated that the amount of sequential dependencies was inversely related to the amount of information transmission from S n to R n . Subsequent studies of absolute identification also found an inverse relation between the amount of information transmission and that of sequential dependencies on previous stimuli and responses (Mori, 1986; Mori and Ward, 1990; Ward and Lockhead, 1970, 1971), although they did not use multivariate information analysis to measure the amount of sequential dependencies. Recently, Mori (1989) extended the approach of McGill (1957) (Equation 15) and Garner (1953) (Equation 16) to measure the amount of sequential dependencies on both Sn., and Rn., by the following equation; U ( R J = U(R n:S a) + U(R n:S n. 1 |S n) + U(R n :R n . 1 |S„) + INT(Rn:Sn.1,Rn.11S J + U(R n |S n ,S n . 1 ,R n . 1) (17), which is identical to Equation 14.3 except for the notation of the variables. Mori (1989) applied Equation 17 to two experiments of absolute identification of line lengths. In Experiment 1, Mori (1989) systematically changed the contrast of the lines in an attempt to manipulate the amount of information transmission U(R„:SJ. The results showed that U(R„:SJ increased as the contrast of the lines increased, and that U(R n :S n . , |S n ), U(R n :R n . 1 |S n ) , and INT(R n:S n. 1,R 1 1.,|S n), all of which measure the amount of sequential dependencies, decreased as U(R n:S n) increased. Thus the results confirmed the inverse relation between the amount of information transmission and that of sequential dependencies on the previous stimulus and response (Garner, 1953; McGill, 1957). In Mori's (1989) Experiment 2, U(R n:S n) was manipulated by the combination of the stimulus range used, stimulus exposure duration, and trial-by-trial feedback. One condition of Experiment 2 was a guessing condition (Mori, 1988) in which the observers were told to identify the length of a briefly presented line, but in fact no line was presented. An inverse relation was again found between U(R n :SJ and U(R„:Sn.11SJ, and INT(RB:S 0. I,R 1 1. I|S 1 I), but not between U(R n :SJ and U(Rn:R„.11SJ. In addition to those results, Mori (1989) found in both experiments that the amount of multiple contingent uncertainty U(R n:S n,S n. 1,R n. l), where U(R n:S n,S n. 1,R n. 1) = UfR.iS.) + U(R n:S n. 11SJ + U(R n:R n. 11SJ + INT(R n:S n. 1,R n. 11SJ (18), was about 2.5 bits, irrespective of variation of the amount of each of the four terms in the right-hand side of Equation 18. Since 25 bits was obtained even in the guessing condition of Experiment 2, Mori (1989) reasoned that 2.5 bits of U(Rn:SB,SB.1,RD.1) would reflect the limited capacity of the observers' response process in absolute identification experiments. Overall, it can be said that multivariate information analysis is a very powerful tool for analyzing sequential dependencies in absolute identification experiments. As Garner and McGill (1956) showed, the structure of information components in multivariate information analysis is analogous to that of variance components in metric analysis, such as analysis of variance and multiple correlation analysis. Especially, Equation 17 has a form very similar to a multiple regression equation which has been frequently used to analyze sequential dependencies in scaling tasks, such as magnitude estimation and cross-modality matching (DeCarlo and Cross, 1990; Jesteadt, Luce, and Green, 1977; Mori and Ward, 1990; Ward, 1979, and many 43 others). In analyzing sequential dependencies in absolute identification experiments, multivariate information analysis is to be preferred to multiple correlation analysis, because multiple information analysis is a nonmetric technique that is applicable to the categorical data obtained from absolute identification experiments. Furthermore, information measures in multivariate information analysis measure the amount, not the degree, of the effect of a previous stimulus and response on the current response. Remember the earlier discussion that the amount of information transmission is the amount of relation, not the degree of relation, and that this property of information transmission is particularly valuable in absolute identification experiments in which the number of stimuli used is manipulated (Garner, 1962). The same argument also holds for multivariate contingent uncertainty, in a form such as Equation 17. Each term in the right-hand side of Equation 17 measures the pure amount of the effect of predictor variables, without respect of the amount of error uncertainty U(R n |S n ,S n . 1 ,R n . 1). It is conceivable, and will be shown later, that U(R n |S n ,S n . „R n.,) increases as the number of stimuli used increases. In order to compare the amount of sequential dependencies in a variety of conditions in which U(R n | S^S^R,^) or error variance may be different, Equation 17 would be preferred to a multiple regression equation which has a structure analogous to Equation 17 but measures the degree of sequential dependencies relative to the size of error variance. Despite its usefulness, however, multivariate information analysis is an underdeveloped technique, mainly because the use of information concepts and measurement in psychology research virtually ceased in the 1970s. In subsequent chapters of this thesis, I will clarify some problems associated with the use of multivariate information analysis and will apply the analysis, in a form similar to Equation 17, to study sequential dependencies in absolute identification experiments under a variety of conditions. / III. DEVELOPMENT OF MULTIVARIATE INFORMATION ANALYSIS In Chapter II, I reviewed some of the studies that used information concepts and measurement to investigate sensory and perceptual processes. At the end of the chapter, I emphasized the usefulness of multivariate information analysis in studying sequential dependencies in absolute identification experiments. Especially, the analysis used by Mori (1989) (Equation 17) has a form similar to that of a multiple regression analysis which has been frequently used to measure sequential dependencies in scaling tasks, such as magnitude estimation. Using Equation 17, Mori (1989) measured the effects of both the previous stimulus (S^) and response (Rn.j) and their interaction on the current response (R^), and the measured amount of those effects was generally consistent with the results of other studies on absolute identification (e.g., McGill, 1957; Ward and Lockhead, 1971). Despite the usefulness of this analysis technique, there have been only a few studies that used multivariate information analysis to measure the amount of sequential dependencies (Garner, 1953; McGill , 1957). In the first section of this chapter, I will report further applications of multivariate information analysis (Equation 17) to the measurement of sequential dependencies in absolute identification experiments. While the results of the new experiments were generally consistent with the previous studies, they also indicated some problems in the use of Equation 17. The remaining sections of this chapter will discuss the problems and plausible solutions for them. As a result, I will propose a practical and reliable way of using multivariate information analysis, in the form of Equation 17, in order to analyze absolute identification data. III. 1. Experiment 1 As mentioned in Chapter II, Garner (1953) found by using Equation 16 that the sequential dependency of R n on Sn.j fl^R^So., \SJ] increased with the number of stimuli. It is plausible that the sequential dependency of R n on R n., would also increase with the number of stimuli used. In order to determine the effect of the number of stimuli used on these sequential dependencies, I conducted absolute identification experiments in which the number of stimuli was varied and Equation 17 was used for the data analysis. In addition to the effect of the number of stimuli used, the relation between U(Rn:Sn) and the amount of sequential dependencies was examined. As mentioned in Chapter II, previous studies of absolute identification found that the amount of sequential dependencies was inversely related to U(Rn:Sn) (McGill, 1957; Mori, 1989; Mori and Ward, 1990; Ward and Lockhead, 1970, 1971). However, except McGill (1957), those studies used 10 stimuli. Thus, it should be important to examine the generality of the inverse relation for different numbers of stimuli used. III.l.l. Experimental method The stimulus modality was frequency of pure tone under a masking and no-masking condition. In the masking condition, the tones were presented in white noise. In the no-masking condition, the tones were presented alone. In both conditions, the number of tones to be identified was systematically manipulated : 4, 6, 10, and 16 tones. A . Stimuli and apparatus. A Hewlett-Packard Vectra ES/12 computer system controlled the presentation of the stimuli and recorded the responses, which were made on a dimly red-illuminated, standard computer type keyboard. The tones were delivered diotically through Koss Pro-4AAA earphones and amplified and gated by an electronic switch. The frequencies of tones used were selected in such a way that 4, 6, 10, and 16 tones were spaced equidistant on a logarithmic frequency scale over the range of 100 Hz to 8000 Hz (cf. Pollack, 1952). It is well known that perceived pitch of pure tone depends on not only the frequency but also on the intensity level of the tone (Stevens and Davis, 1938). To eliminate loudness cues that would aid identification, each observer was asked to match the loudness of the tone of each frequency with the loudness of a 1000 Hz, 60 dB tone before absolute identification experiments. The resulting matches determined tone intensities for that observer. Tone intensities were thus all 60 phons. The tone duration was 500 msec. The intensity of white noise used in the masking condition was 85 dB. The presentation of white noise started 500 msec, before the presentation of the pure tone and terminated with the pure tone. B. Procedure. The observers were given the typical instructions for an absolute identification task. Before each experimental session, the observers were told the number of tones to be used in that session and presented with them and the required responses for each. They were instructed to identify the presented tone on each trial. Feedback as to the current identity of the presented tone was given after each response. Trials were self-paced. For each df the different number-of-tone conditions, each observer made 60 responses to each tone in both masking and no-masking conditions. The order of the number-of-tone and masking/no-masking conditions was randomized for each observer. C. Subjects. Five students of the University of British Columbia, one of them the author, participated. Four were males and one was female. A l l had no difficulty hearing the tones used as stimuli. III.1.2. Results Equations 17 and 19 were used to analyze the data; U ( R J = UCR^SJ + UCR^S^JSJ + U t R ^ l S J + INT(R l l:S I 1. 1,R n. 11SJ + U(R n | S ^ . R ^ ) (17) = U f R . r S ^ . R . . , ) + UCRjSA-.-R,,-.) (19). (for the detailed calculation procedure of the information measures and their interpretation, see Chapter II and Appendix) Since the subjects' individual data closely resembled each other, the information measures were calculated for each subject and averaged over them. Figure 6 and Table 1 present the results. It is clear in Figure 6 that manipulation of U(R n:S n) was successful in that U(R n:S n) was .5 - .6 bits higher for the no-masking condition than in the masking condition in each of the number-of-tone conditions. As found in other absolute identification studies on channel capacity (e.g., Pollack, 1952), U(R n:S n) levelled off as the number of tones increased in both the masking (1.6 bits) and no-masking (2.2 bits) conditions. As for the sequential dependencies, an inverse relation was again observed between U(R n:S n) and the amount of sequential dependencies; U(R B :S n . 1 |SJ and U(R n :R 0 . 1 1SJ were larger in the masking condition than in the no-masking condition. Furthermore, as predicted from the results of Garner (1953), U(R n :S n . J S J and U(R n :R n . 1 |S n ) increased with the number of tones. Unlike Mori (1989), however, INT(R n:S n.„R n. J S J was negative for most of the experiments (Table 1). The value of U(R n:S n,S n.„R n. 1) was dependent on the number of tones used and the presence or absence of masking. U(R n:S n,S n. 1,R n. 1) increased with the number of tones used, and it was smaller for the masking condition than for the no-masking condition, especially in 4- and 6-tone conditions. However, the difference between the masking and no-masking conditions decreased as the number of tones increased. A M O U N T O F I N F O R M A T I O N ( b i t s ) H c 5 O) CO CD m 30 o •n 7 O co O m x •o 3 3 m c CO m o LP «-»• iff1 • D iff 3 z o I > CO > CO 48 Table 1. Mean and standard deviation (S.D.) of INT(R n:S n. 1,R n. 11SJ in Experiment 1. Number of tones Condition 4 6 10 16 No Mask Mean -.019 -.090 -.105 -.239 S.D. .014 .025 .071 .027 Mask Mean -.003 .008 .027 -.188 S.D. .028 .038 .078 .060 III.1.3. Discussion The results are consistent with previous studies with a few exceptions. The amount of sequential dependencies of R n on SaA and R ^ ^(R^S,,., | S J and U(R n:R n. 11SJ] increased with the number of tones used (Garner, 1953). An inverse relation was observed between U(R n:S n) and the amount of the sequential dependencies. However, there are also a couple of problems that would render the measures of Equations 17 and 19 somewhat dubious. First, INT(R n:S 0.i,RB. 1 |S n) was negative for most of the experiments (Table 1). Negative interactions indicate spurious associations, i.e., associations among variables that covary only because they are dependent upon another variable (von Eye, 1984). The second problem is that U(R n:S n . J S J and U(R n :R n . , |S n ) seem too large in the 10- and 16-tone conditions. In the 16-tone condition, U(R n:S n. J S J and U(R n :R n . , |S n ) together amounted to about 2 bits in the masking condition even though U(R n:SB) was only 1.5 bits. These amounts may have been inflated because their calculations were based on fewer observations than required. Although the inflation of such information measures as U(R n :S n ) and U(R„) has been recognized and a few remedies have been proposed (Houtsma, 1983; MacRae, 1970, 1971; Miller, 1954), there are no reports on the inflation of multivariate information measures or its remedy, as far as I know. III.2. Negative interaction The existence of negative interaction terms in multivariate information analysis has been recognized (Attneave, 1959; Pollack, 1953; Garner, 1962). Garner and McGill (1956) proved that an interaction term is negative if the predictor variables in the term are correlated, or nonorthogonal. In cases of INT(R n:S n. 1,R n. , | S J , the predictor variables are and Rn.j, which are in most cases highly correlated. The negative interaction term is not interpretable in the same way as are other information measures, including positive interaction terms (Attneave, 1959; von Eye, 1984). It follows that we should not take into consideration INT(R n:Sn.„Rn., | S„) if it is negative, and that in order to interpret the results in a consistent way, we might as well ignore INT(R n:S n. 1,R n. 11SJ even if it is positive. Therefore throughout the rest of this thesis, I will not use INT(R n:S n. 1,R n. 11S J in the discussion of results when Equation 17 is used for data analysis. As an alternative of Equation 17, we can use another type of multivariate information analysis that does not include such interaction terms as INT(R n:S n.„R n. 11SJ but still measures the effects of S^, and Rn_,. For example, by slightly changing Equation 17, we have U ( R J = U(R n :SJ + U(R n:S n . 1 |S n) + U(R n:R n. 11 Sa,SDA) + U(R„ | S ^ R ^ ) (20), where U(R n:R,, 11 S ,^ , ) = UfR.rR,,., | S J + INT^R^S^R,. , | SJ . U(R n:R n. 1 |S I 1,S n. 1) is the amount of U ( R J that is explained by R^j but by neither S n nor Sn.], and is interpreted as the amount of sequential dependency on R^! with the effects of S n and of excluded (see Equations 12.2 and 12.3 in Chapter II). Since the other terms in Equation 20 are the same as those in Equation 17, Equation 20 can be also rewritten to Equation 19 by replacing the first three terms of the right-hand side of Equation 20 with U(R n:S n,S n. 1,R I 1. 1). Figure 7 presents U(R n:R n . 1 |S n ,S n . 1) from the data of Mori (1989) and those of Experiment 1 of this thesis. It seems that U(R n :R n . 1 IS^S,,.,) is at least as good a measure of the effect of R n., as U(R n :R n . i ISJ is, in the sense that it varies with U(R n:S n) and the number of stimuli in a way that is consistent with previous work. In some ways it is even better, for example, U(R n:R n. 1|S n,Sn.i) is negatively correlated with U(R n:S n) in Mori's (1989) Experiment 2, while U(R n :R n . 1 |S n ) is not (see Chapter II). It is important to note that • U ( R n : R n - 1 I S n ) m U ( R n : R n - 1 I S n , S n - 1 ) • U ( R n : S n ) 1 . 5 1 . 0 0 . 5 L N F L F H N F H F C o n d i t i o n ( a ) M o r i ' s ( 1 9 8 9 ) E x p e r i m e n t 2 0 . 0 M A S K N O - M A S K 1 6 . N u m b e r o f T o n e s ( b ) E x p e r i m e n t 1 Figure 7. UCRniRn-ilS^Sn-i) 51 l^R^R,, . , | Sn,Sn.,) excludes the effect of the previous stimulus (Sn.,), which is correlated with the previous response (R^i). As mentioned before, such multivariate information analysis as Equations 17 and 20 is analogous to a multiple regression equation used to analyze sequential dependencies in scaling tasks (e.g., magnitude estimation, cross-modality matching), and DeCarlo and Cross (1990) recently pointed out that using such correlated dependent variables as R,,., and simultaneously in a regression model could result in biased regression coefficients. The same may happen in Equation 17, that is, U(R n :R n . , | S J may be biased by the correlation of R n . i and Sn. l 5 especially when the correlation is high because of large U(R„:Sn). Since U(R n:R n . 1 |S n ,S n . 1) seems to be a very useful measure of the effect of R n.„ I will use Equation 20 as well as Equation 17 [omitting INT(Rn:Sn.1,Rn.1|S I 1)] for data analyses, unless noted otherwise. III.3. Experiment 2 Even if Equation 20 is used to analyze the 16-tone data in Experiment 1, the measures of sequential dependencies [U(R n:S n. 11SJ and U(R n:R I l. 1 |S n,S n. 1)] are still too large compared with the size of U(R n:S n). It has been pointed out that the amount of information transmission U(R n:S n) is likely to be inflated when its calculation uses a small number of observations (responses) compared with the number of cells in the confusion matrix for the calculation (Houtsma, 1983; MacRae, 1970; Miller, 1954; Miller and Madow, 1954). The inflation problem is more serious for multivariate information measures because the calculation requires larger number of cells in the confusion matrix than does that of bivariate information measures. In cases where the number of stimuli used is N and is equal to that of response categories, there are N 2 cells in the confusion matrix for the calculation of U(R n :SJ , N 3 cells for U(R n:S n . 1 |S n) and U(R n :R n . , |S n ) , and N 4 cells for I N T ^ S , , . , ^ , | S J , U(R„:Rn.1 I S ^ ) , and U(R n:S n,S n. 1,R„ 1). Table 2 shows the number of cells in the confusion matrix for the calculation of the multivariate information measures used in Mori (1989) and Experiment 1. As you can see, since the calculations are based on the same number of observations obtained in an experiment, the discrepancy between the number of observations and that of cells in the confusion matrix gets larger with the number of variables considered in the calculation of information transmission as well as with the number of stimuli and response categories. It is also conceivable that the inflation is bigger when fewer observations are used (Houtsma, 1983; MacRae, 1970). In the following sections, I will discuss the inflation of information measures and remedies for the inflated measures. First, I report results of Experiment 2 in which I replicated the no masking condition of Experiment 1 with a greater number of 52 observations, in order to see how the information measures change as a function of the number of observations. Table 2. The number of cells in confusion matrices for the calculation of information measures in Equations 17 and 20. Number of Stimuli U(RJ U(RB:S„) U C R ^ I S J U f R ^ R . j S . A . , ) UCRAA^R, , , ) 4 4 16 64 256 6 6 36 216 1296 10 10 100 1000 10000 16 16 256 4096 65536 Note - The number of stimuli is equal to the number of response categories. IH.3.1. Experimental method The procedure and stimuli used were identical to those of the no masking condition of Experiment 1, except the tone duration was 100 msec in this experiment. The author was the only observer. Like Experiment 1, the number of tones used was 4, 6, 10, and 16, and the order of the number of tones used was randomized. The observer made 180 responses per tone in the 4-, 6-, and 10-stimulus conditions, and 400 responses per tone in the 16-stimulus condition. III.3.2. Results The data were combined with those of the author from the no masking condition of Experiment 1. The information measures of Equations 17 and 20 were calculated from the data in two ways. First, the data were classified into subsets of 60 observations per stimulus in a chronological order, and the information measures were calculated from each subset. The purpose of this analysis was to check learning or practice effects on the information measures. The results are presented in Figure 8. Second, each subset of 60 observations per stimulus was added to the preceding one(s), and the information measures were calculated from the accumulated data. The results are presented in Figure 9. U^RJ was always near its maximum value [equal to UfS,,)] and showed neither learning effects nor effects of the number of observations used. However, there seem to be some learning effects on the other information measures, especially for 10- and 16-tone conditions (Figure 8). U(Rn:Sn) was increased by up to 0.3 bits in the later sessions, while the measures of sequential dependencies [U(Rn:Sn.1 SJ, U(Rn:Rn.1 SJ and U(Rn:Rn.1|Sn,Sn.1)] were decreased by 0.1 to 0.2 bits. The changes in the information measures over sessions were negligible in the 4- and 6-tone conditions. U(Rn:Sn,Sn.„Rn.1) was strikingly constant for all of the number of tones used. In Figure 9, U(Rn:SJ was constant as the number of observations upon which the measure was based increased. However, other information measures decreased with increasing number of observations, and the size of decrease seemed to depend on the number of tones used in the experiment. U(Rn:Sn., | SJ decreased dramatically in the 16-tone condition (about 0.6 bits) and, to a lesser degree, in the 10-tone condition. There seemed to be minimal changes in the 6-tone condition, and U(Rn:Sn.I | SJ was nearly zero in the 4-tone condition. A similar pattern was observed for U(Rn:Rn.1|S11) and U(Rn:Sn,Sn.1,Rn.1). However, UflLjRn. i|Sn>Sn.i) showed a pattern slightly different from those of the other multivariate information measures. U(Rn:Rn_i | Sn,Sn.,) was nearly zero in the 4- and 6-tone conditions, and there seemed to be no changes in the 10-tone condition. In the 16-tone condition, U(Rn:Rn.1  Sn,Sn.,) first increased from 60 observations to 120 observations per stimulus, then decreased with increasing number of observations. The amount of decrease was less than that of U(Rn:Sn.! | SJ or U(R„:Rn.1 SJ. III.3.3. Discussion It is reasonable to assume that the more measures are inflated by using a small number of observations, the more they will decrease with increasing number of observations until they reach asymptote. The present ~i 1 1 1 1 — — r U(R„:Sn) «o I 2 3 I' I o O Z 1.0 O 0.8 Z 0.6 § 0.4 < 0.2 0.0 .M A... o—e—e—Q 1 1 1 1 1 1 r— A A A A A A • A A . A A • • • • G — 8 — 9 — 0 U(Rn:Sn,Sn-i,Rn-i) _1 I J L _l I 1_ 60 180 300 420 60 180 300 420 ~i 1 1 1 1 1 r UCRnlSn-! IS„) ''A- A-.. A " •- • • • O- ( D — © — © 1 1  T 1 1 1 1 1 r UCRnlRn-! IS„) 'A. A ''A A'' •••• -@ © © ©-o 4 t o n e s • 6 t o n e s A 10 t o n e s * 1 6 t o n e s - i 1 1 r U(R„:R„-i IS^Sn-i) •A A _ i i i _ 60 180 300 420 60 180 300 420 60 180 300 420 NUMBER OF OBSERVATIONS PER TONE Figure 8. The results of Experiment 2. (1) ~t r — i 1 1 1 1 r U(R„:S„) tt I 2 S 1 I o DC O | 1-0 O 0.8 Z 0.6 § 0.4 < 0.2 0.0 A - A A A h o—e—e—e - i — i — i i i 60 180 300 420 U(R„:Sn-i IS„) • ; » » • 'O i o ' o J — i i i_ ~i 1 1 1 1 1 1 r A A G—e—e—e U(Rn:Sn,Sn-i,Rn-i) - i — i — i — i — i i i i _ 60 180 300 420 i 1 1 1 1 1 1 r UCRnlRn-! ISn) • • • Q - 'Q ' O i o i _ i i i _ o 4 t o n e s • 6 t o n e s A 1 0 t o n e s A 1 6 t o n e s -i 1 r UtRnlRn-! ISn^-i ) 60 180 300 420 60 180 300 420 60 180 300 420 NUMBER OF OBSERVATIONS PER TONE Figure 9. The results of Experiment 2. (2) 56 results confirmed those of previous studies in that the inflation (change with increasing number of observations) of information measures is large when a large number of stimuli (e.g., 16) is used, while the inflation is negligible for a small number of stimuli (e.g., 4, 6). Since the amount of decrease of U(R n:S n . ^ S J and U(R n :R B . 