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Investigation of the psychophysical relationship of kinesthetic extent of arm movement Ryan, Martha Lorraine 1971

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AN INVESTIGATION OF THE PSYCHOPHYSICAL RELATIONSHIP OF KINESTHETIC EXTENT OF ARM MOVEMENT by MARTHA LORRAINE RYAN B . P . E . , THE UNIVERSITY OF BR IT I SH COLUMBIA A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF PHYSICAL EDUCATION i n t h e S c h o o l o f P h y s i c a l E d u c a t i o n and R e c r e a t i o n 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 t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF.BRIT ISH COLUMBIA A u g u s t , 1 9 7 1 In 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 the r e q u i r e m e n t s f o r an advanced deg ree a 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 , 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 and s t u d y . I f u r t h e r a g r ee 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 be g r a n t e d by the Head o f my Depar tment o r by 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 not be 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 . Depar tment o f 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, Canada Da te ABSTRACT Forty, volunteer, University of B r i t i s h Columbia, Physical Education students took part i n a study to deter-mine the relationship between the physical stimulus con-tinuum and the psychological continuum of kinesthetic extent of arm movement. The subjects were randomly assigned to one of four conditions. The conditions were chosen to provide tests of three functional c r i t e r i a , outlined by S. S. Stevens (1957), for differentiating between two classes of continua, prothetic and metathetic, into which the majority of sensory modalities naturally f a l l . The task for a l l conditions involved a straight arm movement from the shoulder joint, i n a horizontal plane, towards the midline of the body. Condition I involved the psychophysical ratio scaling method of fractionation and from these data the subjective Kine function, Kine = .1010 S " , for kinesthetic extent of movement was derived. Condition II employed the psychophysical category production method to derive the category scale for kinesthetic extent of movement, which was found to be linear when plotted against the subjective Kine scale values. In Conditions III and IV, the psychophysical ratio scaling method of fraction-ation v/as used and i t was determined that the hysteresis effect was not present for kinesthetic extent of movement. The individual results from each condition supported the hypothesis that kinesthetic extent of arm movement is' representative of the metathetic class. Therefore, the general conclusion, determined from a synthesis of the three tested functional c r i t e r i a , was that one attribute of kinesthesis: extent of arm movement, i s subserved by a metathetic process. i ACKNOV/LEDGMENT To the members of my committee, Dr. R. G. Marteniuk, Dr. K. D. Coutts, Dr. W. G. Davenport, and Mr. R. W. Schutz, I would l i k e to express my .appreciation f o r t h e i r guidance throughout the preparation of t h i s study. A s p e c i a l thanks to my advisor, Dr. Marteniuk f o r h i s continued assistance and encouragement during my academic program. i i TABLE OF CONTENTS CHAPTER PAGE I Statement of the Problem 1 Introduction . . . . 1 The Problem 2 Subproblems 2 D e f i n i t i o n of Terms 3 Hypothesis 5 Limitations 6 Delimitations 6 II Review of the L i t e r a t u r e 7 Psychophysical Laws 7 Functional C r i t e r i a . . . . 9 Methods of Constructing Ratio Scales of Subjective Magnitude . 11 Empirical Evidence of the Prot h e t i c Continuum 12 Empirical Evidence of the Metathetic Continuum 13 Research Related to the Kinesthetic Sense 15 P h y s i o l o g i c a l Evidence Related to Prothetic and Metathetic Continua . . . 17 C o n f l i c t i n g Views 20 Summary 24-I I I Methods and Procedures 25 Subject 25 Apparatus 25 Experimental Conditions 25 i i i CHAPTER Page Condition I 25 Condition II 26 Condition III and IV 26 Procedure 2? St a t i s t i c a l Analysis 28 Condition I 29 Condition II 30 Condition III and IV 31 IV Results and Discussion 33 Results 33 Determining the Kine Function . . . 33 The Category Scale 4-0 The Category Scale Plotted Against the Subjective Kine Scale . . . . 4-4-The Hysteresis Effect 4-7 Discussion., 52 The Kine Function 52 The Category Scale 54-The Category Scale Versus the Kine Scale 55 The Hysteresis Effect 57 A Synthesis of the Individual Results 58 V Summary and Conclusions 61 Recommendations 63 References 64-Appendices A. Calculation of the Kine Scale . 