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On the learning and transfer of multi-cue judgement processes Thorngate, Warren 1971

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ON THE LEARNING AND TRANSFER OF MULTI-CUE JUDGEMENT PROCESSES by WARREN BAYLEY THORNGATE B.A., U n i v e r s i t y o f C a l i f o r n i a , 1966 M.A., U n i v e r s i t y o f B r i t i s h C olumbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f PSYCHOLOGY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1971 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may h e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a D a t e fO A*f';l l?7Z~ ABSTRACT Three experiments employing a multiple cue probability learning situation were conducted to determine how multi-cue judgement processes are learned and transferred. Each subject was required to predict one of two responses (yes or no) to questions ostensibly answered by four types of stimulus persons each described by values on two dichotomous dimensions, marital status (married or single) and sex (male or female). In the pretest subjects were required to predict, without feedback, each stimulus person's responses to two such questions which varied in their relatedness. In the learning phase subjects continued to predict responses to only one of the two questions but after each prediction they were given feedback about its correctness. In the estimation phase subjects were required to estimate the proportion of yes responses given by each of the four types of stimulus person (married males, mar-ried females, single males and single females) and by each of the four stimulus characteristics (marrieds, singles, males and females) on the basis of feedback received in the learning phase. In the posttest sub-jects were again required to predict, without feedback, each stimulus person's responses to the two questions in the pretest. Each of the three experiments varied the proportion of yes responses given by the four stimulus types. Half the subjects in Experiment I learned and estimated proportions that varied as a function of a main effect of one stimulus dimension (an Additive function), the remainder learned and estimated proportions that varied as a function of the in-i i . teraction between both stimulus dimensions (an Interactive function). All subjects in Experiment II learned and estimated proportions that varied as a function of the main effect of one stimulus dimension and the interaction between both stimulus dimensions (a Composite function). And all subjects in Experiment III learned and estimated proportions that varied as a function of the main effects of both stimulus dimen-sions (a Compound function). The results of Experiment I indicated that the Interactive function was learned at an almost identical rate as the Additive function. This supported a class of learning models which assumed that both functions were learned by associating responses with stimulus types or configura-tions, rather than by associating responses with stimulus characteristics or dimensions. The results of Experiment II indicated that the interac-tion component of the Composite function was learned more slowly than its main effect. This supported a class of judgement models which as-sumed that when responses to a stimulus type had not yet been learned, predictions about this type were made by extrapolating from similar stimulus types with learned associations. The results of Experiment III indicated that parameters of the extrapolation process underwent signi-ficant changes over time. Over all three experiments the time taken to estimate the propor-tion of yes responses to each stimulus type was shorter than the time taken to estimate the proportion of yes responses to each stimulus characteristic. However, over all three experiments, the variability of the stimulus type estimates was greater than the variability of the i i i . stimulus characteristic estimates. These results gave additional sup-port to the hypothesis that the functions were learned cell-by-cell rather than dimension-by-dimension. It was hypothesized that in the posttest phase of all experiments the judgement process used to predict responses to the question in the learning phase would be transferred to a second question to the extent that the two questions were related in the pretest. Though attempts were made to vary the strength of the pretest relationship by an a_ priori selection of question pairs, pretest responses to all questions were found to be virtually unrelated to one another. However, in the posttest, the relationships of responses to all question pairs increased, and attained a rank order of strengths originally predicted by the a_ priori selection of questions. A transfer model, similar to the judge-ment model tested in Experiments II and III, was proposed to account for this finding. iv. TABLE OF CONTENTS Chapter Title Page 1 Introduction 1 2 The Learning of Judgement Processes . . 11 3 Learning vs. Judging 22 4 The Extinction of Judgement Processes . . . . . . . 25 5 The Communication and Transfer of Judgement Processes 27 6 The Learning Situation 31 7 Possible Models of Learning and Transfer 39 8 Experiment I 48 9 Experiment II 71 10 Experiment III 95 11 Summary and Speculation 112 Bibliography 120 Appendix 128 V . LIST OF TABLES Table Title Page I The Four Additive Functions. . 34 II The Two Interactive Functions 35 III Q-i'Ss a n d Q-i'Sa P r e t e s t Contingencies of the Additive and Interactive Functions . 54 IV ANOVA Summary Table for Learning Phase of Experiment 1 55 V Average Cell Estimates and Average Marginal Estimates of the Additive and Interactive Functions 58 VI ANOVA Summary Table of Additive and Interactive Function Cell Estimates 59 VII Average Latencies of Cell and Marginal Estimates of Additive and Interactive Functions 60 VIII ANOVA Summary Table of Estimate Latencies 61 IX Variances of Cell and Marginal Estimates of Additive and Interactive Functions 62 X Estimated Sample Sizes on Which Cell and Marginal Estimates Were Based 63 XI ANOVA Summary Table for Posttest of Experiment I . . . 66 XII Observed and Predicted Average Qp -Qo mpr Proportions in Each Trial Block of the Posttest 67 XIla Possible Outcome of Judgements of Optimism Where mM and sM Values Known, mF and sF Values Guessed Blindly . 72 XII lb Possible Outcome of Judgements of Optimism Where mM and sM Values Known, mF and sF Guessed by Generalization 72 XIV Average Proportion of "YES" Responses Given by Sub-jects in the Pretest and in Each of the Five Learning Blocks 79 XV ANOVA Summary Table of Learning of Composite Function Components 82 vi. Table Title Page XVI Average Cell Estimates and Average Marginal Estimates of the Composite Function 84 XVII ANOVA Summary Table of Estimates of the Composite Function 85 XVIII Variances and Estimated Sample Sizes of Cell and Marginal Estimates 86 XIX Posttest mpr Proportions for 0, and 0- With 0- Posttest Proportions Predicted From SimpleModel 87 XX Probabilities of "Yes" to 0, in the Four Compound Functions . 95 XXI Average Proportions of "Yes" Responses Given by Subjects in the Pretest and in Each of the Five Learning Blocks 100 XXII ANOVA Summary Table of mpr Proportions in the p=0.2 and 0.8 Cells of the Compound and Composite Functions . . 102 XXIII ANOVA Summary Table of the Learning of Compound Function Main Effects 103 XXIV Average Cell and Marginal Estimates of the Compound Functions .105 XXV ANOVA Summary Table of Cell Estimates of the Compound Functions 106 XXVI Variances and Estimated Sample Sizes of Cell and Marginal Estimates 107 v i i . LIST OF FIGURES Figure Title Page 1 Average mpr Proportions of Subjects Learning Additive and Interactive Functions. . . . . . . . 56 2 Average Posttest mpr Proportions of Subjects Learning Additive and Interactive Functions . . . 65 3 Average mpr Proportions of Subjects Learning p=0.2 and p=0.8 Cells of the Composite Functions . . . 81 4 Learning Curves of the Composite Function and Its Components 83 5 Flow Chart of a Possible Model of the Judgement Process 90 6 Flow Chart of a Possible Model of the Transfer Process 92 7 Average mpr Proportions of Subjects Learning p=0.2 and p=0.8 Cells of the Compound Functions . . . . 101 8 Learning Curves of the Compound Function and Its Components 104 9 Outline of a Model of the Judgement and Transfer Process 114 vi i i ACKNOWLEDGMENT I wish to thank the members of my dissertation committee for all effort they extended in the reading of this manuscript. Special thanks are given to Dr.s Michael Humphreys, Robert Knox, William Petrusic and Paul Slovic for their invaluable guidance and encourage-ment in the planning and execution of the research reported herein. INTRODUCTION Whenever an individual evaluates a stimulus having more than one attribute or dimension he is said to be making a multi-cue judgement. Multi-cue judgements are a ubiquitous aspect of cognitive behaviour. They occur, for example, each time one judges the trustworthiness of a politician, the possibility of inclement weather, the desirability of a member of the opposite sex, the profundity of a musical composition, the classification of a psychiatric patient or the quality of a disserta-tion. In one form or another multi-cue judgements have occupied the attention of psychologists since the mid-19th century. The first in-vestigations of multi-cue judgements came primarily from the area of perception, and in particular from attempts to specify the cues which affected judgements of shape, size, depth, and color (see, for example, James 1890). The Gestalt movement, with its concern over the means by which cues from multi-cue stimuli were combined, greatly increased per-ception's interest in these judgements (see Kohler 1947; Kafka, 1935). The interest has continued to motivate much contemporary perceptual re-search, manifesting itself in attempts to scale multi-dimensional stimu-l i in Euclidian and non-Euclidian spaces (see Shepard, 1964) and in attempts to determine how perceptual patterns are recognized (see Neisser, 1967). But investigations of multi-cue judgements have by no means been confined to strictly perceptual phenomena. Evaluative or value judge-ments of multi-attribute objects and commodities (theatre tickets, wagers, 2. dinners, stocks, automobiles, etc.) have long been the subjects of ex-perimental inquiry (see Becker & McClintock, 1967). So too have numer-ous predictive and evaluative judgements of people—perhaps the most in-teresting of multi-dimensional stimuli—both in social settings (see Tagiuri, 1968) and in clinical settings (see Kleinmuntz, 1968). Indeed, i t appears that the majority of theoretical and empirical work on multi-cue judgements done within the last fifteen years has come from these "non-perceptual" areas. How are multi-cue judgements made? Attempts to answer this ques-tion—a question originally posed by the Gestaltists—have occupied much contemporary multi-cue judgement research. This research has pursued a common strategy: models of possible judgement processes are first con-structed, and experiments are then conducted to determine how well each model can account for the results. Most multi-cue judgement process models proposed to date can be placed in an elegant conceptual frame-work, the so-called "Lens Model" first proposed by Egon Brunswik (1952, 1956). As currently formulated, models subsumed within this conceptual framework may be partitioned into a hierarchy of equivalence classes. At the top of this hierarchy is a dichotomy between what are termed addi-tive processes and what are termed configural processes (see Hoffman, 1960; Goldberg, 1968). A vociferous controversy has arisen over which of these two generic classes of judgement processes is most character-istic of human judgements. The controversy is an important one, for i t has direct implications not only about the optimality of human judgements and the training of more accurate judges, (see Goldberg, 1968) but also 3. about t he n a t u r e o f p s y c h o l o g i c a l measurement (see B e a l s , K r a n t z and T v e r s k y , 1968). I t seems, t h e r e f o r e , i m p o r t a n t t o r e v i e w and a n a l y z e t h i s c o n t r o v e r s y i n some d e t a i l . But f i r s t i t i s n e c e s s a r y t o b r i e f l y d e f i n e what i s meant by an a d d i t i v e and a c o n f i g u r a l p r o c e s s . C o n s i d e r a j u d g e who i s r e q u i r e d t o e v a l u a t e , t h a t i s , l o c a t e on some c r i t e r i o n d i m e n s i o n , E, each o f a s e t o f m u l t i - d i m e n s i o n a l s t i m u l i , S i s - j , ^2' " * •'•'•j' • • *' k ^ " Assume t h a t t he d i m e n s i o n v a l u e s o f each s t i m u l u s a r e r e p r e s e n t e d by c u r e s such t h a t t he i t h s t i m u l u s , s ^ , can be r e p r e s e n t e d as t h e o r d e r e d n - t u p l e , s i = ( c i , T c i , 2 ' ' ' c i , j ' ' ' * c i , n ^ where the cue c. . r e p r e s e n t s t h e v a l u e o f t h e i t h s t i m u l u s o n d i m e n s i o n j . I f , f o r a l l s. i n S, t h e j u d g e ' s e v a l u a t i o n o f s. can be r e p r e s e n t e d as some w e i g h t e d a d d i t i v e f u n c t i o n o f the e v a l u a t i o n s o f each cue i n s.., t h a t i s , i f : Vs.eS, J ( s . ) = " w. J ( c . .) (1) 1 1 j = l J 1 , J where, J ( s ^ ) i s t h e e v a l u a t i o n o f s t i m u l u s s. such t h a t t h e r e e x i s t s a J ( s ^ ) f o r a l l s t i m u l i i n S, w. i s t h e s u b j e c t i v e w e i g h t g i v e n t o d i m e n s i o n j such t h a t w. i s c o n s t a n t f o r a l l s. i n S, J ( c . .) i s t h e e v a l u a t i o n o f cue c. . such t h a t t h e r e i »J > »J e x i s t s a J ( c .) f o r a l l c. . i n s. (measured at the l e v e l of an i n t e r v a l s c a l e ) then he i s s a i d to be u s i n g an a d d i t i v e judgement p r o c e s s (see Appendix f o r some n u m e r i c a l i l l u s t r a t i o n s ) . 4. Numerous distinctions have been made between different types of additive processes. Anderson (1965, 1970), for example, distinguishes between additive and averaging processes; a judge is said to be using the former i f his " W j S " sum to more than unity but is said to be using the latter i f his "w.s" sum to unity. Linear and non-linear processes have also been distinguished. A judgement process is considered linear i f the evaluations of each stimulus dimension can be represented as a linear function of the cues on that dimension (i.e., i f J ( c • ) = a(c. .) + b). Models of these processes, for example expected value models, have been discussed by Becker and McClintock (1967) and Anderson (1968). A judge-ment process is considered non-linear i f the evaluations of some stimulus dimensions cannot be represented as a linear function of the cues on that dimension. Models of these processes, for example subjective expected utility models, have been discussed by Edwards (1961). Other distinctions are discussed by Rosenberg (1968). Consider again a judge who is required to evaluate each of the set, S, of multi-dimensional stimuli. If his evaluations cannot be represented as some weighted additive function of the evaluations of the cues, then he is said to be using a configural judgement process (see Appendix for some numerical illustrations). For example, a judge would be said to employ a configural process i f he evaluated all stimuli with a value on the kth dimension greater than some constant, t, with different dimension weights than all stimuli with a kth dimension value less than (or equal to) t, that is i f : Vs.eS such that c. b>t, 5. n J(s.) = E w.J(c. .) but 1 j=l J 1' J Vs.eS such that c. .It, (2a) (2b) where (2c) A judge would also be said to employ a configural process i f he evaluted stimuli by multiplying, rather than adding or averaging, the weighted eval-uations of their cues, that i s , i f : Many models of configural processes exist, though explorations of them have not been nearly as extensive as explorations of their additive cousins. Configural models of the type outlined in eq.s 2a) through 2c)--the so-called lexicographic models—are discussed by Coombs (1964) and exemplified by a choice model of Messick and McClintock (1968). Configural models of the type outlined in eq. 3 are discussed, along with others, by Dawes (1964) and Einhorn (1970). Perhaps the most important characteristic of the additive-configural distinction concerns the assumed relationship between cues in the judge-ment process. Additive processes are characterized by cue independence; each cue of a stimulus being evaluated by an additive process is assumed to contribute to the total evalution independently of the other cues. In contrast, configural process are characterized by cue interdependence; each cue of a stimulus being evaluated by a configural process is assumed to contribute to the total evaluation only in' relationship to, or in the context of, the other cues. Hence, whereas addi-Vs.eS, J(s i) = n (3) 6. tive processes produce evaluations of stimuli equal to the (weighted) sum of its parts, configural processes produce evaluations of stimuli either greater than or less than the (weighted) sum of their parts. Of course, phrased in this manner the additive-configural distinction sounds much like the structural-gestalten distinction contructed by the gestalt psychologists some thirty years before the former distinction was formalized (see Kdhler, 1947). It has long been a popular assumption that humans, for the most part, employ configural processes in making multi-cue judgements (see, for example, Asch, 1946; Meehl, 1954). Configural judgement processes have, in particular, been thought to characterize professionals—clini-cians, doctors, court judges, stock brokers, etc.--presumed to be highly trained in making "sophisticated" or "insightful" (if not accur-ate) judgements (see Kleinmuntz, 1968; Meehl, 1954). Arguments have persisted that models of additive processes are somehow too crude, too mechanistic, to characterize the complexities or "humanness" of human judgements (see Goldberg, 1968, for a discussion). The intuitive evi-dence for this argument seems compelling. Individuals, for example, often use such configural sounding phrases as, "going beyond the evi-dence," "looking for hidden meanings," "getting an overall picture," or "putting into context" to describe their own judgement processes. Some, however, have disagreed with this reasoning, and have pro-vided equally compelling arguments for their case (see Anderson, 1968; Wishner, 1960). Though on an intuitive basis such configural sounding phrases as "putting into context" might be used to describe intuitive 7. feelings about many judgements, so too might such additive sounding phrases as "summing up," "adding to (or subtracting from) the total im-pression," or "weighing the evidence." Though existing additive pro-cesses models might be too crude to describe the intricacies of human judgements there is no reason to believe that they could not be expanded and refined to include such intricacies rather than abandoned for a con-figural alternative. Though current additive process models might be considered "mechanistic" there is no reason to believe that configural process models would ultimately forego this feature. And though human judges might consider their judgement processes to be "complex" or "sop-histicated," there is s t i l l no a priori reason to believe that their judgement behaviour could not be accounted for by "simple minded" addi-tive models. Which of the two viewpoints is correct? Can human judgements be described as primarily additive or primarily configural? A simple test of the adequacy of additive and configural models to account for vari-ous multi-cue judgements has been proposed by Hoffman (1960). It relies on correlational analyses to determine the extent to which judgement variability can be accounted for by linear (additive) and nonlinear (both additive and configural) components of regression equations. A similar test has been proposed by Hammond, Hursch and Todd (1964). But perhaps the most elegant test has been proposed by Hoffmann, Slovic and Rorer (1968). Though formally similar to the first two, i t fully ex-ploits the power of analysis of variance techniques (see Anderson, 1970). All that is needed is a judgement paradigm to produce evaluations of multi-dimensional stimuli in a factorial or equivalent design. If a 8. multi-cue judgement experiment were performed within such a paradigm, and i f the appropriate analysis of variance model was applied to the re-sulting judgement data, tnen an examination of the main effects and interactions would, in theory, reveal much about the judgement process. As Hoffman, Slovic and Rorer demonstrated, additive judgement process models would predict that judgements would vary only as a function of dimension main effects, while configural judgement process models would predict that judgements would vary, at least in part, as a function of interactions between dimensions. An impressive number and variety