1 | SJ was larger than the learning effect observed, the inflation should not be attributed to the larger variance in the first session due to the lack of practice. The above results also suggest that while U ( R n : S J seems to be stable in all conditions, the other multivariate information measures of Equations 17 and 20 decrease with increasing number of observations. The size of decrease of those measures is substantial in the 16-tone condition, although they seem to be reaching their asymptotic levels around 460 responses for each tone, which are not enough to fill in all cells in the confusion matrix used to calculate information measures in the 16-tone condition as well as in the other number of tones used (Table 2). Thus, we may not need the same number of observations as those of cells in the confusion matrices in order to obtain reliable estimates of those information measures (Houtsma, 1983). This point will be considered in the next section. III.4. Methods for correction of inflated information measures : Review. The results of Experiment 2 indicate that the information measures are more likely inflated when fewer observations are used and/or many stimuli are used. There are three ways to obtain an uninflated estimate of information measures. The first and simplest solution is to use a large enough number of observations for the information measures to reach their asymptotic level. The second is to use Miller's (1954) correction equation which is based on statistical properties of the information measure (Miller and Madow, 1954). The third is to run computer simulations of absolute identification, estimate the amount of inflation as a function of the number of observations, and apply an appropriate correction to the measures calculated from empirical data. III.4.1. Estimation from a large number of observations. There are two conceivable ways to increase the number of observations for the calculation of information measures. One way is to increase the number of observations for each observer in the experiment, like Experiment 2. The other is to pool individual data in the experiment. Although the former is possibly the simplest solution, it is very difficult in practice because very few individuals can do so (the author performed 57 in 7360 trials altogether in the 16-tone condition alone!). The pooling of individual data is more practical and has been used in studies of a relatively larger number of stimuli (see Houtsma, 1983 for a review). The pooling method has one disadvantage: pooling of individual data is statistically valid only if the individual data are homogeneous. Although we can minimize the individual differences by using well-experienced observers and giving them trial-by-trial feedback, most data would not satisfy the homogeneity requirement. The consequence of violation of the homogeneity assumption in pooling the data is that the estimated noise (or error) is biased, and in the case of an information measure, this would result in underestimating the amount of information transmission. That is, the information measures calculated from pooled data would decrease with the number of individual data to be added to the calculation, if the individual data are heterogeneous. However, the results of Experiment 2 showed that although the information measures decreased with the number of observations, it did not keep decreasing and reached an asymptotic level with a reasonable number of observations. To see how the individual differences would affect the information measures calculated from the pooled data, I pooled different combinations of individual subjects' data from Experiment 1 and calculated various information measures. HI.4.1.1. Combination method The number of ways to select r from p data set is p X (p - 1) X ... X (p - r + 1) p C r = , r! where r! = r x (r - 1) x (r - 2) x ... x 1. There were five observers (p = 5), A , B, C, D, and E in Experiment 1 and they did the same number of trials. When r = 1, the information measures were calculated from each individual's data and then averaged. When r = 2, JCJ = 10 and the possible combinations were (A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), and (D,E). The information measures were calculated from the pooled data of each combination and averaged. Similarly, average measures were calculated for r = 3 and r = 4. When r = 5 , 5 C S = 1 and the all individual data were pooled and the information measures were calculated from the resulting unique confusion matrix. III.4.1.2. Results and discussion CO 3 < s DC o IL z IL o r-Z 3 O s < 3 2 1 0 1 . 2 1 . 0 0 . 8 0 . 6 0.4 0 . 2 0 . 0 - i r - i r U(R„:S„) - i — i — i L . 6 0 1 8 0 3 0 0 i iKRnlSn-i IS„) ffl (D EL U(Rn:Sn,Sn-i,Rn-i) _ l L. 6 0 1 8 0 3 0 0 - i r I UtRnJR.-, IS„) 1. ® m i n f t f t 0 4 c a t e g o r i e s • 6 c a t e g o r i e s A 1 0 c a t e g o r i e s * 1 6 c a t e g o r i e s Vertical bars indicate standard deviations n r UCRnlRn-ilSn.Sn-i) I i i L . . ft ft ft ft ft 1 8 0 3 0 0 1 8 0 3 0 0 6 0 1 8 0 3 0 0 6 0 NUMBER OF OBSERVATIONS PER TONE Figure 10. The information measures of pooled data 00 59 Figure 10 presents the information measures obtained by applying the pooling method discussed above to the no-masking condition data of Experiment 1. The information measures obtained by the pooling method showed a pattern quite similar to those from the single-observer data in Experiment 2 (Figure 9). While U(R n :SJ was strikingly constant without respect to the number of observations used for the calculation, the other information measures [U(Rn:Sn.11SJ, U(Rn:Rn.11SJ, U(Rn:R^, | S A . , ) , and U(Rn:Sn,Sn.1,Rn.,)] decreased with increasing numbers of observations, specially for the 10- and 16-tone conditions. Note also that U(Rn:Rn.11 Sn,Sn-i) of the 16-tone condition first increased from 60 observations to 120 observations per stimulus then decreased slowly. Despite the inflation with a small number of observations used in the experiment, most of the information measures seem to be approaching their asymptotic level with a sufficient number of trials as more individual data are pooled. Thus we may not need to fill in an enormous number of cells in a confusion matrix in order to obtain a reasonable estimate of information measures. Moreover, given the similarity between the information measures calculated from single-subject data and from the method discussed here, it seems that it is the small number of observations, rather than the variance in data, that biases the estimate of information measures (Houtsma, 1983). Of course, this is true only if the variance in data is reasonably small, like that of Experiment 1 used here. Two additional points are important to note. First, a close comparison of Figure 10 with Figure 9 indicates that, despite the similarities, there are small but consistent difference between the information measures obtained by the pooling method and those of the single-observer data. For example, U(R n:S n) was somewhat larger when it was calculated from the single-observer data than when it was calculated by the pooling method, and the pattern was opposite for some of other information measures [e.g., U(Rn:R I J.i|S n,SD. ,)]. Second, the pooling method discussed here has an advantage over a straightforward pooling of all individual data (as when r = 5 here), in that it reveals how information measures change as a function of the amount of data pooled, so that experimenters can appreciate the amount of inflation and have a better idea of asymptotic values as unbiased estimates. III.4.2. Miller's (1954) correction equation. Miller and Madow (1954) showed that the amount of information transmission can be transformed into values of log likelihood ratio statistics L 2 = 2 x l n 2 x N X T (21) 60 where N is the total number of observations and T is the amount of information measure calculated from the observations. The degrees of freedom associated with the transformed L 2 's are identical to those computed for L 2 tests, which is approximately X 2 distributed. Table 3 shows the degrees of freedom associated with the information measures of Equations 17 and 20. From this transformation equation, Miller (1954) provided a correction equation for the bias of information measures caused by small N T* = T - df/(1.3863 x N) (22) where T' is the corrected information measure, and df is the number of degrees of freedom associated with Equation 21. MacRae* (1970) applied the correction equation to information measures reported in the literature and found that most information measures had been overestimated. x Table 3 The degrees of freedom associated with the information measures in Equations 17 and 20. Information measures Degrees of freedom U ( R J (K,-l) U(R a:S n) (Kr-IXK,-!) L K R A - . I S J K,(K r -l)(K,-l) U(R n:R n. 1 |S„) K ^ - l ^ - l ) U(R n:R n . 1 |S n ,S n . 1) K ^ - l ) 2 U(R n:S n,S n. 1,R n. 1) ( K X - I X K , - ! ) Note - K,, K , : the number of stimuli and response categories used, respectively. The correction equation is not without criticism. It has been pointed out that it should be used only if the following two conditions are met (Attneave, 1959; Krippendorf, 1986); (1) The number of observations should be large: ideally it is five times larger than the number of cells in the confusion matrix for the 61 Table 4 Degrees of freedom associated with the information measures of Equations 17 and 20 (df) and five times the number of cells in a confusion matrix for the calculation of information measures (5m). Number of Stimuli U(R n ) U(Rn:S„) U(R n:Sn-i|S n) U(R n :R n . 1 |S n ,S n . 1 ) U(R n:R„. 1 |S n) 1 U(R n:S n )S n . 1 )R n . 1) 4 df 3 9 36 144 189 5m 20 80 320 1280 1280 6 df 5 25 150 900 1075 5m 30 180 1080 6480 6480 10 df 9 81 810 8100 1075 5m 50 500 5000 50000 6480 16 df 15 225 3600 57600 61425 5m 80 1280 20480 327680 327680 Note - The number of stimuli is equal to the number of response categories. calculation of the information measure (Krippendorf, 1986), and at the very least greater than the number of degrees of freedom associated with the correction equation (Attneave, 1959). (2) The confusion matrix should contain few zero-frequency cells. The majority of research does not satisfy these two conditions, and it has been shown that the two conditions are not always necessary for reasonable estimates of information measures. Table 4 shows the degrees of freedoms associated with the information measures calculated in Experiments 1 and 2 and five times the number of cells in the confusion matrix. As you can see, except for U(R„) and U(R n:S n) for the small number of stimuli used, condition (1) is not satisfied in most cases, even \ 62 though the pooling method and single-subject data discussed above give reasonable asymptotic estimates of information measures. Condition (2) is also unlikely to be satisfied in empirical studies, because in order to obtain a reasonably high information transmission [U(Rn:Sn)] ) say 2 bits, the frequency of observations must be large in the diagonal cells and relatively small or zero in off-diagonal cells in the confusion matrix, and this is the case in most empirical studies. Therefore, the correction equation provides reasonably accurate estimates of information measures only for small numbers of stimuli and large numbers of observations, but it may underestimate them when there is a large number of stimuli. This overcorrection tendency was also demonstrated in Houtsma's (1983) computer-simulation study, which I will discuss next. III.4.3. Computer simulation. Since inflation of information measures is mainly due to the limited amount of data available, a practical solution is to run computer simulations of absolute identification with a large number of trials (repetitions) and estimate the asymptotic values of the information measures. Houtsma (1983) ran computer simulations for 125 stimuli/categories by using the model R = S + O where S is an integer chosen with equal probability in the range 1 - 125, and O is a uniformly distributed random integer in the range -A to A. With various values of A, Houtsma (1983) calculated the amount of contingent uncertainty U(R n:S n) as a function of the number of observations from 125 trials to about 25000 trials per stimulus. The results show that U(R a:S n) decreases monotonically with the number of trials to an asymptotic value, and that U(R n:S n) with a large value of A decreases more than that with a small value of A. Houtsma (1983) also showed that Miller's (1954) correction equation underestimates U(R n:S n) if the number of observations is small. In psychology literature, there is little work available on computer simulations of information measures (Houtsma, 1983; MacRae, 1971). Although some analytical solutions for the inflation of information measures can be found in recent work on log likelihood ratio statistics (cf., Gokhale & Kullback, 1978), there is no report available (as far as I know of) on multivariate information measures such as those of Equations 17 and 20. In order to estimate the amount of inflation in multivariate information measures, I ran Monte Carlo simulations of absolute identification, varying the number of stimuli used (response categories) and the number of trials, and calculated the values of the various information measures of Equations 17 and 20. 63 III.5. Computer simulation of absolute identification. Two methods were used to run Monte Carlo simulations of absolute identification. The first method was to use a regression equation model similar to Houtsma's (1983) but also include the effects of the previous stimulus (Sn.!) and response (R^) . As you will see below, however, the preliminary study of simulations by this method suggested that the use of regression-equation model would not be a good way to manipulate the amount of sequential dependencies measured by multivariate information analysis, such as Equations 17 and 20. The second method, which I call the method of matrix manipulation, was to manipulate probability distributions in three matrices, corresponding to the effects of Sn, Sa.v or R n.„ on R n , and to simulate absolute identification responses from the combination of the probability distributions. One of the advantages of this method is that the theoretical values of information measures can be obtained from the probability distributions of the matrices and compared with the simulation results. I used this method to run a series of Monte Carlo computer simulations and obtained the estimates of information measures with various numbers of stimulus/response categories and of trials (repetitions). III.5.1. Regression-equation method Absolute identification responses were simulated by the following model; R n = Sn + a Sn., + fi R n., + a where tt and fi are constants that determine the magnitude of dependency on Sn., and on R^j, respectively, and 0 is normally distributed with mean 0 and variance y2. Thus there are three parameters in the equation a, fi, and y. When a = fi = 0, the equation is similar to that used by Houtsma (1983), except Houtsma (1983) used a uniform distribution for O, while a normal distribution was used here. In a preliminary study, the number of stimuli used (equal to that of response categories ) was 4-32, the number of trials per stimulus was 5 - 500, and a, fi, and y were manipulated independently. The information measures of Equations 17 and 20 were calculated from absolute identification responses . generated with given combinations of a, fi, and y, and they were plotted as a function of the number of stimuli and the number of trials. Although only a limited set of parameter values were used in the simulation, the results are suggestive. Like Houtsma's (1983), U(Rn:Sn) decreased monotonically with the number of trials to an asymptotic level. When |tt| = \fi\ with opposite signs, the decrement of U(Rn:SB) and the asymptotic level depended on the size of 7, while the sizes of a and fi had little effect on U(Rn:Sn). 64 The decrement of U(Rn:Sn) is bigger and the asymptotic level is lower with a large value of y than with a relatively smaller value of 7. However, when |tt| <> \fi\, U(Rn:Sn) was very small even with a small value of y. U(Rn:Sn., ISJ and U(Rn:Sn., | SJ are overestimated even with a = fi = 0 and rapidly decrease to zero with increasing number of trials. Although the size of a was thought to have a direct effect on U(Rn:Sn.1|Sn), the amount of U(Rn:Sn.11SJ is not much affected by a when |tt| = \fi\. The same pattern was found in U(Rn:Rn.1|Sn) and fi. In general, U(Rn:Rn.,|Sn,Sn.1) at first increased with the number of trials and then decreased. Although all possible combinations were not exhausted, it follows from the above results that setting the absolute value of tt equal to that of fi is not a good model of absolute identification for use in such simulations because the magnitude of sequential dependencies measured by multivariate information measures does not change very much with a and fi. Multiple regression analyses of absolute identification data suggest that both tt and fi are positive and that their size would be different. In general, the relation between the values of information measures and the size of coefficients in a regression equation is not clear, because the regression coefficients indicate the direction and form of effects, while the information measures reflect the size of effects, like R 2 . Furthermore, regression analysis assumes interval scales while multivariate information measures require the nominal scales. Although more data should be accumulated before a conclusion is made as to the validity of the regression equation model, it is necessary to find a better way to manipulate the size of dependencies on the previous stimulus and response in simulations to investigate the inflation of multivariate information measures. HI.5.2. Method of matrix manipulation I emphasized in Chapter II that information measures are calculated from a confusion matrix. In the multivariate case, they are calculated from a multiple contingency matrix and their amount reflects relations among the variables in the matrix (von Eye, 1984). For example, in a two-way confusion matrix, there is no functional relation between the two variables if probabilities in all cells in the matrix are equal, and the amount of contingent uncertainty between two variables is zero. If the probabilities are not uniformly distributed, the contingent uncertainty is more than zero, indicating some relation between the variables. 65 Thus we can manipulate the amount of information measures by biasing the probability distribution in the matrix. In this section, I will report results of a series of Monte Carlo computer simulations that was generated by a method of matrix manipulation. One of the advantages of this method is that the theoretical values of information measures can be obtained from the probability distributions in the matrices and compared with the simulation results. III.5.2.1. Method and Programming The programs used here were written in MICROSOFT QUICKC (Version 5.1) with a built-in pseudo-random number generator. Three matrices were used to generate response sequences, namely a stimulus-response ( S ^ R J matrix, a prestimulus-response (S n.,xRJ matrix, and a presponse-response ( R ^ X R J matrix. Each cell represents the probability of a given combination (i, j = 1, k, k is the number of categories), and 2 Pj = 1.0. The theoretical value of U(R n :R n ), U(R n:S n., | S J , and U(R n:R n .! | Sn) [or U(Rn:R„.11 Sn,Sn.!)] can be calculated from each of three matrices, respectively. In each trial, a row was selected from each of three matrices according to the identity values of Sn, S^, and R,,.,, and probabilities of corresponding cells were multiplied and re-scaled to make new probabilities pj in such a way that 2 Pj = 1.0. The stimulus sequence was randomized and each stimulus was used equally often. A discrete approximation to the standard normal distribution was used to create the S n xR n matrix. The proportions assigned to the cells in each row of the S n xR n matrix were approximately normally distributed around the center of the response category equal to the stimulus category. The standard deviation (s.d.) of this distribution, in category units, was manipulated. If the distribution was truncated, mainly in cases of low or high stimulus categories, the truncated proportion was added to the remaining (untruncated) cells according to the original proportion of the remaining cells (cf. Torgerson, 1961). Theoretical values of U(R n:S J can be calculated from the probability distributions of the matrices and compared with those calculated from simulation data generated by the same matrices. Table 5 shows examples of the S n xR n matrix and the corresponding theoretical values of U(R„:Sn). Similarly, a discrete approximation to the standard normal distribution was used to create S n.!XR n and R n - i x R n matrices. The proportions assigned to cells in each row were approximately normally distributed and 66 Table 5 Examples of stimulus-response matrix, (a) s.d. = 0.3 : a theoretical value of UCR^SJ = 3.257 bits Response category Stimulus 1 2 3 4 5 6 7 8 9 10 1 .996 .004 .000" .000 .000 .000 .000 .000 .000 .000 2 .004 .992 .004 .000 .000 .000 .000 .000 .000 .000 3 .000 .004 .992 .004 .000 .000 .000 .000 .000 .000 4 .000 .000 .004 .992 .004 .000 .000 .000 .000 .000 5 .000 .000 .000 .004 .992 .004 .000 .000 .000 .000 6 .000 .000 .000 .000 .004 .992 .004 .000 .000 .000 7 .000 .000 .000 .000 .000 .004 .992 .004 .000 .000 8 .000 .000 .000 .000 .000 .000 .004 .992 .004 .000 9 .000 .000 .000 .000 .000 .000 .000 .004 .992 .004 10 .000 .000 .000 .000 .000 .000 .000 .000 .004 .996 (b) s.d. = 1.0 : a theoretical value of U(R n:S n) = 1.471 bits Response category Stimulus 1 2 3 4 5 6 7 8 9 10 1 .570 .346 .077 .006 .000 .000 .000 .000 .000 .000 2 .257 .424 .257 .057 .005 .000 .000 .000 .000 .000 3 .054 .243 .401 .243 .054 .004 .000 .000 .000 .000 4 .000 .054 .242 .399 .242 .054 .004 .000 .000 .000 5 .000 .004 .054 .242 .399 .242 .054 .004 .000 .000 6 .000 .000 .004 .054 .242 .399 .242 .054 .004 .000 7 .000 .000 .000 .004 .054 .242 399 .242 .054 .004 8 .000 .000 .000 .000 .004 .054 .243 .401 .243 .054 9 .000 .000 .000 .000 .000 .005 .057 .257 .424 .257 10 .000 .000 .000 .000 .000 .000 .006 .077 .346 .570 the standard deviation (in category units) was varied according to the longer of the two intervals from the center of the distribution to the extreme category (1 or n). For example, suppose 10 categories are being used and the stimulus (or response) category chosen is 4. This divides the interval from 1 to 10 into two parts, 1 to 4 and 4 to 10. The longer interval is from 4 to 10. Then one standard deviation is determined by the division of the relevant interval size in category units, 6 in this case (10-4), by a given value, for example 3, so that one standard deviation is 2.0 category. In this case, each category represents a "bin" 0.5 s.d. wide. The proportion assigned to each category is then calculated by multiplying the relevant probability from the discrete approximation to the standard normal distribution corresponding to that category by the number s.d. units it represents (0.5 in this case). This procedure was carried out for each simulation run in each relevant matrix. As in the S^R^ matrix, the distribution was often truncated at the extreme category and the truncated part was added to the remaining part according to the proportion assigned to each cell. Finally, the proportion of each category was scaled in such a way that z p i = 1.0. The size of s.d. (in category units) was manipulated by changing the size of the divisor (D) of the largest interval in the previously described procedure; D was defined by the equation D = 3 / P. As P increases, D decreases and the s.d. increases. As a result, the distribution is flattened and the amount of information measures of the effects of a previous stimulus and/or response is decreased. I call P used for S^xR,, matrices PSV, and P used for R n . 1 xR n matrices PRV. Thus information measures of sequential dependencies on Sn., and R,,., should vary inversely with PSV and PRV respectively. Like S n xR n matrices, theoretical values of information measures of sequential dependencies on Sn., and Rn.1 -[U^R^S,,., | S J , U(R n :R n . 1 |S n ) , and U(R n:R n.i | Sn,Sn-i)l can be calculated from the probability distributions of the matrices and compared with those calculated from simulation data generated by the same matrices. Table 6 shows examples of S^XR,, matrix and corresponding theoretical values of U(R n:S n., ISJ. The numbers of stimulus categories (always equal to that of response categories) used were 4, 6, 10, and 16, and the numbers of trials (repetitions) per stimulus category were 10, 30, 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 700, 800, and 900. Simulations were repeated 10 times for each combination of parameters. The information measures in Equations 17 and 20 were calculated from each simulation run, and the averages and standard deviations of the 10 simulation results from each condition are reported. m.5.2.2. Results 68 Table 6. Examples of stimulus-prestimulus matrix, (a) PSV = 05 : a theoretical value of U(R n:S n.! | S J = 1.328 bits Prestimulus category Stimulus 1 2 3 4 5 6 7 8 9 10 1 .420 .336 .173 .057 .012 .002 .000 .000 .000 .000 2 .258 .342 .258 .111 .027 .004 .000 .000 .000 .000 3 .080 .240 347 .240 .080 .013 .001 .000 .000 .000 4 .004 .054 .242 .399 .242 .054 .004 .000 .000 .000 5 .000 .001 .027 .233 .479 .233 .027 .001 .000 .000 6 .000 .000 .001 .027 .233 .479 .233 .027 .001 .000 7 .000 .000 .000 .004 .054 .242 .399 .242 .054 .004 8 .000 .000 .000 .001 .013 .080 .240 .347 .240 .080 9- .000 .000 .000 .000 .004 .027 .111 .258 .342 .258 10 .000 .000 .000 .000 .002 .012 .057 .173 .336 .420 (b) PSV = 1.5 : a theoretical value of UCR^S^JSJ = 0.162 bits Prestimulus category Stimulus 1 2 3 4 5 6 7 8 9 10 1 .168 .164 .152 .135 .113 .091 .069 .050 .035 .023 2 .153 .158 .153 .140 .120 .096 .072 .051 .034 .021 3 .130 .146 .152 .146 .130 .106 .079 .055 .035 .021 4 .093 .123 .145 .154 .145 .123 .093 .063 .038 .021 5 .047 .082 .122 .155 .168 .155 .122 .082 .047 .021 6 .023 .047 .082 .122 .155 .168 .155 .122 .082 .047 7 .021 .038 .063 .093 .123 .145 .154 .145 .123 .093 8 .021 .035 .055 .079 .106 .130 .146 .152 .146 ,130 9 .021 .034 .051 .072 .096 .120 .140 .153 .158 .153 10 .023 .035 .050 .069 .091 .113 .135 .152 .164 .168 69 This section consists of two parts. The first part reports the simulation resulting from using only one of the three matrices at a time in the simulations to check the convergence of estimates of information measures to their theoretical values. The second part reports the simulation results from using combinations first of only S„xRn and S n.!XR n matrices and then of all of the three matrices. For the sake of space, I will present only the small portion of results that are absolutely necessary for my argument in this thesis. More completed results are discussed in Mori & Ward (1991b) and are available from either author. A. Simulations with a single matrix A . l . S n xR n matrix only. The s.d.'s of S„xRn matrix used were 0.1, 0.3, 0.5, 0.8, 1.0, and 1.5; the theoretical value of U(R n:S n) decreases as s.d. increases. Figures 11 to 13 present estimates of five information measures as a function of the number of trials per stimulus, with s.d. = 0.3 and 1.0. In Figure 11, horizontal lines striking through symbols correspond to theoretical values of U(R n:S n) for a given number of categories and values of s.d. When s.d. = 0.1 or 0.3, the theoretical values of U(R n:S n) are close to those of U(R n ) , which is the maximum value U(R n:S n) can take for a given number of categories (see Chapter II), and the estimates hardly showed any inflation even with a small number of trials (Figure 11). As s.d. increased and the theoretical value of U(R n:S n) decreased, the estimates were generally inflated when a small number of trials (10-30) was used, and they approached their theoretical values after 60-120 trials per stimulus were used. The amount of inflation was large when a large number of categories was used and/or the theoretical value of U(R n:S n) was relatively low, but the amount of inflation never exceeded more than 0.4 bits. Although the standard deviations of 10 simulation increased with the number of categories used, they were relatively small, specially when the number of trials per stimulus was over 120. s . d . - 0 . 