67 B. Individual Geometric Mean Responses 69 i v LIST OF TABLES PAGE I The Group Median Responses Corresponding to Each Stimulus in Degrees, for Condition I.. 34 II Analysis of Variance Table of Log S and Log R Magnitudes 35 III Analysis of Variance Table of S and R Magnitudes 38 IV The Relationship of Psychological Extent of Arm Movement i n Kines to Physical Distance i n Degrees 39 V Analysis of Variance Table of Log S Versus Log Kine 40 VI The Median Responses of the Group for the Category Production Task 42 VII Analysis of Variance Table of Stimulus Categories Versus Median Response Categories 44 VIII A Table of the Subjective Category Scale with the Corresponding Kine Value 45 IX Analysis of Variance Table of the Category Scale versus the Kine Scale 47 X Median Responses for Group III and Group IV Corresponding to the Stimuli i n Degrees. . . . 48 XI Analysis of Variance Table for Two Conditions, Ten Subjects, and Nine Categories of Con-ditions III and IV 50 V LIST OF FIGURES page 1, Graphic Representation of the Relationship ©f half-distance judgments of stimuli distances on log-log coordinates 36 2 Graphic representation of the relationship of half-distance judgments of stimuli distances 1 on linear-linear coordinates . . . 37 3 The Relation of log Kine to log S for Data ' derived by halving 41 4 A Graph of the Median Response Categories on the ordinate against the nine equally-spaced categories on the abscissa 43 5 Graph of the Category Scale plotted against the Scale of Subjective Movement i n Kines. . 46 6 Hysteresis Effect - Graph of Median Re-sponses for Conditions III and IV plotted on the ordinate against the physical stimuli on the abscissa 49 CHAPTER I STATEMENT OF THE PROBLEM Introduction Two psychological laws that purport to explain the relationship between the intensities of stimuli on the physical continuum and their perceived psychological inten-s i t i e s are Eechner's Logarithmic Law (Howes, 1966) and Stevens' Power Law (Stevens, 1957) . Fechner's logarithmic function implies that equal stimulus ratios produce equal sensation differences. Stevens' power function implies that equal stimulus ratios produce equal subjective ratios and i t i s this type of relationship that has been found i n most recent investigations (Stevens 1957, Stevens and Galanter 1957, Stevens 1961, Hoff 1966, Wood 1969). However, despite this fact, Stevens (1957) has described two fundamental classes of continua into which the majority of sensory mod-a l i t i e s naturally f a l l . The class distinction i s due prim-a r i l y to differences i n the nature of discrimination over a range of perception. The Class I or prothetie are quanti-tative i n nature and seem to answer the question 'how much?* whereas the Class II or metathetic are qualitative and seem to answer the question 'what kind?' or 'where?'. Prothetic continua are thought to be additive at the physiological .level -2-whereas metathetic continua are thought to be substitutive. The psychological scale for Class I may be represented by Stevens' power function. A linear function or a power function with an exponent of approximately one usually represents the psychological scale for Class II. The present study was an attempt to determine the relationship between physical movement and the perception of that movement. By applying Stevens' techniques (Stevens, 1957) to the kinesthetic stimuli and responses i t i s hoped that the relationship between the two continua can be de-termined. Further, this method w i l l allow some implications to be made about the basic physiological mechanisms under-lying kinesthetic judgments. The Problem The problem of this investigation was to determine the relationship between a physical stimulus continuum and the interpreted psychological continuum for one attribute of the kinesthetic sense: extent of arm movement. Subproblems 1. To derive the subjective Kine scale function from the ratio scaling method of fractionation. 2. To derive a category scale for extent of arm movement by the category scaling technique of category pro-duction. 3. To determine the relationship between the ratio and category scales. 4. To determine i f a hysteresis effect i s present for kinesthetic extent of arm movement. Definition of Terms . Kinesthesis - a term describing the sense of movement, de-rived from two Greek words, meaning "to move" and "sensation". The sensations from joint receptors provide awareness of limb positions, rate of movement, -extent of movement, and direction. (Adams, 1968) Absolute Threshold - denotes either stimulus (lower) thres-hold or terminal threshold. The former i s the stim-ulus value that evokes a positive response denoting recognition or sensation for 50$ of the t r i a l s . The la t t e r , the higher l i m i t of sensation, i s generally considered the point beyond which the stimulus i s no longer sensed. Difference limen of Just Noticeable Difference (J.N.D.) - i s the quantity of stimulus increase or decrease which w i l l be correctly noticed on 50$ of the t r i a l s . The J.N.D. i s a measure of keenness of discrimination. Continuum - a closely graded series, one step merging im-perceptibly into the next, the whole forming a line signifying changes i n a single direction. (Guilford, 1954) Prothetic - one of two general classes into which perceptual continua divide themselves. - an additive process on which discrimination i s based at the physiological l e v e l . - an example i s l o u d n e s s , where p r o g r e s s i o n i s made a l o n g t h e c o n t i n u u m b y a d d i n g e x c i t a t i o n t o e x c i t -a t i o n . ( S t e v e n s , 1957) M e t a t h e t i c - one o f t h e two g e n e r a l c l a s s e s i n t o w h i c h p e r -c e p t u a l c o n t i n u a d i v i d e t h e m s e l v e s . - a s u b s t i t u t i v e p r o c e s s on w h i c h d i s c r i m i n a t i o n i s b a s e d a t t h e p h y s i o l o g i c a l l e v e l . - an example i s p i t c h , where p r o g r e s s i o n i s made a l o n g t h e c o n t i n u u m b y s u b s t i t u t i n g e x c i t a t i o n f o r e x c i t a t i o n . ( S t e v e n s , 1957) S c a l i n g - method o f d e t e r m i n i n g i n f o r m a t i o n abou t t h e p s y c h -o l o g i c a l a s p e c t s o f a s e n s o r y m o d a l i t y t h r o u g h o u t t h e e x t e n t o f i t s r a n g e . P s y c h o l o g i c a l s c a l i n g t e c h -n i q u e s e x p r e s s t h e i r s c a l e v a l u e i n t e r m s o f t h e p s y c h o l o g i c a l d i m e n s i o n . ( G u i l f o r d , 1954) R a t i o P r o d u c t i o n - a p s y c h o p h y s i c a l method o f s c a l i n g i n w h i c h t h e s u b j e c t i s p r e s e n t e d w i t h a s t i m u l u s and i s r e -q u i r e d t o s e t h i s r e s p o n s e as a r a t i o , e . g . o n e - h a l f o r d o u b l e t h e s t i m u l u s . A s u b j e c t i v e r a t i o s c a l e may be d e t e r m i n e d f r om t h e s t i m u l u s and r e s p o n s e v a l u e s . C a t e g o r y P r o d u c t i o n - a p s y c h o p h y s i c a l s c a l i n g method i n w h i c h t h e s u b j e c t i s p r e s e n t e d w i t h s t i m u l u s c a t e g o r i e s , s u c h as 1 and 9 , and i s r e q u i r e d t o p r o d u c e a s c a l e o f i n t e r v a l number s , f o r e x a m p l e , t h e c a t e g o r i e s 1 t h r o u g h 9 . Prom t h e s e r e s p o n s e v a l u e s a s u b j e c t i v e c a t e g o r y s c a l e may be d e r i v e d . - 5 -jTractionatiom - one of the psychophysical scaling methods of ratio production i n which the subject i s required to set the stimulus as a fraction, e;g. % of the standard. Standard Movement - the physical stimulus distance presented, to which the subject makes his response. Kine - operationally defined for the present study as the psychological unit for a distance of 10 degrees on the subjective scale of movement, for kinesthetic extent of arm movement. Hysteresis - a lagging behind. It describes what happens when apparent sense distances between successive stimuli are judged i n different ascending and des-cending orders. (Stevens, 1957) Ascending Stimuli - operationally defined as stimuli presented as increasingly longer distances. Descending Stimuli - operationally defined as stimuli pres-ented as increasingly shorter distances. Hypothesis A working hypothesis was developed i n terms of the functional c r i t e r i a , devised by Stevens (1957) , different-i a t i n g Class I or prothetic and Class II or metathetic con-tinua. Several c r i t e r i a must be used since no single test w i l l determine which class a specific sense modality w i l l be-long to. Behavioral studies and physiological research pro-vided evidence which aided i n the formulation of the hypothesis. -6-It was hypothesized that the subjective judgment of extent of arm movement, one attribute of the kinesthetic sense, was characteristic of Class II or the metathetic continuum. Specifically: 1. The Kine function obtained by the ratio scaling tech-nique i s a power function with an exponent of approx-imately one. 2. The function derived from the plot of the subjective category scale against the subjective Kine scale values, determined by the ratio scaling technique, i s linear. 3. The hysteresis effect i s not present when ascending subjective values are plotted against the corres-ponding descending subjective values. Limitations 1. This study was limited to forty volunteer undergrad-uate and graduate physical education students at the University of B r i t i s h Columbia. Delimitations 1. Extent of movement was the only aspect of kinesthesis being investigated. 