of multi-cue judgement experiments have been undertaken to test for additive and configural judgement pro-cesses (see Goldberg, 1968; Hammond & Summers, 1965; Anderson, 1968, for reviews). Goldberg and Rorer (1968), for example, have studied how clinicians evaluate psychiatric patients when given a set of diagnostic test scores. Slovic (1969) has studied how stockbrokers evaluate com-panies on the basis of indicators from Standard and Poor reports. Slovic & Lichtenstein (1968) have studied how subjects evaluate by different means the attractiveness of gambles. Hoffman, Slovic and Rorer (1968), have studied how radiologists estimate the probability of ulcer malig-nancy given a synopsis of roentgenological symptons. Anderson (1962, 1965), and Wishner (1960) have studied how subjects evaluate other per-sons on the basis of a string of descriptive adjectives. And Thorngate (1971) has studied how subjects judge the moods of other persons given a number of facial cues. In the vast majority of such studies (and in all those cited above) a remarkably consistent finding has emerged: virtually all of the accountable variance in the judgements has come from what amounts to th main effects of the stimulus dimensions and not from their interactions Statistically significant interactions have often occurred (see, for example, Slovic, 1969; Thorngate, 1971) but they have seldom, i f ever, accounted for more than a small fraction of the total judgement vari-ance. Indeed, as Anderson (1969) has pointed out, many of these inter-actions could be artifactual. Ratings resulting from a process de-scribed in equation 1, for example, could produce significant interac-tions i f floor and ceiling effects were present in the rating scale employed. In light of these findings i t is tempting to conclude that multi-cue judgements are, in most instances, the result of additive rather than configural processes. At best, however, this conclusion appears premature. Unfortunately, in practice, the ANOVA model (or its multipl regression equivalent) is so robust that i t can "mask" or "cover-up" many configural processes with its main effects (see Goldberg, 1968; Green, 1968; Hoffman, 1968; Yntema and Torgerson, 1961). As a result, the small interactions dismissed by some as artifactual may instead in-dicate underlying configural process structurally unlike any subsumed under equation 1. The interpretation of interactions, especially those which are re-latively small in comparison to main effects, has thus remained equivo-cal. Perhaps as a reaction to this fact, current research on multi-cue judgement processes has diversified in its approach. The bulk of 10. this research has continued to focus upon what may be termed "static" or "structural" questions, that is questions about judgement processes as-sumed to: 1) already exist, and 2) remain unchanged, at least during the time that they are experimentally scrutinized. But a growing body of research has begun to focus upon what may be termed "dynamic" or "functional" questions, that is questions about how and why judgement processes: 1) come to exist in the first place, and 2) change (if at all) with time or circumstance. Little is yet known about the dynamics of multi-cue judgement pro-cesses. If more were known about them, a resolution or at least a re-analysis of the additive-configural controversy might result, and our breadth of understanding of these processes would no doubt increase greatly. A review of what is known and what needs to be known about multi-cue judgement dynamics therefore seems in order. 11. THE LEARNING OF JUDGEMENT PROCESSES How are multi-cue judgement processes learned? Can the learning process itself account for the large percentage of multi-cue judgement variance due to cue main effects? As one might expect, a number of paradigms may be employed to attempt to answer these questions. One of them, a straightforward application of a common learning paradigm (see Bruner, Goodnow and Austin, 1956) has been used almost exclusively in the relevant research done to date. In this paradigm, subjects are re-quired to learn a predetermined relationship between stimulus cues and values on some criterion dimension. The relationship is selected and varied by the experimenter; he may, for example, require some subjects to learn an additive relation and some an interactive (i.e., configural) relation. The speed and/or accuracy with which these relationships are learned may then be compared, and the comparisons, in turn, may be used for testing different notions about the learning process. The paradigm has a number of limitations. Nonetheless, research done with i t has produced some intriguing results and i t seems appropriate to review and analyze them in some detail. The majority of studies concerned with the learning of judgement processes have required the subject to learn one of a number of additive functions, functions of the form shown in equation 1 (above) which varied only in the nature of their f .'s. Hammond and Summers (1965) and J Summers and Hammond (1966), for example, found that linear f.'s (i.e., linear cue-criterion relationships) were more quickly learned than non-linear f.'s. Bjorkman (1965) also found that linear f.'s with a posi-12. tive slope (i.e., a positive correlation between the cues and criterion) were more quickly learned than linear f.'s with a negative slope. But these studies, while interesting in their own right, have unfortunately been of limited value in answering questions about the learning of addi-tive and interactive functions. Few comparisons of the learning of ad-ditive and interactive functions exist and most have not been undertaken within a judgemental context. A discussion of three of the more rele-vant comparisons is presented below. In a study of concept formation Neisser and Weene (1962) defined a number of logical relations mapping two-dimensional stimuli into a di-chotomous criterion, then determined and compared the rate at which each relation was learned. Five basic logical relations were defined: Affir-mation, Conjunction, Disjunction, Conditional, and Biconditional. To illustrate these, let: C-| = one stimulus dimension (e.g., colour), c 1 5 ~ C i = two values of C-|, C£ = a second stimulus dimension (e.g., shape), C2»~C2 = two values of C2> J = the criterion or judgement dimension, and y (=1) and n (=0) = the two criterion values. The five basic relations may then be summarized as: RELATION 13. LOGICAL REPRESENTATION TABULAR REPRESENTATION c2 ~C2 Affirmation y «—» c ] c. 1 1 c l 0 0 Conjunction y < — ( c 1 A c 2) c2 ~c2 c. 1 0 c l 0 0 Disjunction y » (c ] v c 2) c2 ~C2 c. 1 1 ~ c l 1 0 Conditional y > (~c] v c 2) C2 ~C2 c l 1 0 ~ c l 1 1 Biconditional y C(c1 A c 2) C2 ~c2 V (~c1 A ~c 2)] c. 1 0 ~ c l 0 1 Neisser and Weene noted that the relations formed a hierarchy of cogni-tive complexity. Affirmation relations were least complex since a criterion "yes" was based upon the presence of only one cue. Conjunc-tion, Disjunction and Conditional relations were more complex since, for each, a criterion "yes" was based upon one logical connective between two cues. Biconditional relations were most complex since a criterion "yes" was based upon two logical connectives between four cues. Be-cause of this difference in complexity, Neisser and Weene predicted 14. that an Affirmation relation would be more easily learned than either a Conjunction, Disjunction, or Conditional relation which, in turn, would be more easily learned than a Biconditional one. This prediction was well supported by their results. Essentially identical results were al-so obtained in a partial replication of the study (Haygood and Bourne, 1965). Neither Neisser and Weene, nor Haygood and Bourne placed their studies in a multi-cue judgement context. But the logical relations with which they concerned themselves seem roughly translatable into ad-ditive and configural terms. In particular, the Affirmation relation seems completely additive; its tabular representation (above) resembles one which would result only from a main effect on the C-| dimension. In contrast, the remaining four relations appear to be configural; their tabular representations resemble those which would result from various types of interactions between the c^  and c 2 dimensions. The Conjunction, Disjunction and Conditional relations have, in ANOVA terms, both a main effect and an interaction component. The Biconditional re-lation has, in ANOVA terms, only an interaction component. As a result, the results of both Neisser and Weene and of Haygood and Bourne suggest that additive relations are easier to learn than configural ones. In addition, their results suggest that the greater the interactive com-ponent of a configural relation the more difficult is the relation to learn. A somewhat more direct comparison of the rate at which additive and configural relations are learned was reported by Summers, Summers and 15. Karkau (1959). Subjects in their experiment were required to predict the age of blood cells on the basis of two visual cues. Each cue re-presented one of eight values on either of two dimensions: 1) the size of the cells, and 2) the amount of non-cellular matter in the background. All 8 x 8 = 64 combinations of cues were constructed and randomly pre-sented to subjects ten times, once within each of ten 64-trial blocks. Subjects predicted the age of each combination as i t appeared and fol-lowing each prediction on trial blocks 2 through 9 the "true" age was revealed. True ages were determined by one of the four following pre-selected functions (simplified for illustrative purposes): FUNCTION NAME EQUATION TABULAR EXAMPLE Additive Age = (c, + c 9 - 2) + e c, = 1 c, = 2 Multiplicative Age = (c-,c? - 1) + e 0 1 1 2 c 2 = 1 c 2 = 2 0 1 1 3 Exponential Age = (c, - 1) + e c, = 1 c, = 2 0 0 16. c l Exponential Age = (c 9 - 1) + e c 9 = 1 c 9 = 2 Control c c c 0 1 0 3 (where = the value of dimension 1, = the value of dimension 2, e = a random error component with mean = 0). Summers ejt. al_. did not report the proportion of age variance due to random error. But the nonerror variance distributed itself as follows: for the Additive function, 100% was due to cue main effects and 0% to cue interaction; for the multi-plicative function, 88% was due to main effects and 12% to interaction; and for each Exponential function, 77% was due to main effects and 23% to interaction. Results indicated that, for all subjects, correct predictions of ages determined by any of the four equations increased substantially over trials. However, the rate at which accuracy increased differed significantly between equations. Most of the learning rate difference was due to the retarded performance of subjects attempting to predict ages derived from the Exponential Control equation. Subjects who pre-dicted ages derived from either the Additive, Multiplicative, or Exponen-tial equations increased the accuracy of their predictions at an almost identical rate to an almost identical asymptote (by visual inspection— Summers e_t. a/k did not report the relevant statistical tests). Because of the Exponential Control group's slower learning rate the pooled aver-age of the Exponential equation's learning rate was significantly slower than the Additive or Multiplicative equation's learning rates. But the 17. deficit seemed small and could have resulted from the fact that subject's age estimates on trial block 1 (where no feedback was given) were found to have a strong additive bias. At best all we may conclude from this study is that the learning of a configural relation was not faster than the learning of a completely additive one. Perhaps the most direct comparison of the rates at which additive and configural relations are learned has come from Brehmer (1969). On each of 400 learning trials in his study, subjects were shown two lines of varying length—each representing a value on one of two dimensions— and required to predict the length of a third. Following each prediction the "correct" length was revealed. This length was completely deter-mined by one of four functions: FUNCTION NAME EQUATION TABULAR EXAMPLE x 2 = 1 x 2 = 2 2 3 3 4 x 2 = 1 x 2 = 2 1 2 2 4 Additive y = a + b(x1 + x 2) Multiplicative y = a + b(x^x2) X , = 1 X , -2 x, - 1 x, - 2 18. Ratio y = a + b(x,/x9) x, = 1 x, = 2 1 1/2 2 1 Difference y = a + b|x,-xJ x 9 = 1 x9 = 2 0 1 1 0 (where x^  is the line length of dimension 1, x 2 is the line length of dimension 2, y is the "correct" line length, a and b are constants). As a result, the percent of variance in y due to the dimension interaction was 0% for the Additive function, 10% for the Multiplicative function, 50% for the Ratio function, and 100% for the Difference function. In line with the results of Summers et. al_., Brehmer's results showed that by the end of the learning session all four functions were about equally well learned. There was also a significant difference in the rate at which each was learned. But curiously enough the Ratio function and not the Difference function was most difficult to learn. The correlation between subject's predicted y values and the y values attributable to the Ratio equation's main effects (a measure of predic-tive accuracy) increased slowly and reached an asymptotic level of .95 only after the 150th t r i a l . The correlation between subjects' predicted y values and the y values attributable to the Ratio equation's inter- action increased even more slowly and reached a rather meager asymptotic 19. level of .70 only after the 350th t r i a l . The Multiplicative function was the next most difficult one to learn. Though the Multiplicative function's interaction was learned as slowly as that of the Ratio func-tion its main effects were learned much more quickly: the correlation between subjects' predicted y values and the y values attributable to its main effects reached an asymptotic level of about .95 by the 50th t r i a l . As expected the Additive function was learned more quickly than either the Ratio or Multiplicative functions. But, most astonishingly, there was no significant difference between the rate at which the Addi-tive and Difference (i.e., interactive) functions were learned! (From visual inspection, since Brehmer also failed to report the relevant statistical test). Both of these functions were learned quickly: the correlation between S_'s predicted y values and the correct y values ex-ceeded 0.70 for both functions by the 50th trial and asymptoted at about 0.85 by the 100th t r i a l . Brehmer's results, then, indicate that: 1) configural relations with both an additive and interactive component are more difficult to learn than additive relations and 2) configural relations with both an additive and interactive component are more difficult to learn than configural relations with no additive component. Indication 1) seems to coincide with the results of Neisser & Weene (1962), Haygood and Bourne (1965), and Summers ejt. al_. (1969). However, indication 2) seems con-trary to Neisser and Weene's, and Haygood and Bourne's results. Why the contradiction? Perhaps some task or instructional variable could ac-count for i t . For example, Neisser and Weene used strings of letters as 20. stimuli, and varied the correct "yes"--"no" response as a function of the values each figure possessed on certain physical dimensions (color, shape, numerosity, etc.). In order to learn configural relations i t is necessary (though not sufficient) to attend to the "interrelations" of stimulus dimension values. Intuitively, neither Neisser and Weene's nor Haygood and Bourne's stimuli seem nearly as conducive to comparing or in-terrelating dimension values as do Brehmer's stimuli: two parallel lines of different length. Indeed, perhaps the most salient feature of such lines is their difference, so any function based on this dimension com-parison (for example, Brehmer's Difference function) should be quickly learned. More generally then, task and instructional variables may pro-duce what could be crudely called additive or configural "sets" which may, in turn, impede or facilitate the learning of certain functions. If such sets do exist, our first requirement for further research in the area should at least be to: 1) attempt to measure them (as Summer's ejt. al. did), and 2) provide some room for them in any function learning model. Brehmer's results also indicated that when relations having both additive and configural components are being learned, the additive com-ponent is learned much more quickly than the configural one. This find-ing may give us an important clue about the nature of multi-cue judgement learning. It may indicate, for example, that subjects first detect a function's "cruder" or "more general" aspects (as exemplified by main effects) and then look for its more "subtle" characteristics (perturba-tions of the main effects produced by interactions). To phrase i t 21. another way, they may first learn the "general rules," of a function and then concentrate on memorizing its "exceptions." Unfortunately, no other study to date besides Brehmer's has looked for the differential rate of learning additive and configural aspects within a function. It seems worthwhile therefore, to also require that further research attempt to look for and replicate Brehmer's results, and, i f replicable, to account for i t in some learning model. 22. LEARNING VS. JUDGING In most multi-cue judgement and multi-cue learning paradigms, judge-ment and learning processes are inferred from the judges' overt responses. But do judges "know" more about the situation they evaluate than they re-veal in these overt responses? Is i t possible, for example, for a sub-ject to be fully aware of the configural aspects of a situation he eva-luates, yet base his evaluations on an additive process? If a judge fails to learn a particular function or parts of a particular function is his failure actually the result of input or learning (i.e., perceptual or memory) limitations, or both? These questions are, as one might ex-pect, exceedingly difficult ones to answer which perhaps accounts for the fact that, at least in the multi-cue judgement literature, virtually no attempts to answer them have been reported. There are, in fact, only a few indirect clues available on which to postulate an answer. It is known that subjects can learn simple rela-tionships merely by observing sequences of stimuli. Reber and Millward (1968), for example, found that in a single cue probability learning situation subjects who first merely observed stimuli paired with their criterion values, then later predicted the criterion values as each stimulus was presented responded almost identically to those subjects who had been predicting the criterion values all along. Similarly, in a number of other studies i t has been shown that subjects can quite accurately estimate various summary statistics (e.g., means, proportions, variances) of relationships merely by observing them (see Peterson and Beach, 1967, for a review). It is known that a subject will respond to 23. simple probabilistic functions in widely varying manners or strategies depending on the costs and rewards of employing each (see Atkinson, Bower and Crothers, 1965, Chapt. 5 for a review). As was mentioned earlier, i t is known that additive functions or processes give fairly accurate renditions of many configural ones (see Yntema and Torgerson, 1960; Green, 1968). Finally, i t is thought that many additive processes are computationally easier to employ than configural ones (see Neisser and Weene, 1962; Breiman, 1964).1 In light of these clues i t would seem quite possible that a subject could: (1) easily recognize and learn by observation both the main effect and interactive components of a relation he is to judge, but (2) decide that the cost of any cognitive effort required to include both components in the judgement process would be more than the cost of any judgement errors accrued by including only the main effect component in the judgement process, and hence (3) make judgements which appeared as though he was unable, or slow, to learn the interactive components of the relationship. This hypothesis may sound far-fetched, but there seems to be a good deal of at least anecdotal evidence to support i t . Most everyone "knows", for example, that the effectiveness of a leader, or the future success of student, or the quality of a University are all based upon complex interactions of many factors (i.e., on configural Vhis may not always be true. Einhorn (1970) has demonstrated that certain configural processes, for example lexicographic processes, may be easier to employ than most additive processes save those which weigh only one cue. 24. processes) many of which are also "known." Yet i t would appear that leaders, or students, or universities, are usually judged by much less interactive processes: a leader is effective to the extent he speaks well, a student is a good graduate school risk to the extent that his GPA and GRE indices are high, a University is good i f i t has a large en-dowment, but i t is better i f i t also has many library books and best i f i t also has some famous professors. Such simplistic judgement processes may s t i l l be configural, but an additive model would provide a better account of them than i t would of the knowledge on which they were based. Perhaps the most straightforward, i f naive, means of determining whether judges do know more about situations they evaluate than is re-vealed in their evaluations would be simply to ask them. If, for ex-ample, subjects were required to estimate various summary aspects about a situation they were learning or had learned to judge, then by compar-ing their estimates with their judgements and with the situation itself one could presumably obtain some clues about the extent to which their judgements reflected their knowledge of the situation. A third require-ment for further research should then be to obtain the summary estimates necessary to make these comparisons. 