3 U(Rn:Sn) to 3 5 z 2 o 5 1 h s - a - e - B - s - e - e - e - B B B « ~ B -© - © - © - © - © © H © - © - • © - - © - - © - -oc 0 2 2.0 0 1.5 1 1 0 < 0.5 0.0 U(R n:S n-1 ISi) s . d . - 1 . 0 U(Rn:Sn) SH..B..S..S..B..B-H--S--B--B B—--B—a-Categories • 4 O 6 • 10 • 16 2.0 1.5 1.0 0.5 > m m m see 2 0 0 400 6 0 0 8 0 0 1000 0.0 U(Rn:Sn-1 ISi) SI f© • ° B • a a a l l i i f M • • • Vertical bars indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 NUMBER OF TRIALS PER STIMULUS Figure 1 1 . The results of simulations with Sn X Rn matrix only ( 1 ) IL 0 1.5 H 1 1 0  < 0.5 U(Rn:Rn-1 ISh,Sn-"i) 0.0 jw s »s • •»n a • • • • 2.0 15 1,0 0.5 S . d . - 1 . 0 U(Rn:Rn-1 ISh) Categories • 4 • • i O 6 • 10 fen • 16 U(Rn:Rn-l ISh,Sn-l) a a Vertical bars Indicate standard deviations 200 400 600 800 1000 0 200 400 600 800 1000 NUMBER OF TRIALS PER STIMULUS Figure 12. The results of simulations with Sn X Rn matrix only (2) (0 z o CC O LL U. O =5 o 5 4 3 2 s . d . - 0 . 3 U ( R n : S n a , S h - l ' , R n - i") ^ B B B H H B B B B B B B a • ®©@©eeeee©@ e e 0 4 3 2 1 0 s . d . - 1 . 0 U ( R n : R n , S n - 1 , R n - l ) a s 1 E B e B B B B H 5 © @ © © @ © @ © © © © -Categories • 4 O 6 • 10 • 16 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Vertical bars Indicate standard deviations 13. The results of simulations with Sn X Rn matrix only (3) Since only a S n xR n matrix was used in the simulations, theoretical values of U(R n:S n . j |S„), U(R n :R n . , ISJ, and U(R„:R n. 1|S n,S n. 1) are all zero. When s.d. = 0.1 or 0.3, the estimates of those information measures were almost zero from the point of 10 trials per stimulus (Figures 11 and 12). Remember that those information measures are components of the residual of response information [U(R„)] after the effect of S n [U(R n:SJ] is excluded (see Chapter II), and that the estimates of U(R n:S n) were very close to U(R n) with s.d. = 0.1 or 0.3. Thus no amount of U(R n) was left for the other information measures. For larger s.d.'s, where U(R n:S n) was small, estimates of all the three information measures were inflated for small numbers of trials. Although they approached zero as the number of trials increased, the amount of inflation and the number of trials for the estimates to reach zero was larger as the number of categories increased. U(R n:R n . 1 |S n ,S n . 1) seemed to approach zero more slowly than U(Rn:SB.11SJ or U(R n :R n . , | S J did and never reached zero in some cases (Figure 12). Since U(R n:S n,S n.„R n. 1) measures the net effect of Sn, S„. j , and Rn., on R n and among them only the effect of S n [U(R„:SJ] was manipulated in the simulations here, the theoretical value of U(R n:S n,S n. 1,R n. 1) is equal to that of U(R n:S n). For small numbers of trials per stimulus, estimates of U(R n:S n,S n.i,R n. 1) were all very close to U ( R J (Figure 13). As the number of trials increased, the estimates decreased to their theoretical values. Like U(R n:S n), the amount of inflation and the number of trials for the estimates to reach their theoretical values were larger as the number of categories increased. A.2. S n.!XR n matrix only. The PSV's of S n.,xR n matrix used were 0.3, 0.4, 0.5, 0.7, 1.0, and 2.0; the theoretical value of U(R n:S n. 11SJ decreases as PSV increases, but it does not change very much with the number of stimulus-response categories. Figures 14 to 16 present estimates of five information measures as a function of the number of trials, for 6 and 16 stimulus-response categories with PSV = 0.3, 0.7, and 2.0. The standard deviations of all the estimates were again very small and almost negligible for more than 120 trials per stimulus. In Figure 14, theoretical values of U(R n:Sn.j | S J are shown by horizontal lines (PSV = 0.3, 0.7, and 2.0 from top to bottom). Like U(R„:Sn) of simulations with the S^R^ matrix only (A.l), estimates of U(R n:S n. 11SJ approached theoretical values as the number of trials increased, and it took more trials to reach the relatively low theoretical values than it did to reach high theoretical values (Figure 14). As can be readily seen, the inflation problem was more serious for a large number of categories (e.g., 10 or 16) than for 74 a small number of categories (e.g., 4 or 6). The estimates were more inflated and they took more trials to reach their theoretical value for 16 categories than for 6 categories, and some estimates for 16 categories seemed never to reach their theoretical values (PSV = 2.0; filled square). Also note that some estimates of U(R n:S n. 11S J for a small number of categories were underestimated for a very small number of trials per stimulus (PSV = 0.3; filled circle). In these simulations, theoretical values of U^R^SJ , U(R n:R n . 1 |S I l), and U(R n :R n . , |Sn,Sn-i) are all zero-Estimates of U(R 0 :SJ were inflated for a small number of trials, and the amount of inflation was large when a large number of categories (e.g., 16) was used (Figure 14). The pattern was essentially the same for estimates of U(R n:R n.] ISJ, but the inflation was much more severe than for L ^ R ^ S J , so severe U(R n:R n . 11SJ did not reach zero at all when 16 categories were used (Figure 15). Notably, the values of PSV had very little effect on the estimates of both U(R„:SI1) and U(R„:Rn.1 \SJ. However, estimates of U(R n :R o . 1 |S n ,S 0 . 1 ) showed a very different pattern from those of U(R n:S a) and U(R n :R n . 11SJ (Fig. 13). The estimates of U(Rn:Rn.11 Sn,S„.,) first increased with the number of trials then decreased slowly. This 'up-turn' pattern was more noticeable for a large number of categories and/or a larger value of PSV. As a result, the estimates of U(RD:Rn.11 Sn,Sn.,) were generally more inflated than those of U(R n :SJ or U(R n :R n . 11SJ, and the estimates with large PSV's and large numbers of categories did not reach zero even when 900 trials per stimulus were used. The theoretical value of U(R n:S n,S n. 1,R n. 1) is equal to that of U(R n:S n., | S J in the simulations here [for a reason similar to that discussed previously for the simulation by a S n xR n matrix only (A.l)]. The estimates were close to U(R n ) for a very small number of trials used, and decreased to their theoretical values as the number of trials increased (Figure 16). Thus, like those of U(R n:S 0 . 1 |S n), the estimates of U(R n:S n,SD.„R n. 1) were more inflated and more trials were required to reach asymptote when the theoretical values were low than when the theoretical values were high. When the theoretical values were very small relative to the maximum values (e.g., 10 and 16 categories), the estimates did not reach them even when the estimates were based on 900 trials per stimulus. 2 . 0 1 . 5 h S 1.0 h z 2 0 . 5 6 c a t e g o r i e s U ( R n : S n ) — U . O IT i L or n o LJL* CC 0 . 0 2 3 U ( R n : S n - l IQi) A <£-® i» ft a ®-Q.@- -@- J $ -PSV • 0.3 O 0.7 • 2.0 L U ( R n : S n - 1 IS," 1 Y ® ® e ®@ © e e J • • *• - -Vertical bars Indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 14. The results of simulations with Sn-1 X Rn matrix only (1) 6 c a t e g o r i e s U ( R n : R n - l l S h ) _ 2 co I* CC 0 2 3 UL o Z 3 O 5 1 < U ( R n : R n - 1 I S h , S n - l ) ® 1 6 c a t e g o r i e s I U(Rn:Rn"-11%) PSV • 0.3 O 0.7 • 2.0 Vertical bars Indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 15. The results of simulations with Sn-1 X Rn matrix only (2) 5? (0 •M JO 5 o IL z IL o o s < 6 c a t e g o r i e s 1 6 c a t e g o r i e s 5 4 h U ( R n : S n , S n - l , R n - l ) 3 r 2 1 e J PSV • 0.3 O 0.7 • 2.0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Vertical bars Indicate standard deviations F i g u r e 1 6 . T h e r e s u l t s o f s i m u l a t i o n s w i t h S n - 1 X R n m a t r i x o n l y ( 3 ) 78 A 3 . R n . ,xR n matrix only. The PRV's used for the R^xR,, matrix were 03, 0.4, 0.5, 0.7, 1.0, and 2.0, and the theoretical value of U ( R n : R n . 1 | S J decreases as PRV increases. Since no previous stimulus effect was implemented in these simulations, theoretical values of (R„:Rn.i | Sn,Sn.,) should be equal to those of U(R n :R n . , |S n ) . Figures 17 to 19 present estimates of five information measures as a function of the number of trials, for 6 and 16 stimulus-response categories with PRV = 0.3, 0.7, and 2.0. In Figure 18, theoretical values of (R 0:R n .i |S n ,S n; 1) and U(R n:R n., | S J are shown by horizontal lines (PRV = 03, 0.7, and 2.0 from top to bottom). When small PRV's were used (e.g., 03, 0.4), most estimates of U(R n :R n . 1 | S J hovered around the level below their theoretical values, and the standard deviations were large for a small number of categories used (filled circles in Figure 18). This 'aberrant' pattern was due to the fact that those PSV's created very strong correlations between successive responses, so strong for a small number of categories that the same response category was repeated in a long sequence, resulting in low response information [U(R„)]. The repetition of the same response was particularly devastating for simulations with a small number of trials, for it would cause zero response information. In such cases, the estimates of U(R n :R n . 1 | S J would be lower than their theoretical values, because U(R n:R n. 11SJ is limited by U(R n ). If the response category is occasionally changed in a sequence, it would raise the amount of response information abruptly from zero, which caused large standard deviations. As the number of categories increased, those problems became less severe because different response categories were chosen more frequently in one simulation run. For larger PRV's (e.g., 0.7, 2.0), correlations among responses were not strong and response categories were altered frequently, resulting in high or maximum response information. In such cases, the estimates of U(R n :R n . 1 |S n ) showed the same pattern as U(R n:SD. 1 |S n) did in simulations with S n . iXR n matrix only: they took more trials to reach relatively lower theoretical values than they did to reach higher theoretical values; they were more inflated and they took more trials to reach their theoretical value for 16 categories than for 6 categories, and some estimates for 16 categories (e.g., PRV = 2.0) seemed never to reach their theoretical value. 2 . 0 1 .5 To 5 1 . 0 6 c a t e g o r i e s i i i U ( R n : S n ) IL O U ( R n : S n - l I S i ) ^ t t i i m a s s PRY • 0.3 O 0.7 • 2.0 U ( R n : S „ - i i Vertical bars Indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 17. The results of simulations with Rn-1 X Rn matrix only (1) 6 c a t e g o r i e s _ 2 to U ( R n : R n - 1 ISh) oc o s * i ct <3> ©~@ @ © @ @ IL O r- 2 h Z 3 O 3 1 U ( R n : R n - 1 I S h , S n - 1 > W i l l i ft r • • • • |L r r * «• - -* * * -gr 1 6 c a t e g o r i e s U ( R n : R n - 1 ISh) 0 3 ® © T5~ PRV • 0.3 O 0.7 • 2.0 U ( R n : R n - 1 I S i , S n - t ) Vertical bars indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 18. The results of simulations with Rn-1 X Rn matrix only (2) CO 3 £ 5 6 c a t e g o r i e s cc o IL IL O z o 4 h 3 U ( R n : S n , S h - i , R n - 1 ) 1 E I 4 . ^ $ f @ @ @ @ © © © 4 J • • • o i i i 1 i 1 6 c a t e g o r i e s PRV • 0.3 O 0.7 • 2.0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Vertical bars indicate standard deviations Figure 19. The results of simulations with Rn-1 X Rn matrix only (3) 00 As in simulations with the Sn.,xRn matrix only, the estimates of U(Rn:Rn.11 S^S,^) showed an 'up-turn' pattern (Figure 18). For smaller PRV's and categories (e.g., PRV = 0.3 and 6 categories), the estimates were kept low and the standard deviations were relatively larger, presumably for the same reason as described above for those of U(R n:R n. 11SJ. In other cases, the inflation of the estimate was substantial, and they did not reach or even come close to their theoretical values, especially when a large number of categories was used. Estimates of both TJ(R„:Sn) and U(R n:S n., | S J , whose theoretical values are zero, were inflated for a small number of trials, and the amount of inflation was large when a large number of categories was used and/or a large P R V was used [that is, the theoretical value of U(R n:R 0 . 11SJ was small] (Figure 17). Also, the amount of inflation was larger for the estimates of U(R n:S n . I | S J than for those of U(R n:S n). As observed in previous sections, estimates of U(R n:S n,S n. 1,R n. 1) approached those of U(R n:R n . 11SJ very slowly, and some of them (with a large number of categories) seemed to keep decreasing and never to reach an asymptotic level even when 900 trials per stimulus were used (Figure 19). B. Simulations with a combination of matrices B . l . S n xR n and S n.,xRD matrices The values of s.d. and PSV used in these simulations were the same as those used in the previous simulations, except for the omission of s.d. = 0.1, which generates a maximum value of U(R„:SJ [equal to U(R n)] and thus does not allow the manipulation of S^XR,, matrix to have any effect. Figures 20 to 22 present estimates of five information measures as a function of the number of trials per stimulus, for 6 and 16 categories with s.d. = 03 and 1.0 and PSV = 0.3 and 2.0. As observed in most of the previous simulations reported here, standard deviations of the estimates were very small. Estimates of all the information measures reported here were more or less determined by the size of s.d. and PSV. Like those of U(R n :SJ with the S n XR n matrix only, the estimates of U(R n:S n) decreased as the value of s.d. became larger, in the same way their theoretical values do (Figure 20). Although all the estimates of U(R n:S J reached their asymptotic levels as the number of trials increased, their asymptotic level was equal to their theoretical value only when PSV = 2.0 (squares in Figure 20), and the other estimates decreased as PSV decreased. This was due to the fact that the smaller the PSV, the stronger the effect of S,,.! on R n , which lowered U(R n:S n). Although inflation of the estimates was observed for all numbers of 83 categories tested here, U(R n :SJ reached its asymptotic values for a relatively small number of trials (60-120 trials per stimulus). Estimates of U(R n :S n . 1 |SJ were larger as larger s.d. and/or smaller PSV were used (Figure 20). The estimates were also inflated with a small number of trials used, and they decreased to asymptotic levels as the number of trials increased. The asymptotic levels, however, were not the theoretical values calculated solely on the basis of PSV, unless s.d. was equal to or larger than 1.5. The size of s.d. seems to determine the uppermost level of the estimates of U(Rn:Sn.11SJ, which were obtained with PSV = 0.3 (circles in the figure); they decreased as s.d. decreased. This is predictable because UXR^SJ and U(R n:S n . , | S J were two of the components making up response information and they would be inversely related given a constant amount of response information. With a given s.d., the asymptotic levels of estimates of U(Rn:Sn.j | S J were lowered as PSV increased. The number of trials for U(R n:S n., \SJ to reach an asymptotic level was equal to or sometimes fewer than those with a SB. 1xR n matrix only, because the estimate from 10 trials per stimulus, which was the highest estimate with given parameters,was lower in most of simulations here than in those with a S n.,xR n matrix only. Whereas in the simulations with a S n . 1 xR n matrix only the estimates from 10 trials per stimulus were roughly the same or close irrespective of the size of PSV, in the simulations with S n xR n and S„.1XR0 matrices they were lower for smaller PSV. For the same reason, the amount of inflation was generally smaller in the simulations reported here than those with a S n . 1xR n matrix only. Although estimates of U(Rn:R„.1 ISJ were generally smaller than those of U(RD:S n. 11SJ, they showed a very similar pattern to those of U(R n:S n. 1 |S n) described above; the estimates were larger as larger s.d. and/or smaller PSV were used (Figure 21). This is probably because stimuli and responses were highly correlated in most simulations, except those with a smaller number of categories and smaller PSVs (e.g., 6 categories and PSV = 0.3), where the estimates were very small and decreased to near zero as the number of trials increased. CO 3 5 cc O LL UL O H Z O < 6 c a t e g o r i e s 2 w U(Rn:Sn-i l%> < j > © @ ® @ @ © © © © © e 0) i l l • • • 5 4 3 F 2 1 6 c a t e g o r i e s U(Rn:Sn) 0 3 ? ™ H B B B B B B B B B B B . ^ 9 9 0 s e 9 9 0 9 © © © U(Rn:Sn-l ISi) Ii ®®@©©@©©© © © © 0 200 400 600 800 1000 B B I I i i s.d. PSV • 0.3 0.3 • 0.3 2.0 o 1.0 0.3 • 1.0 2.0 Vertical bars indicate standard deviations 0 200 400 600 800 1000 NUMBER OF TRIALS PER STIMULUS Figure 20. Simulations with Sn X Rn and Sn-1 X Rn matrices (1) t 2.0 1.5 co 6 c a t e g o r i e s U ( R n : R n - l l S r i ) 2 0.5 h ec o.O 2 2.0 Z IL O 1.5 H Z 2 i.o o 5 < 0.5 0.0 m ® • g fi f U ( R n : R n - 1 I S h , S n - 1 > ksai§§g88s § i § 2.0 1 6 c a t e g o r i e s 1.5 \ 1 . 0 h U ( R n : R n - 1 I S O OD 0.5 0.0 2.0 a a B a B B B B B B 1.5 1.0 0 . 5 $ 0.0 U ( R n : R n - 1 I S h , S n - l ) So © © © • • s e s.d. PSV • 0.3 0.3 • 0.3 2.0 O 1.0 0.3 • 1.0 2.0 Vertical bars indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 21. Simulations with Sn X Rn and Sn-1 X Rn matrices (2) 00 CO A ~ 5 z o o u. z UL o Z o s < 6 c a t e g o r i e s U ( R n : S n , S n - 1 , R n - l ) fe: 2 [ ^ • • i i i i i i i i i i B s = 1 h H e e a e a e H H e 1 6 c a t e g o r i e s U ( R n : R n , S n - l , R n - l ) § i l t l | e e e g s.d. PSV • 0.3 0.3 • 0.3 2.0 O 1.0 0.3 • 1.0 2.0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N U M B E R O F T R I A L S P E R S T I M U L U S Vertical bars indicate standard deviations Figure 22. Simulations with Sn X Rn and Sn-1 X Rn matrices (3) Since UfR^R^lS^S,,. ,) excludes the effects of not only S n but Sn.„ it is predicted that the estimates would be smaller than those of U(R n:R n., |SJ , which reflected high correlations between stimuli and responses. The prediction seems to be confirmed (Figure 21); although most of the estimates of U(R n:R n. 11 Sn,Sn.,) showed an 'up-turn' pattern as observed in other simulations, they were generally smaller than those of U(R„:RB. l |S n). This is particularly true for those with smaller s.d.'s and P R V ' s (s.d. = PRV = .3), where the estimates of U(Rn:R„.11SJ stayed at non-zero levels but those of U(R n:R n . , | Sn,Sn.,) were almost zero even at 10 trials per stimulus. In addition, the size of PSV had little effect on them (e.g., 16 categories). Like other simulations reported above, estimates of U(R n:S n,S n. 1,R n.,) were also inflated with a small number of trials, and they decreased to their asymptotic levels as the number of trials increased (Figure 22). The amount of inflation and the number of trials for the estimates to be asymptotic were larger as the number of categories was increased and/or the asymptotic levels were lower. B.2. Combination of all three matrices It was predicted from the results of previous sections that simulations with a combination of S n xR n , S^ xR n , R n l x R n matrices would yield estimates of information measures that changed in a systematic way as the values of s.d., PSV, and PRV change. Thus, instead of exhausting all possible combinations of the values of the three parameters, I decided to use parameter values with which simulations are predicted to (1) generate estimates of information measures in the most demonstrative manner, and (2) yield estimates that would capture the characteristics of empirical data (e.g., Experiments 1 and 2 of this thesis). Figures 23 to 27 present estimates of the five information measures for 6 and 16 categories, both with s.d. = 0.3, 1.0, PSV = 0.3, 2.0, PRV = 0.3, 0.7, and 2.0 (in cases where there was no substantial difference in estimates between PRV =, 0.7 and 2.0, the estimates with PRV = 2.0 are omitted from the presentation for the sake of clarity of the figures). As observed in the previous sections, standard deviations of information measures are generally relatively small, especially after 120 - 180 trials per stimulus. For a given number of categories, estimates of U(R n:S n) were large when (1) s.d. was small, (2) PSV was large, and/or (3) PRV was large (Figure 23), which was predicted from the results in previous sections. Although the value of s.d. had a more substantial effect on the estimates than that of PSV or PRV did, the degree to which the parameters had dominant effects on the estimates depended on the values of the other 88 parameters. For example, when PSV = 0.3, there was no room for PRV to have a very powerful effect on U(R n:S n) (filled symbols). As in the previous simulations, estimates of U(R n :SJ were inflated with a small number of trials, although the amount of inflation was rather small (0.1 - 0.3 bits) and U(R n :SJ reached asymptotic levels with 30-60 trials per stimulus. For a given number of categories, estimates of U(R n:S n . 11SJ and U(R n:R n . , | Sn) were generally large when (1) s.d. was large, (2) PSV was small, and (3) PRV was small [Figure 24 for U(R n:S n., | Sn), and Figure 25 for U(R n:R n . 11SJ], which was again consistent with results from simulations described in previous sections. Thus most of the estimates of U(R n:S n., | S J and U(R n:R n., |S I 1) were inversely related to those of U(R n:S n) with given parameter values. When PSV was small for a small number of categories (e.g., PSV = 0.3 for 6 categories), the values of s.d. and PRV had little effects on the estimates of U(R n :S n . 1 1SJ (filled symbols in Figure 24). As the number of categories increased (e.g., 16 categories), the estimates of U(R n:S n-i | Sn) varied with the values of s.d. and PRV. In most cases, the estimates did not change substantially when PRV increased from 0.7 (triangles) to 2.0 (squares). The estimates of U(R n :R n . , | S J showed a pattern similar to those of U(R n :S n . 1 |SJ , particularly when s.d. was small (Figure 25). This is because U(R n:S n) was relatively large with values of s.d.'s used here, which resulted in relatively high correlations between Sn.j and R n . j . When a large value of s.d. was used for a small number of categories (e.g., s.d. = 1.0 for 6 categories), PRV had much greater effects on the estimates of U(R n :R n . 1 1SJ than PSV did. Most estimates of both U(R n :S n l | S n ) and U(R n :R n . 1 1SJ were inflated with a small number of trials and decreased to their asymptotic levels. The amount of inflation increased greatly as the number of categories increased; for 16 categories, some estimates dropped from about 1.5 bits to almost zero (open squares at s.d. = 1.0 in Figures 24 and 25). For a small number of categories, however, some estimates of U(R„:RD. 11SJ increased as the number of trials increased and hovered around a relatively high level (open circles for 6 categories in Figure 25). They also showed relatively large standard deviations. As observed in the simulations with the R^xR, , matrix only, this is caused by the fact that those parameter values, specially very small PSVs, created very strong correlations between successive responses, so strong for a small number of categories that the same response category was often repeated for long sequences. s . d . CO 3 5 s or. O u. IL o H Z 3 O 2 < ^ 2 0 . 3 6 c a t e g o r i e s B S B S B B B B B B H B A A A d s A A A A A A A A ^ feiSttSISSSS I 2 S 0 5 4 3 2 j-1 • 1 6 c a t e g o r i e s A A A A A A A A A A A A fift£fi£fiffi£ft £ ft 4 3 h 2 1 0 s . d . * 1 . 0 • i i 6 c a t e g o r i e s a a s a a i B B S B m B e A A A A A A A A A A A A A ^ A A A ^ A 1 6 c a t e g o r i e s a B B AAA B B B B B A A A A A B B B A A A keeeeeeeee * • • PSV PRV • 0.3 0.3 A 0.3 0.7 • 0.3 2.0 o 2.0 0.3 A 2.0 0.7 • 2.0 2.0 Vertical bars indicate standard deviations 0 200 4 0 0 6 0 0 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 23. Simulations with three matrices (1). U(Rn:Sn) 00 <$(D@<I><i>®@©<§ © @ © A A A A A A A A A A A A 0.0 fafa B i i B i B e a i a I a £ 2.0 Z IL 0 1.5 h 1 1 0 fc s « 0 . 5 1 6 c a t e g o r i e s £ ± A * A A A A A A A • « © © © © © © © © © © © 0.0 * M l A A ' A A A ' A A A A A A A 1.5 1.0 0.5 0.0 2.0 1.5 S.d. " 1 . 0 6 c a t e g o r i e s <» U ™ * 4 A ± 4 « A A A 1 6 c a t e g o r i e s 1.0 0.5 H o.o * * * * * * * * * A A A P •••••••• • • • CD ®®@©e©© © © © a A A A A A A A A A 9 B H B B H B A A PSV PRV • 0.3 0.3 A 0.3 0.7 • 0.3 2.0 o 2.0 0.3 A 2.0 0.7 • 2.0 2.0 Vertical bars Indicate standard deviations 2 0 0 400 6 0 0 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 24. Simulations with three matrices (2). iKR^Sn.-ilS,,) s 1.5 , 101" § 0.5 s . d . • 0 .3 i i i 6 c a t e g o r i e s S I S E A A & A A A A A A A A A CC 0.0 a =* a ° ° *=••-> ™ •* " * n £ 2.0 U. O 1.5 h 1 6 c a t e g o r i e s > a o ^ I A A ^ A A A A ® @ @ © @ 0 © © @ © © @ 0 2 0 0 400 6 0 0 8 0 0 1000 1.5 1.0 0.5 0.0 2.0 1.5 1.0 0.5 0.0 S . d . - 1.0  i i i i 6 c a t e g o r i e s <|<]>$$$<£<|)<D<I> CD <£ Q J " ' B I I I I I I I I I I 1 6 c a t e g o r i e s 4 * * A * * A A A A A A m A A N A A A A H A A A A A A A A H g H - a . a g g J L J L PSV PRV • 0.3 0.3 A 0.3 0.7 • 0.3 2.0 o 2.0 0.3 A 2.0 0.7 • 2.0 2.0 Vertical bars Indicate standard deviations 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 25. Simulations with three matrices (3). U(R n:Rn-ilSn) 1.5 s . d . - 0.3 6 c a t e g o r i e s ~ 1-0 h CO Z 0.5 O cc o . o * a * £ 1.5 z LL O H 1.0 H z 3 O ? 