2. The psychophysical scaling techniques and the dis-tinction between metathetic and prothetic continua, as defined by Stevens (1957). were delimitations of the investigation. CHAPTER II REVIEW OP LITERATURE Early psychophysicists were challenged to define a theory or formula which would explain the relationship be-tween the psychological continuum and the physical continuum for environmental stimuli. Prom the beginning of psycho-physical law, by Weber i n 1830, psychophysics has passed through various stages. There i s s t i l l much controversy i n psychophysics but the most recent developments include the refinement of s t a t i s t i c a l procedures and scaling procedures. Psychophysical methods are being applied i n such problems as: human engineering, considerations of working comfort and efficiency; and advertising, the receptivity of potential customers to certain products. This study attempted through the use of psychophysics, to gain further insight into the kinesthetic sense. Psychophysical Laws The f i r s t actual psychophysical law was Weber's Law, which states that the just noticeable difference increase of a stimulus i s a constant fraction of the stimulus. This was the basis from which Fechner formulated his law i n 1877. Fechner's Law deals with the relationship of -8-sensory intensity to physical stimulus intensity. It i s stated as: S = K log R where S = magnitude of sensation or psychological unit R = magnitude of physical standard K = constant to be empirically determined. Thus a logarithmic increase in the stimulus intensity results i n a linear increase i n sensation. S. S. Stevens, on whom most of the recent psycho-physical information i s based, disagreed with Fechner*s Law, saying that i t definitely did not hold for a l l sensory modal-i t i e s . S. S. Stevens (1957) proposed the relationship bewteen stimulus intensity and psychological sensation approximates a power function. This means that equal stimulus ratios tend to produce equal sensation ratios. Steven's Bower Law i s S = KRn where S = the sensory magnitude, psychological unit R = magnitude of the physical stimulus K = constant to be empirically determined n = exponent This power function relationship i s descriptive of data obtained by the ratio scaling methods. Ekman (1961) said that i n the present stage of psychophysical development we are concerned i n particular with quantifying human exper-ience on a rel a t i v e l y simple l e v e l , for example, measuring subjective brightness or pitch. The measurement technique which i s used in most of the work has been developed by S. S. Stevens over a number of years. In support of his law, Stevens (1961) provides a lengthy l i s t of twenty-two sensory -9-continua and the representative exponents of their power functions, determined by the ratio scaling method. The ex-ponent represents the rate of increase of the function when a physical stimulus value i s plotted against the psycho-logi c a l value. The exponents range from .5 for brightness to 3.5 for ele c t r i c shock. The entire l i s t i s represent-ative of the prothetic continua, because he says the pro-thetic continua always produce a power function by the ratio scaling technique, whereas the metathetic may not. Functional C r i t e r i a According to Stevens (1957) there are several func-tional c r i t e r i a which differentiate between metathetic and prothetic continua and which may be used to test them exper-imentally. 1. The Subjective Size of the Just Noticeable Difference. Fechner proposed that scales can be constructed by counting off J.N.D.'s. Since the J.N.D.'s increase l i n e a r l y as a function of the log of the stimulus, Fechner proposed that equal stimulus ratios correspond to equal sensation d i f f e r -ences. However Stevens (1962) implies that this may be true for metathetic but i n the prothetic continuum equal stimulus^ ratios correspond to equal sensation ratios. /. Class I - J.N.D. not equal i n subjective size. Class II - J.N.D. approximately equal i n subjective size. 2. The Category Rating Scale. This i s obtained when a subject judges a set of stimuli i n terms of a set of cate--10-gories labelled either by numbers .or. adjectives. The category scale for a prothetic continuum i s con-cave downward when the values are plotted against the values from a ratio scale of subjective magnitude whereas for a metathetic continuum the category scale i s less concave and usually linear. The chief factor that produces non l i n e a r i t y i n category scales of Class I i s the variation i n the subject's sen s i t i v i t y to differences. Near the lower end, discrimin-ation i s good, the categories tend to be narrow and by con-sequence the slope of the function i s steep. Near the upper end, where a given stimulus i s less easy to detect, categ-ories broaden and the slope declines. In Class II, where sensitivity, measured i n subjective units, remains rel a t i v e l y constant, i t i s ordinarily possible to produce category scales that are l i n e a r l y related to sub-jective magnitude. 3. Time-Order Error. This refers to the fact that the second of the two equal stimuli tends to be judged greater than the f i r s t . On Class I continua, time-order error may be char-acte r i s t i c of the judgments. In Class II, time-order error i s not expected, nor i s i t generally found. Time-order error i s usually small, a fraction of a just noticeable difference. What time-order error seems to depend on i s the same basic process that makes category -11-seales on the Class I continua turn out to be concave down-ward, namely the asymmetry of sensitivity. The time-order error i s very d i f f i c u l t to measure and i n testing, i t i s usually avoided by random presentation. 