25. THE EXTINCTION OF JUDGEMENT PROCESSES In the standard multi-cue judgement learning paradigms subjects ob-tain feedback about the correctness of their judgements with great haste and regularity. In normal multi-cue judgement situations, however, such feedback is usually much harder to come by. For example, i t may take years for a voter to determine the wisdom of his judgements about politi-cal candidates, or for an admissions committee to determine the wisdom of admitting (or even worse, rejecting) a student into their graduate programme. What happens to judgement processes when no validating feed-back is available? Do configural judgement processes or interactive as-pects of judgement processes extinguish more quickly than their additive counterparts, i f indeed they extinguish at all? Can any differential extinction rate account for the preponderance of multi-cue judgement variance due to cue main effects? Here again, research directly relevant to answering these questions is virtually non-existant. Even indirectly relevant research is rare and the results inconsistent. Reber and Millward (1968) found that in a standard no-cue probability learning situation (i.e., where the probabi-lity of events to be predicted was constant and independent of subjects' responses) learned responses became slightly less consistent or reliable after validating feedback was eliminated than while i t was available. However, Bjorkman (1969) and Brehmer and Lindberg (1969) found that in a single-cue probability learning situation (i.e., where the probability of events to be predicted varied as a function of the value of a one-dimensional stimulus) learned responses became slightly more consistent or reliable after validating feedback was eliminated than while i t was 26. available. Apparently only Goodnow (1954) has studied the effects of eliminating feedback on learned multi-cue judgement processes. Like Bjorkman, and Brehmer and Lindberg, Goodnow found that response con-sistency increased after the elimination of validating feedback. But this finding was only based on responses conditioned to additive func-tions, so no comparison between post feedback changes in additive and configural processes could be made. A fourth requirement, therefore, for further research in the area should be to determine whether there is any differential facilitation (or extinction) of learned additive and configural judgement processes after validating feedback is eliminated. 27. THE COMMUNICATION AND TRANSFER OF JUDGEMENT PROCESSES Though learning experiments have demonstrated that some simple multi-cue judgement processes can be learned by directly exposing the learner to multi-dimensional stimuli, eliciting a judgement response from him, and providing validating feedback, there is no reason to assume that all--or even most—judgement processes are obtained in this way. There are, in fact, at least two other means of learning to make judge-ments which are probably equally as common as this "direct exposure" method. First, one may presumably obtain judgement processes by communi-cating with other judges. Certainly much of our formal education con-cerns itself with the rote memorization of prefabricated judgement pro-cesses; we become proficient at judging the correctness of many word spellings, for example, by recalling our 3rd grade teacher's incantation " i before e, except after c . . ." Many clinical judgement processes are, no doubt, similarly obtained. Indeed, a clinician may learn to judge the meaning of MMPI or WAIS test scores simply by reading their "how to diagnose" manuals! Second, one may presumably obtain a judgement process by "borrowing" or transferring them from other situations. Such analogies or metaphors abound. Judgements about the personality or behavior of oneself or others are often made with the aid of biological or physical metaphors. Thus, a person participating in a demonstration may be judged "mentally i l l " by anyone who assumed that this "abnormal" behavior, like abnormal behavior of the viscera, was a sign of disease (see Szaaz, 1960). And if the demonstration was followed by some social change, the demonstra-28. tor may be judged to be responsible by anyone who assumed that social changes, like physical motion, must have a single immediately proceed-ing cause (see Heider, 1958). Of course, judgements of oneself or of others are also commonly made with personal or interpersonal analogies. Thus, a person may judge another's reactions to things around him by "projecting," by assuming that the other will respond to stimuli in a manner similar to himself. How are multi-cue judgement processes communicated or transferred? Are configural processes or configural aspects of processes more d i f f i -cult to communicate or transfer than their additive counterparts? Once more, l i t t l e directly relevant research has been done to answer these questions however, what l i t t l e research does exist is most intriguing. In an interesting study on interpersonal learning of judgement processes through communication, Earle and Miller (1969) first taught some subjects a linear relation between the values of a stimulus dimen-sion (A) and a criterion (C), and others a U-shaped relation between another stimulus dimension (B) and the criterion (C). Then they paired linear with linear, U-shaped with U-shaped, and linear with U-shaped subjects to discuss and jointly predict C on the basis of cues from A and B. It was found that as joint predictions progressed linear subjects "picked up" one another's functions with about the same speed and accu-racy as did the U-shaped subject pairs. However, when linear subjects were paired with U-shaped subjects the latter picked up significantly more about the former's relation than vice-versa. Both the linear and U-shaped functions were, of course, main effects so the experiment offers us no data for a direct comparison between the effectiveness of communi-29. eating additive and configural relations. However, to the extent that U-shaped and configural functions may be seen as more "cognitively com-plex" than linear or additive functions, one could speculate that the configural ones might, in fact, be more difficult to communicate than the additive ones. Such a speculation seems worthy of empirical test. The transfer of multi-cue judgement processes from one situation to another presumably depends on the similarity of the situations. Simi-larity, of course, has been a particularly nasty concept to deal with in psychology and controversy over its nature has been rampant for some time (see Hume, 1758; Hall, 1967, chapters 14-15; Tversky, 1969). Wallach (1958) in reviewing some of the controversy, has suggested there are at least two different types of similarity: stimulus and response. Stimu-lus similarity is defined in terms of the communality of stimulus elements (or dimensions) between two situations. Response similarity is defined in terms of the communality of response elements (or dimensions) between two situations. Unfortunately, neither the effects of stimulus similari-ty nor response similarity upon their transfer of multi-cue judgement processes are known to have been explored. A fifth requirement for further research in the area should therefore be to begin these explora-tions. By way of summary, listed below are some of the aspects of the learning of multi-cue judgement processes which further research should explore: 1) The learning of additive and interactive (configural) functions with controls for learning "sets" or response dispositions, 30. 2) the learning of additive and interactive aspects of "mixed" func-tions (functions having both main effect and interactive components) with controls for learning sets or response dispostiions, 3) estimations of additive, interactive, and mixed functions and the relationship between the estimates and subsequent judgements, 4) changes (if any) in judgements of additive, interactive and mixed functions, once validating feedback is eliminated and 5) the transfer of judgements of additive, interactive, and mixed functions to situations of varying similarity. It would, of course, be difficult to extensively explore all of these five aspects of judgement process learning in one, or two or even a dozen experiments. The three experiments reported below, however, represent a start in this direction. 31. THE LEARNING SITUATION2 The first Experiment was undertaken to: a) measure and control for response predispositions prior to learning, then b) compare the learning rates of a completely additive function and a completely interactive function, c) compare verbal estimates of two (cell and marginal) aspects of these functions, d) determine and compare any changes in judgements of these functions following the elimination of feedback, and e) deter-mine the extent of judgement transfer to two situations thought to differ in response similarity. An experimental situation was needed which would be structured enough to provide data necessary for these undertak-ings, simple enough for subjects to master in a reasonable length of time, and representative enough of a non-laboratory situation at least to capture and retain subject's attention. A two-cue probability learn-ing situation, presented to subjects as an experiment in person percep-tion (specifically in judging the emotions of others) seemed to meet these requirements. On each of 400 judgement trials (ten 40-trial blocks) subjects were read a two word description of a stimulus person. There were four such two word descriptions: S-j = married male S2 - married female S^  = single male S^  = single female These four descriptions were constructed from all possible pairings of 2 The reader is advised to skip to the Method section of Experiment I for a more complete description of the procedure employed. 32. two dichotomous dimensions, a marital status (married-single) dimension and a sex (male-female) dimension. An equal number of exemplars--ten--of S.j, Sr,, S3, and S^  appeared in each 40-trial block, but the order in which they appeared was randomized. Following the presentation of a des-cription (S-j . . ., S^ ) subjects were required to judge whether the person so described would answer "yes" or "no" to the question: 0^  = Are you usually more optimistic than pessimistic? was used as the criterion dimension for the pretest, learning, esti-mation and posttest phases of the experiment. It was selected because previous pilot testing had shown l i t t l e tendency to differentially guess yes or no to any of the four stimulus persons, hence l i t t l e tendency to respond to either the main effects or the interaction of the two stimu-lus dimensions. The Pretest. Trials 1-80 (trial blocks 1 and 2) constituted the Pretest of the Experiment. On each Pretest trial subjects not only guessed at a stimulus person's answer to 0^ , but also guessed at the stimulus person's answer to one other question, either: 0^  = Are you usually more humorous than serious?, or 0^3 = Are you usually more sociable than unsociable? Half of the subjects guessed at 0^  and 0^ . The remainder guessed at 0^  and 0^ . During the Pretest no feedback about the correctness of their guesses was given to subjects. All three questions 0^ , 0^  and 0^  were derived from Semantic Differential Rating scales (optimism, sociability, 33. and humour, respectively). Osgood, Suci and Tannenbaum (1958) had found optimism and sociability to have a moderately high positive correlation, but optimism and humour to have virtually no correlation. Because of this i t was hoped that by pairing 0^  with and with Oq two levels of response similarity would result. The pretest made i t possible to examine subjects' base rate of guessing "yes" to 0^  , and 0^  given each of the four types of stimulus persons. It also made i t possible to determine the extent to which they guessed the same answer to and 0^  or to 0^  and 0^ , and thus to estir mate the strength of the relationship between these guessing responses for use in subsequent analyses of transfer. Learning. Trials 81 - 280 (trial blocks 3 through 7) constituted the Learning phase of the experiment. On these trials subjects guessed only at the stimlus person's answers to 0^ . Following each guess sub-jects were told whether i t was "correct" or "wrong." The subject's task was to make as many correct guesses as possible. The correctness of a guess was determined by comparing i t to the answer given by the particu-lar stimulus person being judged which, in turn, was determined by one of four additive probabilistic functions, or one of two interactive pro-babilistic functions. The four additive functions are summarized in Table I; cell and marginal entries indicate proportions of "yes" answers to Q,. 34. Functions: A2 A3 A4 Tabular M F M F M F M F Summary: m .8 .8 .8 m .2 .2 .2 m .8 .2 .5 m .2 .8 .5 s .2 .2 .2 s .8 .8 .8 s .8 .2 .5 s .2 .8 .5 .5 .5 .5 .5 .8 .2 .2 .8 Table I: The Four Additive Functions (m = married, s = single, M = male, F = female) Each of the four additive functions was constructed so that the pro-portion of "yes" answers to varied as a main effect of one and only one of the two stimulus dimensions; functions A-j and A, varied yes pro-portions as a function of marital status, functions A_3 and A^ varied yes proportions as a function of sex. These, of course, were the simplest cases of additive functions. They were chosen for the experiment be-cause their cell entries were the same in number and magnitude--two 80% and two 20%--as those of the interactive functions (see below). This was considered a necessary control to test learning rate differences be-tween additive and interactive functions. More complex additive func-tions were constructed in Experiment II and III. The four additive func-tions A.j - A^ were, in fact, only variants of one basic additive function (tie four variants served only as controls for any response biases.) Be-cause of this, data obtained from each was pooled where appropriate. The two interactive functions are summarized in Table II; again cell and marginal entries indicate proportions of "yes" answers to Q,. 35. Functions: I 1 V M F M F Tabular m .8 . 2 .5 m . 2 .8 .5 Summary: s . 2 .8 .5 s .8 . 2 .5 .5 .5 .5 .5 Table II: The Two Interactive Functions, (m = married, s = single, M = male, F = female) The interactive functions were constructed so that the proportion of yes answers to 0^  varied only as the interaction of the two stimulus di-mensions. The interactive functions, like the additive ones, were vari-ants of one basic interactive function and served to control for response biases. Again, the data obtained from each were pooled where appropri-ate. A subject who wished to increase the number of correct guesses he made beyond chance level (50%) could only do so by "mimicing" with greater or lesser consistency the relation mapping the stimuli into the correct answers. For example, a subject judging an A^  function would be expected to make correct guesses 50% of the time if he guessed randomly, 68% of the time i f the guessed "yes" to 80% of the married persons and to 20% of the single persons (a so-called "matching" strategy) and 80% of the time—the greatest accuracy possible—if he guessed "yes" to all of the married persons and to none of the single persons (a so-called "maximizing" strategy). To the extent that subjects came to mimic functions A, - A., the variability of their judgements would be account-36, able in ANOVA terms to the main effect of the one relevant dimension. Similarly, to the extent that subjects came to mimic functions Lj or L>, the variability of their judgements would be accountable in ANOVA terms to the interaction of the two dimensions. Thus, to the extent that subjects came to mimic an additive or interactive function they would be behaving as though they were additive or configural processors, re-spectively. For this reason, the learning phase made i t possible to determine at what rates and to what asymptotes these additive and con-figural processes would be learned. Estimations. Immediately following the learning trials subjects estimated eight proportions characterizing the function to which they had been exposed. In particular, subjects estimated the proportion of: 1) married males, 2) married females, 3) single males, 4) single females, 5) married people, 6) single people, 7) all males, and 8) all females v/ho had answered "yes" to Q_^. Estimates 1) - 4) were, therefore, made about the four types of stimulus persons or the cells of the tabular re-presentations (see Tables I and II). Estimates 5) - 8) were made about the four characteristics of stimulus persons or the marginals of the tabular representations. Estimation latencies, as well as estimated proportions, were also determined. 37. Just as data from the Learning phase could provide one index of how well the additive and configural processes were being learned, data from the Estimation phase could also provide an index. For to the ex-tent that subjects accurately estimated the cell proportions of the Additive and Interactive functions, the variance of their estimates would respectively be accounted for, in ANOVA terms, by a main effect or interaction. Variances in the guessing proportions of the Learning phase and in the estimated proportions of the Estimation phase could be compared. The comparison could theoretically reveal any differences between the learning of a function and the judgements made of i t (see chapter 3. Both the cell and marginal estimates and the cell and marginal estimate latencies were compared in order to test some hypotheses con-cerning the way in which subjects stored or remembered contingencies of the additive and interactive functions. These hypotheses were inspired by some characteristics of the learning models presented in the next section, where they are explained in greater detail. The Posttest. Immediately after subjects made their estimates they began the Posttest (judgement trials 281 - 400). The Posttest was identical to the Pretest in format: subjects guessed at answers to the same questions at which they had guessed in the Pretest (either Q-j and or 0^  and 0^ ) and received no feedback about the correctness of their guesses. However, the Posttest was made 40 trials (one 40-trial block) larger than the Pretest in order to better estimate any systematic changes in the learned additive and configural processes used to guess at ansv/ers. The Posttest, of course, not only made i t possible to 38. determine the subjects' propensities of guessing "yes to 0^  for each type of stimulus person, but also made i t possible to determine subjects' propensities of guessing "yes" to 0^  and 0^ for each type of stimulus person. These latter determinations could be used to test a simple model of transfer outlined in the next section. 