0.5 |-1 6 c a t e g o r i e s 0,0 llff ft i t s s sa s s s see s . d . - 1.0 i i i 6 c a t e g o r i e s flOD Q> <£ <D * * * * * * * A A A PSV PRV • 0.3 0.3 • 0.3 0.7 • 0.3 2.0 o 2.0 0.3 A 2.0 0.7 • 2.0 2.0 Vertical bars indicate standard deviations 0 2 0 0 4 0 0 600 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 26. Simulations with three matrices (4). U(Rn:Rn-ilSn,Sn-i) VO t o ^ 2 to 4-* 1 1 2 CC 0 2 5 IL O H Z 3 O s < s . d - - 0 . 3 • i i 6 c a t e g o r i e s S B S B S B B B H 8 S i i O I I I I I 1 6 c a t e g o r i e s 4 A A A A A A A A A A A A f i § i S * 8 8 § § 8 • i 8 3 h 2 1 s . d . - 1 . 0  i i i i 6 c a t e g o r i e s fc. * * * A A A A A A A A A 1 [K> 0 5 <D <T) ffi © ® <D © A A A A A A A 1 6 c a t e g o r i e s 4 2 1 0 " " 2 2 2 2 I PSV PRV • 0.3 0.3 A 0.3 0.7 • 0.3 2.0 o 2.0 0.3 A 2.0 0.7 • 2.0 2.0 Vertical bars Indicate standard deviations 2 0 0 4 0 0 6 0 0 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 27. Simulations with three matrices (5). U(Rn:Sn,Sn-i,Rn-i) 94 When s.d. was small (e.g. s.d. = 0.3), estimates of U(R n:R n . , |S„,Sn-i) w e r e much smaller than those of U(R 0 :R n . 1 |S n ) (Figure 26), because U(R n:R n., |Sn,Sn-i) excluded the effects of S^,. When s.d. was large (e.g., s.d. = 1.0), the estimates were about as large as those of L ^ R ^ R ^ J S J , and they showed the up-turn pattern, as those in previous simulations. Like those of U(RB:S I 1), estimates of U(R n:S n )S n. 1,R n. 1) were mainly determined by values of s.d., although those of PSV and PRV had some effects on them (Figure 27). In general, the estimates decreased to their asymptotic levels. Compared with those from the other simulations reported earlier, the asymptotic levels were high, so that the amount of inflation and the number of trials for the estimates to reach asymptote were small. III.5.2.3. Comparison of simulated and empirical information measures In this section, I compare information measures of simulation data generated by the matrix-manipulation method and those of the pooled data from Experiment 1 (no mask conditions only; see Figure 10) and the single-observer data from Experiment 2 (Figure 9). Simulations were done using a combination of S nxR n , S n.,xR n, and R n . ,XR n matrices. For each number of categories, one set of the values of s.d., PSV, and PRV were chosen from a variety of different sets, and among them it gave the best fit by eye simultaneously to all of the five empirical information measures reported here. This "fitting-by-eye" procedure was also used by Houtsma (1983) and Tan et al. (1989) to correct the inflated estimates of UfR^tSJ. It should be ideal to use a more objective method to choose a set of parameters with which the simulation results fit the empirical ones (e.g., least square fitting), although it is quite difficult to develop such a method because five different curves should be fitted simultaneously. I will discuss this issue later in this chapter. Figures 28 to 31 superimpose estimates of U(R„:SJ, U(Rn:Sn.11SJ, U(R„:Rn.1  S^S,,.,), and U(R n:S n,S n. 1,R n. 1) from the empirical data on those from the simulation data for each number of stimulus-response categories used, with parameters used in the simulations (s.d., PSV, and PRV) indicated in the legends [estimates of U(R n:R n . , | S J were omitted because they were always quite similar to those of U(R n:S n. 1 |S 1 1)]. Overall, the estimates from the simulation data seem to fit the empirical data quite well. The fit is specially good for the estimates of U(R n :SJ and U(R n:S n,S n. 1,RB. 1) with all of the numbers of categories used (except 16-category data in Experiment 1). For U(R n:S n), this is simply because there is no large inflation of the estimates reported here. The estimates of U(R n:S n., | S J [and U(R n:R n_, | SJ] from simulation data fitted 95 the empirical data reasonably well, especially for 4 and 6 categories (Figures 28 and 29). Although the fit of U(R n :S n . 1 |SJ is not particularly good for 10- and 16-category data, the estimates from the simulation data decrease with increasing number of observations, in a way very similar to those from the empirical data (Figures 30 and 31). However, there are some discrepancies between the estimates of U(R n:R n . , |Sn,Sn.,) from the simulation data and those from the empirical data. For the results from 4- and 6-category conditions, both simulation and empirical results showed almost no inflation for U(R n:R n . , | Sn,S0.,) (Figures 28 and 29). This is simply because the absolute identification task with those small number of categories was very easy in the experiments reported here and the subjects' performance were close to perfect, so that there were few errors from the very first data point [see also estimates of U(R n:S n. 11SJ and U(R n :R n . j ISJ]. As the number of categories increased, the inflation of U(Rn:Rn.11 S^S,,.,) becomes evident, and the discrepancies between the estimates from the simulation data and those from the empirical data increase (Figures 30 and 31). This is specially so for the estimates from the pooled data (open circles in the figures). In terms of the number of categories used, the fitting of the simulation estimates to the empirical ones is reasonably good for 4 and 6 categories. This is simply due to the fact that there is no large inflation of any of the information measures in the experiments reported here. However, as the number of categories increases, the estimates of the information measures are inflated, and it becomes difficult to find a set of parameters for the simulation with which all of the information measures are inflated in the same way as are those from the empirical data. It is also important to note that the estimates from the simulation data [except U(Rn:S n)] keep decreasing after the maximum number of observations obtained in the empirical data (300 - 460 observations per tone). The amount of decrement is larger with a large number of categories used (ejg., 16). When I discussed the results of Experiment 2 and the pooling method, I concluded that the information measures seemed to be reaching asymptotic values at the maximum number of observations. However, the simulation results show that they would still decrease further by 0.2 to 0.5 bits. Therefore, in order to obtain an uniiiflated estimate of the information measures, it may not be sufficient to have each subject make a large number of observations or to use pooled data. (0 a 1 z o oc O 0 1.0 U ( R „ : S n ) J U L Q Q Q o Experiment 1 • Experiment 2 Z 0.8 O 0.6 H 3 0.4g-O < 0.2 h U ( R n : R n -1 I S h , S n - l ) 1.0 0.8 h 0.6 0.4 0.2 0.0 3 U ( R n : S „ - 1 I S i ) hSfff ign ' i T—r -U ( R n : R n , S n - 1 , R n - l ) 2 F Q Q S 5 Q _ _ : s.d. PSV PRV 0.3 2.0 2.0 0.3 2.0 1.0 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 0 200 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 28. Simulation and empirical results (1). 4 categories U ( R n : S n ) £ Q Q Q Q ; +* 3 1 w 1 o I • CC 1.0 o LL Z 0.8 IL ° 0.6 H 3 0.4 h O < 0.2 -o Experiment 1 • Experiment 2 0.0 U ( R n : R n -1 I S h , S n - l ) 9gft-nr-U ( R n : R n , S n - l , R n -1> ^ Q Q Q Q • ^ ^ — —— — — — s.d. PSV PRV 0.35 1.5 1.5 0.4 2.0 1.0 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 29. Simulation and empirical results (2). 6 categories vO U ( R n : S n ) A A W I Z o o Experiment 1 • Experiment 2 3 2 U ( R n : R n , S n - l , R n - 1 > ^ f t r f e — :  s.d. PSV PRV 0.5 2.0 0.8 0.6 2.0 0.9 0 2 0 0 4 0 0 600 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 30. Simulation and empirical results (3). 10 categories VO 00 (0 £ 1 o 5 o I L z IL o U ( R n : S n ) O Experiment 1 • • Experiment 2 o < 1 0 U ( R n : R n ,Sn-1 , R n - l ) s.d. PSV PRV 0.8 2.0 0.7 - - 0.95 2.0 0.6 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 N U M B E R O F T R I A L S P E R S T I M U L U S Figure 31. Simulation and empirical results (4). 16 categories 8 100 III.5.2.4. Discussion Simulations with the matrix-manipulation method was quite successful in that simulations with a single matrix generally yielded estimates of information measures that eventually reached their theoretical values calculated from the matrix with a given parameter value. This is a great advantage over the methods using a linear equation with an error term (e.g., Houtsma, 1983), which do not, at least explicitly, provide theoretical values of information measures with a given parameter(s). Although some estimates of U(RD:R n., | S J in simulations with the R„.1xRn matrix only did not come very close to their theoretical values, this is attributed to the fact that the manipulation of R n . 1 xR n matrix changed not only the value of U(R n :R n . 1 |S n ) but also that of U ( R J . While I did not calculate the theoretical values in simulations with a combination of matrices, the simulations yielded estimates of information measures that changed in a systematic way as the parameter values changed (Results Section B). From the results of the simulation, we can appreciate how much the information measures are overestimated, or inflated, with a given number of trials and number of stimulus/response categories, and roughly how many trials will be required for the estimates to reach their theoretical or asymptotic value. In most of the simulation estimates reported here, the standard deviations of 10 repetitions of each simulation were very small, which is consistent with Houtsma (1983). This is specially so when the number of trials is large. Although relatively large standard deviations of the estimates of U(R n :R n . 1 |S n ) and U(R n:R n. 1 |S n,S n_ I) were sometimes observed for a small number of categories (Figures 18 and 24), they can be attributed to strong correlations between successive responses, too strong to be observed in most empirical data. Given the small standard deviations even for a small number of trials, Houtsma (1983) argued that the major problem resulting from using a small number of trials to estimate the information transmission U(R n :SJ was not the variance of the estimate, but its inflation. A l l of the information measures reported here showed some amount of inflation in their estimates with a small number of trials. In general, the inflation of the estimates increases with (1) the number of categories used and/or (2) the number of variables involved in the calculation of the measure. It is clear from the simulations and empirical data that the inflation problem is more severe in a 16-category experiment than in a 4- or 6-category experiment. The inflation is larger for U(Rn:Sn,Sn.j,Rn.,) (where 4 variables are involved in the calculation) than U(R n:S n) (where only 2 variables are involved in the calculation). Miller's (1954) correction equation (Equation 22) also recognized the effects of the number of categories and the number of variables on the estimation of information measures, because df in the equation reflects both factors; df increases with the number of categories and/or the number of variables. As mentioned before, however, Miller's (1954) correction equation is not applicable to most empirical data, and it has a tendency to overcorrect, or underestimate, the information measures (Houtsma, 1983). In his simulations, Houtsma (1983) showed that Miller's (1954) equation was inappropriate for correcting the estimates of U(R n :SJ calculated from 125-category data. The simulation reported here indicates that the equation is also inappropriate when many variables are involved in the calculation of the information measures, such as the multivariate information measures reported here. For example, in the simulation with a R n xS n matrix only (Results Section A. l ) , the estimate of U(R n:S n )S n. 1,R n. 1) is 2.9 bits (s.d. = 1.0) for 10 categories with 30 trials per stimulus (Figure 13). Since df used in the equation is 8991 and N is 300, the corrected estimate turns out to be -18.7 bits! Therefore, although Miller's (1954) correction equation correctly recognizes the inflation of information measures caused by the number of categories and the number of variables involved in the calculation, its applicability is limited andjt should be used with caution. Houtsma (1983) found in his 125-category stimulations that the number of trials required for asymptotic estimates of U(R n:S n) was larger when the asymptote was relatively low than when it was high. The same pattern was observed in the simulations reported here for not only U(R n:S n) but also U(R n:S 0. 11SJ, U(R n :R n . 1 ISJ, and U(RB:S n,S n.„R n. 1). Although estimates of U(R n:S n) in our simulations reached their theoretical value or asymptote with 60-120 trials because the amount of inflation of the estimates were generally small, fewer trials were required to reach a relatively high asymptote than to reach a low asymptote. The dependency of the number of trials on the level of asymptote was more apparent for U(R n:S n., |S B), U(R n :R n . J S J , and U(R n:S n,S n.„R n. 1). Since the number of trials required for an asymptotic estimate was roughly correlated with the amount of inflation observed for the estimate, we can conclude that the number of trials required for an asymptotic estimate of information measures is dependent on the number of categories and the number of variables involved in the calculation. While estimates of most of the information measures monotonically decreased to their asymptotic levels as the number of trials per stimulus increased, those of U(R n:R n.! | Sn,Sn.,) showed a different pattern; they increased at first then decreased slowly. This 'up-turn' pattern is not an artifact of the simulations used here, because the estimates from the empirical data (Figures 30 and 31) also show the "up-turn" pattern. Although 102 I do not have a definitive answer to this puzzling pattern, I suspect that this is because U(Rn:R„., |Sn,Sn-i) >s actually the sum of two information measures, U(R n:R n . 1 |S n) and INTT(RI1:Sn.1,RD.1 | S J (Section III.2). As you have seen, U(R n:R n. 11S J monotonically decreases as the number of trials increased, while INT(Rn:Sn.i,Rn. ! | S J may behave in a different way. The combination of the two information measures that change in a different way as a function of the number of trial could possibly generate such an unexpected pattern as that ofU(R n :R n , |S n ,S n . 1 )-Finally, the estimates of the information measures from the simulation data seemed to fit reasonably well with those from Experiments 1 and 2 [except U(Rn:R„., | Sn,Sn-i)]- As mentioned before, both estimates from the simulations and from the empirical data clearly indicate that the estimates are inflated with a small number of trials, especially for 10 and 16 categories. In practice, the overestimation of U(R n:S n) would be small and negligible unless an unusual number of stimulus-response categories is used (e.g., Houtsma, 1983; Tan et. al., 1989). However, the overestimation of such multivariate information measures as those of Equations 17 and 20 cannot be ignored even if the number of stimulus-response categories is a usual one. III.6. Evaluation of correction methods In Section III.4, I reviewed three methods of obtaining an uninflated estimate of information measures. The simplest way is to use a large number of observations, either by increasing the number of observations for each observer or by pooling individual data in the way discussed in Section III.4.1. The others are to apply Miller's (1954) equation to the obtained information measures, and to estimate the amount of inflation from the results of computer simulations (e.g., Houtsma, 1983). In Section III.5, I reported results of a series of computer simulations from which the information measures of Equations 17 and 20 were obtained as a function of the number of categories and the number of trials. From the discussion in Sections III.4 and III.5, it is now certain that Miller's (1954) equation is inappropriate, at least for most information measures obtained from absolute identification experiments. An ideal alternative is to have each observer make a number of observations large enough for the information measures to reach asymptotic values (e.g., Tan et al., 1989; also Experiment 2 of this thesis). However, this is not always possible in empirical studies, specially when the data are analyzed by multivariate information measures which require a very large number of observations. In cases where only a marginal number of observations is available from each observer, the pooling method could assist in making 103 reasonable estimations of asymptotic values of information measures. However, I found in Section III.4 that it may not be sufficient to use the pooling method only, because the simulation results showed that estimates of some information measures would decrease further after the maximum number of observations available for the pooling method. The size of the decrement is substantial for a large number of categories (e.g., 16). Thus, as a practical way to obtain an uninflated estimate of information measures, I propose a combinatory use of the pooling method and the computer simulations. As I demonstrated in Section III.5.2, simulations should be done by a combination of S nXR n , S^xR^ and R^xR,, matrices, for which a set of parameter values (s.d., PSV, and PRV ) should be chosen in such a way that they give the best simultaneous fit to the empirical information measures by eye. From the fitted curve, a further decrement of the information measures can be estimated and the projected asymptotic values will be taken as "corrected" estimates of the "information measures. Before I apply this method to the information measures from the empirical data, I must mention three problems in the use of the correction method. First, the "fitting by eye" procedure is in a sense primitive and it may be better to use more objective procedures (e.g., least-square fitting). However, it is very difficult to develop such procedures for the studies reported in this thesis, because the procedures have to deal with three or more curve fittings of different information measures simultaneously. The fact that each of the three parameters (s.d., PSV, and PRV). in the simulation can change the estimates of all of the information measures makes the curve fitting more difficult. Since the fitting-by-eye procedure was successfully used in the correction of U(R n:S n) (Houtsma, 1983; Tan et al., 1989), I will use that procedure in this thesis and leave the development of more sophisticated procedures to a future project. Second, as you may have noticed in Section HI.5.2, the simulation results did not yield a reasonable fit to U(R n:R n . 11 S^S^) for 10 and 16 categories used, so it is difficult to estimate their asymptotic levels. In such cases, I will calculate U(R n:R n . 11 Sn.S'n.i) from the corrected estimates of other information measures by using the following equation U(Rn:Rn.11SD,S^) = U(R n:S n,S n. 1,R n. 1) - U(R n:S n) - U(R n:S n . 11SJ (23). The value of U(R n :R n . i | Sn,Sn.,) calculated by Equation 23 is thus an estimate from other estimates, and it may not be as accurate an estimate as would be obtained from a reasonable fit of simulation results. However, the estimates by Equation 23 should be better, I think, than uncorrected estimates or those from an unsatisfactory fit. Therefore throughout the rest of this thesis, I will use Equation 23 when I am not able to obtain a reasonable fit of simulation measures to empirical values of U(R n:R n. 11 Sn,Sn-i)> and I will make a note of using it. Third, since mdividual subjects' data are pooled to obtain an uninflated estimate of the information measures, commonly-used statistical tests, such as t tests and analyses of variance, are not available to interpret the results. In previous studies, Miller's (1954) transformation equation (Equation 21) of information measures to L 2 statistics was used to test the significance of information measures (McGill, 1957). However, my simulation results (also Houtsma, 1983) have shown that Miller's (1954) correction equation (Equation 22), which was derived from Equation 21, is inappropriate for correcting the estimates of information measures, specially those of multivariate information measures, suggesting that Equation 22 may not be appropriate for testing the significance of information measures. Although statistical tests are not available to test the significance of each information measure, the simulation results showed that the information measures are remarkably stable (with a few exceptions); in general, the standard deviations of 10 simulations are very small (0.010 to 0.015 bits) for all of the information measures, specially when the number of trials per stimulus is greater than 120. In all of the experiments in the present study, as you shall see, the total number of observations in pooled data will be at least 300 observations per stimulus to estimate each information measure. For example, there were a total of 360 observations per stimulus in each condition of Experiment 1. In such cases, it seems reasonable to assume that the (uninflated) estimates of information measures obtained by the correction method are stable, within a range of plus or minus 0.010 to 0.015 bits. Thus as an alternative to the commonly-used statistical tests, I will use 0.020 to 0.030 bits (corresponding to 2 standard deviations of 10 simulation runs) as an index of a reliable difference in the (uninflated) estimates of the information measures. For example, Table 7 presents uninflated estimates of information measures obtained by applying the correction method to data from Experiments 1 and 2 (the uncorrected estimates of the information measures [averages of information measures calculated from individual subject data] of these and subsequent experiments can be found in Appendix 2). Applying the above index, none of the information measures of sequential dependencies for 4 stimulus/response categories are reliably different from zero, except U (R n :R n . i | S^S,,.,) in the mask condition of Experiment 1. It also seems that for each number of stimulus/response categories used in Experiment 1, U(R n :S n . 1 |S B ) in the mask condition is not different from that in the no-mask condition. As you shall see in subsequent chapters, those results fit quite well with the overall pattern of the results obtained in the present Table 7 Corrected estimates of information measures in Experiments 1 and 2. Number of Stimuli UCR^SJ U(R n:S n-i |S n) UCR^R^JSJ U ^ R ^ l S A i ) U(R n:S n )S n . 1 !R n . 1) Experiment 1. Mask Condition 4 1.039 .023 .023 .049 1.111 6 1.288 .046 .078 .045 1.379 10 1.300 .162 .140. .361' 1.823 16 1.281 .260 .255 .749* 2.290 Experiment 1. No Mask Condition 4 1.818 .018 .017 .010 1.845 6 1.971 .065 .065 .040 2.076 10 1.967 .131 .093 .062* 2.160 16 1.766 .260 .196 .510' 2.536 Experiment 2 4 1.990 .002 .002 .000 1.992 6 2.272 .029 .029 .021 2.322 10 2.260 .124 .093 .160' 2.544 16 2.283 .187 .195 .232' 2.736 * The estimate was obtained by using Equation 23. 106 study and previous studies (Treisman, 1985; Ward & Lockhead, 1971). III.7. Discussion of Chapter III The results of Experiment 1 raised two important problems that we must face when we use multivariate information measures as an analysis method of absolute identification performance : negative interaction and inflation of information measures. Since a negative interaction is impossible to interpret in the same way as are the other information measures, I have decided to discount INT(R n:S n. 1,R n.i | S J from the discussion of results in further studies, even if it is positive. I also suggested another type of multivariate information analysis (Equation 20) which does not include interaction terms. The inflation problem is more difficult to solve, for there have been few studies on multivariate information measures and no decisive solution is available. I discussed the possible correction methods in Sections III.4 to III.6, including the results of my computer simulations (Section III.5.2). Based on the discussion, I proposed a combinatory use of the pooling method (Section III.4.1) and the computer simulations (Section III.3.2) in order to obtain uninflated estimates of information measures. In the next chapter, I will apply the correction method to information measures obtained from a variety of absolute identification experiments. / 107 IV. DYNAMIC ASPECTS OF ABSOLUTE IDENTIFICATION P E R F O R M A N C E In this chapter, I report results of absolute identification experiments, in terms of multivariate information measures (Equations 17 and 20) that are corrected in the way discussed in the previous chapter. My main concern is with sequential dependencies in absolute identification and factors affecting them. There is a growing recognition among psychophysicists that sequential dependencies of observers' responses reflect dynamic aspects of their performance and should be incorporated in any model of psychophysical judgments (e.g., Ward, 1979; DeCarlo and Cross, 1990). Absolute identification is no exception. In fact, sequential dependencies in absolute identification responses had been recognized long before psychophysicists paid attention to those in scaling tasks (Cross, 1973; Ward, 1973). In early work, Garner (1953) and McGill (1957) used multivariate information analysis to measure the amount of sequential dependencies in absolute identification experiments. These and other results stimulated Ward and Lockhead (1971) to propose the general principle of psychophysical judgment that the less stimulus information the observer gets, the more likely his or her current response will be to depend on previous stimuli and responses. The principle was later tested and confirmed by Mori (1989) using multivariate information analysis (Equation 17). Mori (1989) found that the amount of sequential dependencies of R„ on Sn., and R^, was inversely related to the amount of information transmission U(R n:S n) (also Garner, 1953; McGill, 1957). Although the information measures of sequential dependencies reported by Mori (1989) might be inflated because of the small number of observations used for a relatively large number of stimulus/response categories, the corrected information measures from Experiment 1 (Table 7) also showed an inverse relation between U(R n :SJ and the amount of sequential dependencies. Figure 32 presents the results. For all the numbers of tones used, U(R n :R n . 