4. Hysteresis. This describes what happens when the apparent sense-distances between successive stimuli are judged i n different ascending and descending order. For ex-ample, i n judging loudness i n ascending order, the subject sets the bisecting level some 5 to 8 decibels higher than i n descending order. It i s as though the loudness the subject hears lags behind what he should hear as he goes up and down the scale. As a result of this lag the graphs of the ascend-ing and descending functions exhibit a "hysteresis loop", the ascending path different from the descending. E i s l e r and Ottander (1965) said that hysteresis i s most l i k e l y due to different subjective zeros i n the psychological function for ascending and descending series. Stevens (1957) says that hysteresis occurs i n Class I continua but not in Class I I . 5. Prothetic continua produce a power function when using the methods of ratio production and magnitude production whereas metathetic may not. Methods of Constructing Ratio Scales of Subjective Magnitude For the purpose of c l a r i t y i n the following review of literature, a summary of the various methods for construct-ing ratio scales of subjective magnitude was included. Stevens (1957) said that although the methods are i n the develop--12-mental process, i n one form or another they a l l require sub-jects to make quantitative judgments of subjective events. 1. Ratio estimation - The subject i s required to name an i l l u s t r a t e d ratio between two stimuli. 2. Ratio production (fractionation and multiplication) -The subject i s required to set the stimuli as some ratio, e.g. half or double, of the standard. 3. Magnitude estimation - The subject i s required to assign numbers (e.g. i n inches) to a series of stim-u l i presented to him. 4. Magnitude production - The subject must adjust the stimuli to produce the particular magnitude. The difference between category production and mag-nitude production i s that i n category production the subject i s asked to produce a scale of interval numbers, e.g. 1 through 9 , whereas i n magnitude production the subject i s asked to reproduce the stimulus in inches, lbs., etc. Empirical Evidence of the Prothetic Continuum A large number of studies have investigated many sense modalities i n order to determine an appropriate function for the physical stimulus and subjective response relation-ship. The majority of modalities investigated have been shown to be members of the Class I or prothetic continua. Class I i s prothetic because i t includes among other things, magnitudes l i k e heaviness, loudness, brightness, etc. for which discrimination appears to be based on an additive mech-anism at the physiological l e v e l . As mentioned previously, - 1 3 -Stevens (1961) has provided a lengthy l i s t of examples with exponents varying from . 5 to 3 . 5 . For example, Stevens says that i n order to make a sound seem half as loud as another the physical energy must be reduced by about 90$ and this required reduction i s approximately the same regardless of the level of the starting point. Also, i n order to make one l i f t e d weight seem half as heavy as another, the orig-inal weight must be reduced by about 38$ and this percentage reduction i s approximately constant over a wide range of stimuli. The approximate constancy of the percentage re-duction corresponding to a given ratio demonstrates that the sensation S i s proportional to the stimulus R raised to the power n. Power functions have been i l l u s t r a t e d by different experimenters i n a wide range of areas such as t a c t i l e stim-ulation of forearm (Gilbert, 1969) , magnitude estimation of average length (Miller, 1970) , various factors of l i f t e d weights (Ross, 1968) , isometric muscular contractions for force and duration ( J . C. Stevens, 1970) , apparent heaviness and size-weight i l l u s i o n ( J . C. Stevens, 1970) , olfactory intensity (Mitchell, 1968, 1970) and taste (Gregson, 1 9 6 6 ) . Empirical Evidence of the Metathetic Continuum The Class I I i s metathetic because i t includes pitch, position, etc. for which discrimination behaves as though based on a substitutive mechanism at the physiological l e v e l . The number of studies on sensory modalities representative of metathetic continua are rather limited and not as clearly defined as the prothetic continua. Stevens and Galanter (1957) - I n -state that the metathetic continua, being a qualitative measure and being concerned with what and where, have to be forced into scalable continua as they do not f a l l naturally into unitary scales as do the quantitative aspects of things. Pitch, (Ekman, 1961) measured i n mels, i s a good example of a continuum of this sort. The mel scale of subjective judg-ment, constructed by the method of fractionation