39. POSSIBLE MODELS OF LEARNING AND TRANSFER What sort of models would account for the learning of the Additive and Interactive functions? No models of multi-cue judgement process learning were known to exist. There were many models of related learning phenomena, particularly within the areas of discrimination, concept and probability learning (see Martin, 1963; Luce and Galanter, 1963; Estes, 1964 for reviews). But, unfortunately, most of these related models were constructed to account for learning in situations which differed in critical ways from that of the present experiment. For example, most discrimination and concept learning situations with which models had been concerned had determinate rather than probabilistic outcomes, while most probability learning situations with which models had been con-cerned did not employ multi-dimensional discriminative stimuli. Attempts to adapt these models to the present situation proved exasperating; ada-ptations of many were found to be impossible without gross distortion, while adaptations of some (especially probability learning models) were found to be so numerous that any psychological meaning was lost. It was therefore decided to begin from "scratch" and to construct the following conceptual framework, general enough to include the assumptions of most learning models but specific enough to account for any difference in the learning rates of the Additive and Interactive functions. The Conceptual Framework. Consider a subject who, on learning trial n, is confronted with a stimulus person having marital status "a" and sex "b." Assume that because of selective or unselective attention, hypo-thesis testing or whatever the subject could sample one of the four sub-sets of the pair of cues (a,b) that is either: 40. a only the marital status cue, or b only the sex cue, or ab both the marital status and sex cue, or neither cue in guessing how the stimulus person responded to 0^ . Let, g = the subject's guess about the stimulus person's answer to 0^ , and G = the stimulus person's answer to 0^  . Assume that the probability of guessing g to the stimulus person (a,b) on trial n is dependent upon which of the four cue subsets had been sampled on that t r i a l . In particular, let: Pn[g|(a,b)] = pn(g|a)pn(a)+pn(g|b)pn(b)+pn(g|ab)pn(ab)+pn(g|-)pn(-) (4) where, for xe{a,b,ab,-}: PnCg|(a,b)] = the probability of guess g on trial n given stimulus person (a,b), P n(gIx) = the probability that g was made when x was sampled on trail n, Pn(x) = the probability that x was sampled on trial n, and where for all x and x'e{a,b,ab,-} such that xj'x', Equation 4 constitutes the conceptual framework, within which the following learning assumptions are made. Learning Assumptions. In order to describe trial-by-trial changes in the responses to the Additive and Interactive functions let, p (x and x') = 0. n (5) P-|(g|x) = the initial probability of g given x, p(G|x) = the probability of answer G given x, and 41 . k = the number of trials (k<n) on which x was sampled. X — We define the probability of guess g given x on trial n as, Pn(g|x) = f[p-,(g|x)9 p(G|x), k x], (6) where f is a monotonically increasing or decreasing function, that i s , where: (-FCP-, (g|x), p(G|x), Jx]-0.5| < |f Cp-, (g |x), p(G|x), kx]-0.5| (7) for all k >j , and where X X ftp^glx), p(G|x),o] = p^glx) (8) for all p(G|x), and f[0.5, 0.5, k ] = 0.5 (9) X for all k*;n. Finally, let the expected value of k on trial n be given by, n x E(k ) = E p.(x). (10) x i=l 1 These learning assumptions seem to be met by all known probability learning models (see Estes, 1964). When employed within the conceptual framework they predict that any difference in the learning rates of the Additive and Interactive functions would be due to the sampling proba-bilities, p.(x). To illustrate, consider two possible outcomes of the learning of the married Male contingency of the A-j and L| functions. Re-call f i r s t , that in the A^  function (see p. 33 ), p(YES|mM) = 0.8, p(YES|m) = 0.8, p(YES|M) = 0.5, and p(YES) = 0.5, (11) and that in the L| function (see p. 34), 42. p(YES|mM) = 0.8, p(YES|m) = 0.5, p(YES|M) = 0.5, and p(YES) = 0.5. (12) Let p stand for all probabilities associated with function and p' stand for all probabilities associated with function. Assume that prior to the learning phase: P-,(y|x) = p](y|x) = 0.5, and (13) P1(x) = pj(x) (14) (reasonable assumptions i f the appropriate experimental controls are em-ployed). In the first case assume that no difference is found in the (m,M) learning rates of the A^  and Lj functions. This would imply that, Pn[y|(m,M)] = P|;[y|(m,M)], for all n>l (15) which, in terms of eq. 4 would imply that, Pn(y|m)pn(m)+pn(y|M)pn(M)+pn(y|mM)pn(mM)+pn(y|-)pn(-) = p;(y|m)p;(m)+p;(y|M)p (^r.i)+pr;(y|mM)p(;(mri)+p|;(y|-)pr;(-). (16) From eq.s 9 and 11 the above reduces to, Pn(y|m)pn(m)+0.5+pn(y|mM)pn(mM)+0.5 = 0.5+0.5+p^ (y|mM )p^ (mM)+0.5, which in turn reduces to: Pn(y|m)pn(m)+pn(y|mM)pn(mM) = p^ (y|mM)p^ (mM)+ 0.5 (17) In the second case assume that the (m,M) learning rate of the A-| function is found to be greater than that of the function, i.e., that: Pn[y|(m,M)]>P|;[y|(m,M)] (18) for some n>l. Using a reasoning similar to that employed in the first case this would imply that: 43. Pn(y|m)pn(m)+pn(y|ml-i)pn(mM)>p^(y|mM)p^(mM). (19) Little can be inferred from eq.s 17 and 19 unless one additional as-sumption is made, namely that, Pn(x) = p'(x). (20) for all n>l. If this were true then from eq.s 6 and 10, Pn(y|mM) = p;(y|mM). (21) As a result eq. 17 would hold i f , Pn(y|m)pn(m) = 0,5 which would be true when, pn(m) =0 for all n trials (22) and eq. 19 would hold i f , Pn(y|m)pn(m)>0.5 which would be true when, p (m)>0. for all n trials. (23) Similar derivations can be made for the corresponding contingency of any Additive and Interactive function. In terms of an average Additive and an average Interactive learning curve, then, the following can be stated: 1) i f there is no difference in the learning rates of the Additive and Interactive functions, then pn(a)+pn(b) = p^(a)+p^(b) = 0; and hence (24) pn(ab)+pn(-) = p^(ab)+p'(-) =1; 2) i f the learning rate of the Additive functions is greater than that of the Interactive functions, then pn(a)+pn(b) = p^(a)+p^(b)>0. (25) Equations 24 and 25 have important psychological implications. By 44. eq. 24, the lack of a learning rate difference would imply that, when sampled, both cues in the domain of the Additive and Interactive functions are sampled together as a group or gestalt. By eq. 25, the existence of a learning rate difference would imply that, when sampled, at least one of the cues in the domain of the Additive and Interactive functions is sometimes sampled alone. Since a pair of cues designates a cell of a stimulus matrix and a single cue designates a dimension value of a stimu-lus matrix, a further interpretation is possible. The lack of a learning rate difference would imply that learning occurs cell-by-cell, while the existence of a learning rate difference would imply that learning occurs, at least in part, dimension-by-dimension. The power of these implications, of course, rests upon the assump-tion of eq. 20. A stimulus (cell) contingency of any Interactive func-tion can presumably be learned only by sampling both cues (from eq.s. 9 and 12). But a stimulus contingency of any Additive function can presumably be learned either by sampling both cues or by sampling the one cue associ-ated with the function's main effect (from eq.s 9 and 11). Unless eq. 20 is assumed true, the lack of a learning rate difference could occur even i f , pn(a) = P;(ab) for all n>l, and the existence of a learning rate difference could occur even i f , P (a) = 0. Kn v ' for all n>l (by eq.s 17 and 19). Though the assumption of eq. 20 seems reasonable, i t is unfortunately not directly testable. It is indirectly testable, however, from the data 45. of the Estimation phase. To the extent that stimulus contingencies are learned cell-by-cell, an estimate of a dimension or marginal probability could presumably be made only by: 1) estimating the two cell probabi-lities associated with the marginal, and 2) combining the two resulting values. If, for example, a subject had learned that p(YES|mM) = 0.8, and p(YES|sM) = 0.2, but had not learned that p(YES|M) = 0.5, then an estimate of the Male contingency could presumably be made only by estimating both the (YES|mM) and (YES|sM) probabilities and appropri-ately combining them. Two results should occur. First, the marginal probabilities should take longer to estimate than the cell probabilities. Secondly, because the marginal estimates would be based on more observa-tions (or data) than the cell estimates, the former should have less variance than the latter. In contrast, to the extent that the stimulus contingencies were learned dimension-by-dimension, an estimate of a cell probability could presumably be made only by: 1) estimating the probabilities of the two marginals which defined i t , and 2) combining the two resulting values. For example, i f a subject had learned that, p(YES|m) = 0.8 p(YES|M) = 0.5, but had not learned that, p(YES|mM) = 0.8, then an estimate of the married Male contingency could presumably be 46. made only by estimating both the (YES|m) and (YES|M) probabilities and appropriately combining them. In this case opposite results from those above should occur. First, the cell probabilities should take longer to estimate than the marginal probabilities. Secondly, the cell estimates should have less variance than the marginal estimates. As mentioned above, the cell contingencies of the Interactive func-tions can only be learned by sampling cues together, that is.by learning cell-by-cell. As a result, estimates of cell contingencies should be faster but more variable than marginal estimates in the Interactive func-tions. To the extent that the Additive functions were also learned cell-by-cell the same result should occur, but to the extent that they were learned dimension-by-dimension, the opposite result should occur. A Simple Model for Predicting Transfer. How would i t be possible to account for any change in or 0^  guessing behaviour from the Pretest to Posttest? A simple means presents itself, i f i t is assumed that a change in this behaviour would be a function of: 1) the change in Q_-| guessing behaviour and, 2) the relation (assumed constant from Pretest to Posttest) between Gq and 0^ , or 0q and Q^ . In particular, the pro-bability of guessing g that a stimulus person (a,b) would answer G to 0^  or to 0^  is defined by the equations: p(g2|a,b) = p(g2|g1)p(g]|a,b)+p(g2|g])p(g1|a,b), or (26a) p(g3|a,b,) = p(g3|g])p(g1|a,b,)+p(g3|g1)p(g1|a,b), (26b) where: g.j = guess "g" given to question Q;., g. = guess "not-g" given to question 0., 47. pCg^ lg-j) = the probability of guessing g to Q_. given that "g" was guessed to 0^  , and p(g^ |g-j) = the probability of guessing g to given that "not-g" was guessed to . The quantities p ( g 2 | g 1 ) , p ( g 2 i 9-,), p (g 31 g-,) and p(g21 g1) can, of course, be estimated from the Pretest, simply by calculating the proportion of "yes" guesses to 0^  or 0^  which follow a "yes" guess and a "no" guess to C^j over the 80 Pretest trials. The quantities p(g^|a,b) and p(g-j |a,b) can be determined for each of the three 40-trial blocks in the Posttest of the Additive condition and of the Interactive condition. Predictions can then be made of the Posttest values of p(g2|a,b) and p(g^|a,b) in order to determine how closely this simple model approxi-mates the actual probabilities. 48. EXPERIMENT I Method Subjects. Eighteen paid volunteers (5 males and 13 females) from two undergraduate social psychology courses at the University of British Columbia served as subjects. Each was randomly assigned to one of the two experimental conditions and completed the experiment in one 11/2 hour session. Most subjects indicated interest in the experimental task, and were able to complete i t successfully, although some commented that the Posttest became rather boring. Two subjects, one in each of the Addi-tive and Interactive conditions, indicated no interest in the task: one admitted after completing i t that she had not listened to the Instructions and didn't understand what she was supposed to do, the other admitted that she became so bored about halfway through the Learning phase that she decided to give up and guess randomly from then on. These two Sub-jects produced results quite deviant from the others; by the end of the Learning phase, neither had shown any improvement in learning or guess-ing accuracy. It was therefore decided to exclude their data from all analyses. Materials. A large pool of 3 x 5 cards on which were printed either "Married" or "Single," and either MALE or FEMALE, and either YES or NO (the answer to 0^ ) was first constructed. From this pool, 40 cards were drawn to represent one of the four Additive functions or one of the two Interactive functions. For example, to represent the A^  function, 8 MARRIED MALE YES, 2 MARRIED MALE NO, 49. 8 MARRIED FEMALE YES, 2 MARRIED FEMALE NO, 2 SINGLE MALE YES, 8 SINGLE MALE NO, 2 SINGLE FEMALE YES, and 8 SINGLE FEMALE NO cards were drawn. Printed on two additional 3 x 5 cards were either 0^  and 0^  or 0^  and 0^ . Each subject held one of these cards during the Pretest and Posttest to remind him of the questions he was judging. Printed on one additional 3 x 5 card was 0^ . Each subject held this card during the Learning phase to remind him of the question he was learning. During the Estimation phase, a stopwatch was used to record estimation latencies. Procedure. Upon arriving, subjects were seated at a table facing away from the experimenter. Since only the function to be learned was varied in this Experiment, all subjects were read the same instructions. These were as follows: Recently a questionnaire was given to a large number of people enrolled in University Extension psychology courses. The pur-pose of this questionnaire was to determine, amongst other things, how a person's martal status and sex related to their feelings about a number of issues. An equal number of married males, married females, single males and single females were asked about their moods, social l i f e , personal philosophy, etc. Two of the questions asked were: 0, Are you usually more optimistic than pessimistic? yes or no. §2 Are you usually more humorous than serious? yes or no. (U (alternate) Are you usually more sociable than un-sociable? yes or no. 50. On each of these questions about half of all the respondants answered "yes" to either one of them or both of them, and a-bout half of all the respondants answered "no" to either one of them or both of them. It was found, however, that the propo-rtion of "yes" and "nos" depended on whether the persons were married males, married females, single males or single females. I want to get some idea of how accurate a judge you are of the answers these people gave to the two questions I just read. Here are the two questions again [experimenter hands subject copy of the questions printed on a 3 x 5 card]. I am going to present you one-by-one with information about the marital status and sex of a large number of the people who answered these questions. After I read you a particular person's marital status and sex I want you to guess as best you can, f i r s t , whether this person said "yes" or "no" to Question 1), and second, whether this person said "yes" or "no" to Question 2). For ex-ample, I may call out "married female" and you will then try to guess whether this married female answered "yes" or "no" to the question "Are you usually more optimistic than pessimistic?" and whether this married female answered "yes" or "no" to the question "Are you usually more humorous than serious?" [altern-ate: "Are you usually more sociable than unsociable?"]. You should call out your guesses as you make them, so I should hear from you two words, either "no, yes;" "yes, yes;" "no, no;" or "yes, no." Again you will be judging an equal number of single females, married males, married females, and single males. They will be presented to you in a completely random order. You will make your judgements until I tell you to stop. For this part of the experiment I will not tell you whether your judgements are correct or incorrect. Later you will be told this, but right now I am interested in how you guess these people will respond when you don't have any feedback. Do you have any questions? Following these instructions the experimenter ran through the rando-mized deck of 40 cards without replacement, called out each stimulus person as they appeared, and recorded both the stimulus person's attri-butes and the responses given by the subject. The deck would then be re-randomized out of the subject's view and run through again in the identi-cal manner. Immediately after the 80 pretest trials the following in-structions would be read: 51. We have now finished the first part of the experiment. In the second part I will continue calling out the marital status and sex of persons you are to judge. But this time I want you to guess only whether each person said "yes" or "no" to the first question, that is , to the question "Are you usually more optimistic than pessimistic?" Here is a copy of the question [experimenter takes the original 3 x 5 card from subject and gives him the 3 x 5 card with only QJJ printed on i t ] . Also, in this part of the experiment after I have called out the marital status and sex of a particular person and after you have guessed whether he or she said "yes" or "no" I will tell you whether your guess was either "correct" or "wrong." I am interested here in seeing how your guesses change when you have this feedback. Again try to be as accurate in your guesses as you can This part of the experiment will be a l i t t l e longer than the f i r s t . Do you have any questions? Immediately after these instructions were given the experimenter con-tinued to run through the re-randomized 40 card deck in the manner of the Pretest. But this time "Correct" was called out each time the sub-ject's guess corresponded to the answer printed on the 3 x 5 card, "Wrong" was called out otherwise. (This, of course, provided the schedule of reinforcement dictated by the function the subject was to learn.) Again, the deck was re-randomized after each complete run through i t . Each of the 200 trials lasted about 8 seconds. Stimulus values, responses and correct criterion values were recorded. Immediately after the Learning trials subjects were told: We have now finished the second part of the experiment. I now want to get some idea of your feelings about the propor-tion of different people who said yes to the first question. I will call out classes or types of people. For example, I may call out "married females" or simply "females." After each I want you to estimate the percentage or proportion (from 0% to 100%) of that type which I just called out who said "yes" to the question. Try to be as accurate as you can. Do you have any questions? 52. The experimenter then called out one of the four stimulus types or one of the four stimulus characteristics (their order was randomized for each subject). Estimation latencies were taken from the time the second word in each phrase was called out until the time the proportion was called back. (Pre-testing showed this latency to range from about 2 to 10 seconds). Both the latencies and proportions were recorded. Finally, Subjects were told: We will now begin the last of the experiment. This part will be exactly like the first part in every way. Here are the two questions you guessed at in the first part [experimenter hands back to the subject the 3 x 5 card which both questions printed on i t ] . Again, as soon as I call out a particular person's marital status and sex you will call back your guesses about how this person answered the first question and the second question. Again, I will not tell you whether your guesses are right or wrong. Do you have any questions? The three randomized blocks of 40 post-test trials would then be given to the subject in exactly the same manner as the two blocks of 40 Pretest trials. Following the Posttest, subjects were paid $2.00, de-briefed and excused. Results Responses made by each subject during the Pretest, Learning, Estima-tion and Posttest phases of the experiment were first summarized in tabular form. A more popular response (or mpr) index was then derived for each subject on each 40-trial block of the Pretest, Learning and Post-test phases as a measure of how "well" responses were reflecting the function to be learned. The mpr index was computed by averaging the number of "yes" responses given during a 40-trial block to the stimulus types whose more popular answer to 0q was "yes" with the number of "no" 53. responses given to the stimulus types whose more popular answer to was "no." To clarify, consider a subject who was to learn the Interac-tive function . In the function Lj the more popular answer to 0^  given by married Males and single Females was "yes" and by married Females and single Males, "no." If i t were assumed that in one 40-trial block the subject gave 7 "yes" responses to the 10 married Males, 9 "yes" re-sponses to the 10 single Females, 4 "no" responses to the 10 married Females, and 8 "no" responses to the 10 single Males presented to him, then his mpr index for this 40-trial block would be: (7/10 + 9/10 + 4/10 + 8/10) / 4 = 0.70. The mpr index was, in effect, merely an extrapolation of the standard response index, p, used in most two response probability learning studies where p had been, by convention, defined as the probability of predicting the more popular of two events: or (see Atkinson, Bower & Crothers, 1965). Like P, the mpr index is binominally distributed and has a range from 1.0 to 0.0. A subject who responded randomly in the Experiment would have an expected mpr of 0.5. One who responded in a manner contrary to the function he was to learn would have an expected mpr less than 0.5. One who employed a "matching" strategy (see Atkinson et. al., 1965) would have an expected mpr of 0.8. And one who employed a "maximizing" strategy would have an expected mpr of 1.0. The Pretest. In order to determine whether any differential response bias existed prior to learning an Additive or Interactive function an average mpr proportion was computed for each. The overall average Pre-test mpr proportion of the 8 subjects learning an Additive function was 0.47. Similarly, the overall average Pretest mpr proportion of the 8 54. subjects learning an Interactive function was also 0.47. Prior to learn-ing, therefore, neither group had any propensity to respond to the stimu-l i in a manner resembling the functions which they were to learn. In order to determine the Pretest relationship between the questions Qq - and 0q - response contingencies were calculated. The probabi-lity of guessing "yes" to 0^  (humour) given a "yes" or a "no" to 0q was computed from the 80 Pretest trials for each subjects guessing 0q - Qg in the Additive and Interactive conditions. Similarly, the probability of guessing "yes" to (sociability) given a "yes" or a "no" to 0q were computed for the 0q - CL^  subjects. Group Averages for the 0q - Qg Additive, 0q - 0^  Additive, Oq - 0^, Interactive, and 0q - 0^  Interactive subjects are shown in Table III. 02 23 2-2 y n n y n y n °Hn .61 .39 .66 y .34 ^ n .60 .40 .68 .59 .41 .59 0 y .41 ^ n .55 .45 .57 .50 .50 .47 .53 .32 .47 .53 .43 .57 .43 .58 .42 ADDITIVE .56 .44 .54 .46 INTERACTIVE .50 .50 Table III. 0q - Qg and Qq - Pretest Contingencies of the Additive and Inter-active Functions. Since the experimental conditions in the Pretest were identical for all Additive and Interactive functions i t was decided to pool the continge-2 ncies of these functions. The x indices of association between the 2 pooled Oq - Q2 contingencies (x = 7.6, df = 1, p<.01) and the pooled 0q -2 Q- contingencies (x = 8.6, df = 1, p<.01) were both statistically signifi-55. cant. However the strength of association of both contingencies were ex-tremely low. (O^ -Clg =0.10, 0^ -Q.j =0.11; see Hays 1963, p. 604) and did not appear to differ markedly from each other. In particular, the strength of association between and seemed to be less than that obtained by Osgood et. al_. (1958). The manipulation of the strength of association, therefore, did not appear to be successful. Learning. In order to determine overall changes in responses dur-ing the learning phase, the mpr proportions of all subjects learning an Additive function were averaged for each 40-trial block, as were the mpr proportions of all subjects learning an Interactive function. These averages are shown in Figure 1. Insert Figure 1 about here A 2 x 5 (Function by Trial Blocks) ANOVA was performed on these mpr proportions to test for a learning effect and for differences in a learning effect between the Additive and Interactive subjects (see Kirk, 1968). A summary of the Analysis is presented in Table IV. Source df MS F Between S 15 121.93 Function 1 46.51 0.62(n.s.) Error 14 75.42 Within S 64 525.03 Trial Blocks 4 505.73 37.28(p<.001) TB X F 4 5.73 0.42(n.s.) Error 56 13.57 Table IV. ANOVA Summary Table for Learning Phase of Experiment I 1.00 0.90 . 