1 |S B ) and U(R n:R n. 1 |S n,S n. 1) increased when U(R n :S J was decreased in the mask condition. The increment of U(R n:R n . 11SJ and U(Rn:Rn.11 S^S^j) was specially large for the 10- and 16-tone conditions. Although U(R n:S n., | S J did not show an inverse relation with U(R n :S n ), at least as consistently as U(R n:R n.] | S J and U(R n :R D . I | Sn,SD.,) did, this would be probably because trial-by-trial feedback was given in Experiment 1, and the effects of Sn.i on R n would be confounded with the effect of feedback on U(R n :S n . 1 |S n ). Other factors affect sequential dependencies in absolute identification responses. For example, the results of Experiments 1 and 2 showed that the amount of sequential dependencies varied with number of & c 3 CO ro • o o -1-fl> o Sf Q . • • • • 3 O i 0) 3 <D fi) CO 9 to m x TJ <D 3 00 m O -n H O z m CO c m a A M O U N T O F I N F O R M A T I O N ( b i t s ) O O O O O O b ^ ro to ^ i n o - » to • o> c 3D 3 • a a* CO o o o o o • • • • • o - A ro to * o o cn b • * o • 0> c 3 a • 30 n i • CO 3 o • - 0) • 555 z o I > CO > CO 80T 109 stimulus/response categories used. In Figure 32, U(Rn:S,,.i | S„), U(R n :R n . i | S J and U(R n:R n . 1 |S n ,S B . 1) increase as the number of tones used increases. The increase of these information measures cannot be explained by an inverse relation between U(R n:S n) and the amount of sequential dependencies, because U(R n:S n) was fairly constant for each of the mask and the no mask conditions, without respect to the number of tones used. The same pattern was observed in the results of Experiment 2 (all the information measures were corrected), which are presented in Figure 33. Garner (1953) is the only other study (as far as I know) that investigated the effects of the number of stimuli used on the amount of sequential dependencies and found that U(R n:S n. 11SJ increased with increasing number of stimuli used (he suggested that the effects of R ^ also increased). However, the Garner's (1953) estimates of U(R n:S n . 1 |S n) were probably substantially inflated for the larger numbers of tones used (e.g., 20). In addition, his estimates pf U(R n:S n) from the same data decreased as the number of stimuli increased, suggesting that the effect of the number of stimuli on TJ(Rn:Sn., | S J would be confounded with the inverse relation between U(R n :SJ and U(R n:S n . , | S J . Giving trial-by-trial feedback also influences sequential dependencies in absolute identification responses. Ward and Lockhead (1971) found that the dependency of R n on Sn.[ was larger when feedback was given than when it was not, and that the dependency on R n., was smaller when feedback was given. Ward and Lockhead (1971) interpreted these .results as arising from the observer's response processes (strategies) in an absolute identification task. When a presented stimulus (SJ does not give the observer enough information to choose a single response category, he or she uses information from other sources, such as S n . i and R ^ j , in order to make a "correct" response. Ward and Lockhead (1971) assumed that the observers use the most reliable and available information under given conditions. When trial-by-trial feedback is given, the observer tends to use this "true" value for S^i as an information source. When feedback is not given, the best information available (except S J is R,,.,, the observer's own previous response. Thus, the presence or absence of feedback was supposed to influence the observer's use of S „ . i and/or R n., as an information source additional to Sn, causing the changes in sequential dependencies observed by Ward and Lockhead (1971). Siegel (1972) and Treisman (1985) proposed a similar type of response processes regarding the effects of feedback on sequential dependencies. rV.l. Experiment 3 CO < CC O UL UL O H Z D O < 4 . 6 . 1 0 . 1 6 . 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 U ( R „ : S „ - i I S „ ) 4 . 6 . 1 0 . 1 6 . 0 . 0 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 4 . 6 . 1 0 . 1 6 . ~i r U ( R „ : R „ - i I S „ ) 4 . 6 . 1 0 . 1 6 . N U M B E R O F T O N E S U S E D Figure 33. Corrected information measures of Experiment 2 I l l While the results of Garner (1953) and Experiments 1 and 2 of the present study demonstrated the effects of the number of stimulus/response categories on the amount of sequential dependencies, all the experiments were conducted with auditory stimuli (sound intensity in Garner, 1953, and sound frequency in Experiments 1 and 2). Thus one could argue that these effects of the number of stimulus/response categories could be observed only for absolute identification of auditory stimuli. However, if the effects are found for a very different sensory continuum, such as brightness, they will be attributable to more general psychophysical processes and not to sensory or decision processes unique to the auditory system. For that purpose, I conducted absolute identification experiments in which luminance levels of light circles were used as stimuli to be judged. The number of luminance levels was 10 or 16 in this experiment, because I found in Experiments 1 and 2 that the amount of sequential dependencies was very small and sometimes negligible in the 4- and 6-tone conditions and that the effects of the number of tones (and response categories) used were clearly demonstrated in the 10- and 16-tone conditions. IV.1.1. Experimental method A . Stimuli and apparatus. A Hewlett-Packard Vectra ES/12 computer system controlled the stimulus presentation and recorded the observer's responses, which were made on a dimly red-illuminated, standard computer keyboard. Light was presented on a circular L E D (6 mm in diameter) mounted at about eye level on the surface of the chamber wall facing the observer (the viewing distance was about 80 cm). The luminances were measured by PHOTO R E S E A R C H SPOTTER M O D E L UBD 1/2 (PR-1500). The luminances of the light were selected in such a way that 10 and 16 luminances were spaced roughly equidistant on a logarithmic scale over the range of 0.002 - 50 Footlamberts (1 Footlambert = 1 / 3.426 cd/m2). The duration of light presentation was 500 msec. Subjects sat in the dark in a sound-attenuation chamber while making their judgments. B. Procedure. Every observer participated in both 10- and 16-luminance conditions. The observers were given typical instructions of an absolute identification task. Before each experimental session, the observers were told the number of luminances to be identified in that session and presented with them and the required response for each. A trial consisted of a 500-msec warning sound, a 600-msec pause, a light presentation, a self-paced response period, and a 2-second feedback presentation. The inter-trial interval was 1 second. The observers were dark-adapted for 4 minutes before they were presented with any light. Observers performed in a 1-hour practice session in each of the 10- and 16-luminance conditions, and they were also given 50 to 80 practice trials before each main session. Every observer made 60 judgements per stimulus in both 10- and 16-luminance conditions. The order of the experiments was counterbalanced across the observers. C. Subjects. Five male students of the University of British Columbia, one of them the author, participated. Al l had normal or corrected-to-normal vision. IV.1.2. Results. First, the amounts of multivariate information measures were calculated for each subject and condition. Since the subjects' individual results closely resembled each other, I pooled the data and applied the correction method to estimate the uninflated information measures. Table 8 presents the results. Table 8 Results of Experiment 3. Number of .U(R n:S a) U(R n:S n., | S J UiRj^ | S J U(R n:R n . 1 |S n ,S n . 1) U(R B:S.,S l l r l,R 1, 1) luminances 10 2.002 .089 16 1.872 .256 .093 .257 .045 .400 2.160 2.528 Note - The estimates of U(Rn:Rn.11 Sn,Sn.,) were obtained by using Equation 23. 113 Unexpectedly, U(Rn:S„) decreased (by 0.15 bits) as the number of luminances increased. A l l of the other information measures increased with increasing number of luminances. Like those of Experiments 1 and 2, the estimates of U(R n :S n . , |SJ were almost identical to those of U(R n:R n., ISJ. IV. 1.3. Discussion As observed in Garner (1953), as the number of luminances increased, U(R n :S n) decreased and the information measures of sequential dependencies [U(Rn:SB., | SJ , U(R n:R n . , |S I 1), and U(R n :R n . j | Sn,Sn.,)] increased. Thus the results could be explained by the inverse relation between U(R n:S n) and the amount of sequential dependencies. However, I calculated the total amount of sequential dependencies on Sn., and Rn.] [U(R n :S n . 1 ,R B . I |SJ = U(R n:S n,S n.„R n. 1) - U(R„:Sn)], and it increased by 0.518 bits from the 10-luminance to 16-luminance condition. Since the size of increase of U(Rn:S„.1,Rn.1  Sn) was substantially larger than that of decrease of U(R n:S n) (0.15 bits) with increasing number of luminances used, the inverse relation between U ( R n : S J and the amount of sequential dependencies could not be the only factor causing the changes in the amount of sequential dependencies in the two conditions. It is thus concluded that there were some effects of the number of luminances used on the amount of sequential dependencies in this experiment, even though it would be confounded by the inverse relation between U(R n:S n) and the amount of sequential dependencies. IV.2. Experiment 4 In this experiment, I examined the inverse relation between U(R n:S n) and the amount of sequential dependencies in a situation where U(R n:S n) is higher than 2.5 bits. Experiment 1 and previous studies that found an inverse relation between U(R n:S n) and the amount of sequential dependencies (McGill, 1957; Mori, 1989; Ward and Lockhead, 1971) used modalities that would yield 2.5 bits of U(R n:S n) at most. It could be said that the results of those studies would be generalizable only to situations in which U(R n:S n) is less than or equal to 2.5 bits. I think it particularly important to examine the generality of the inverse relation between U(R n :S J and the amount of sequential dependencies in situations where UfR^S,,) is more than 2.5 bits, because there is a suggestion that sequential dependencies would disappear if U(R n:S 0) is more than 2.5 bits (e.g., Mori, 1989). To obtain a high amount of U(R n:S n), I conducted an experiment involving absolute identification of 16 pointer positions along a horizontal line. In a previous study, Hake and Garner (1951) obtained 3.14 bits to 114 3.19 bits as U(R n:S n) for 10 to 50 pointer positions on a horizontal line. Although MacRae (1970) later noted that those estimates of U(R n:S n) were likely to be inflated because a relatively small number of observations were used to calculated them, other studies also obtained a high amount of U(R„:Sn) for absolute identification of visual positions (for a review, see Garner, 1962). In order to examine the inverse relation between U(R n:S n) and the amount of sequential dependencies in this experiment, I manipulated U(R n:S n) by changing the size of a range within which the positions were uniformly spaced (see below). It has been reported for a variety of modalities (e.g., sound intensity, Luce et al., 1976) that U(R n:S n) increases as the range of stimuli used is increased. It was thus expected that changing the size of the stimulus range would affect U(R n:S n) for absolute identification of pointer positions, too. IV.2.1. Experimental Method A . Apparatus and stimuli. A l l presentations were made on a CRT monitor (NEC MULTISYNC II) controlled by a Vectra ES/12 microcomputer (Hewlett Packard). The viewing field was viewed through a 17 cm x 24 cm window in a black board that covered the other parts of the display. The viewing field consisted of a 0.4 cm x 16 cm rectangle (called a line) centered on the field and a 0.4 cm x 0.1 cm filled square (called a pointer) presented on the line, all of which were drawn in white on a black background There were 16 pointer positions along the line. In the 'small range' set, the 16 pointer positions (defined by the center of the pointer) were uniformly spaced in 0.31 cm steps at the center of the line. In the 'large range' set, they were spaced in 0.93 cm steps. The leftmost position of the marker was identified as T and the rightmost position as '16', and the intermediate positions were identified accordingly. The other presentations, such as instructions, a subject's response, and feedback, were also made on the display and drawn in dim but visible red. The viewing distance was about 60 cm. B. Procedure. Every observer participated in both small- and large-range conditions. The observers were given the standard instructions of an absolute identification task. They were seated in a dimly lighted room and presented with 16 pointer positions used in the experiment. A trial consisted of a warning buzzer, a 115 500-msec pause, an 800-msec presentation of a pointer, a free-paced response period, and a 1-second feedback period. The inter-trial interval was 1 second. One session consisted of 320 trials (20 trials per position), and there were 3 sessions for each of the small- and large-range conditions. A l l the observers practised for 200-400 trials before the experimental sessions started, and they all participated in the large-range condition first. C. Subjects. Six students of the University of British Columbia, one of them the author, participated in the experiment. Three of them were females. Al l had normal or corrected-to-normal vision. IV.2.2. Results. As for Experiment 3, information measures of Equations 17 and 20 were first calculated for each subject and condition. Since the subjects' individual results closely resembled each other, I pooled the data of each condition and applied the correction method discussed in Chapter III. Table 9 presents the resulting unbiased estimates of information measures. Table 9 Results of Experiment 4. • Stimulus UCR.iSJ U(R n:S n . 1 |S n) U(R n :R n . , | S J U(R n :R n l |S n ,S I , 1 ) U(R n:S n,S n. 1,R n. 1) range Large 3.378 .059 .059 Small 2.127 .141 .146 Note - The estimates of U(R n:R n . 1 |S n ,S n . 1) were obtained by using Equation 23. .030 .260* 3.467 2.528 116 The manipulation of the stimulus range had a substantial effect on U(R n:S n) and, to a lesser degree, on the amounts of sequential dependencies. Reducing the stimulus range lowered U(R n:S n) by 1.2 bits. The amount of sequential dependencies [U(R n:Sn.i|S n) ) U(R n :R n . 1 |S,J, and U(R n :R n . 1 IS^S,,.,)] were larger for the small stimulus range than for the large stimulus range. For the large stimulus range, U(R n :SJ was more than 3.3 bits and was even larger than that obtained by Hake and Garner (1954) in their 20- and 50-position conditions. The information measures of sequential dependencies were all very small. U(Rn:Sn.11 Sn) was identical to U(R n :R n . 1 |S D ) , which may be due to the fact that Sn.! and R n., were highly correlated. Although the small stimulus range yielded a substantially smaller amount of L ^ R ^ S J than did the large stimulus range, it was still comparable to the highest value of U(R n:S n) in the previous experiments. While in • I this condition U(R n:S n., | S J took a value very similar to that of U(R n:R n . , | S J , both of them were much smaller than UtR^R,,., | SJS^-IV.2.3. Discussion. The purpose of this experiment was to examine the inverse relation between U(R n:S n) and the amount of sequential dependencies in a situation where U(R n :SJ is higher than 2.5 bits. The results may be said to fulfil the purpose. While U(R n:S n) decreased from 3.3 bits for the large stimulus range to 2.1 bits for the small stimulus range, U(R n:S n., | SJ , U(R n:R n. 1|S n), and U(R n:R n . 1 |S n ,S n . 1) were all larger for the small stimulus range than for the large stimulus range. For the large stimulus range, although U(Rn:R„.j | Sn,Sn.1) was negligible, U(Rn:Sn., | S J and U(Rn:R„.11SJ seemed to show small but noticeable dependencies on Sn., and R,,.,. IV.3. Experiment 5 This experiment investigated the effects of trial-by-trial feedback on the amount of information transmission U(R n:S n) and on the amount of sequential dependencies. As mentioned earlier, not only does giving feedback improve U(Rn:Sn), but also it affects sequential dependencies (Braida and Durlach, 1972; Chase et al., 1983; Eriksen, 1958; Siegel, 1972; Ward and Lockhead, 1970, 1971). Mori (1989) examined the effects of feedback by using multivariate information transmission (Equation 17), and the results were somewhat ambiguous. When the discriminabilities among stimuli (line lengths) were low, the presence or absence of feedback had little effect on U(R„:Sn) and U(R n:S n., | Sn), and there was a slight increase in U(R n :R n . , | S J with feedback. However, when the stimulus discriminabilities were improved, giving feedback increased U(Rn:S„) and decreased U(Rn:S„.1  Sn) and U(Rn:R„.1 |S0), although the changes were small. Besides the fact that Mori (1989) did not correct the information measures, the ambiguity seems to have come from two sources. First, Mori (1989) used a between-subject design and the performance differences in the presence or absence of feedback were obscured by the individual differences. Second, trial-by-trial feedback helps observers maintain the identification function from stimulus to response sets that the experimenter has induced before the experiments. In the absence of feedback, the observers tend to forget the "true" identification function and use a biased function that maps the stimuli to the responses in an inappropriate way. In order to control those factors obscuring the effect of feedback on absolute identification performance, this experiment was conducted with (1) a within-subject design and (2) a sequence-alternating method, in which sequences of trials with feedback given and those with feedback absent were alternated within a session. The sequence-alternating method was designed to help the observers to maintain the 'true' identification function when feedback was not given. In this experiment, the stimuli to be judged were sound intensity levels, which were identical to those used in one of Braida and Durlach's (1972) experiments in which they found a 20 % difference in the measure of observer's sensitivity (d') between the sessions with feedback given and those without feedback. If the sequence-alternating method was as effective as originally designed, and the feedback effects found by Braida and Durlach (1972) was due to the fact that then-observers did not maintain the true identification function in the no-feedback sessions, the effects of feedback on U(R n :S n ) would be smaller in the sequence-alternating methods than those found by Braida and Durlach r (mi). I also manipulated U(R n :SJ , in order to see whether the effects of feedback on absolute identification performance would differ for high U(R n:SD) and low U(R„:Sn), as Mori (1989) found in his Experiment 2. The manipulation of U(R n :SJ was done by changing the range of sound intensity levels. As mentioned in the previous section, it is known that the observer's resolution of sound intensities increases with the range of intensity levels to be judged (Braida and Durlach, 1972; Weber et al., 1977). IV.3.1. Experimental method. A . Apparatus and stimuli. The apparatus was identical to that used in Experiments 1 and 2. The stimuli were 10 intensity levels of a 1000 Hz, 500-msec pure tone. There were two sets of stimuli. In the 'large range' set, the intensity levels ranged from 50 dB to 86 dB in 4-dB steps. In the 'small range' set, the intensity levels ranged from 77 dB to 86 dB in 1-dB steps. The sets of tones were identical to those used in Braida and Durlach's (1972) Experiments 1 and 4, respectively. B. Procedure. Sequences of trials with feedback given (FEEDBACK sequence) and those with feedback absent (NO-FEEDBACK sequence) were alternated in a session. The FEEDBACK sequence was always given first and the NO-FEEDBACK sequence was then given. The number of alternations were 5 and the number of trials in one sequence was 30. Thus one session consisted of 300 trials, half of which were in F E E D B A C K sequences and another half were in NO-FEEDBACK sequences. Observers were instructed to identify 10 stimuli and to make responses in a consistent way in both FEEDBACK and NO-FEEDBACK sequences. They were presented with 10 tones before they started identification trials. Trials were self-paced. There were 4 sessions for each of the large- and small-range sets of tones, and each observer made 60 judgments for each tone of each set of stimuli under each of feedback and no-feedback conditions. The order of the large- and small-range conditions was counterbalanced across observers. Each observer did half-an-hour practice before each session. C. Subjects. Six students of the University of British Columbia, one of them the author, participated in this experiment. A l l had no difficulty hearing the tones used as stimuli. Four were males and two were females. F/.3.2. Results. The individual data for each condition were pooled and the correction method discussed in Chapter III was applied to the information measures calculated from the pooled data. Table 10 presents the corrected, uninflated estimates of the information measures. 119 For both stimulus ranges, there was very little difference between U(R n :SJ of the F E E D B A C K sequence and that of the NO-FEEDBACK sequence. As observed by Braida and Durlach (1972) and Weber et al. (1977), the manipulation of the stimulus range had a substantial effect on U(R n:S n); reducing the stimulus range lowered U(R n:S n) by more than 0.5 bits. I also found an inverse relation between U(R n:S n) and the amount of sequential dependencies; the estimates of U(R n:S n., | SJ , U(R n:R n. 11SJ, and U(R n:R n. 11 Sn,Sn.,) were all larger for the small stimulus range than for the large stimulus range. Table 10 Results of Experiment 5. Sequence U(Rn:S„) U(R n :S, 1 |S n ) U(R n:R n. 11SJ U(R n :R n . 1 1S^ , ) U(R n:S n,S n. 1,R n. 1) Large range 0.142 0.137 0.222 1.788 0.098 0.133 0.301 1.821 FB 1.424 NFB 1.422 Small range FB 0.887 0.179 0.175 0.423 1.489 NFB 0.858 0.116 0.206 0.495 1.469 Note - FB = F E E D B A C K sequence, NFB = NO-FEEDBACK sequence. The estimates of U(R n :R n . i | Sn,Sn. ) were obtained by using Equation 23. Although the differences between the amount of sequential dependencies of the F E E D B A C K sequence and that of the NO-FEEDBACK sequences were very subtle, they showed a consistent pattern for both stimulus ranges. While U(Rn:S„.1 | S J was larger in the F E E D B A C K sequence than in the N O - F E E D B A C K sequence, U(R n:R n. 1 |S n,S I 1. 1) was smaller in the F E E D B A C K sequence. U(R n:R n . 11SJ was relatively unchanged in the FEEDBACK and the NO-FEEDBACK sequences. Note also that U(R n :R n .! | S J was almost equal to U(R n:S n . 