and equi-section, was found to be linear. Although the category scale i s essentially linear i t i s distorted at the ends by land-marks and at the lower end by di f f e r e n t i a l f a m i l i a r i t y . Judg-ments based on position actually provide some of the clearest examples of Class II continua. Visual position (azimuth) was examined by G. A. Miller (1961) by requiring the subjects to judge the position of a pointer which could be made to appear at the top edge of a cardboard. The plot revealed a linear category scale. Volkmann (1950) did further experi-ments on azimuth position of stimuli using the method of magnitude estimation. He concluded that judgments of pos-i t i o n measured by magnitude estimation, yield a function that i s essentially linear also. Rogers (Stevens, 1961) carried out an experiment on the apparent inclination of lines and found, by the bisection method, that the magnitude scale was linear, not including the landmarks of the endpoints. The unit of inclination (enc) plotted against the physical measure of the stimulus i n degrees was found to be a linear function. - 1 5 -Research Related to the Kinesthetic Sense There i s a limited amount of research on the psycho-physics of kinesthesis and since there are several aspects of kinesthesis, the number on kinesthetic extent of movement i s p r a c t i c a l l y non existent. A study by Brown, Knaft and Rosenbaum ( 1 9 4 8 ) found that subjects had a tendency to overshoot shorter distances and undershoot longer ones i n a kinesthetic positioning task. This might tend to contradict the metathetic continuum hypoth-esis. However, according to Stevens ( 1 9 5 7 ) , on both types of continua, alterations i n the form of category scales may be produced by a variety of factors, including stimulus spacing, or relative frequency of presentation, landmarks and d i f f e r -ential familiarity. By means of a series of experiments i t i s possible to achieve a stimulus spacing that w i l l neutral-ize these other effects and produce a 'pure' category scale. The only comparative study to the present one was done by Ronco i n 1 9 6 3 . He studied amplitude of movement, and attribute of kinethesis, by comparing the category production scale against the scale of subjective movement i n Kines. Since the task used was a straight arm extension forward from the shoulder, the Kine function was formulated in inches, with on Kine corresponding to five inches. A linear function was obtained when the category production scale was plotted against the subjective Kine scale. Ronco suggested that since the linear plot indicates that sen s i t i v i t y i s compar-atively uniform over the length of the scale, that amplitude -16-of movement may be c l a s s i f i e d as metathetic. Hoff (1966) carried out an extensive study to deter-mine the psychophysical scales for five attributes of kines-thesis: weight, force, speed, extent of movement, and thick-ness. The task for extent of movement was similar to that of Ronco (1963). Ronco obtained a power function exponent of 1.05 whereas Hoff, by the fractionation method, reported an exponent of 1.21 for extent of movement. Since Hoff found an exponent of .9697 for thickness and exponents of 1.4-3, 1.56, and 1.54 for weight, force, and speed of movement respectively, she suggests that thickness and extent of move-ment may represent the metathetic continuum whereas weight, force, and speed of movement are prothetic. Wood (1969) investigated rate of movement, an attribute of kinesthesis, by comparing the category scale to the sub-jective Kine scale, determined by magnitude and ratio pro-duction. She obtained different power function exponents (1.18 and .844) for each method. When the category scale was plotted against the Kine scale, a concave downward curve was determined. Also, the hysteresis effect was found when ascending and descending functions were compared. The con-clusion from these results, i n keeping with Stevens' (1957) c r i t e r i a , was that kinesthetic rate of movement i s represent-ative of the prothetic continuum. The approximately linear Kine function exponents, 1.05 and 1.21, derived by Ronco and Hoff respectively, along with Ronco's finding that a linear function v/as obtained when the -17-category scale was plotted against the subjective Kine scale, provide empirical evidence suggesting that kinesthetic ex-tent of movement i s representative of the metathetic class. Physiological Evidence Related to Prothetic and Metathetic  Continua To explain the existence of prothetic and metathetic continua Stevens (1957) proposed that the sense receptors mediating prothetic continua work differently from those mediating metathetic continua. A prothetic continuum re-quires a quantitative receptor mechanism, whereas a meta-thetic continuum requires a substitutive qualitative one. According to Wood (1969) receptors mediating a pro-thetic continuum respond i n increasing numbers, while;, an i n -crease i n the stimulus