0.80 . 0.70 0.60 . 0.50 0.40 . Additive Interactive Pretest Learning Blocks Figure 1. Average mpr Proportions of Subjects Learning Additive and Interactive Functions 57. As can be seen from Table IV, the overall learning effect for the Additive and Interactive functions was highly significant. However there was no significant difference in the rate at which each was learn-ed, nor was there a significant learning-by-rate interaction. There was, therefore, no evidence to support the hypothesis that the Additive functions were learned more quickly than the Interactive functions. In terms of the conceptual framework and the learning and sampling assu-mptions proposed in chapter 7, this result implied that, Pn(a)+pn(b) = p^(a)+P;(b) = 0, and hence that Pn(ab)+pn(-) = p;(ab)+p'(-) = 1.0, (27) for all n trials. Estimates. In order to easily compare group estimate averages of the Additive and Interactive conditions, both the estimates and the esti-mate latencies of the A2, A^, and A^  functions were transformed to appear as those of A^  and the estimates and estimate latencies of the I 2 func-tion were transformed to appear as those of I-j. For example, since a row-row transformation of A2 made i t appear as A^  the (s,M) estimates of the A2 function were averaged with the (m,M) estimates of A-j, the (s,F) estimates of A2 with the (m,F) or A^ , the (m,M) estimates of A2 with the (s,M) of A-| and the (m,F) estimates of A2 with the (s,M) of A-|. Of course, i t was possible to transform I 2 to I-j by either a row-row or a column-column transformation, so the former was arbitrarily chosen. The cell and the marginal Additive function estimate averages and the cell and marginal Interactive function estimate averages are shown in Table V. 58. Additive Function Interactive Function Required to Estimate: Actual Estimates Actual Estimates Married Male .80 .68 .80 .71 Married Female .80 .69 .20 .38 Single Male .20 .29 .20 .31 Single Female .20 .39 .80 .78 Married People .80 .74 .50 .55 Single People .20 .42 .50 .54 All Males .50 .56 .50 .53 All Females .50 .62 .50 .48 Table V. Average Cell Estimates and Average Marginal Estimates of the Additive and Interactive Functions (n=8 per estimate). A 2 x 2 within subject ANOVA was performed on the cell estimates of the additive function to determine how much of the estimate variance was due to the main effect to be learned. Similarly, a 2 x 2 within subjects ANOVA was performed on the cell estimates of the Interactive function to determine how much of the estimate variance was due to the interaction to be learned. These analyses are summarized in Table VI. 59. Additive Function Interactive Function Source df SS F p SS F p Subjects Marital Stat. M.S. x s.w.g. Sex Sex x s.w.g. M.S. x Sex M.S. x Sex x s.w.g. 7 1 7 1 7 1 7 2098.4 8256.1 35.64 (p<.001) 1621.4 45.1 0.66 n.s. 473.4 210.1 0.45 n.s. 3208.4 1061.7 0.8 0.00 n.s. 730.5 19.5 0.05 n.s. 2337.7 12600.8 41.42 (p< .001) 2129.5 Total 31 15912.9 18880.5 Table VI. ANOVA Summary Table of Additive and Interactive Function Cell Estimates From Table VI i t can be seen that the estimates of the cell con-tingencies of the Additive and Interactive functions did, in fact, signi-ficantly reflect the main effect and the interaction to be learned. In-deed, 52% of the total estimate variance of the Additive functions was due to the main effect to be learned, and 67% of the total estimate variance of the Interactive functions was due to the interaction to be learned. There was therefore no reason to believe that the contingencies of the Additive functions were any more reliably estimated than those of the Interactive functions. To further test the conclusion of eq. 27 in the manner outlined in chapter 7, latencies for each cell estimate and marginal estimates were first calculated. Average cell and marginal estimate latencies of the Additive and Interactive functions are presented in Table VII. 60. Estimate of: Additive Interactive married Male 2.7 3.3 married Female 3.0 3.7 single Male 3.6 3.3 single Female 3.0 2.6 married People 2.9 3.3 single People 3.9 4.0 Males 3.7 3.6 Females 5.0 3.6 Table VII. Average Latencies (in seconds) of Cell and Marginal Estimates of Additive and Interactive Functions (n=8 per latency). If eq. 27 were correct,latencies of cell estimates should be shorter than those of the marginal estimates in both the Additive and Inter-active functions. An analysis of variance of the functions and esti-mate types should, therefore, produce a significant estimate type main effect but no significant function by estimate type interaction. A summary of this analysis is presented in Table VIII. (Each subject con-tributed two pieces of data for this analysis, an average cell esti-mate latency and an average marginal estimate latency.) 61. Source df SS F P Between 15 1069.0 Function 1 8.3 0.1 (n.s.) Error 14 1060.7 Within 16 251.6 Estimate Type 1 49.3 3.5 (p<.10) Funct. x E. Type 1 6.6 0.4 (n.s.) Error 14 196.3 Total 31 1320.6 Table VIII. ANOVA Summary Table of Estimate Latencies. It can be seen from Table VIII that the function-by-estimate type interaction was not significant and that the difference between the average cell estimate latency (3.02 sec.) and the average marginal estimate latency (3.75 sec.) was significant at the p=0.10 level. At conventional levels, of course, this main effect was only marginally significant. One reason for this may have been the nature of the distri-bution of latency differences. Like many distributions of latencies i t was highly skewed; one subject, in particular took an average of 2.6 seconds longer to estimate the cell proportions than to estimate the marginal proportions because one cell estimate took about 14 seconds to make. To reduce this skewedness, all latencies were subjected to a square root transformation. A t-test performed on these data was con-sidered to be significant (t=2.05, p<0.06, df=15). Latency predictions derived from eq. 27 were therefore confirmed. Recall from Chapter 7 that i f the Additive and Interactive functions 62. were learned cell-by-cell (as implied by eq. 27) then an estimate of a marginal contingency would presumably require the estimates of both cell contingencies which defined i t . If the estimate of a cell contingency was based on sample size s, then the estimate of a marginal contingency would presumably be based on a sample size larger than s. As a result, the variance of the sampling distribution of marginal estimates should be smaller than the variance of the sampling distribution of the cell estimates. To test this hypothesis the between subject variances about each cell and each marginal mean in Table V were computed for the Addi-tive and Interactive functions using the standard formula: 2 n - 2 est. o =i [p.-p] p i=l 1 n-1 where p.= the ith subject's estimates proportion, p was the average of the estimated proportions and n was the number of subjects in the sample (in these cases n=8). The variances are presented in Table IX. Required to Estimate: Additive Interactive Married Male 3.7 1.3 Married Female 3.2 3.1 Single Male 1.4 1.6 Single Female 2.4 2.9 Married People 1.1 0.6 Single People 2.7 1.7 Males 1.0 0.2 Females 1.1 1.4 Table IX. Variances (o ) of Cell and Marginal Estimates of Additive and Interac-_2 tive Functions (all variances x 10 " ). 63. It can be seen from Table IX that 6 of the 8 smallest estimate variances occurred in the marginals and not the cells. There was, un-fortunately, a problem in trying to test for differences between these variances: since the estimates presumably came from binominal distri-butions, estimate variances would be dependent upon the estimate pro-portions in a manner given by the formula, 2 _ est.a =p(l-p), p m where m was the average sample size on which the estimate average (p) was based. To overcome this problem the sample size m was estimated from the above equation for each cell and marginal contingency of the Additive and Interactive functions. Predicting lower variance of marginal estimates was thus equivalent to predicting higher values of m in the marginals. The m values of the cells and marginals for the Additive and Interactive functions are presented in Table X. Required to Estimate Additive Interactive Married Males 5.9 16.7 Married Females 6.6 7.5 Single Males 14.7 13.8 Single Females 10.0 6.1 Married People 17.3 43.8 Single People 9.0 14.8 Males 23.4 118.6 Fema1es 21.9 17.5 Table X. Estimated Sample Sizes on Which Cell and Marginal Estimates Were Based 64. It may be seen from Table X that 7 of the 8 largest sample sizes came from the marginals and not from the cells, a proportion found to be statistically significant (p<.01) by Fisher's Exact Test. This re-sult could have been an experimental artifact; of the eight marginals to be estimated, six had an actual value of 50%—no doubt a favourite number for subjects who, for one reason or another, were making a proverbial stab in the dark. Such an explanation, for example, may have accounted for the low variance around the "all males" estimate average of the In-teractive function resulting in its surprisingly high estimated sample size of 118.6. Yet the results did not refute the notion that marginal estimates were obtained by combining information from two cell samples. Indeed, in conjunction with the latency data, the result even tended to give some credence to i t , and thus gave further support to the conclusion of eq. 27. The Posttest. In order to determine what happened to guessing behaviour in the Additive and Interactive conditions following the elimi-nation of feedback, the average mpr indices of the Additive condition and of the Interaction condition were computed for each of the three Post-test trial blocks. These are shown in Figure 2. Insert Figure 2 about here A 2 x 3 (function by trial-block) between-within ANOVA was then per-formed on the Posttest mpr indices. It is summarized in Table XI. 1.00 0.60 J 0.50 0.40 J !> Z 1V r r , r L e a r n i n g 5 1 2 3 P o s t t e s t F i g u r e 2. Average P o s t t e s t mpr P r o p o r t i o n s o f S u b j e c t s L e a r n i n g A d d i t i v e and I n t e r a c t i v e F u n c t i o n s 66. Source SS df F P Between 2024.8 15 Function 130.0 1 0.96 n.s. Error 1894.8 14 Within 234.0 32 Trials 4.9 2 0.36 n.s. FXT 40.0 2 2.96 n.s. Error 189.1 28 Table XI. ANOVA Summary Table for Posttest of Experiment I. It can be seen from Table XI that there was no significant diffe-rence between the guessing behaviour of the Additive and Interactive sub-jects following the elimination of feedback. There was also no signifi-cant interaction between these two variables. The divergence of the two graphs shown in Figure 3 resulted primarily from the behaviour of one subject in the Interactive condition, who—for reasons unknown—plummet-ted from a mpr proportion of 0.90 in the last learning trial block to an average mpr proportion of 0.31 in the Posttest. In general, however, i t was concluded that the elimination of feedback did not significantly alter guessing behaviour either between or within the Additive and Configural functions. In order to determine the effects of transfer, each subject's ave-rage mpr proportion was computed for 0^  or 0^  on each Posttest trial block in the same manner as i t was for Oq. Since no significant diffe-rences existed between the strength or direction of the Oq-Q^  and Q-j-Qg relations in the Pretest, Posttest Cu and 0~ mpr proportions given by 67. subjects in the Additive condition were averaged, as were the 0^  and Q3 mpr proportions given by subjects in the Interactive condition. These averages were used as the data "observed" against which the predictions of the simple model were tested. Predictions of the quantities: p (mpr to O^ ) and p (mpr to 0^ ) were derived from the simple model using the Pretest estimates of p. (mpr to C^ /mpr to 0^ ) and p (mpr to O^ /mpr to 0^ ) and the Posttest estimates of p (mpr to 0^  ). Both the predicted and actual proportion of 0^  and 0^  more popular re-sponses in each Posttest trial block are summarized in Table XII. Additive Interactive Posttest Trial Block observed predicted observed predicted 1 0.67 0.55 0.58 0.53 2 0.61 0.54 0.56 0.53 3 0.56 0.54 0.58 0.53 Table XII. Observed and Predicted Average 0~-0~ mpr Proportions in Each Trial Block of the Posttest. As can be seen in Table XII, the predicted transfer was consiste ntly less than the obtained transfer in both the Additive and Interactive con-ditions. Apparently either the O^ -O^  or the Q^ -Qg (or both) relation-ships had not remained constant from Pretest to Posttest but had increased 68. 2 in magnitude during this time. To test this notion, a x index of asso-ciation between Pq-G^  a n d between Oq-Qg w a s computed respectively for Qq-Qg and Pq-Qg contingencies from: 1) the last 40 Pretest trials, and 2) the first 40 Posttest trials. The Pq -Q^ relation increased in 2 strength from these Pretest to Posttest trials (Pretest X =2.4, x=0.09; 2 Posttest X =15.8, A=0.22). However, the increase was not significant (X2(Qq x Qo x Trials)=0.88, p>.50, df=l; see Winer p. 629-631). The Oq-Qg relation also increased in strength from these Pretest to Posttest trials (Pretest X2=3.0, x=0.10; Posttest X2=30.6, x=0.31). This increase 2 was significant (X (Oq x ^ x Trials)=7.26, p<.01, df=1). Hence, the Oq-Pg relationship became stronger than the Pq-Qg relationship, a result in agreement with Osgood ejt. a]_.'s (1958) finding that optimism and sociability were more strongly related than optimism and humour. Discussion The major finding of this experiment was that no significant dif-ference existed in the rates at which the Additive and Interactive func-tions were learned, a finding which concurred with that of Brehmer (1969). The finding, of course, had all the classic stigmas associated with a failure to reject the null hypothesis. As a result, i t could always be argued that the non-significance merely reflected a lack of discrimi-nating power in the experimental task. Both in this Experiment and in Brehmer's, the Additive functions were learned slightly faster than in Interactive ones. If more "complex" learning tasks were employed (for example, ones involving functions with a domain of far more than just two stimulus dimensions) a significant learning rate difference might very well be obtained. 69. Yet, as shown in chapter 2, learning rate differences have been pro-duced by varying the function in situations as simple as that of this Ex-periment. The lack of a learning rate difference in this Experiment im-plies that, at least in "simple" situations, all functions should be learned at the same rate. Why then were the five functions of the Neisser and Weene (1962) study summarized in chapter 2 not learned at the same rate? And why were the Multiplicative and Ratio functions of the Brehmer (1969) study not learned as quickly as the Additive and Difference func-tions? Equation 27 would simply not predict these results. Indeed, none of the findings of this Experiment shed much light on the answers. Though the functions to be learned in this Experiment provided a powerful means for distinguishing alternative learning strategies, they had at least one obvious limitation. The event probabilities, p[G|(a,b)] in both the Additive and Interactive functions were what may be termed, "information equivalent." Since all cells had an event probability of either 0.2 or 0.8, the same amount of uncertainty--0.71 bits--was con-tained in each (see Coombs, Dawes and Tversky, 1970, chapter 9). Feed-back about each cell, therefore, contained an equivalent amount of infor-mation. This was done to insure that each cell contingency would be learn-ed at the same rate. But i t produced a learning situation which appeared to be unrepresentative of most (see Brunswik, 1956). If criterion values were being learned cell-by-cell, i t seemed im-portant to determine what the characteristics of a judgement process would be when the cell contingencies were not learned at the same rate. Different learning rates should result i f the amount of uncertainty in each cell contingency were varied. Experiment II, therefore, attempted 70. to vary the amount of uncertainty in the cell contingencies in a manner which made i t possible to compare the results with those of Experiment I. 71, EXPERIMENT II Consider a subject who had learned the correct criterion value asso-ciated with some, but not a l l , of the possible combinations of a set of stimulus dimension values. How might he behave i f asked to judge a stimulus whose criterion value he had not yet learned? Let us say, for example, that a subject had learned that 80% of married males and 20% of single males were optimistic, but had not yet learned the proportions of married and single females so inclined. What might we expect the subject to do if asked to judge the optimism of a married or single female? He could, of course, judge "blindly." But i f we were to take the principles of stimulus generalization and discrimination seriously (see Hall, 1965 and Luce and Galanter, 1963 for reviews) we ought to expect some extra-polation of what was known about married and single males in judging their female counterparts. In particular, i t seemed reasonable to assume that since a married female had one characteristic in common with a married male (namely marital status) but no characteristic in common with a single male, a judgement about her optimism should be more simi-lar to a judgement of the married male than to a judgement of a single male. By similar reasoning, a judgement of a single female's optimism should be more like that of a single male than that of a married one. If this were to occur what characteristics would the resulting pro-cess have? Let us first consider a limiting case, namely that in which a subject did not generalize but instead guessed blindly. Assuming a "blind guess" resulted in 50% of both married and single females being judged optimistic (a reasonable assumption i f , as in Experiment I, sub-jects were told that "50% of all people were . . . " and i f , as in 72. Experiment I, response biases were controlled), a tabular summary of judgement propensities would appear thusly (see Table Xlla), M F 80% 50% 20% 50% 50% 50% Table XIla. Possible Outcome of Judgement of Optimism Where mM and sM Values Known, mF and sF Values Guessed Blindly In ANOVA terms this summary would characterize a judgement process whose accountable variance came half from the marital status main effect and half from the marital status-by-sex interaction. Let us now consider a subject who did not guess blindly but instead judged the optimism of married females to be much like that of married males, and the optimism of single females to be much like that of single males. A tabular summary of his judgement propensities might appear thusly (see Table XHIb): M F B0% 70% 20% 30% 50% 50% Table XII lb. Possible Outcome of Judgements of Optimism Where mM and sM Values Known, mF and sF Guessed by Gene-ral ization In ANOVA terms this summary would characterize a judgement process whose accountable variance came almost completely from the marital status main effect and very l i t t l e from the marital status-by-sex interaction. In-deed, from this example i t was possible to extrapolate a general rule 73. about judgements of stimuli constructed from two dichotomous dimensions: to the extent that a subject generalized in the manner outlined above from the cell(s) whose criterion value(s) were known in order to judge the cell(s) whose criterion value(s) were not known, his judgements would be more accountable to main effects than to interactions. Generalizations, therefore, would tend to produce what would look more like an additive than configural process. (Special conditions of this rule have been pre-sented by Coombs, 1964, and Tversky, 1969.) What remained to be deter-mined was the extent to which this generalization process would occur. Experiment II was designed to make this determination. Subjects in Experiment II were required to learn to judge the 0^  (optimism) answers of married males, married females, single males and single females in exactly the same manner as Experiment I. However, in Experiment II, the functions relating each of the four stimulus types to the answers were varied. In particular, subjects in Experiment II were required to learn one of eight Composite functions obtained by averaging one of the Additive (A-j-A^ ) functions with one of the Interactive (I..