1 |S n) in the FEEDBACK sequence, which was observed in some of the previous 120 experiments (e.g., Experiment 3). IV.3.3. Discussion. The results were generally consistent with predictions and previous studies. The sequence-alternating method was very effective in maintaining the same amount of U(Rn:Sn) of absolute identification in the F E E D B A C K and the NO-FEEDBACK sequences. Since U(Rn:Sn) was the same in the F E E D B A C K and N O - F E E D B A C K sequences, the differences between the amount of sequential dependencies of the two sequences should reflect the effects of giving feedback on sequential dependencies, not the inverse relation between U(R n:S n) and the amount of sequential dependencies. The pattern of the sequential dependencies was consistent with that obtained in previous studies (e.g., Ward and Lockhead, 1971), in that the dependency on a previous stimulus [U(Rn:Sn., |SJ] is larger when feedback is given than when it is not, and that the dependency on a previous response [U(R0:Rn.1 IS^S^)] is smaller in the presence of feedback (Ward and Lockhead, 1971). Although U(Rn:Rn., ISJ was not lowered in the FEEDBACK sequence in this experiment, this would be probably because the stimulus and the response were highly correlated in the F E E D B A C K sequence. This also explains that U(Rn:RD.1 ISJ was almost equal to U(R n:S n.i | S J in the F E E D B A C K sequence. IV.4. Experiment 6 The purpose of this experiment was to further investigate the effects of feedback on absolute identification performance, by using the sequence-alternating method. Although the results of Experiment 5 proved that the sequence-alternating method was very effective in maintaining the same amount of U(R n:S n) of absolute identification with and without feedback, the obtained amount of U(R n :SJ was somewhat lower, even for the large stimulus range used, than those obtained in Experiments 1 to 4 (except the masking condition in Experiment 1). There are at least two types of explanation for the low U(R n:S n) obtained in Experiment 5. One is that although the purpose of using the sequence-alternating method was to help the observers to maintain the identification function when feedback was not given, the alternation of F E E D B A C K and NO-FEEDBACK sequences might have disturbed their performance in the F E E D B A C K sequence, resulting in a decrease of L^R^SJ in the F E E D B A C K sequence, rather than an increase of U(R n :SJ in the NO-FEEDBACK sequence to the same level as in the F E E D B A C K sequence. The other explanation is that sound intensity is a difficult modality for absolute identification, in the sense that U(R n:S n) of sound intensity is quite often lower than that for other modalities, unless observers are given substantial practice (e.g., Braida and Durlach, 1972). For example, Garner (1953) obtained about 1.8 bits of U(R n:S n) for 10 stimuli, and Mori and Ward (1990) obtained 1.666 bits for a stimulus set similar to that of the large stimulus range used in Experiment 5. In order to determine whether the low U(R n:S n) in Experiment 5 would be due to the use of sequence-alternating method or that of sound intensity as stimuli, in this experiment I used sound frequency as a stimulus modality, because I obtained relatively high U(R n:S n) (about 2 bits ) of sound frequency in Experiments 1 and 2. If a low amount of U(R n:S n) is again obtained in this experiment by using the sequence-alternating method, this means that alternating the two types of sequences would somehow disturb the observer's performance in absolute identification, resulting in lowering U(R n:S n) in the F E E D B A C K sequence. For the second purpose of this experiment, I examined the effects of the number of stimulus/response categories in the FEEDBACK and the NO-FEEDBACK sequences. Although the results of Experiments 1, 2, and 3 showed the amount of sequential dependencies increased with increasing number of stimulus/response categories used, the experiments were conducted with feedback. It is conceivable that the number of stimulus/response categories would not affect the amount of sequential dependencies without feedback. Furthermore, the effects of feedback on the amount of sequential dependencies could change as the number of stimulus/response categories increases. In order to test these possibilities, I used sets of 6 tones, 11 tones, or 16 tones (as in the previous experiments, the number of response categories was equal to that of tones used). IV.4.1. Experimental method A . Stimuli and apparatus. The apparatus was identical to that used in Experiments 1 and 2. The frequencies used as stimuli ranged from 100 to 8000 Hz in equi-distant logarithmic steps, and the observers matched the loudness of each frequency with that of 1000 Hz 60 dB tone to produce 60 phon intensities for all frequencies. The tone duration was 500 msec 122 B. Procedure. The procedure was identical to that of Experiment 5, except the number of tones to be judged and the number of trials in a sequence (the number of trials in a sequence was the same in the F E E D B A C K and the N O - F E E D B A C K sequences). The numbers of trials in a sequence were 36 trials for the 6-tone condition, 33 trials for the 11-tone conditions, and 32 trials for 16-tone condition. Every observer participated in all the conditions. The observers had a 100- to 300-trial practice session before they started in each condition. Each observer made 60 judgments per tone for each condition, and there were 2 sessions for the 6-tone condition, 4 sessions for the 11-tone condition, and 6 sessions for the 16-tone condition. The order of the conditions was counterbalanced across the observers. C. Subjects. Five students of the University of British Columbia, one of them the author, participated in this experiment. A l l had no difficulty hearing the tones used as stimuli. Four were males and one was a female. IV.4.2. Results Like other experiments, the corrected, uninflated estimates of information measures were obtained by the correction method discussed in Chapter III. Table 11 presents the results. The obtained amounts of U(R n:SD) were 1.8 - 2.2 bits and were comparable to those obtained in the no-mask condition of Experiment 1 and of Experiment 2. For all the numbers of tones used, U(R n :S n) was slightly (0.05 - 0.15 bit) larger in the FEEDBACK sequence than in the NO-FEEDBACK sequence. As observed in Experiments 1 and 2, the amount of sequential dependencies [U(R n:S n. 1 |S n), U(R n :R n . , | S J , and U(R n:R n.,|S I 1,S n. 1)] increased with increasing number of tones, in both F E E D B A C K and NO-FEEDBACK sequences. For all the numbers of tones used, the pattern of sequential dependencies in the F E E D B A C K and the NO-FEEDBACK sequences was quite similar to that of Experiment 5, in that (1) U(R n:S n . 1 ISJ of the F E E D B A C K sequence was larger than that of the NO-FEEDBACK sequence, (2) U(R n :R n . , | Sn,Sn.,) of the. F E E D B A C K sequence was smaller than that of the NO-FEEDBACK sequence, (3) U(R n:R n . 11SJ was relatively unchanged in the FEEDBACK and the NO-FEEDBACK sequences. In addition, (4) in the. FEEDBACK sequence U(Rn:Sn.1|Sn) was always larger than U(Rn:Rn.1 SJ, and (5) the pattern was the opposite in the NO-FEEDBACK sequence. Table 11 Results of Experiment 6. Sequence UCR.rSJ. U(Rn:Sn.i ISJ U^R^ISJ U(RB:RB.1|S1I>S1,1) U(Rn:SB,SB.1,Rn.1) 6 categories FB 2.051 0.049 0.043 0.013 2.113 NFB 1.995 0.031 0.042 0.069 . 2.095 11 categories FB 2.001 .168 ,142 .153 2.322 NFB 1.818 .116 .141 .358 2.292 16 categories FB 2.204 .239 .195 ;199" 2.642 NFB 2.040 .157 .195 .430" 2.627 Note - FB = FEEDBACK sequence, NFB = NO-FEEDBACK sequence. The estimates of U(Rn:Rn.1|Sn,Sn.1 ) were obtained by using Equation 23. IV.4.3. Discussion. The results of Experiment 6 clarified two important points. First, since the obtained amounts of U(Rn:S„) were reasonably high and were comparable to those obtained in Experiments 1 and 2, the low U(Rn:Sn) obtained in Experiment 5 seems to have arisen from the use of sound intensity in that experiment. Moreover, the sequence-alternating method was again very effective in maintaining the same amount of U(R n:S n) of absolute identification with and without feedback. For demonstration purposes, I performed a repeated-measure t test on U(R n:S n) of the FEEDBACK and the NO-FEEDBACK sequences, treating the five experimental conditions of Experiments 5 and 6 (the small and the large ranges of Experiment 5 and the 6-, 11-, and 16-tone conditions of Experiment 6) as five different subjects. The t test showed no significant difference in U(R n :S J between the FEEDBACK and the NO-FEEDBACK sequences [t(4) = 2.376, p > .05]. Thus, it can be said that the sequence-alternating method helped the observers to maintain the experimenter-induced identification function (e.g., Ward, 1972) in an absolute identification task with no feedback. Second, the pattern of sequential dependencies observed in Experiment 6 was consistent with previous studies (e.g., Garner, 1953; Ward and Lockhead, 1971). The amount of sequential dependencies increased with increasing number of tones used. The dependency on a previous stimulus [U(Rn:Sn.! | SJ] was larger in the F E E D B A C K sequence than in the NO-FEEDBACK sequence, and that the dependency on a previous response [ L ^ R ^ R ^ | S^S^)] is smaller in the FEEDBACK sequence. To support this observation on the changes in the sequential dependencies, a two-factor repeated-measure analysis of variance was performed on the amount of sequential dependencies obtained from the five conditions of Experiments 5 and 6, factors being type of sequences [FEEDBACK and NO-FEEDBACK] and type of information measures [U(R n:Sn. J S J . U f R ^ R ^ j S J and U(R n:R n., IS^S^)] 2 . There were significant main effects for the type of sequences [F(l, 4) = 8.48, p < .05] and for the type of information measures [F(1.03, 4.12) = 7.85, p < .05], and their interaction [F(1.29, 5.18) = 13.56, p < .05]. Subsequent post-hoc tests showed significant effects of the types of sequences on U(R B:S n . 1 |S n) [F(l, 4) = 23.949, p < .001] a n d ^ ( R ^ R ^ J S ^ ) [F(l,4) = 12.135, p < .05], but not on U(R n :R n . , |S„) (p > .10). Although the results of these analyses only serve for demonstration purposes, they suggest that the pattern of sequential dependencies was different in the F E E D B A C K and the NO-FEEDBACK sequences, in a way consistent with previous studies (e.g., Ward and Lockhead, 1971). IV.5. Experiment 7 I have found that the number of stimulus/response categories is one of the major factors affecting the observers' absolute identification performance, in terms of U(R n:S n) and the amount of sequential dependencies. Although the number of response categories was always equal to that of stimuli used in the previous experiments, there is a suggestion that it is the number of response categories, rather than that of 125 stimuli used, that affects the observers' absolute identification performance. As mentioned in Chapter II, Eriksen and Hake (1955) did no-feedback absolute identification experiments in which the number of stimuli used differed from that of response categories permitted to the observers. Eriksen and Hake found that U(R n ) was more dependent on the number of response categories than on that of stimuli used, indicating that the observers had a strong tendency to use all responses permitted even if the number of stimuli used was fewer than that of responses. When the number of responses was equal to or greater than that of stimuli used, U(R n :SJ was the same irrespective of the number of stimuli. It is also conceivable that the number of response categories could have effects on sequential dependencies, in a way different from those of the number of stimuli used. For example, the number of response categories would affect the dependency on R n., more than that on Sn.„ and the number of stimulus categories would affect the dependency on Sn_, more. However, Eriksen and Hake did not analyze any type of sequential dependencies in the data. In order to distinguish between the effects of the number of response categories and those of the number of stimuli used, I used the sequence-alternating method, in which the number of stimuli used is equal to that of response categories in the FEEDBACK sequence but is reduced in the NO-FEEDBACK sequence. The questions I addressed in this experiment are (1) whether Eriksen and Hake's (1955) results will be replicated in the sequence-alternating method, (2) whether the amount of sequential dependencies on S„., and Rn.j will be affected more by the number of stimuli used or by the number of response categories. IV.5.1. Experimental method. A . Apparatus and stimuli. The apparatus was identical to that used in some of the previous experiments. The stimuli were 16 tones ranging from 100 - 8000 Hz in equi-distant logarithmic step, and the observers matched the loudness of each of 16 tones with that of 1000 Hz 60 dB tone to produce 60 phon intensities for all stimulus frequencies. B. Procedure. The procedure was identical to that of Experiment 6, except that in the NO-FEEDBACK sequence the number of tones used was fewer than that of response categories. There were 3 conditions in respect to the combination of the number of tones and response categories in the NO-FEEDBACK sequence; Condition 1. 6 tones and 11 response categories. 126 Condition 2. 11 tones and 16 response categories. Condition 3. 6 tones and 16 response categories. As mentioned before, the same number of tones and response categories was used in the F E E D B A C K sequence in each condition ; 11 tones were used in the F E E D B A C K sequence in Condition 1, and 16 tones were used in Conditions 2 and 3. The sets of tones used in the three conditions are listed in Table 12. The number of trials in the FEEDBACK and NO-FEEDBACK sequences was made roughly the same; 33 trials in the F E E D B A C K sequence and 36 trials in the NO-FEEDBACK sequence in Condition 1, 32 trials and 33 trials in Condition 2, and 32 trials and 36 trials in Condition 3. In all conditions, the F E E D B A C K sequence was given first, and the NO-FEEDBACK sequence was then given. The two sequences were repeated 5 times in one session, so that one session consisted of 325 - 345 trials. Table 12 Three sets of stimuli used in Experiment 7. V A set of stimuli Condition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 FB X X X X X X X X X X X NFB X X X X X X 2 FB X X X X X X X X X X X X X X X X NFB X X X X X X X X X X X 3 FB X X X X X X X X X X X X X X X X NFB X X X X X X Note - FB = F E E D B A C K sequence, NFB = NO-FEEDBACK sequence. The observers were bstructed that the same number of tones was used in both the F E E D B A C K and NO-FEEDBACK sequences, and that they should keep their performance consistent in both sequences. During the experimental sessions, none of them noticed that the actual number of stimuli presented was smaller in the NO-FEEDBACK sequences than in the FEEDBACK sequences. Every observer performed in all three conditions. Each observer made 60 judgments per tone in the no-feedback sequences in each condition. The order of the conditions was counterbalanced across the observers. C. Subjects. Six students of the University of the British Columbia participated in this experiment. A l l had no difficulty hearing the tones used as stimuli. Five were males and one was a female. IV.5.2. Results The results of the NO-FEEDBACK sequences in Experiment 6 serve as control conditions where the same number of tones and response categories were used. The information measures were calculated from the NO-FEEDBACK sequences for each of the three conditions of the present experiment, and they were corrected by the method discussed in Chapter III . Figures 34, 35 and 36 present the results, along with those of Experiment 6 for comparison. It can be readily seen that U ( R J depended more on the number of response categories than on the number of tones used (Figure 34). When U ( R J was plotted against the number of tones used, it is only when 6 tones were used that U ( R J was about 0.15 bits smaller than would have obtained if all 11 or 16 response categories had been used. The estimates of U(R n :SJ were about 1.8 to 2.1 bits for all the conditions (Figures 35 and 36). It seems that U(R n :SJ slightly decreased as the number of tones used increased (Figure 35), whereas with a fixed number of tones, it was 0.2 to 0.3 bits larger for 16 response categories than for 6 or 11 response categories used (Figure 36). 11 RESPONSES • 6 STIMULI Number of Tones Used Number of Response Categories Figure 34. The results of Experiment 7 (1). U(R n) 0 . 0 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 U ( R „ : R „ - i IS^Sn-!) 11 RESPONSES 16 RESPONSES 6 1 1 6 1 1 1 6 6 1 1 6 1 1 1 6 Number of Tones Used Figure 35. The results of Experiment 7 (2). 7 CO 0 E 6 1 1 1 6 1 1 1 6 O 0 . 3 0 . 0 6 1 1 1 6 1 1 1 6 0 . 3 0 . 2 0 . 1 0 . 0 6 1 1 1 6 1 1 1 6 0 . 0 • 6 TONES 11 TONES 6 11 1 6 1 1 1 6 Number of Response Categories Figure 36. The results of Experiment 7 (3). 131 Regardless of the number of tones used or response categories, U(Rn:Sn., | SJ was smaller than U(R n:R n. ^ S J or U(R n:R n. 1|S n,S n. 1), as observed in other no-feedback absolute identification experiments (e.g., Experiment 6). With a fixed number of response categories (Figure 35), U(Rn:Sn., |S0) slightly but steadily-increased as the number of tones used increased. When 6 tones were used (Figure 36), U(Rn:Sn.11SJ increased with an increasing number of response categories, but the changes were negligible. With 11 tones used, there was no increase of U(Rn:Sn., | SJ from 11 to 16 response categories. In general, U(Rn:Rn.11S J and U(Rn:RI1.1 |Sn,Sn.,) increased as the number of tones used and/or response categories increased. A relatively large increase of U(Ra:R0.11SJ and U(R n:R n., | Sn,Sn.,) was observed when the number of tones increased from 6 to 11 with 16 response categories (Figure 35), and when the number of response categories increased from 6 to 11 with 6 tones used (Figure 36). There was also a small increase of the two measures when the number of response categories increased from 11 to 16 with 11 tones used. IV.5.3. Discussion There were two purposes of this experiment, both of which concerned the effects of the number of response categories and stimuli used on the observer's absolute identification performance. The first one was to replicate the results of Eriksen and Hake (1955) that showed the effects of the number of response categories on U ( R J and U(Rn:Sn). This experiment did replicate Eriksen and Hake's (1955) results; U ( R J depends more on the number of response categories than that of tones used; U(R n:SJ was constant or slightly increased with 16 response categories. It seems that the observers of this experiment also had a tendency to use all responses permitted even if the number of stimuli used was fewer than that of responses, and that they tried to identify a presented tone in the NO-FEEDBACK sequence, at least as much as did the observers in previous experiments in which the number of stimuli used was equal to the number of response categories. The second purpose of this experiment was to examine the effects of the number of response categories on the amount of sequential dependencies, separately from that of the number of stimuli used. As expected, the dependency on [U(Rn:Sn.! ISJ] increased only with an increasing number of tones used. On the other hand, the dependency on R,^ [U(Rn:Rn.,|Sn) and U(Rn:Rn.1|Sn,Sn.1)] depended on both the number of tones used and the number of response categories used. Except for U(R n:R n. 1 |S n) with 11 response categories used (Figure 35), U(Rn:R„. 11SJ and U(Rn:Rn.11 Sn,Sn.,) increased when the number of tones used and/or the 132 number of response categories increased from 6 to 11. When the number of tones used and the number of response categories increased from 11 to 16, their effects on U(R n:R n., | SJ and U(Rn:Rn.11 S0,Sn.,) were relatively small or negligible. IV.6. Discussion of Experiments 1 to 7. IV.6.1. The effects of U(R n:S J and the number of stimulus/response categories The results of Experiments 3 and 4, along with those of Experiments 1 and 2 and previous studies, have established two factors affecting the amount of sequential dependencies in absolute identification responses: (1) the amount of information transmission U(Rn:S„) (McGill, 1957; Mori, 1989; Experiments 1 and 4) and (2) the number of stimulus/response categories used (Garner, 1953; Experiments 1, 2, and 3). Some explanations have been proposed for the effects of these two factors on the amount of sequential dependencies. For the explanation of the effect of U(R n :SJ, Ward and Lockhead (1971) argued that the less stimulus information the observer gets [measured by U(Rn:S I1)], the more likely his or her response is to depend on previous stimuli and responses, resulting in the increase of the amount of sequential dependencies. Mori (1989) confirmed Ward and Lockhead's (1971) argument by using Equation 17 to measure the amount of sequential dependencies as well as U(R n:S n). Furthermore, Mori (1989) found that U(R n:S n,S n. 1,R n. 1) was constant at about 2.5 bits regardless of the changes in the size of its components [e.g., U(R n:S n)]. From his results, Mori (1989) suggested the idea of a limited capacity response process; that is, the observer's response process in an absolute identification task can utilize only a limited amount of information to make a response, and this amount might be 2.5 bits. If this amount of information can be obtained from S n alone, R n will be determined solely by S„. Otherwise, the response process tends to get information from other sources, such as Sn., and Rn.j, in order to make up for the lack of information from Sn, thus causing sequential dependencies. But the total amount of information that the response processes can utilize would not exceed 2.5 bits. However, the results of Experiments 1 to 4 clearly disconfirmed the idea of a limited-capacity response process. Not only did the size of U(Rn:Sn,Sn.„Rn.1) vary as the number of stimulus/response categories changed (Experiments 1 to 3), but also it varied across experiments even when the same number of categories was used (Experiments 1 and 4). Moreover, there is still a small but noticeable amount of sequential dependencies when U(R n:S n) is more than 2.5 bits (Experiments 4). Since Mori (1989) did not 133 correct any of the estimates of information measures, the seemingly constant size of U(Rn:Sn,Sn.„Rn.,) must have been an artifact caused by the inflation of the information measures. However, the rejection of a limited-capacity response process does not necessarily mean that Ward and Lockhead's (1971) argument is also invalid. Their argument predicted an inverse relation between U(R n:S n) and the amount of sequential . dependencies, which was confirmed by the results of Experiments 1 and 4. As an explanation for the increase of U(R n:S n. 1 |S n) with the increasing number of stimulus/response categories, Garner (1953) pointed out that as the number of stimulus/response categories increases, the complexity of making judgments increases, and that in a complex judgmental situation the response is dependent on many factors in addition to the current stimulus. Since Sn.i is one such factor, the amount of sequential dependencies on Sn_i increases with the increasing complexity of making judgments, which is induced by the increasing number of stimulus/response categories. This explanation for the effects of the number of stimulus/response categories can be expressed in informational terms as follows. The dependency of the response on the factors other than the current stimulus is U(R n |S B ) , the residual of U ( R J after the effect of Sn [U(Rn:Sn)] is removed. To be more explicit, I use Equation 5 in Chapter II with a different notation: U ( R J = U(R„:Sn) + U(R n | S a ) (24). Figure 37 illustrates the relation among the three information measures in Equation 24. In Figure 37, a horizontal dashed line divides U(R„) into U(R n:S n) and U(R n |S n ) . It is clear in Equation 24 (and Figure 37) that with U(R n:S n) held constant, U(R n |S n ) increases as U(R„) increases. Remember that in most absolute identification experiments, including Experiments 1 to 4 of the present study, U ( R J is always near its maximum value, which is equal to the logarithm (to the base of 2) of the number of stimulus/response categories (Chapter II), so U(R n ) increases as the number of stimulus/response categories increases. It has also been found in the present study as well as in the literature (e.g., Miller, 1956) that U(R n:S J first increases to a certain level as the number of stimulus/response categories increases, and then remains constant at that level (a dashed line in Figure 37), which is usually called the channel capacity for the stimulus dimension involved (see Chapter II). As a result, when Ufll^S,,) remains constant, U(R n | S J usually increases with an increasing number of stimulus/response categories. It is also important to note that U ( R n | S J includes the information measures of the sequential dependencies of R n on S n .j and R„. i , which can be easily shown by using Equations 20 and 24: 0 2 4 8 1 6 N U M B E R O F S T I M U L U S / R E S P O N S E C A T E G O R I E S Figure 37. The effect of the number of categories i N U M B E R O F S T I M U L U S / R E S P O N S E C A T E G O R I E S Figure 38. The effect of U(Rn-Sn) 136 U(RJ = U(Rn:SIJ) + U(Rn:Sn.,|Sn) + U(RB:RB.1|SB,S11.1) + I K R J S ^ R , . , ) (20) = U(R„:Sn) + U(R n |SJ. Therefore, U(Rn | SJ = U(Rn:Sn.1 SJ + U(Rn:Rn., | Sn,Sn.,) + U(R„ | Sn,SQl,Rn.t) (25). In other words, U(Rn | SJ consists of the amount of sequential dependencies of Rn on Sn., and R^ plus U(Rn|Sn,Sn.1,Rn.1), the amount of U(RJ that is not explained by any of Sn, S ,^, or Rn.,. In Figure 37, the amount of sequential dependencies is represented by the size of the area between the dotted line and the dashed line. It can be readily seen in Figure 37 that, as U(Rn|Sn) increases, the amount of sequential dependencies increases, as long as U(Rn|Sn,Sn.„Rn. 1) does not increase as much as U(RJSn) does. Figure 37 represents U(R„ | S^S^R,,.,) as being a constant proportion of U(Rn | SJ, which is accurate for all of the date presented in this thesis. An inverse relation between U(Rn|S„) and the amount of sequential dependencies can be explained in a similar way and is illustrated in Figure 38. In Equation 24, with a fixed amount of L^RJ, realized by a fixed number of stimulus/response categories, U(Rn|Sn) increases as U(Rn:Sn) decreases. In Figure 38, the decrease of U(Rn:Sn) [or the increase of U(Rn|SI1)] is shown by an arrow from the dashed line to the horizontal solid line. As mentioned above, the amount of sequential dependencies (the area between the broken line and the horizontal solid line in Figure 38) increases as U(Rn|Sn) increases, if U(Rn | S^S^R,^) does not increase substantially. Therefore, the two factors affecting the amount of sequential dependencies, the amount of U(Rn:SJ and the number of stimulus/response categories, could be summarized by noting their dependence on the amount of U(Rn|Sn). To show the relation between U(Rn | SJ and the amount of sequential dependencies, in Figure 39 I plotted the information measures of sequential dependencies [UfR^S,,., |SJ, U(Rn:Rn.! |SJ, U(Rn:RB.1|Sn,Sa.1), and U(Rn:Sn.1,Rn.1 SJ] as a function of U(Rn | SJ, treating each different experimental condition in Experiments 1 to 4 as a single data point. There are 16 points in Figure 39; four number-of-tone conditions (4, 6, 10, and 16) from each of the mask and the no-mask conditions in Experiment 1, four number-of-tone conditions (4, 6, 10, 16) in Experiment 2, the 10- and 16-luminance conditions in Experiment 3, and the small and large range conditions in Experiment 4. All of the information measures of sequential dependencies seem to increase linearly with U(Rn|Sn). For example, in the mask condition of Experiment 1 (filled circles in Figure 39), the four information measures generally increase with an increasing amount of U(R n |SJ. An 0 . 3 S 0 . 2 2 O 0 . 1 5 U(Rn:Sn-i l% > * 3 • O • o 0 . 0 * £ 0 . 3 IL o £ 0 . 2 3 O < 0 . 1 0.0 U(Rn:Rn-1 ISh > * o it O o ' 1b. • E X P . 1 MASK J o E X P . 1 NO MASK ^ E X P . 2 • E X P . 3 • E X P . 