magnitude on a metathetic continuum would cause a different population of receptors to be activ-ated with no increase i n the number of receptors responding. According to Ekman (1961) the power law i s empirical and descriptive and per se does not t e l l us anything about the mechanisms responsible for the transformation i t describes. In general, these mechanisms w i l l have to be investigated on the physiological level but very l i t t l e has been done about i t so far. Ekman says i t i s well known that peripheral re-sponse processes often bear a logarithmic relation to stim-ulation. Since the f i n a l subjective outcome i s often a power function of stimulation, we have to expect that the central process generating the subjective response i s an antilogar-ithmic function of the peripheral process. -18-Mountcastle, Poggio, and Werner (1963), c a r r i e d out i n v e s t i g a t i o n s i n order to y i e l d r e s u l t s permitting one to recognize laws governing the transformations between per-i p h e r a l sensory and c e n t r a l neural events. Studies of t h i s kind deal with two q u a n t i f i a b l e v a r i a b l e s , a p h y s i c a l stim-ulus continuum and a continuum of neural a c t i v i t y measured i n some u n i t s appropriate to the nature of the neural r e s -ponse. The studies (Mountcastle et al) were c a r r i e d out on unanesthetized, deafferented-head monkeys, employing a r e -quired neuromuscular blocking agent. The continuum of neural a c t i v i t y was measured from e l e c t r i c a l s ignals of impulse d i s -charge of sing l e c e l l s of the ventrobasal nuclear complex of the thalamus. The p h y s i c a l stimulus continuum was the j o i n t (elbow and knee) r o t a t i o n , c o n t r o l l e d with an instrument allowing independent v a r i a t i o n s of speed of r o t a t i o n and the extent of the angle, which activated the c e l l . A f t e r a lengthy discussion, the general point made was that a given neuron can only change i t s rate of f i r i n g , i . e . i t can only vary ' i n t e n s i v e l y ' . Mountcastle et a l reported that change i n p o s i t i o n must be s i g n a l l e d by v a r i a t i o n s i n the s p a t i a l patterns of a c t i v i t y i n a large group of neurons, i . e . vary 'extensively'. Simultaneously, a l l neurons activated at a given moment w i l l discharge along monotonically i n c r e a s i n g functions, reaching t h e i r maximal f i r i n g rates at the l i m i t of movement. Therefore the p a r t i c u l a r movement made i s s i g -n a l l e d 'extensively' while the degree of movement i s s i g n a l l e d ' i n t e n s i v e l y ' as well by i n c r e a s i n g the rate of discharge of -19-the relevant group of c e l l s . The main concern of this study (Mountcastle et a l , 1963) was an investigation of the stim-ulus-response relation measured along the intensive continuum, as i t was viewing primarily single neurons. However, both 1 intensive' and 'extensive 1 variations of neuron acti v i t y are necessary for perception of joint position. Physiological evidence seems to suggest that position-ing i n kinesthesis operates on the matathetic principles. That i s , an increase i n the stimulus magnitude would cause a different population of receptors to be activated. There i s considerable evidence that receptors i n the joint capsule are stimulated by many aspects of movement and limb position, i . e . the attributes of kinesthesis. However, i t seems that not a l l aspects of kinesthetic perception act similarly. Skoglund (1956) conducted a series of experiments on knee-joint innervation of the cat. As a result he class-i f i e d three kinds of sensory receptors of the joint. Type 1 are slow adapting Golgi-type endings, associated with the ligaments, which signal the exact position of the joints. They also record the direction of movement. Type 2 are slow adapting Ruffini (spray) endings which l i e i n the capsule and signal the direction and speed of movement. Type 3 are Vater Pacini (paciniform) corpuscles of rapid adaption which are very sensitive to quick movements and independent of their direction. Skoglund considers the second and third type to be additive and subserve acceleration. According to this c l a s s i f i c a t i o n extent of movement, involving position, would require type one receptors which are considered to act -20-s u b s t i t u t i v e l y . Although not as e x p l i c i t l y defined, a number of studies (Gardner, 1 9 6 6 ) , (Boyd and Roberts, 1 9 5 3 ) , ( W i l l i a m s , 1 9 6 9 ) and (Smith, 1 9 6 9 ) describe several kinds of j o i n t r e -ceptors acting e i t h e r a d d i t i v e l y or s u b s t i t u t i v e l y , and responsible f o r d i f f e r e n t aspects of movement. I t i s gen-e r a l l y agreed that j o i n t receptors of p o s i t i o n are of the slow adapting type and have a l i m i t e d angular range. A metathetic continuum d e f i n i t i o n would seem to follow