-^) functions. One subject, for example, was required to learn the continge-cies formed by averaging the matrices of functions A., and 1-j, that is: M A l + !1 bo .8 1 .2 .8 .2 = m .8 .5 .65 .2 .8 s .2 .5 .35 .50 .50 .50 (28) Another subject was required to learn the contingencies formed by avera-ging the matrices of functions A-j and Ig, s t i l l another learned the average of Ag-I-j , etc. 74. These Composites had a number of interesting features. First--as exemplified in matrix 28 above—each Composite contained two cells (p=.8 and p=.2) whose proportion of "yes" answers to 0^  were equal to those of the original Additive and Interactive functions. Since Experiment I had indicated that learning was occurring cell-by-cell i t was therefore hypo-thesized that: H.| = the cells with contingencies p=.8 and p=.2 would be learned as quickly as those of the first experiment. Second, each Composite contained two cells whose proportion of "yes" answers to were equal to the overall proportion of "yes" answers (i.e., p=.5). There were numerous reasons to believe that, because of this, the p=0.5 contingencies would take longer to learn than the p=0.2 or 0.8 con-tingencies. Some of the reasons were theoretical. It was known from probability theory, for example, that since the variance of a binomial distribution was smaller when p=0.2 or p=0.8 than when p=0.5, hypo-theses about the probability of a "yes" answer given by stimulus person type (a,b) would be expected to reach any given confidence interval with fewer observations i f p(yes|a,b)=0.8 or 0.2 than i f p(yes|a,b)=0.5 (see Hays and Winkler, 1970, chapter 8). If, therefore, subjects were attempt-ing to learn the contingencies by some form of hypothesis testing (a reasonable assumption; see Bruner e_t. al_. 1956, chapter 7; Edwards and Phillips, 1964), fewer trials would be needed to learn the former con-tingencies than the latter. The remaining reasons were empirical. Friedman, Burke, Cole, Keller, Millward, and Estes (1963) had reported that in a two-choice probability learning task with shifting even pro-babilities, the learning rate, e(l<e<0), became significantly smaller as 75. the absolute magnitude of the shift decreased. For example, when the event probability shifted from p(E^)=0.5 to p(E^)=0.9, e was estimated to be 0.62; but when i t shifted from p(E^)=0.5 to p(E^)=0.6, e was esti-mated to be 0.07. By extrapolation, therefore, i t seemed reasonable to assume, again, that the p=0.2 and p=0.8 contingencies would be learned more quickly than p=0.5 ones. One nasty problem remained. If a subject had not yet learned a p=0.5 contingency, all known conceptualizations of the learning process (see Atkinson, Bower, and Crothers, 1965) would predict that he would guess "yes" with probability p=0.5. Yet, i f a subject had learned a p=0.5 contingency, most of these conceptualizations would also predict that he would guess "yes" with probability p=0.5. There was, therefore, no simple means of determining from the subject's responses whether or not he had learned the p=0.5 contingency. What could be said, however, was this: a) If the p=0.5 cells were learned at the same rate as the p=0.2 and p=0.8 ones, then the probability of a "yes" judgement given these cells should remain at about p=0.5 throughout the learning phase, but b) i f the p=0.5 cells took longer to learn than the p=0.2, p=0.8 ones, and i f subjects did generalize from the cells whose contingen-cies they knew to those whose contingencies they did not know, then, p("yes"|p=0.5 cell adjacent to p=0.8 cell)> p("yes"|p=0.5 cell adjacent to p=0.2 cell). As a result, i f a) (above) were true, the main effect and interaction of 76. the Composite would be learned at the same rate. But, i f b) (above) were, true, then: FL: the main effect of the Composite would be learned more quickly than its interaction. Estimations of Cell and Marginal Contingencies. Subsequent to the learn-ing phase, subjects in Experiment II were also required to estimate cell and marginal proportions in the same manner as the subjects of Experiment I. In light of the results of Experiment I, i t seemed appropriate to en-tertain at least two hypotheses about the estimations: H3 = estimation latencies for the cells would be smaller than estimation latencies for the marginals, but H4 = the variance of cell estimates would be larger than the variance of the marginal estimates. The Composite functions provided perhaps the better test of these two hypotheses than the Additive and Interactive functions of Experiment I, for they could not only be tested between all four cells and all four marginals (as in Experiment I) but also between the two cells with p=0.5 contingencies and the two marginals with p=0.5 contingencies. These latter tests would allow for better control of any confounding effects which resulted from the estimation of proportions of differing magnitude. Transfer of the Judgement Process. In order to better examine the nature and extent of any transfer of the judgement process from one question to another i t was felt that a stronger a priori relationship should exist between the two questions than had existed between Oq and Qv, or Oq and 0^ in Experiment I. It was therefore decided to present subjects in Experiment II with: 77. 0^  = "I am usually more optimistic than pessimistic", and = "I am usually more happy than sad", both before and after the learning phase of the Experiment. 0^  was chosen to replace 0^  and 0^ because Osgood e_t. al_. (1958) had found optimism and happiness to be much more highly correlated than either optimism and humour or optimism and sociability. The results of Experiment I indicated that the strength of the re-lationship between 0^  and 0^  and between Q-j and increased from the Pretest to the Posttest. If this increase were somehow due to the learning of the Additive or Interactive functions themselves (or perhaps to the expected increase in predictive accuracy which came with the learning) then a few interesting predictions could be made. Firstly, since the Composite functions contained two cells whose proportions (p=0.8 and p=0.2) were similar in magnitude to those of the Additive and Interactive functions (and which, of course, when learned could increase a subject's predictive accuracy) i t was hypothesized that: Hg = the strength of the relationship between 0^  and would in-crease in the p=0.8 and p=0.2 cells of the Composites from Pretest to Posttest. Second, since the remaining two cells of the Composites contained pro-babilities (p=0.5) which precluded any increase in accuracy (subjects would always be expected to get 50% correct regardless of whether or not they had learned the p value) and which presumably took longer to learn, i t was also hypothesized that: Hg = any increase in the strength of the P^-O^ relationship in the two p=0.5 cells of the Composites from Pretest to 78. Posttest would be less than the corresponding in-crease in the p=0.8 and p=0.2 cells of the Composites. Method Subjects. Eight paid volunteers (6 males and 2 females) from two social psychology courses served as subjects. Each was required to learn one of the 8 Composite functions in the same manner as subjects in Experiment I. Each completed the Pretest, Learning, Estimation and Possttest phases of the experiment in one 1 1/2 hour session. Materials. The materials of Experiment II were the same as those used in Experiment I. The only exception was, of course, that the 40-card sets were drawn to make up one of the Composite functions rather than one of the Additive or Interactive functions. For example, 8 MARRIED MALE YES, 2 MARRIED MALE NO, 5 MARRIED FEMALE YES, 5 MARRIED FEMALE NO, 2 SINGLE MALE YES, 8 SINGLE MALE NO, 5 SINGLE FEMALE YES, and 5 SINGLE FEMALE NO cards were selected to represent the A-j -1 -j Composite described in matrix 28, above. Procedure. The procedure of Experiment II was identical to that of Experiment I with one exception: since no significant effects were found across the 120 trials of the Posttest of Experiment I, it was deci-ded to run the Posttest of Experiment II for only 80 trials. Results To facilitate analyses the matrices of each subject's responses given in the Pretest, Learning, Estimation and Posttest phases were transformed, where appropriate, to appear in the form of the A-| — I -j Com-posite (as was done to the estimates of Experiment I). The average number of yes responses given by subjects to the four stimulus persons during the 80-trial Pretest and during each of the five 40-trial Learning blocks are presented in Table XIV. The predominance of yes responses in the Pretest seemed to result from a weak response bias. It was not con-sidered large enough to affect the results. Required to Learn Pretest Learning Block 1 Block 2 Block 3 Block 4 Block 5 1.80 .50 .60 .60 .65 .54 .79 .72 .82 .70 .89 .74 .92 .66 1.20 .50 .62 .62 .52 .51 .39 .34 .24 .44 .19 .48 .14 .54 Table XIV. Average Proportion of "YES" Responses Given by Sub-jects in the Pretest and in Each of the Give Learn-ing Blocks. In order to test hypothesis H-. an mpr index was calculated for the first column of the Pretest matrix and of each of the Learning Block matrices presented in Table XIV. (For example, the mpr index of Learning Block 1 was computed as [0.65 + (l-0.52)]/2=0.56). A graph of these indices is presented, along with a graph of the mpr indices of the Addi-tive and Interactive functions of Experiment I, in Figure 3. 80. Insert Figure 3 about here As can be seen in Figure 3, the rate of learning the p=0.8 and p=0.2 contingencies of Experiment II was virtually identical with the rate of learning the p=0.8 and p=0.2 contingencies of Experiment I. It there-fore seemed reasonable to conclude, by visual inspection, that these rates did not significantly differ. In this weak sense, appeared to be confirmed. The data relevant to a test of hypothesis Hg were derived by com-puting for each subject on each learning block two mpr indices: 1) a main effect component mpr index, computed by comparing each subject's response matrices to the Additive function from which the Composite was composed, and 2) an interaction component mpr index, computed by com-paring each subject's response matrices to the Interactive function from which the Composite was composed. For example, a subject who was re-quired to learn the A^-I-j Composite and who gave 7 YES responses to 10 mM stimuli, 6 yes responses to 10 mF stimuli, 2 YES responses to 10 sM stimuli, and 5 yes responses to 10 sF stimuli would receive a main effect component mpr index of: 7 + 6 + 8 + 5/40 = 0.65, and an interactive component mpr index of: 7 + 6 + 2 + 5/40 = 0.50. The average main effect mpr, interaction mpr, and the combined mpr for the 80-trial Pretest and for each 40-trial Learning Block are shown in Figure 4. 1.00, 0.90 0.80. 0.70 0.60J 0.50 0.40. Additive (Exp. I) // / Interactive (Exp. I) Composite Pretest 1 2 3 4 Learning Blocks Figure 3. Average mpr Proportions of Subjects Learning p=0.2 and p=0.8 Cells of the Composite Functions 82. Insert Figure 4 about here A 2 x 5 (mpr source x learning block) ANOVA was performed on the main effect component and interaction component mpr data. It is summa-rized in Table XV. Source SS df F P Between 230.2 7 Within 2239.3 72 Components 314.1 1 3.74 p<.10 C x sub. 587.2 7 Trials 338.1 4 5.03 p< .01 T x sub. 470.2 28 C x T 196.4 4 4.13 p< .01 C x T x sub. 333.3 28 Total 2469.5 79 Table XV. ANOVA Summary Table of Learning of Composite Function Components. From Table XV i t may be seen that subjects increased their Compo-site mpr significantly over trials, form which i t may be concluded that they did, in fact, learn the Composite function. It also may be seen that there was a marginally significant difference in the learning of the main effect and interaction components and a significant components-by-trials interaction. An examination of Figure 6 indicated that most of the rise in the main effect mpr occurred in the first 80 learning trials (B-j and B ^ ) while most of the rise in the interaction mpr occurred in the last 120 learning trials ( B ~ thru B r ) . It is this difference in 1.00., 0.90 0.80 J Pretest 1 2 3 4 5 Learning Blocks Figure 4. Learning Curves of the Composite Function and Its Components 84. learning rate which seemed to produce the aforementioned statistical significance. It was, therefore, concluded that hypothesis H2 was con-firmed, that the main effect of the Composite was learned more quickly than the interaction. Estimates. As mentioned previously, the cell and marginal estimates were first transformed to appear in the A-j -1 -j Composite format (with the 0.8 cell to be learned occurring in row 1, column 1 of the matrix, etc.). The estimates were averaged over subjects and appear in Table XVI. Required to Actual Estimated Estimate: Proportion Proportion Married Male .80 .84 Married Female .50 .55 Single Male .20 .35 Single Female .50 .49 Married People .65 .67 Single People .35 .44 Males .50 .52 Females .50 .52 Table XVI. Average Cell Estimates and Average Marginal Estimates of the Composite Function (n=8 per estimate) A 2 x 2 ANOVA was then performed on the cell estimates to determine the distribution of accountable variance across the marital status main effect (the one to be learned), the sex main effect (the "irrelevant" dimension) and the marital status-by-sex interaction. A summary of the ANOVA appears in Table XVII. 85. Source SS df MS F Between S 998.5 7 Within S 16470.0 24 Marital Status 5886.1 1 5886.1 16.85 (p<.01) Marital Status x s.w.g. 2445.4 7 349.3 Sex 406.1 1 406.1 1.07 (n.s.) Sex x s.w.g. 2650.4 7 378.6 Marital Status x Sex 3698.0 1 3698.0 18.70 (p<.01) Marital Status x Sex x s.w.g, 1384.0 7 197.7 Total 17468.5 31 Table XVII. ANOVA Summary Table of Estimates of the Composite Function It may be seen from Table XVII about 32% of the total variance was ac-countable to the main effect to be learned and about 21% of the total was accountable to the interaction to be learned. Both proportions were highly significant, and though the main effect seemed slightly better estimated than the interaction i t was concluded that by the end of the Learning phase subjects had learned both components of the Composite function quite well. In order to test hypothesis H3, two t-tests were performed on 1) the arithmetic difference and 2) a square root transformation of the ari-thmetic difference of subjects' average cell estimate latency (3.32 sec.) and average marginal estimate latency (4.02 sec). Both the t^^^ value (t=2.20, p<.07, df=7) and the t ^ . f f value (p=2.72, p<.03, df=7) were considered sufficiently different to warrant similar t-tests on the difference in average estimate latencies of the p=0.5 cells (3.31 8 6 . sec.) and the p=0.5 marginals (4.05 sec). Similar t values were ob-tained (t d i f f=2.15, p<.08; V d i f f = 2-52, p<.04; df=7) and i t was con-cluded that rig was confirmed, that cell estimates took less time to make than marginal estimates. Hypothesis H^  was tested by computing the between-subject estimation variances of each cell and each marginal then deriving estimates of the sample sizes on which these estimations were based (in the manner des-cribed in Experiment I). These variances and sample sizes are presented in Table XVIII. Required to Estimated Estimated Estimate Variances Sample Size Married Male 1.80 7.6 Married Female 5.80 4.3 Single Male 0.71 32.0 Single Female 5.05 4.9 Married People 1.59 13.9 Single People 1.74 14.2 Males 1.08 21.6 Females 1.70 14.7 Table XVIII. Variances (a ) and Estimated Sample Sizes of Cel? and Marginal Esti-mates (n=8_per estimate; all vari-ances x 10" ) It may be seen in Table XVIII that 3 of the 4 largest estimate variances and smallest sample sizes came from the cells. Also, estimates of the l»=0.5 coll proportions were estimated to have from 3 to 5 times as much variance and from 1/3 to 1/5 the sample size as the p=0.5 marginal pro-portions. Obviously, the small size of the experimental sample on which these statistics were derived precludes any firm statement about hypo-87. theses H^ . But in conjunction with the corresponding results of Experi-ment I, i t was concluded the was--in all likelihood—correct. Transfer. In order to best estimate transfer effects, all appropriate analyses were performed over the 80 posttest trials. Table XIX presents the average additive and average interactive mpr proportions obtained for 0^ , 0^  in the Pretest and the additive and interactive mpr pro-portions for predicted by the sample model presented in chapter 7. Obtained Predicted Si % additive mpr .62 .59 .51 interactive mpr .58 .56 .51 Table XIX. Posttest mpr Proportions for 0, and Q. With 0. Posttest Proportions Predicted From ^ Simple Model It can be seen from Table XIXthat the simple model again under-estimated the degree of transfer of both the main effect and interaction, indicating that the relationship between 0^  and 0^  became stronger from pretest to Posttest. This, in fact, was the case. In the Pretest the the P^-0^ relationship was, like the P^ -O^  and 0^  -Q3 relationships, quite weak (x =3.1 ,x=0.10). In the Posttest the P^ -Q^  relationship strengthened 2 2 considerably (x =60.1 ,X=0.43). This increase was significant (x (0^x 0i4xTrials)=20.7, p<.01; see again, Winer, 1962, p. 629-631). It was 88. therefore concluded that hypothesis H^  was correct. Hypothesis Hg was tested by computing the strength of the Posttest (q-O^ contingencies for the p=0.5 cells and for the p=0.2 and 0.8 cells. The strength of Oq-0^  Posttest contingency for the p=0.2 and 0.8 cells 2 was found to be : X =34.1, or x=0.47; for the p=0.5 cells i t was found 2 to be: X =33.4, or x=0.46. The difference between these strengths was minimal; hypothesis Hg could therefore not be supported. Discussion The major result of this Experiment, that subjects learned the main effect of the composite function faster than the interaction com-ponent, replicates the result of Brehmer (1969) summarized in the Intro-duction. The learning rate of the p=0.2 and p=0.8 contingencies, the estimation latencies, and the estimation variances all supported the hypothesis that learning occurred cell-by-cell and not dimension-by-dimension. The estimates themselves indicated that by the end of the Learning Phase, subjects had learned all four cell contingencies. But the learning results indicated that during the Learning Phase, subjects were responding to the p=0.5 contingencies by generalizing from the p= 0.2 or p=0.8 contingency of the adjacent cell. All these results give evidence to a conceptualization of the judge-ment process sketched in the Introduction of this Experiment. By way of a more cogent exposition, consider a subject who, on trial i is pre-sented with stimulus s. having dimension values (a,b) and required to judge s. on dimension D. The results of Experiment I and II suggest that the flow chart in Figure 5 may capture the essence of the subject's 89. judgement process. Insert Figure 5 about here It should, of course, be recognized at the outset that the flow chart in Figure 5 is only one of a number that may account for the results of Experiment I and II. However, as a heuristic device i t seems worthy of some discussion. The principle features of the model are these: 1) A subject, in making a judgement about s^  must first attend to s^  as the stimulus of concern, 2) He will then address his memory for information regarding the parti-cular relation between s. and D; 3) If this information is retrieved he will respond appropriately, i f i t is not, he will begin searching his memory for D values of "similar" s^  's; 4) If he finds one (or more) of these values he will respond according to some rule (r^, r^), i f he does not, he will presumably guess (rule r 3 ) ; 5) Regardless of how the judgement is made, any feedback concerning the correct D value associated with the stimulus s. will be address-ed to the memory of s.. (Whether or not i t invariably gets into memory is, of course, another matter and one in which type of feed-back obtained, the length of the judgement to feedback interval, etc. would no doubt play an important part.) The judgement process model may be considered in two parts, respec-tively labelled A and B, above and below the dotted line of Figure 5. INPUT s ^ U . b ) 1 Memory ~7K r Attend to. i YES |^ Is D value of s.j known? YES ^ Respond NO > SNAFU * Feedback? NO Memory Is D value of S j = (a' ,b) known? YES > Derive D value by rule r-j YES YES > NO NO . Derive D value by rule J YES NO Derive D value by rule r^ NO Respond Figure 5. Flow Chart of a Possible Model of the Judgement Process. 