4 U(RnlSn) Figure 39. Sequential dependencies as a function of U(RhlSh) (1) -4 138 inverse relation between U(R n:S 0) and the amount of sequential dependencies (Experiment 4, open squares) is also described by a linear relation between U(R n | Sn) and the amount of sequential dependencies. It thus seems that the two factors affecting the amount of sequential dependencies, the number of stimulus/response categories and the amount of U(R n :SJ , are well described by a linear relation between U(R„ | S J and the amount of sequential dependencies. To statistically confirm the trend seen in Figure 39,1 calculated Pearson moment-product correlations of those information measures with U(R n | SJ , the number of stimulus/response categories, and U(Rn:S„), again treating 16 different experimental conditions as different data points. The results are presented in Table 13. It is clear in Table 13 that each of the information measures is highly correlated with U(R n |S n ) . Although the information measures are also highly correlated with the number of stimulus/response categories, the correlation coefficients are not as high as those with U(R n |S n ) . As expected, the correlation coefficients with U(R n:S n) are all negative although they are not statistically significant. As a tentative conclusion, the amount of sequential dependencies increases linearly with U(R n |S n ) , which can be manipulated by the number of stimulus/response categories and/or U(R„:Sn). Table 13 Correlation coefficients of information measures of sequential dependencies with factors affecting them. (1) Experiments 1 to 4 (n = 16). Factors U(R n:S 0. 11SJ U(R n :R n . 1 |S n ) U(R n:R n . 1 |S n ,S n . 1) U(R n:S n,S n. 1,R n. 1) L K R J S J Number of categories U(R n :SJ 0.916*** 0.925*** 0.885*** 0.910*** 0.822*** 0.830*** 0.715*** 0.759*** -0.144 -0.160 -0.305 -0.263 Note - Bonferroni-adjusted probabilities were reported for the correlation coefficients of U(R n:S n . 1 |S n), U(R n :R n . 1 | S J , and U(R n :R n . 1 |S n ,S n . 1 ). * p < .05; ** p < .01; p < .001. 139 As Garner (1953) pointed out, in complex, difficult judgment situations, the response is dependent on factors additional to the current stimulus and, among those factors, the previous stimulus and response would provide the most useful information to make a response (Mori, 1988, 1989; Ward and Lockhead, 1971). In other words, the observer's response process in an absolute identification experiment tends to get information from Sn.! and R n.„ unless S n provides all the necessary information to make a response. As the complexity of a judgmental situation increases, as induced by increasing the number of stimulus/response categories and/or lowering U(R n:S n), the observer's response becomes more dependent on Sn., and R ^ . Therefore, the linear increase of the amount of sequential dependencies with U(R n | S J fits quite well with the idea that sequential dependencies in an absolute identification task reflect the operation of the observer's response process in the task (Mori, 1988, 1989; Mori and Ward, 1990; Ward, 1979; Ward and Lockhead, 1971). IV.6.2. The effects of trial-by-trial feedback Although the results of Experiments 1 to 4 suggest that the amount of sequential dependencies increases linearly with U ( R n | S J in absolute identification responses, there are also other factors affecting the amount of sequential dependencies. For example, the presence or absence of feedback has been known to affect the pattern of sequential dependencies as well as U(R n :SJ (e.g., Braida and Durlach, 1972; Siegel, 1972; Ward and Lockhead, 1971). In Experiments 5 and 6,1 systematically examined the effects of giving trial-by-trial feedback on the amount of sequential dependencies. In both experiments, I used the sequence-alternating method, which was very successful in maintaining almost the same amount of U(R n:S n) in the F E E D B A C K and the NO-FEEDBACK sequences. In both experiments, I found that the dependency on Sn., [U(R n:S n. j lSJ] was larger when feedback was given (FEEDBACK sequences) than when it was not (NO-FEEDBACK sequences), and that the dependency on R,,., [U(R n:R n., | S^S,,.,)] was smaller when feedback was given. The results were consistent with previous studies (Mori, 1989; Ward and Lockhead, 1971). Although U(R n :R n . 11SJ, another measure of the dependency on R n . ] ( was not lowered in the N O - F E E D B A C K sequences, this was probably because the stimulus and the response were highly correlated in the N O - F E E D B A C K sequences. Experiments 5 and 6 also showed the effects of U(R n:S n) (Experiment 5) and the number of stimulus and/or response categories (Experiment 6) on the amount of sequential dependencies. Following the discussion of Experiments 3 and 4, their effects can be summarized by the amount of U(R n | S J varying with A M O U N T O F I N F O R M A T I O N ( b i t s ) O ro o o • • CO O o ro o CO < < • > ip CO "Tjin O-S rn>< n o m o ) rjoO) m > O 75 oS rn>< y ™y •n o men rjocn m > S o 141 these factors. Given the effects of feedback mentioned above, it is expected that there would be some interaction between U(R n | S J and the presence or absence of feedback. Figure 40 plots U(R n:S 0. 11SJ, U C R ^ R ^ J S J , U(R n:R n . , |S n ,S n . 1), and UCR^S^.R^JSJ from various conditions of Experiments 5 and 6, as a function of U ( R n | S J . It is clear in Figure 40 that, although U(Rn:Sn_, | S J generally increased with U(Rn|Sn), it did not show as clear a linear trend as observed in Figure 39. The lack of a linear trend was mainly caused by the effect of feedback on UfX . iS^ lSJ . U(R n:S n.! ISJ was always larger in the F E E D B A C K sequences (filled triangles in Figure 40) than in the NO-FEEDBACK sequences of the same condition (open triangles), while U ( R n | S J was almost the same in both sequences because U(R n:S n) was kept constant by the sequence-alternating method. Interestingly for U(Rn:Sn., ISJ of Experiment 6 (inverse triangle), the difference between the F E E D B A C K and the NO-FEEDBACK sequences is larger the larger the amount of U ( R j S n ) . Although U(R n :R n . 1 |S n ) and U(R n:R 0 . 1 |S n ,S n . 1) are also affected by feedback, their differences in the F E E D B A C K and the NO-FEEDBACK sequences are relatively small. Finally, U(R n:S n.„R n. 1 | S J increases fairly linearly with U(R n | S J in the same way as in Figure 39. Table 14 presents Pearson product-moment correlations of the four information measures from the various conditions in Experiments 5 and 6 with U ( R n | S J , the number of stimulus/response categories, and U(R n:S B). Despite the small sample size (n = 10), the correlation coefficients with U(R n |S 0 ) agreed with the trend seen in Figure 40. The correlation between U(R n:S n., | S J and U(R n | S J is not significant. Interestingly, U(Rn:Sn.11SJ is only (significantly) correlated with the number of stimulus/response categories, suggesting that the number of stimulus/response categories might itself have had some direct effect on U(R n:S n. 11SJ. However, this does not necessarily mean that U(R n:S n. 11SJ depends more on the number of stimulus/response categories than on U ( R n | S J , because U ( R n | S J [or U(R n:S n)] was relatively unchanged by using the sequence-alternating method in Experiments 5 and 6. On the other hand, the other three information measures, particularly U(Rn:S n. 1,R n. 1|SJ, are highly correlated with U(R„|SJ. As observed in the results of Experiments 1 to 4 (Table 13), their correlation coefficients with U(R n |S n ) are higher than those with the number of stimuli. It follows from the above discussion that both U(Rn|Sn) and trial-by-trial feedback affect the amount of sequential dependencies, but in a different way. The size of U(R n |S t t ) seems to determine the total amount of sequential dependencies on and R„.„ that is, U(R n:S n . 1 ,R n . 1 |S n). In the discussion of Experiments 3 and 4,1 contended that as the complexity (or difficulty) of a judgmental situation is increased by increasing 142 Table 14 Correlation coefficients of information measures of sequential dependencies with factors affecting them. (2) Experiments 5 to 6 (n = 10). Factors U(R n:Sn-i ISJ U(R n:RD., |Sn) U(R n:R n., I S ^ ) U ^ S ^ ^ R , , . , ) U ( R n | S J 0.641 Number of categories 0.833* U(R n :SJ -0.023 Note - Bonferroni-adjusted probabilities were reported for the correlation coefficients of U(R n:S n . 1 |S n), U t R ^ R ^ I S J , and U(Rn:Rn.1|Sn,S„.1). * p < .05; »* p < .01; *** p < .001. the number of stimulus/response categories and/or lowering U(R n:S n), the observer's current response tends to depend more heavily on Sn., and R n.„ because they provide the most useful additional information in such a situation. Thus, the amount of sequential dependencies increases with U(R n | S J , which could be thought of as an information measure of the complexity of a judgmental situation. On the other hand, giving trail-by-trial feedback determines the relative contribution of S„., and R n., to R n . As an explanation of the effect of giving trial-by-trial feedback on the sequential dependencies, Ward and Lockhead (1971) argued that an observer tries to use the most reliable and available source to make a correct response, and that the feedback value associated with Sn., is such a source when feedback is given. On the other hand, if no feedback is given, the most reliable source of additional information is Rn.„ the observer's best guess as to the identity of Sn.!, and so is used. This explains the changes in the pattern of sequential dependencies, that is, the dependency on Sn.i [U(R n :S n J | SJ] is larger in the FEEDBACK sequence that in the NO-FEEDBACK sequence, and the dependency on R n - 1 [U(R n:R n., | S^S,^)] is smaller in the F E E D B A C K sequence. The results of Experiments 5 and 6 showed that the effect of giving trial-by-trial feedback was particularly large on U(R n:S n . , |S n), so large that U(R n:S n., | no longer showed a simple linear trend with U(R n |S n ) . 0.909 0.894 0.950 0.811* 0.480 0.666* -0.373 -0.647 -0.547 Moreover, remember that the difference between U(Rn:Sn.11SJ for the F E E D B A C K sequences and that for the NO-FEEDBACK sequences was larger the larger the number of stimulus/response categories used (inverse triangles in Figure 40). This suggests that the observer's tendency to use (or the previous feedback) in the F E E D B A C K sequences was enhanced by the increasing complexity of the judgment situation. The same type of effect was also observed for U(R n:R n., |Sn,Sn.,), but to a lesser degree. However, the effects of feedback do not seem to be strong enough to change the linear relation between U(R n | S J and U(R n:S n. 1,R n. 11S J . In other words, giving trial-by-trial feedback changes the pattern of sequential dependencies, but not the total amount of sequential dependencies [U(R n:Sn.„Rn.11SJ] that is determined by U ( R J S J . As a final comment on the results of Experiments 5 and 6, they demonstrated the usefulness of the sequence-alternating method. The sequence-alternating method was very effective in maintaining almost the same amount of U(R n:S n) in the F E E D B A C K and the NO-FEEDBACK sequences. Importantly, the constant amount of U(R n:SB) in the F E E D B A C K and the NO-FEEDBACK sequences suggests another function of feedback in the absolute identification task, that is, to help the observers to maintain the true identification function for making judgments. While some models of absolute identification performance emphasize a trial-by-trial role played by feedback (e.g., Treisman, 1984), others assume that feedback strengthens the memory of perceptual anchors that are used as references for judgments (e.g., Chase et al., 1983). Later, I will discuss in detail these models and their assumptions about the function of feedback in the absolute identification task. IV.6.3. The effects of response categories and the number of stimuli used In Experiment 7,1 tried to separate the effects of the number of response categories on an observer's absolute identification performance from that of the number of stimuli used, because there is a suggestion (Eriksen and Hake, 1955) that it is the number of response categories, rather than that of stimuli used, that most affects the observer's performance. For this purpose, I used the sequence-alternating method, in which the number of stimuli used was equal to that of response categories in the F E E D B A C K sequence but was reduced in the NO-FEEDBACK sequence. As observed in Eriksen and Hake, I found that U(R„) depended more on the number of response categories than on the number of stimuli used (Figure 34), and that U(R n:S n) remained constant as long as the number of response categories was larger than or equal to that of 144 stimuli used (Figures 35 and 36). According to Eriksen and Hake, those results can be explained, at least in part, by the observer's response process in the absolute identification task. In the absolute identification task, the observers have a strong tendency to use all response categories permitted even if the number of stimuli used is fewer than that of response categories, and at the same time they try to maintain a one-to-one correspondence between stimuli and responses (the identification function) as much as they do when the number of stimuli used is equal to that of response categories, thus maintaining the same performance level [ U ^ S J ] . The measures of sequential dependencies showed an interesting but puzzling pattern. The dependency of R n on Sn., increased only with an increasing number of stimuli used. On the other hand, the dependency of R n on R n.! showed reliable changes only when the number of stimuli used and/or the number of response categories increased from 6 to 11. There was only a small or negligible increase of the dependency on Rn.j when the number of stimuli used and/or the number to response categories increased from 11 to 16. While the effects of the number of response categories on the dependency on R„., can be explained by a linear relation between U(R n |S n ) and the amount of sequential dependencies in the way discussed in the previous sections, the effects of the number of stimuli used on the dependency on Sn., and, to a lesser degree, on Rn_1 cannot be explained by increasing U(R n | SJ , for in this experiment U(R n) changed with the number of response categories, not with the number of stimuli used (Figure 34), and U(R n:S n) remained constant irrespective of the number of stimuli used or the number of response categories. Thus the changes in the number of stimuli did not change U(R n | SJ . Although the above results may suggest an important point, it is hard to interpret the results because the size of the effects on the amount of the amount of sequential dependencies was subtle and there were some inconsistencies in the results [e.g., no difference between U(R n:R n., | S J for 6 tones used and that for 11 tones used with 11 response categories; see Figure 35]. In order to obtain larger effects, it may be necessary to run an experiment in which the difference between the number of response categories and that of stimuli used are larger than those used in Experiment 7, for example, 6 stimuli and 25 response categories. However, there are two problems with using those numbers of stimuli and response categories. First, the observer may notice during the experiment that the number of stimuli used is different from that of response categories, even if the sequence-alternating method is used. Secondly, a very larger number of trials and subjects will be required to obtain an uninflated estimate of the information measures. The inflation problem is even harder 145 to solve if the sequence-alternating method is used, because only about half of the observations can be used for the analysis. Thus it seems very difficult to run such an experiment, at least at this point. IV.7. Notes 1. The physical size of the presentation was calculated from the number of single dots (0.31 mm x 0.5 mm) in the presentation. 2. I applied a sphericity test before the repeated-measures analysis of variance, and Hyunh-Feldt corrected degrees of freedom and tail probability were reported whenever a sphericity assumption (see Vasey and Thayer, 1987) was shown to be violated. 3. The simulations used to correct the information measures of this experiment were done with the same combinations of the numbers of stimuli and response categories as those used in the experiment (e.g., 6 stimuli and 11 response categories). ( V. General Discussion The results of Experiments 1 to 7 suggest that an observer's absolute identification responses are greatly influenced by response processes. Especially, the pattern of sequential dependencies in absolute identification responses may be attributed to the operation of response processes (e.g., Garner, 1953; Ward and Lockhead, 1971)A In this chapter, I will discuss the implications of these results in the context of two recent models that predict a wide range of phenomena in absolute identification experiments (Braida and Durlach, 1988; Treisman, 1985). Although the two models have in common their assumption of Thurstonian models of absolute identification and their use of signal detection theory (Green and Swets, 1966), they emphasize different aspects of absolute identification performance. I will also discuss the differences between the amount of sequential dependencies and the degree of sequential dependencies. Throughout this thesis, I have used the amount of sequential dependencies, measured by multivariate information analysis, and it is possible that I might have obtained a different pattern of results if I had measured the degree of sequential dependencies (e.g., multiple correlation coefficients). As you will see, however, the degree of sequential dependencies, measured by the proportion of information measures (Krippendorf, 1986), shows a pattern essentially identical to that of the amount of sequential dependencies, at least as long as they are calculated from the data of the present experiments. Finally, I will summarize the present work and discuss future directions it could take. V . l . Models of absolute identification performance. In this section, I will discuss two recent models of absolute identification: Braida and Durlach's (1988) model of intensity resolution and Treisman's (1985) model of criterion-setting. The two models are common in two aspects. First, they are based on a Thurstonian model for absolute identification. In the Thurstonian model, which is illustrated in Figure 41, the effects of repeated stimulus presentations are described by a normal distribution on a sensory continuum with a mean and variance determined by the physical magnitude of the stimulus. Note that the distributions corresponding to the physical stimuli overlap each other, indicating that the presented stimuli will be sometimes confused each other and judgmental errors will occur. The second aspect common in the two models is that they are both written in terms of signal detection theory (Green and Swets, 1974). As in a Thurstonian model, signal detection theory usually assumes that the "1" "2" "3" "4" CATEGORY Figure 41. Thurstonian model and signal detection theory - 0 sensory effects of repeated stimulus presentations are normally distributed. In the theory, an observer's performance is described by two measures. One is d' (or sensitivity), which is the difference between the means of two normal distributions corresponding to two stimuli (Figure 41). A high value of d' means that the two stimuli are easy to distinguish or identify. The other measure is bias (/}), or the response criterion, by which the observer decides which stimulus was presented on each trial. In Figure 41, for example, a presented stimulus is judged as "1" whenever its sensory effect falls to the left of /}„ and it is judged as "2" whenever its sensory effect falls between /?j and /J2, and so on. Combined with a Thurstonian model, signal detection theory offers a way in which d's and /J's are calculated from absolute identification data and an observer's absolute identification performance is modelled. Making additional assumptions, Braida and Durlach's (1988) and Treisman's (1985) models have proven to be very successful in explaining a wide range of phenomena observed in absolute identification experiments. V . l . l . Braida and Durlach's (1988) model of intensity resolution. Braida and Durlach's (1988) model, which I briefly described in Chapter II, is strictly a model of resolution of sound intensity and tries to account for sensitivity to sound intensity differences measured in various psychophysical tasks (absolute identification, fixed and roving 2IFC discrimination). There are two main components in the model: sensory noise and memory noise. The sensory noise is specific to a given stimulus set and subject, and it is independent of the task in use. The memory noise, on the other hand, accounts for the requirement of memory in a given task, and it is assumed that differences in sensitivity measures obtained in different psychophysical tasks can be attributed to differences in memory noise. Braida and Durlach (1988) assumed that the memory noise comes from two sources; sensory-trace mode and context-coding mode. The sensory-trace mode maintains the image or trace of the sensation of a previously presented stimulus. The context-coding mode encodes the general context of sounds in the experiments (range, an identification function), allowing the observer to compare sensory traces of a sound to the context. It is assumed in the operation of the context-coding mode that the sensation corresponding to each stimulus intensity is coded by estimating its position relative to perceptual anchors, which in most cases are just outside the sensation corresponding to the extremes in a given stimulus set. According to Braida and Durlach (1988), only the context-coding mode operates in absolute identification tasks, because the inter-trial (stimulus) interval is too long (usually 5 to 8 seconds) for the 149 sensory-trace mode to operate. In their early work, Purks, Callahan, Braida, and Durlach (1980) calculated from absolute identification data sensitivity (d') and bias (fi) as a function of S^, and found that d' was not much affected by Sn.„ while fi was systematically changed by S„.,. From the results, Purks et al. (1980) concluded that the sequential dependencies in absolute identification responses are due to the changes in fi caused by a previous stimulus, but not by changes in d' (also, Braida and Durlach, 1988). Since the primary goal of Braida and Durlach's (1988) model is to explain the changes in d' in various conditions (tasks) and stimulus parameters, their model does not try to account for sequential dependencies in absolute identification responses, such as those observed in the present experiments, because they believed that the sequential dependencies reflect changes in fi, not in d'. Their model also assumes that sensitivity is independent of the number of stimulus/response categories used, and the model makes no prediction for the effects of the number of stimulus/response categories used on an observer's identification performance. In their latest account for the function of feedback (Chase et al., 1983), a previous stimulus and its associated feedback serve to strengthen, or make less variable, perceptual anchors in the stimulus set. Since the perceptual anchors are used to encode the sensation corresponding to each stimulus in the context-coding mode, less variable perceptual anchors increase sensitivity, measured by d' [or U(R n:S n)]. Thus, according to Chase et al. (1983), feedback will help to maintain U(R n :SJ at a high level as long as feedback is given every trial. However, our results of Experiments 5 and 6 suggest that the effect of feedback on U(R n:SB) lasts longer than Chase et al. (1983) assumed. U(R n:S n) can be maintained at a high level even if feedback is not given every trial, as long as a sequence of about 30 trials with feedback is alternated with that without feedback. The long-lasting effect of feedback on U(R n:S n) is particularly well demonstrated in Experiment 5, in which I found no difference in U(R n:S n) between FEEDBACK and N O - F E E D B A C K sequences in absolute identification of sound intensity levels that were almost identical to those used in Braida and Durlach (1972), who showed 20 % improvement of sensitivity (d') by giving feedback. Thus, Braida and Durlach's (1988) model, although very quantitative and comprehensive in the domain of sound intensity resolution, only deals with sensitivity measures and factors affecting them, and does not make, at least explicitly, predictions for the type of sequential dependencies obtained in the present study. Although the model predicts that feedback improves sensitivity [d' and U(R n :SJ] , the results of my experiments have shown that the effects of feedback are longer-lasting than the model assumes. 150 V.1.2. Treisman's (1985) model of criterion setting Treisman's (1985) model of criterion setting is a general model of psychophysical judgments, ranging from detection and discrimination to absolute identification to ratio judgments and confidence rating (Treisman and Williams, 1984). Treisman's (1985) model assumes that the setting of response criteria (shifting and maintenance) is a primary source of variance in absolute identification performance, and it describes the mechanisms of criterion'setting. Figure 42 illustrates Treisman's (1985) model. In the model, there are three systems responsible for criterion setting in an absolute identification task; reference, stabilization, and tracking. The reference system sets the initial positions of response criteria (called reference criteria in Figure 42) corresponding to the stimuli used in the task, according to the general properties of the task (presentation probability, stimulus parameters, prior experience with similar tasks and stimuli, and so on). The other two systems shift the response criteria on a trial-by-trial basis. The stabilization system monitors the presented stimuli and shifts all of the response criteria every trial towards the sensation corresponding to the previous stimulus. In Figure 42, for example, the sensation corresponding to stimulus 3 presented on trial n-1 falls to the right of response criterion Z 3 , causing all of the response criteria to shift toward it. Treisman (1985) assumed that the size of a criterion shift caused by stabilization is a function of the distance between the sensation and the criterion; the larger the distance, the larger the size of the shift. The tracking system 'tracks' the observer's responses and shifts the criteria away from the previous response, so that the immediately previous response (R^) is more likely to be repeated. Using the example of Figure 42, the response on trial n-1 was "4" because the sensation of S„.i fell to the right of Zj, and all of the response criteria are thus shifted to the left for the next trial. As with the stabilization systems, the size of a criterion shift is assumed to be a function of the distance between the sensation and the criterion. Since the model assumes that all the three systems contribute to the setting of the response criteria on each trial, the effective value of each criterion is expressed as the weighted combination of the three systems, and the relative contribution of the three systems depends on the weight attaching to each system. In Figure 42, the (relative size of) weight changes the size of criterion shifts caused by the stabilization and tracking systems. The response category given to the stimulus on trial n is chosen according to the relative positions of the sensory effect of the stimulus and the relevant response criterion(a) (resultant criteria in Figure 42) on the sensory continuum. 