91. Part A functions to process s^  as a gestalt or configuration; Part B functions to process s^  as the member of a set of configurations which vary in one (or more) attributes. It is important to note that s^  is processed through Part A f i r s t , i t advances to Part B only i f no "solu-tion" can be found at Part A. According to this model, then, generali-zation—the phenomenon produced in Part B--is invoked only as an expedi-ent and is used in the judgement process only until sufficient feedback concerning the "true" D value of s. is available to be captured in long-term memory. The Pretest-Posttest increase in the strength of the O^ -p^  rela-tionship parallels those of the P^ -Qg and -Q^  relationships in Experi-ment II. All three relationships were exceedingly weak in the Pretest, but in Posttest the 0^ -Cj^  relationship was stronger than the 0^ -0^  relationship which in turn was stronger than the 0^ -Qg relationship. The resulting rank order of these Posttest strengths was the same as the rank order of P^ -p^ * Q.-|~P.3 a n c ' P-i'Pg c o r r e l a t i ° n s reported by Osgood et. al_. (1958). This perplexing finding may, interestingly enough, be accounted for by a simple variant of the judgement process model outlined in Figure 6. Consider again a subject required to judge stimulus s.=(a,b) on dimension D. A flow chart of this transfer process is presented in Figure 6. Insert Figure 6 about here The essential features of the model in Figure 6 are these: 1) A subject must first attend to stimulus s.; 2) He will then address his memory for information regarding the particular relation of s. to D; 92. INPUT s.=(a,b) Memory I* YES NO SNAFU NO Memory Is the value of s. on any "similar1 dimensions to D (eg D' ,D") known? NO Derive D value with rule r» Respond Guess Figure 6 . Flow Chart of a Possible Model of The Transfer Process. 93. 3) If this information is retrieved he will respond accordingly; i f i t is not, he will search his memory for s^'s value on dimensions correlated with D (i.e., D', D", etc.), 4) If he finds one or more such values, he will combine them by some rule (r^) to produce a judgement, if he does not find at least one such value, he will presumably guess (randomly) to produce a judgement. 5) Regardless of the means used to produce a judgement, any feedback concerning the D value of s.. will be addressed to the memory of s.. The similarity of the judgement and transfer models seem obvious. They differ only with respect to what is used to judge similarity. The judgement model concerns itself with stimulus comparisons, with compari-sons between targets (stimuli). The transfer model concerns itself with response comparisons, with comparisons within targets. How can this transfer-model explain the Pretest-Posttest rise in the strength of the 0^-0^ relation? Consider a subject in the Pretest who was required to judge the happiness (0^) of some stranger described only by two words: single Female. Recall that this task was designed to be ambiguous, to ensure that subjects would have l i t t l e a priori knowledge about the moods and/or personality of such a sketchy stranger. Assuming this were true (and there is good reason to believe i t was) what might a subject do to judge the single Female's happiness? According to the above model, he would first consult his memory for any information about the happiness of single Females. Failing to retreive this information, he would then search for information about some "similar" feature(s) of single Females, for example, optimism (CO. And failing to retreive 9 4 . this, he would randomly guess. As a result, any a priori relation be-tween happiness and optimism would not reveal itself in the Pretest data. Finally, consider the same subject required to judge the happiness of a single Female in the Posttest. Again, according to the above model, he would first search his memory for information about the happiness of single Females. As in the Pretest, this search should be in vain. Therefore, he would again search for information about some "similar" feature(s) of single Females. In the Posttest this information should be available, since during Learning the subject has presumably learned about the propensity of single Females to be optimistic. Whatever rule (r^) the subject had for deriving a judgement of happiness could therefore be employed and, assuming the rule was some function of the 0^ -0^  rela-tion, the Q-j-p^  relation should now appear in the Posttest data. Thus, the judgement and transfer models, though perhaps not neces-sary to account for the major results of Experiments I and II, appear at least sufficient to do so. Further tests of these models therefore seem in order, in particular tests of hypotheses concerning the nature of the combination rules r-j, . . ., r^. Experiment III was designed to make some of these tests. 95. EXPERIMENT III The additive functions of Experiment I were formed by placing the two p=0.8 (and p=0.2) contingencies in a row or column of the outcome matrix, and the Interactive functions of Experiment I were formed by placing the two p=0.8 (and p=0.2) contingencies in a diagonal of the outcome matrix. Together, these functions exhausted all possible eel 1 -allocations of the p=0.8 and p=0.2 contingencies. The composite func-tions of Experiment II were formed by placing the two p=0.5 contingencies in a row or column of the outcome matrix. These were not exhaustive of all possible cell allocations of the p=0.5 contingencies. Four more allocations remained (two from placing the p=0.5 cells in the major dia-gonal and two from placing them in the minor diagonal). It was these allocations which formed the functions to be learned in Experiment III. More formally, the four remaining allocations were derived by averaging th A-j and A^ , the A-| and A^, the a n d Ag, and the k^ and A^  functions of Experiment I. The resulting compound functions are shown in Table XX. M F m .8 .5 .65 .5 .8 .65 .5 .2 .35 .2 .5 s .5 .2 .35 .2 .5 .35 .8 .5 .65 .5 .8 .65 .35 .35 .65 .65 .35 .35 .65 A l " A3 A l " A4 A2 " A3 A2 " A 4 Table XX. Probabilities of "yes"to 0^  in the Four Compound Functions. It can be seen from Table XX that in ANOVA terms, the contingencies of each Compound resulted equally from a marital status main effect and sex main effect. The Composite functions of Experiment II were, in one 96. sense, nothing more than the Interactive functions of Experiment I with a main effect "tacked on" to each. Similarly, the Compound functions of this Experiment were nothing more than the Additive functions of Experiment I with an extra main effect "tacked on" to each. This fea-ture made possible a number of cogent comparisons between the results of this Experiment and those of Experiments I and II. What predictions could be made on the basis of Experiments I and II about the learning and transfer of the Compound functions? Since both previous experiments had indicated that functions were learned cell by cell , i t was again hypothesized that contingencies of equivalent "un-certainty" of "signal strength" would be learned at an equivalent rate, regardless of the function in which they were found. More specifically, i t was again hypothesized that, H-j: the p=0.2 and p=0.8 cells of the Compund functions would be learned of the same rate as those of the Composite func-tions of Experiment III. As a corollary to i t was further hypothesized that because the com-pound and composite functions had the same cell entries, H^ : the "overall" learning rate of the Compound functions (the average learning rate of the two main effects) would be the same as the "overall" learning rate of the Composite func-tions (the average learning rate of the main effect and interaction). Predictions equivalent to those of Experiment II were also made regarding the post-learning estimations of the Compounds. Specifically, i t was again hypothesized, that since learning was assumed to be occurring cell 97.. by cell, H3: estimate latencies of the cell proportions would be shorter than the estimate latencies of the marginal proportions; and H^ : estimate variance of the cell proportions would be greater than the estimate variance of the marginal proportions. What tests or refinements of the judgement model outlined in Figure 5 could be derived from the results of Experiment III? Recall that the judgement model assumed that i f the criterion dimension value of a stimulus was not known an attempt would be made to estimate one by "extra polating" from any known criterion dimension values of similar stimuli. No specific predictions were made about the rate at which each main effect of the composites would be learned. But, in terms of the judgement model, a comparison of the learning rates of each main effect could po-tentially reveal much about the nature of the rules (r-j, r 2 and r^) of extrapolation. To illustrate, consider a subject required to learn the A^ -A^  func-tion. If the sex dimension and the marital status dimension were not used to extrapolate from the mM and sF cells to the mF and sM cells, then we would expect, Pn(g|sM) = 0.5 for all n trials. In this case the mpr of the sex main effect (the sex mpr) and the mpr of the marital status main effect (the marital status mpr) would increase at the same rate. Similarly, i f the sex and marital status dimensions were being used to extrapolate but were being weighted equally, then we would report, Pn(g|mF) = Pn(g|sM) 98. for all n trials. Here too, increases in the mpr's of the sex and mari-tal status main effects should be equal. However, i f the sex dimension were weighed more than the marital status dimension when used to extrapolate, then as long as Pn(g|mM)>pn(g|sF), we would expect that Pn(g|sM)>pn(g|mF). As a result, the rate of growth of the sex mpr would be greater than that of the marital status mpr. Conversely, i f the marital status dimension were weighed more than the sex dimension when used to extrapolate then as long as Pn(g|mM)>pn(g|sF), we would expect that Pn(g|mF)>pn(sM). As a result, the rate of growth of the marital status mpr would be greater than that of the sex mpr. Finally, i f on some trials the marital status mpr was greater than the sex mpr but on other trials the sex mpr was greater than the marital status mpr, then we could infer that weights given to each dimension were changing over time. What tests or refinements of the transfer model outlined in Figure 6 could be derived from the results of Experiment III? As in Experiment II, subjects in this Experiment were required to judge the answers to 0^  and 0^  during the Pretest and Posttest. According to the model, there-fore, we should predict that, Hg: in the pretest no significant relationship should exist between the judgements of 0^  and 0^  answers. 99. Furthermore, since the model predicted that any significant a priori pVj-O^ relationship would begin to appear in the data as the contingencies of the function mapping the stimuli into 0^  answers were learned, and since the model assumed nothing about the nature of this function, i t was also hypothesized that, Hg: in the Posttest the strength of the Q.1-Q4 relationship would be equal to that obtained in the Posttest of Experiment II. Method Subjects. Eight paid volunteers (4 males and 4 females) from two social psychology courses served as subjects. Though the male female ratio in this Experiment was less than that of Experiment II, i t was felt that for the task undertaken these groups were comparable. Each was required to learn one of the 4 compound functions (2 subjects per function) in the same manner as subjects in Experiment II. Materials. With the obvious exception of the different proportions of cards used to construct the 40-card decks to represent each compound all materials in Experiment III were identical to those of Experiments I and II. Procedure. Since Experiment III differed from Experiment II only in the nature of the function to be learned, the procedure of the former was identical to the procedure of the latter. Results To facilitate analyses the appropriate row-row, or column-column transformation(s) was performed on the A^ -A^ , Ag-A^  and Ag-A^  compound functions to make them appear as the A,-A_ compound functions. Since no TOO. row-column transformations were performed, the marital status, and sex dimensions maintained these marginal proportions and hence, unlike the composites of Experiment II, made the learning and estimation of each main effect easily comparable. A summary of the average number of yes responses given by subjects to the four stimulus persons during the 80-trial Pretest and during each of the five 40-trial learning blocks is presented in Table XXI. Required to Learn Pretest Learning Block 1 Block 2 Block 3 .80 .50 .50 .20 Block 4 .80 .41 .79 .14 .61 .49 .70 .56 Block 5 .92 .50 .74 .14 .58 .71 .46 .42 .68 .49 .50 .29 .71 .35 .71 .20 Table XXI. Average Proportion of "Yes" Responses Given by Subjects in the Pretest and in Each of the Five Learning Blocks In order to determine any difference in the learning rates of the compound and composite functions, the average mpr index of the 0.2 and 0.8 cells of the compoundwas first computed for the Pretest and for each Learning trial block. A graph of these indices is presented in Figure 7, along with a graph of the corresponding indices of the Composite func-tions. Insert Figure 7 about here A 2 x 5 (function-by-trial block) ANOVA was then performed on these indices, as a simultaneous test of hypotheses and r ^ . A summary of this analysis is given in Table XXII 101. cu o a . </> cu S-s_ CL o Q -Q J o 1.00, 0.90 0.80J 0.70 0.60 0.501 0.40J Pretest Figure 7. Composite (Exp. II) Compound 2 3 Learning Blocks Average mpr Proportions of Subjects Learning p=0.2 and p=0.8 Cells of the Compound Functions. 102. Source SS df MS . F Between 504 15 Function Error 3.2 500.8 1 14 3.2 35.77 Within 865 64 Trials FXT Error 433.4 3.2 428.4 4 4 56 108.35 0.8 7.65 14.2 (p<.01) Total 1369 79 Table XXII. ANOVA Summary Table of mpr Proportions in the p=0.2 and 0.8 Cells of the Compound and Compo-site Functions. It may be seen in Table XXII that though there was a highly signi-ficant learning effect, there was no significant difference in the rate at which the p=0.2 and 0.8 cells of the compound and composite functions were learned. Because of this, hypothesis H-| could not be rejected. And since the "overall" mpr index of the compound and composite functions, as defined in hypothesis H2, was equal to a linear transformation of the mpr index of the p=0.2 and 0.8 cells; specifically since, mpr ("overall") = 0.5 mpr (0.2 and 0.8 cells) + 0.5, it was concluded that H2 could not be rejected as well. In order to test for differences in the learning rates of the marital status main effect and the sex main effect within the compound functions as marital status mpr and a sex mpr was first computed for each subject from their responses in the Pretest and in each learning block. The average marital status mpr's and average sex mpr's are 103. shown in Figure 8. Insert Figure 8 about here It can be seen from Figure 8 that the marital status mpr rose more quickly than the sex mpr during the first learning block but then de-creased in the second and third learning blocks while the sex mpr in-creased sharply. A 2 x 5 ANOVA (dimension-by-trial blocks) was performed to determine i f the difference in learning rates was significant. It is summarized in Table XXIII. Source SS df MS F Between 345 7 Within 1034 9 Dimensions 183 1 183 1.74 (n.s.) Trials 379 4 94.8 5.96 (p<.01) D X T 472 4 118.0 4.92 (p<.01) Error 1856 63 D x s.w.g. 737 7 105.3 T x s.w.g. 446 28 15.9 D x T x s.w.g. 673 28 24.0 Total 3235 79 Table XXIII. ANOVA Summary Table of the Learning of Compound Function Main Effects Table XXIII shows that both the overall learning rate of the Compound func-tion and the dimensions-by-trials interaction were significant. The in-teraction seemed particularly interesting, for as mentioned in the Intro-duction i t indicated that the marital status dimension was being weighted L O G S 0 . 9 0 J 0.804 Pretest 1 2 3 4 5 Learning Blocks Figure 8. Learning Curves of the Compound Function and Its Components 105. more heavily than the sex dimension early in learning but less heavily than the sex dimension later in learning. The average cell and marginal estimates of the compound functions are presented in Table XXIV. Required to estimate Actual Estimates married males .80 .69 married females .50 .49 single males .50 .52 single females .20 .25 married people .65 .59 single people .35 .39 males .65 .61 females .35 .37 Table XXIV. Average Cell and Marginal Estimates of the Compound Functions. In order to determine the sources of variability in the cell estimates of the compoundsa 2 x 2 (marital status-by-sex) ANOVA was performed. A summary of this analysis is presented in Table XXV. 106. Source SS df F i P Between 1559 7 Within 11826 24 Marital Status 3301 1 10.97 p<.02 M.S. x s.w.g. 2105 7 Sex 4348 1 47.88 p< .01 Sex x s.w.g. 636 7 M.S. x Sex 116 1 M.S. x Sex x s.w.g. 1320 7 Total 13385 31 Table XXV. ANOVA Summary Table of Cell Estimates of the Com-pound Function. It can be seen from Table XXV that the cell estimates resulted from a significant marital status main effect and a significant sex main effect. Both main effects of the compound functions, therefore, seemed to be well learned. The average latencies of estimating a cell of the compound functions and a marginal of the compound functions were computed for each subject. Over all subjects the average cell estimation latency was 3.71 seconds and the average marginal estimation latency was 4.68 seconds. A t-test between the latencies (and between a square root transformation of the latencies) proved to be marginally significant ( t ^ . 1 . 