1 "1 «2 • " 3 ' 1?A* Zi z 2 z 3 Figure 42. Illustration of Treisman's (1985) model CATEGORY PHYSICAL STIMULUS REFERENCE CRITERIA STABILIZATION TRACKING I I I I I Rn-1 - 4 i I I I I I RESULTANT CRITERIA In Treisman's model (1985), sequential dependencies are caused by the operation of the stabilization and tracking systems; the stabilization system produces sequential dependencies on SB.ly while the tracking system produces sequential dependencies on Rn.,. Although Treisman (1985) did not discuss specifically an inverse relation between U(R 0:S n) and the amount of sequential dependencies, one possible explanation is that changes in U(R n:S n) change the way in which the three criterion-setting systems in the model contribute to the setting of response criteria. When U(R„:Sn) is high, it means that the current locations of response criteria, which are set by the reference system at the beginning of the experiment, are appropriate, so that there is no need for the stabilization and tracking systems to shift the response criteria every trial. In such situations, the weight attached to the reference system will be large, and the weights attached to the tracking and reference systems will be relatively small. As a result, the amount of sequential dependencies on Sn.] and R n., will be small. When U(R n:S n) is lowered by presenting a masking white noise with tones to be judged (Experiment 1) or reducing the stimulus range (Experiments 4 and 5), it indicates that the current location of response criteria is no longer appropriate, and it is necessary to shift the response criteria to adjust their locations in accordance to the evidence provided by past experience, such as the previous stimulus and response. To realize this in the model, the weight attached to the stabilization and tracking systems will be increased and the size of criterion shifts caused by the two systems will become larger, causing a larger amount of sequential dependencies on Sn.j and Rn.j. Treisman (1985) discussed the effects of the number of stimulus/response categories, but only in terms of U(R n :S n ) . Using computer simulations of his model, Treisman (1985) showed that the level-off of U(R n :S J with increasing number of stimulus/response categories (channel capacity) can be simulated by using constant variance of the sensory distributions and placing the criteria closer together as the number of stimulus/response categories increases. Rephrasing Treisman's (1985) explanation, the opportunity for U(R n :S n ) to increase with increasing number of stimulus/response categories is subverted by the opportunity for a presented stimulus to distribute its sensory effects over a greater number of response criteria. Although Treisman (1985) did not mention the possible effects of the increase of the number of stimulus/response categories on sequential dependencies, the likelihood that the choice of a response category is affected by criterion shifts will increase as the sensory effect of a presented stimulus is distributed over a greater number of response criteria, because the model assumes that all of the response criteria will be shifted every trial by the stabilization and tracking system. Therefore, Treisman's (1985) model can explain the effects of the 153 number of stimulus/response categories on the amount of sequential dependencies. Treisman's (1985) model assumes that the function of feedback in absolute identification is to enable selective tracking; when feedback is given, the tracking system uses a previous response only if it is equal to or close to the feedback value associated with it. In most cases, however, an observer's responses are not equal to their associated feedback values. "When feedback is given, many responses will be ineffective for tracking, allowing the effects of stabilization to bulk relatively larger. But in the absence of feedback, all responses will be effective, producing a larger ... dependency [on R n . t]. This 'release of tracking' is what we see in the no-feedback [results]." (Treisman and Williams, 1984). Since Treisman's (1985) model also assumes that the stabilization system operates no matter whether feedback is given or not, the change in the pattern of sequential dependencies on SnA and R,,., arises from the change in the relative contribution of the operation of the stabilization system and that of the tracking system (also Treisman and Williams, 1984). Thus, the dependency on Rn.„ produced by the tracking system, is smaller when feedback is given than when it is not, and the dependency on Sn.j, produced by the stabilization system, is larger when feedback is given (Experiments 5 and 6). Moreover, selective tracking can also explain why the difference between the amount of sequential dependencies in the FEEDBACK and NO-FEEDBACK sequences becomes larger as the number of stimulus/response categories increases: As the number of stimulus/response categories increases, a greater number of responses will be ineffective for the tracking systems because they are not equal to or even close to the associated feedback values, making the (relative) contribution of the stabilization system higher and that of the tracking system lower. As in Braida and Durlach's (1988) model, however, Treisman's (1985) model does not recognize the long-lasting effects of feedback on the maintenance of the identification function [or reference criteria in Figure 42]. In the model, the effects of feedback on an observer's performance is short-lived, while the results of Experiments 5 and 6 suggest that the effects of feedback on U(R n:S n) lastij at least as long as about 30 trials. In Treisman's (1985) model, there is no specific mechanism that can explain the observer's tendency to use all the response categories equally often even if the number of stimuli used is fewer, or the constancy of U(R n :S n) without respect to the number of stimuli or response categories (Experiment 7; and also Eriksen and Hake, 1955). However, the former effect can be explained by the operation of the stabilization system. In the stabilization system, the response criteria are shifted toward the mean of the distribution of sensory 154 effects of each presented stimulus, tending to prevent the repetition of the same responses to the stimulus. Thus the overall effect of the operation of the stabilization system is to allow all the response categories to be used equally often. Overall, Treisman's (1985) model of criterion-setting can explain the results obtained in this thesis very well, with a few exceptions (e.g., long-lasting effect of feedback). This is partly because the ultimate goal of Treisman's (1985) model is to predict an observer's absolute identification performance, not sensitivity, and partly because in order to predict absolute identification performance, Treisman (1985) used a mechanism of criterion-setting that was originally motivated by the need to explain sequential dependencies (Treisman and Williams, 1984). V.2. The amount of sequential dependencies and the degree of sequential dependencies. In Chapter II, I discussed the difference between the amount of relation and the degree of relation, and-how statistical techniques can be classified into those that measure the amount of relation and those that measure the degree of relation. For example, the correlation coefficient measures the degree of relation between two variables because it is the ratio of two variances, while the contingent uncertainty [U(A:B)J measures the amount of relation between two variables, because it is the amount of information that is explained by the relation between them, without respect to how much information is noj explained by the relation (Garner, 1962). We can extend the same argument to the multivariate case, that is, multivariate information analysis and, for example, multiple correlation analysis. When both analyses are used to measure sequential dependencies of psychophysical judgments, the multivariate information analysis measures the amount of sequential dependencies, and the multiple correlation analysis measures the degree of sequential dependencies. To be more precise, I use Equation 20: U(R„) = U(R n :S J + U(R n:SB. 1 |S n) + U(Rn:Rn.11 S„S^) + U(R n |S n ,S n . 1 ,R n . 1) (20). In Equation 20, U(R n :S n . 1 |S n ) and U(R n:R n. 1 |S I 1,S 0. 1) measure the amount of sequential dependencies on Sn., and Rn.i respectively, without respect to the size of U(R n) or U(R n |S n ,S n . 1 ,R n . 1 ). To measure the degree of sequential dependencies, a multiple correlation analysis has a structure similar to that of Equation 20, with R n being the dependent variable and Sn, Sn.j and Rn_i the independent variables, and it measures the sequential dependencies on and Rn_, by partial correlation coefficients associated with and R n.„ respectively. However, the partial correlation coefficients measure the degree of sequential dependencies, 155 because they are thought of as the ratio of variance explained by the effect of Sn., or R n., to the total variance ofR n . Throughout this dissertation, I used multivariate information analyses (Equations 17 and 20) to measure the amount of sequential dependencies, and I have based my entire discussion on the results of the amount of sequential dependencies. If I had used other analysis methods that measure the degree of sequential dependencies, however, I might have obtained a different pattern of the results. Since multiple correlation analysis cannot be applied straightforwardly to absolute identification data because the data are categorical, I calculated the proportion of the information measures of sequential dependencies [U(R n :S n . i |SJ, U(R n :R n . J S J , U(R B:R n . 1 |S I 1 ,S n . 1), and U(Rn:Sn.„Rn.11SJ] in the total amount of response information [U(RJ] and used the proportion as the measure of the degree of sequential dependencies. The proportion of an information component in the total amount of information is sometimes called the index of predictability (Krippendorf, 1986). Although the proportion of information measures may not be totally analogous to such measures as the partial correlation coefficient, it should give us some indication of what could be obtained if measures of the degree of sequential dependencies are calculated from the data. Figure 43 presents the proportion of the amount of information measures from the various conditions of Experiments 1 to 7, as a function of U(R n |S n ) , in the same way as the amount of sequential dependencies was presented in Figures 39 and 40. As is obvious in Figure 43, the proportion of information measures show a pattern quite similar to that observed in Figures 39 and 40. The proportion of information measures all increase with U(R n |S B ) , and some of information measures show a linear trend with U(R n |S , J . In Experiment 5 and 6, the proportion of U(R n :S B . ) |SJ is larger in the F E E D B A C K sequences (filled triangle) than in the N O - F E E D B A C K sequences (open triangles). Therefore, it is safe to say that I can make the same conclusion about the results of Experiments 1 to 7 on the basis of the proportion of information measures, and that other measures of the degree of sequential dependencies would show a pattern similar to that observed in Figures 39, 40, and 43, at least for the data reported in this thesis. V.3. Conclusions and future directions As Garner (1962) argued, sequential dependencies are ubiquitous in any aspect of human behavior. In psychophysics, there is a growing recognition that sequential dependencies should be incorporated into any model of psychophysical judgments. However, techniques for the analysis of sequential dependencies are still s cc o Li. o o p cc o flu O 0 . 0 8 0 . 0 6 0 . 0 4 0 . 0 0 0 . 0 8 U ( R „ : S „ - 1 I S , ) „ • _ X • E X P . 1 M A S K o E X P . 1 N O M A S K ^ r E X P . 2 - E X P . 3 • E X P . 4 A E X P . 5 F E E D B A C K A E X P . 5 N O F E E D B A C K • E X P . 6 F E E D B A C K v E X P . 6 N O F E E D B A C K x E X P . 7 0 . 0 0 > 0 U ( R n l S h ) Figure 43. Proportion of sequential dependencies 0\ underdeveloped. In this dissertation, I used multivariate information analysis to measure sequential dependencies in absolute identification responses. Because of its nonmetricity, multivariate information analysis is very useful in analyzing categorical data, such as those of an absolute identification experiment. Although information measures are known to be inflated when there are few observations relative to a large number of variables involved in the calculation (Houtsma, 1983), there is no previous research that dealt with the inflation problem of multivariate information measures. Thus, I ran extensive computer simulations of absolute identification and calculated the multivariate information measures as a function of the number of observations and the number of variables used in the calculation. The results, along with a method of pooling individual data, were used to correct the information measures from empirical data. In order to investigate factors affecting the sequential dependencies in an absolute identification task, I conducted seven experiments. The results of the seven experiments can be summarized as follows; (1) . There are two major factors affecting the amount of sequential dependencies in absolute identification responses : the amount of information transmission U(R n:S J (Experiments 1, 4, and 5) and the number of stimulus/response categories (Experiments 1, 2, 3, 6, and 7). Both can be interpreted as the effects of increasing the complexity of the judgmental situation, measured by U(R„ | S J . The amount of sequential dependencies generally show a linear trend with U(R ! i |S n ) . (2) . Giving trial-by-trial feedback also has a substantial effect on the amount of sequential dependencies (Experiments 5 and 6). The dependency on S M [U(Rn:S a.1|Sn)] is larger when feedback is given than when feedback is not given, and the dependency on Rn_, is smaller when feedback is given. The effect of feedback is particularly large on U(R n:SD., |S n), and the increase of U(R n:S n., |S n) by giving feedback is larger for a larger number of stimulus/response categories used. (3) . Alternating a sequence of about 30 trials with feedback with a similar sequence without feedback yields almost the same amount of U(R n:S n) in both sequences (Experiments 5 and 6). (4) . When the number of stimuli used is fewer than that of response categories (Experiment 7), the observers try to use equally all the response categories permitted, causing U(R n ) to depend more on the number of response categories than that of stimuli used. In the same situation, U(R n:S n) is constant regardless of the number of stimuli and response categories. (5) . While U(R n :S n . i |S n) depends more on the number of stimuli than on the number of response categories, U(R n :R n . , | S J and U(R n:R n. 11S^S^) increases as the number of stimuli and/or the number of 158 response categories increases. However, the changes in the amount of sequential dependencies are relatively small. (6). The same pattern of sequential dependencies was obtained by using the proportion of information measures of sequential dependencies in U ( R J , suggesting that the results would not be changed very much by using other measures of the degree of sequential dependencies. The above results are quite consistent with Treisman's (1985) model of criterion setting. Treisman's (1985) model assumes that the variability in an observer's response is caused by the trial-by-trial variability of sensory effects of stimuli and the trial-by-trial setting of response criteria, and that the criterion-setting mechanisms generate sequential dependencies on Sn., and Rn.,. Although we may not need to investigate sequential dependencies in order to predict an observer's sensitivity [d' or U(R n:S J ] measured in an absolute identification task (Braida and Durlach, 1988; Purks et al., 1980), we have to deal with sequential dependencies to understand an observer's actual responses, or behavior, in the task. Finally, there are several directions in future research along the lines of this dissertation: (1) . As I mentioned earlier, it is necessary to develop an objective method to fit empirical estimates of information measures to simulation results, instead of using the "fitting-by-eye" method. (2) . My computer simulations and other studies clearly indicated that Miller's (1954) correction equation, and possibly his transformation equation, of information measures are inappropriate in most empirical research. Since enough data are now available from the simulations and empirical research, it should be possible to develop a new correction equation of information measures. If successful, a new correction equation would be of great use in empirical research. (3) . Although Treisman's (1985) model could explain most of the results obtained in the present experiments, it did not predict the long-lasting effects of feedback on U(R n:S n) in the sequence-alternating method (Experiments 5 and 6), and neither did Braida and Durlach's (1988) model. A tentative hypothesis is that feedback reinforces a long-term memory of the identification function. Before theorizing about the effects, it is necessary to establish how long-lasting the effect is, by systematically changing the number of trials in one sequence. (4) . Since Treisman's (1985) model is a general model of psychophysical judgments, it is possible that sequential dependencies in tasks other than absolute identification will show a pattern similar to that obtained in this dissertation. Among those tasks, the category judgment task is methodologically very close to that of absolute identification, although the category judgment task is regarded as that of interval measurement. It would be interesting to run category judgment experiments under conditions identical to those used in the absolute identification experiments of this thesis, and to see whether the pattern of the sequential dependencies in the category judgment experiments, measured by multivariate information analysis, will be different from that obtained in this dissertation. In addition to this, since category judgment data can be analyzed by multiple correlation analysis because the data are on an interval scale, the results of multivariate information analysis can be compared to that of multiple correlation analysis from the same data. References 160 Attneave, F. (1959). Applications for information theory to psychology. New York : Holt. Beebe-Center, J. G., Rogers, M . S., & O'Connel, D. N . (1955). Transmission of information about sucrose and saline solutions through the sense of taste. Journal of Psychology. 39, 157-160 Berliner, J. E., & Durlach, N . I. (1973). Intensity perception. IV. 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A. , & Siegel, W. (1972). Absolute judgment and paired-associate learning: Kissing cousins or identical twins. Psychological Review. 79, 300-316. Siegel, W. (1972). Memory effects in the method of absolute judgment. Journal of Experimental Psychology. 94, 121-131. Shannon, C. E. (1948). The mathematical theory of communication. Bell System Technical Journal. 27, 379-423; 623 - 656. Stevens, S. S., & Davis, H . (1938). Hearing: Its psychology and physiology. New York : Wiley. 164 Tan, H . Z., Rabinowitz, W. M. , & Durlach, N. I. (1989). Analysis of a synthetic Tadoma system as a multidimensional tactile display. Journal of the Acoustical Society of America. 86, 981-988. Torgerson, W. S. (1961). Distances and ratios in psychophysical scaling. Acta Psychologica. 19, 201-205. Treisman, A . (1986). Properties, parts and objects. In K. Boff, L. Kaufman, & J. Thomas (Eds.), Handbook of perception and human performance. New York : Wiley. Treisman, M . (1985). 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Repeated magnitude estimations with a variable standard: Sequential effects and other properties. Perception and Psvchophvsics. 31. 53-62. Ward, L. M . (1979). Stimulus information and sequential dependencies in magnitude estimation and cross-modality matching. Journal of Experimental Psychology: Human Perception and Performance. 5_, 444-459. Ward, L. M . (1991). Informational and neural adaptation curves are asynchronous. Submitted for publication. Ward, L. M . , & Lockhead, G. R. (1971). Response system processes in absolute judgments. Perception and  Psychophvsics. 9, 73-78. 165 Appendix The amount of information transmission from Sn to R n is defined by U f R ^ S J = U(SJ + U ( R J - UCS^RJ (A.1), where U(SJ = -sp L log 2 P j U ( R J = - s pj log 2 pj U(S n,R J = - z py log 2 P y and p. and p j are the probability of stimulus category i and of response category j, respectively, and py is the joint probability of stimulus category i and response category j. U(SJ and U f R J measure the amount of information of S n and of R n , respectively, and U(R n,S i) the amount of joint information of S n and R n . Let us denote M as the number of total observations in a data set, m- as that of presentations of stimulus category i , nij as that of occurrences of response category j, and my as the number of joint occurrences of stimulus category i and response category j. And let us define T = log 2 M , Tj = (1/M) 2 nij log 2 m., Tj = (1/M) z m- log 2 nij, Ty = (1/M) z my log 2 my, so we can rewrite Equation A.1 as UCR.rSJ = T - T. -Tj + Ty (A.2) ( McGill, 1954 ). Next, we extend this equation to include three inputs, Sn, S^,, R ^ . We denote m k and nij as the number of occurrences of category k of the previous stimulus (S^) and that of occurrences of category 1 of the previous response (R^), respectively. We have the amount of information transmission from Sn.1 to R n as U(R I >:Sn.1) = T - T j - T k + T j k (A.3). But since this measure includes the effect of Sn, we compute U(R n:S n. 1) for each value of S n and then average them together in order to eliminate the effect of Sn. Then we have U(R n :S n . 1 |S n) = T i - T y - T . k + T y k (A.4), which is the information transmission from only S„., to R n and is interpreted as the amount of the effect of Sn.! on R„ (with the value of S n held constant). Similarly, the information transmission from only R n 4 to R n , or the amount of the effect of R ^ on R n with the value of S n held constant, is measured by U(R n :R n . 1 |S n ) = T . - T y - T i l + Ty 1 (A.5), and the amount of the effect of R ^ on R n with the value of S n and held constant is 166 U d ^ R , . , ISA.,) = T . k - T y k - T i k , + T j j k l (A.6). The amount of the net effect of Sn, S^, and R ^ on R n is obtained by U(Rn:Sn,Sn.i,Rn-.) = U(R n :SJ + U ^ S , , . ISJ + UiR^ | Sn,Sn.,) (A.7), and U ( R J can be expressed as follows: UCRJ = UCR^SJ + UCR^S^ISJ + U ^ R . J S A . O + U(Rn|Sn )Sn-i.Rn-i) (A.8), where U(R n | S^S^.R,,.,) means the residual of U(R n ) after the effects of Sn> Sn.u and R,^ are excluded. Appendix 2 Uncorrected estimates of information measures. (1) Experiments 1 to 3 Number of Stimuli U C R ^ S J U f R A J S J I K R ^ I S J U ^ R ^ I S ^ ) U ( R , U . W Experiment 1. Mask Condition 4 1.211 .060 .058 .055 1.326 6 1.456 .174 .165 .175 1.805 10 1.513 .522 .466 .494 2.529 16 1.574 1.009 .949 .762 3.345 No Mask Condition 4 1.853 .029 .029 .010 1.892 6 2.040 .132 .118 .029 2.200 10 2.093 .356 .344 .239 2.689 16 2.018 .865 .825 .585 3.469 Experiment 2*. 4 1.990 .002 .002 .000 1.992 6 2.272 .046 .046 .004 2.322 10 2.260 .163 .138 .121 2.544 16 2.283 .245 .217 .289 2.817 Experiment 3 10 2.111 .337 .347 .259 2.707 16 1.989 .839 .823 .631 3.459 * the estimates of information measures were calculated from the total number of observations (240 observations per tone for the 4-, 6-, 10-tone conditions, and 460 observations per tone for the 16-tone condition) in Experiment 2. Uncorrected estimates of information measures. (2) Experiments 4 to 6 Condition UCR^SJ U(R„:Sn., | S J UCR^R,., | S J U(Rn:R„.1 \S„S^) U ^ i S ^ S ^ R , , . , ) Experiment 4 Large Range Small Range 3.438 2.266 .236 .654 .237 .651 .053 .508 3.726 3.428 Experiment 5. FB NFB 1.543 1.546 .520 .502 Large Range .524 .521 .549 .571 2.613 2.618 FB NFB 1.024 .995 .723 .652 Small Range .698 .715 .870 .948 2.617 2.595 Experiment 6. FB NFB 2.113 2.068 .117 .105 6 categories .116 .109 .034 .089 2.264 2.262 FB NFB 2.123 1.951 .457 .443 11 categories .440 .463 .290 .470 2.870 2.863 FB NFB 2.321 2.187 .735 .711 16 categories .702 .717 .405 .584 3.462 3.481 169 Uncorrected estimates of information measures. (3) Experiment 7 Condition U C R ^ S J U(R n :S n . , | S J U (R a :R B . I | S J U ^ R , . , | S ^ i ) U(R n :S n ) S, , i ,R, , i ) 1 2.110 .212 .367 .310 2.631 2 2.284 .502 .660 .549 3.334 3 2.240 .276 .593 .444 2.960 Mori, S. (1986a). Sequential dependencies in judgment of horizontal line lengths : Application of attention band theory. The Japanese Journal of Psychology. 57, 261-265. (in Japanese). Mori, S. (1986b). Quantitative analysis of sequential dependencies by means of multivariate information transmitted. Brief Reports from the Laboratory of Psychology. Kyoto University. NO. 24. Mori, S. (1988). Two response processes in a guessing task. Perception & Psychophysics. 44, 50-58. Mori, S. (1989). A capacity-limited response process. Perception & Psychophysics. 46. 167-173. Mori, S. & Ward, L. M. (1990). Unmasking the magnitude estimation response. Canadian Journal of Psychology. 44, 58-68. 

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