8 8 , p=0.10; V d i f f ~ ^ * ^ ' P=0.05; df=7). A t-test between combined cell estimation latencies of the compound and composite functions and the combined margi-nal estimation latencies of the compound and composite functions proved 107. to be highly significant (t=2.91, p<.01, df=15). It was therefore con-cluded that hypothesis was correct. In order to test hypotheses H^ , between subject estimate variances of each cell and each marginal of the compounds were computed. The sample size of each estimate was then estimated in the manner described in Experiment I. Both the variances and the estimated sample sizes appear in Table XXVI. Required to Estimate: Variance of Estimates Estimated Sample Size married male 9.8 12.7 married female 14.0 11.9 single male 18.1 9.9 single female 10.2 12.5 married people 2.4 38.4 single people 1.9 68.8 males 5.0 9.4 females 3.2 24.6 Table XXVI. Variances (a ) and Estimated Sample Sizes of Cell and Marginal Estimates (n=8 per estimate; all variances x IO"2) It may be seen from Table XXVI that all 4 cell variances were larger than the marginal variances and 3 of 4 cell sample sizes were larger than the marginal sample sizes. Since all three experiments attempted to test the notion that the cell estimates would have higher variances and hence, be based on smaller sample sizes than the marginal vari-ances, i t was decided to test by pooling the estimate data from Ex-periments I, II and III. The grand median sample size over all three Experiments was 14.0. Thirteen of the 16 cell estimates were based on 108. sample sizes estimated to be greater than this median; 13 of the 16 margi-nal estimates were based on sample sizes estimated to be smaller than 2 this median. The difference was found to be significant (x =12.5, p< .001, df=l). It was therefore concluded that was correct. Hypothesis Hg was tested by computing the strength of association between responses to 0^  and to during the Pretest. No significant 2 association was found (p=0.08; x =1.86, association between and 0^  in the Pretest could therefore not be rejected. Hypothesis Hg was tested by computing the strength of association between responses to 0^  and 0^  in the Posttest. This association was found to be highly significant (p=0.35; x =40.19, p<.001, df=l). The increase in the strength of asso-ciation from Pretest to Posttest was also found to be highly significant (x2=Q_x 0^  x Trials=12.09, p<.001, df=l). Finally, no significant dif-ference was found between the strength of the Posttest Pq-Q^ relation in Experiments II and III. The hypothesis that the strength of the Pq-Q^ relationship in the Posttest of Experiment III would be equal to that of Experiment II could not, therefore, be rejected. Discussion As mentioned in the Introduction, the Compound functions of Experi-ment III and the Composite functions of Experiment II were, in one sense, nothing more than the Additive and Interactive functions of Experiment I with an extra main effect "tacked on" to each. The lack of a significant difference in the learning rates of the compound and composite functions was, therefore, analogous to the lack of a significant difference in the learning rates of the Additive and Interactive functions. Because of 109. this, the learning results of Experiments II and III when examined both separately and together gave support to the notion that the functions were being learned cell-by-cell and not dimension-by-dimension. Of course the estimation latencies and variances also supported this notion. The cell estimate latencies of the functions in Experiment III were, like those of Experiments I and II, shorter than the marginal estimate latencies. And the variances of cell estimates in Experiment III were, like those in Experiments I and II, larger than the variances of marginal estimates. Both the latency and the variance differences were found to be highly significant when the relevant data were pooled from all three experiments. The significant dimensions-by-trials interaction found to occur in the learning phase of Experiment III further supported the notion that some form of extrapolation was being made from the p=0.8 and p=0.2 con-tingencies to the p=0.5 contingencies. The interaction itself could not be immediately explained. However, i t was noted that early in learn-ing greater weight was given to the marital status cue--the first cue read to subjects on each trial--while later in learning greater weight was given to the sex cue--the second cue read to subjects on each t r i a l . The similarity of sequences may have been coincidental, but i t may also have indicated that different extrapolations were being tried and were being tried in some orderly "left-to-right" or "first-cue-to-last-cue" fashion. Such a hypothesis would, at least, be worthy of some future test. The significant dimensions-by-trials interaction also indicated a tendency to use only one cue at a time in attempting some extrapolation. n o . The Compound functions themselves were the result of equally weighted main effects. However, the responses to these functions revealed a tendency to weigh the main effects unequally, first by preferring the marital status dimension and later by preferring the sex dimension. Of course, the expected number of correct guesses in the compound functions was not affected by this unequal weighing; had i t been, an equal weigh-ing might have resulted. Numerous explanations of this differential weighing could be postulated; i t might indicate, for example, that a lexicographic rather than additive extrapolation process was being employed. The phenomenon therefore, seemed worthy of attempts at re-plication, since more data was needed to eliminate some of the possible explanations. Despite the failure of Experiment III to uncover a simplistic ac-count of the extrapolation process postulated to exist in the judgement model outlined in Figure 5, the model itself could certainly not be re-jected. The increase from Pretest to Posttest in the strength of the Oq-Q^  relationship of Experiment III to a level insignificantly diffe-rent from that of Experiment II gave support to the transfer model out-lined in Figure 6. And, as mentioned previously, the judgement and transfer models were structurally similar. There was, therefore, good reason to believe that the structure of the judgement model was like-wise supported. Obviously no alternative judgement or transfer models were proposed, aside from those rejected in Experiment I. Conclusions about the adequacy of these judgement and transfer models, therefore, had to remain tentative. Together they could rationalize at least the gross aspects of the data in all three experiments. Together they had at least some intuitive appeal. These criteria seemed sufficient for a second look at the judgement literature and in particular a second 1 at the additive-configural controversy. 1 1 2 . , SUMMARY AND SPECULATION Subjects in all three experiments were required to learn the response contingencies of four types of stimulus persons each described by a pair of cues representing values on two dichotomous dimensions. The results of Experiment I supported the notion that these four contingencies were each being learned as a separate unit or "gestalt" and, in contrast, were not being learned by independent associations of their dimensional com-ponents. Thus, for example, the optimism of a married female was found to be judged by first recalling the optimism of married females rather than by first recalling separately the optimism of married people and the optimism of females. The results of Experiments II and III further supported this notion and, in addition, supported the notion that judge-ments of stimuli whose response contingencies had not yet been learned were affected by similar stimuli whose response contingencies had been previously learned and by similar responses whose contingencies with the stimuli had been previously learned. Thus, for example, i f a subject were required to judge the optimism of a married female but could not recall the optimism of married females but could recall the optimism of married males or single females, then he would tend to base his judgement upon the knowledge of at least one of these similar stimuli. Or, for example, if a subject were required to judge the happiness of a married female but could not recall the happiness of married females but could recall the optimism of married females, then he would tend to base his judgement upon the knowledge of this similar response. In brief the re-sults of the three experiments gave support for (or, at least, could be rationalized by) the model outlined in Figure 9. 113. Insert Figure 9 about here The generality of this model could, of course, be questioned. It was derived from a situation requiring judgements of only four different types of stimuli. Had a greater number of different stimulus types or configurations been employed, for example, a number greater than 7-2 (see Miller, 1956), subjects may have abandoned any cell-by-cell learning procedure and adopted a dimension-by-dimension one. But i f the dimensions or dimension values had been increased beyond such a magic number would a dimension-by-dimension learning procedure be abandoned for some third alternative? And i f so, what would the alternative be? It would seem equally feasible to imagine that, as the number of cells proliferated, a subject would not abandon a cell-by-cell learning procedure but would instead simply resign himself to learning the criterion values of a few cells and judge the remainder by some form of extrapolation. Future re-search should resolve some of these issues. Assuming for the moment that the model outlined in Figure 9 does have some validity and generality, how can i t account for the large pro-portion of judgement variance reported in the Introduction to be due to stimulus main effects? Recall that the model assumes: (a) that stimuli are associated with criterion values as gestalts or configurations, (b) that judgements of stimuli whose criterion values are known are made "from memory," but (c) that judgements of stimuli whose criterion values are not  known are made by some form of extrapolation from those which 114. Stimulus Memory 3 1 I YES Is the response to this stimu-lus knwon? YES *| Respond NO * SNAFU Feedback? Memory T 4, Is a similar response to this stimulus known? Memory t i Is the response of a similar stimulus known? YES Extrapolate and Respond * Guess YES xtrapolate and Respond NO * Guess Figure 9. Outline of a Model of the Judgement and Transfer Process 115. are known. Therefore, according to the model judgements which were accountable to main effects would occur for two reasons: (1) they would occur i f all the stimulus criterion values were known, and i f these criterion values, in fact, were account-able to main effects (from assumption b above); or (2) they would occur i f only some of the stimulus criterion values were known and i f the extrapolation rule used to judge the remainder were additive (from assumption c above). More philosophically, additivity of judgements may reflect complete know-ledge of an additive "world" or incomplete knowledge of a world which could be either additive or interactive. The first explanation—though possible—seems improbable, especially i f i t is believed that most situ-ations to be judged are at least partially interactive and are sufficient-ly "rich" or complex to prevent complete knowledge of all contingencies within them. The second explanation seems more reasonable but needs closer examination. Consider an "expert" judge who, as a member of a departmental gradu-ate admissions committee, is required to evaluate the graduate potential of applicants described by 7 indices: two Graduate Record Exam scores (verbal and quantitative), a Miller Analogies Exam score, a grade point average, and three letters of recommendation. Assuming that he can make 7 distinctions on each of these indices, 7 7 or 823,543 distinguishable stimuli could potentially come before his scrutiny. But the indices would undoubtedly be correlated and many potential applicants with low scores on all indices would probably not apply. He should, therefore, see some 116. configurations of indices more often than others, in particular from the large 7 dimensional matrix of possible applicants he should see primarily those from the "upper" corner. If our judge is typical, he will reject most of the applicants, and unless the rejects write him of his great insight or foolishness (highly unlikely) he will not obtain feedback about the correctness of these decisions. Of those he accepts, about half will reject him, and, like the original rejects, will probably not keep him informed as to their future success or failure. The residue will presumably come, but each will be expected to have highly similar index configurations. A year later some may excell, some may flunk and some may quit. If the judge is conscientious and has a good memory, he may recall the indices of the good and the bad and, according to the model outlined in Figure 9 learn to associate their success with their configurations. As a result, of the original 823,543 possible configurations the judge might come to associate criterion values with perhaps only 10 or 15 of them, i f that. If he were ever confronted with these same 10 or 15 configurations again he could hopefully produce "corrected" judgements which would reflect any and all of the interactions inherent in the function actually mapping each of these index configurations into the criterion dimension. If for example, he were allowed to accept or reject the same 10 or 15 students again, both his accuracy and his "configura-l i t y " (assuming i t actually existed in nature's function) would no doubt greatly increase. Such is the value of hindsight. The following year he may be confronted with the same judgemental situation. Some of the new applicants may have identical configurations of indices to those who came 117. before; i f so, the model would predict that they would be judged as such; if not, the model would predict they would be compared to the known con-figurations and a decision reached by some form of extrapolation. At the end of this second year, therefore, the judge may at best come to have knowledge of 5 or 10 or 15 additional configurations. Duplicate configu-rations may, of course, not have duplicate success—important indices may be missing from the 7 he is given. If this were so, the judge would again have to extrapolate (though within a configuration rather than be-tween configurations) or give up in frustration. But assume that the judge persists and over the years begins to accumulate, little-by-little, information about a set of popular confi-gurations. This information would make his task easy. Those configura-tions which often repeat themselves could be judged from memory, thus avoiding all cumbersome interpolation techniques. In the great tradition of the Viennese clinician he might often exclaim, "Ah, I have seen this type before!" It is solid clinical wisdom. If a new configuration of indices confronted him, he might attempt some fancy interpolation, one which would preserve much of any configurality of his judgements. "Ah!," he might exclaim, "Indices A, B and C have a configuration like those the case of Mr. X who flunked; indices D, E, F and G have a configuration like those of the case of Mr. Y who was outstanding; by majority rule I will accept him." Any A x B x C and the D x E x F x G interactions which the judge has learned would thus remain intact, and he could quite validly maintain some claim to being a configural processor. For he would have, in a very real sense, a "pocket" of knowledge, learned re-sponses to a small corner of the original 823,543 cell matrix. And to 118. the extent that the function relating the cells in this pocket to actual success were interactive, so too would be his judgement process. Enter the judgemental researcher. If he attempted to determine the judge's evaluation process he would no doubt want to use the most sophisticated design available; a fractional replicated factorial design would do nicely. Configurations of indices would be strategically select-ed to equally represent all cells of the 7 dimensional matrix. They would, of course, not be representative of the judge's ecology (see Brunswik, 1956), but the design would have great statistical power. If the researcher selected, say, 100 configurations or cells to be judged, only a few would be expected to fall within the pocket of the judge's memory. These few could, of course, be judged as configurations. But what of the others? It would seem reasonable to assume that the further removed each cell was from those within the judge's knowledge, the greater would be the tendency to analyze each index value of the cell independently. The extrapolation process might, indeed, appear quite Coombsian (Coombs, 1964) with the judge's pocket of knowledge acting as an "ideal point" or reference point around which judgements of the outer cells were unfolded. If this were the case, additivity should predominate. An analysis of variance of the judgements would then, be expected to show that the judgement variance was predominantly due to main effects. Q. E. D. Such is one rationalization for the major finding of much judge-mental research which is implied by the model outlined in Figure 9. Perhaps the rationalization appears far-fetched (even though i t appears perfectly reasonable to the author). However, i t at least points to the 119. need for considering two characteristics of the judge: (1) the judge's rule of extrapolation or interpolation, and (2) the judge's knowledge of the judgemental situation. The first characteristic has been the major focus of most judge-mental research to date. The second has unfortunately received far less attention, and that which i t has received has been primarily concerned with judgemental accuracy and not with the judgement process itself (see Goldberg, 1968). But the model implies that the judgement process can-not be fully understood unless both characteristics are examined. Much is known about how different extrapolation rules can influence the additivity of judgements (e.g., Coombs, 1964; Tversky, 1969). Little is known about how the additivity of judgements is affected by the number and relationship of the cells with known criterion which form the basis of extrapolation. It can be shown that for any given extrapolation or interpolation rule the number and relationship of these cells will in fact influence the additivity of judgements (Thorngate, 1971). It may also be true that the number and relationship of these cells will actually influence the extrapolation rule itself. Experimental investi-gations of the effects of both the quantity and quality of knowledge about the judgemental situation upon both the extrapolation rule and resulting judgement process are currently being conducted by the author. 120. REFERENCES Anderson, M.H. Application of an additive model to impression forma-tion. Science, 1962, 138, 817-818. Anderson, N.H. Averaging versus adding as a stimulus-combination rule in impression formation. Journal of Experimental Psychology, 1965, 70, 394-400. Anderson, N.H. A simple model for information integration. In R.P. Abelson, E. Aronson, W.J. McGuire, T.M. Newcomb, M.J. Rosenberg, and P.H. 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Journal of Experi- mental Psychology, 1966, 71_, 751-757. Summers, S.A., Summers, R. and Karkau, V. Judgments based on different functional relationships between interacting cues and a criterion. The American Journal of Psychology, 1969, 82_, 203-211. Szaaz, T. The myth of mental illness. American Psychologist, 1960, 15, 113-118. Taguiri, R. Person Perception. In Lindsey, G. and Aronson, E. (Ed.s), Handbook of Social Psychology, Vol. I l l , (2nd Edition), Mass.: Addison-Wesley Publishing Co., 1968. Thorngate, W. Facial cues and judgements of emotion. Tech. Report No. 71-2, Department of Psychology, University of Alberta, 1971. Thorngate, W. Judging multi-cue stimuli by extrapolation: The influence of the number and relationship of reference points upon the judge-ment process. Paper in preparation, University of Alberta, 1971. 127. Tversky, A. Intransitivity of preferences. Psychological Review, 1969, 76, 31-48. Wallach, M.A. On psychological similarity. Psychological Review, 1958, 65_, 103-116. Winer, B.J. Statistical principles in experimental design, New York: McGraw H i l l , 1962. Wishner, J. Reanalysis of "impressions of personality." Psychological Review, 1960, 67, 96-112. Yntema, D.B. and Torgerson, W. Man-computer cooperation in decisions requiring common sense. I.R.E. Transations on human factors in electronics, HFE-2 (1961), 20-26. 128. APPENDIX In order to illustrate various additive and configural processes, consider a judge required to evaluate the quality of a term paper given two cues: 1) the number of pages of the paper and 2) the number of references in its bibliography. Assume that the evaluation is given by a percentage grade so that for each of a set of these, T_{t^j, tg, • • •, t j , . . ., t^} the grade given t. will have a range, 0<g(t.j)<100. Assume also that the number of pages, p, and the number of references, r, will each range from 0 to 50, that i s , that: 0<p..<30, and 0<r..<30. Below are listed some possible process he can employ, an equation exempli-fying each, and an illustrative tabular example. process equation tabular example 1inear additve (w =.70, wr=.50) g(ti)=.70(pi+10)+.50(ri+20) 10 20 30 10 29 34 39 p 20 36 41 46 30 43 48 53 36 41 46 34 41 48 linear averagm (w =.70, w =.30 9(ti)=.70(pi+10)+.30(ri+20) 10 r 20 30 10 23 26 29 26 p 20 30 33 36 33 30 37 40 43 40 30 33 36 129. p r o c e s s e q u a t i o n t a b u l a r example n o n l i n e a r a d d i t i v e g ( t i ) = 5 ( p ^ + l ) + 5 ( r | + l ) (w =.50, w r=.50) C o n f i g u r a l ( m u l t i -p l i c a t i v e ) g ( t i ) = ( p i - i o ) ( r r i o ) C o n f i g u r a l ( l e x i c o - g ( t . ) = 0 , i f p ^ l 5 g r a p h i c ) g ( t . Hp^+r., I f p>15 10 20 30 10 41 48 53 47 20 48 55 60 54 30 53 60 65 59 47 54 59 10 r 20 30 10 0 2 4 3 20 2 20 40 21 30 4 40 80 41 3 21 41 10 r 20 30 10 0 0 0 0 20 30 40 50 40 30 40 50 60 50 23 30 37 

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