t h i s type of p h y s i o l o g i c a l d e s c r i p t i o n . C o n f l i c t i n g Views A great deal of controversy has been associated with whether or not there i s a single "psychological law of judg-ment" and whether there i s some explanation f o r the discrep-ant r e s u l t s which are found when d i f f e r e n t methods are used to estimate sensory magnitude: such as r a t i o estimation and c a t e g o r i z a t i o n . E i s l e r ( 1 9 6 5 ) has dealt extensively with t h i s problem and proposed a model whereby e i t h e r the magnitude estimation or the category scale can be derived. E i s l e r concluded that subjects have two d i f f e r e n t scales, methodologically, char-acterized as open-ended (magnitude scale) and closed (cate-gory s c a l e ) . Stevens ( 1 9 5 7 , 1 9 6 6 ) believed that there was only one unbiased scale and that was some form of the magnitude or r a t i o s c a l e . He also argued that a category scale i s quite d i f f e r e n t from a d i s c r i m i n a t i o n scale (Stevens, I 9 6 0 ) . In -21-his opinion reverting to category or discrimination scales was possible, but unacceptable, strategy. Poulton (1968) and Ross and BiLollo (1968) attacked the problem of psychophysical scales at a more fundamental l e v e l . On empirical as well as theoretical grounds, these investigators argued that i t i s not appropriate to describe the judgment process i n terms of a psychophysical function or behavior of sense organs unless consideration i s given to the mechanisms of response learning, response bias, experi-mental conditions, etc. Eyeman and Kim (1970), to carry out the previous idea, developed a model for examining the "errors" made i n psychophysical judgments, with the hope of character-izing the response patterns that emerge from the data. They reported that the structural relationship between the stimulus and response patterns appears to be a function of the place-ment of the standard stimuli and the order of magnitude of the numerical values which the subjects were asked to use as the standard stimulus levels. Partitioning of judgment error, by analysis of variance prior to the investigation of the psychophysical law, i s suggested as a useful method for greater understanding and isolation of the effects of many of the experimental factors known to affect Psychophysical judg-ments. Anderson (1970) discusses a procedure of functional measurement that may be helpful i n the study of psychophys-i c a l judgment. Whereas a great deal of psychophysical i n -vestigation attempts to f i r s t establish a scale of measure-ment from which quantitative laws of behavior may be developed. -22-Anderson's view i s that the development of the behavior theory and of the scale are integrally and intimately re-lated. Psychophysical scaling, i n particular, becomes part of psychophysical judgment. Technically the main features of functional measurement are reliance on f a c t o r i a l designs, quantitative response measures, and a monotone rescaling procedure. He applies functional measurement theory to four experimental situations i n psychophysical scaling. The multi-pli c a t i v e judgment model for ratio-setting tasks i s the one •:. which would apply to the present study. For this task the response i s on the physical stimulus scale and this i s not, in general, an adequate psychological scale. However, the monotone rescaling procedure of functional measurement pro-vides a way to obtain a v a l i d scale of the dependent response variable. It aims to relate the underlying psychological value of the response to the psychological value of the stim-ulus. Functional measurement thus leads to increased emphasis on judgmental processes and decreased emphasis on scaling per se. Anderson claims that i t s value i s i n providing a simple, unified approach to scaling and a clear rationale and basis for the general study of judgment. However he con-cludes that his model i s only on paper and needs more serious practical consideration. Helson (1964) provides the strongest criticism of the metathetic-prothetic division. He says, opposed to the div-i s i o n of sensory continua into prothetic and metathetic types i s the fact that curves f i t t i n g magnitude and quality judg--23-ments are often the same. There i s much evidence showing that curves are more affected by conditions of judgment than by the difference between so-called metathetic and prothetic dimensions (Helson, 1964). In support of this Garner (1954) has stated that judgments of loudness, which involve mag-nitude, are affected by composition of variables judged and by anchors; and judgments of pitch, which involve quality, are affected i n the same way by these qualities. Helson (1964) says i t appears from several lines of evidence that subjects adjust to the series and background stimuli presented, to the type of judgment or operation which constitutes the method, and to many other factors i n psycho-physical experiments. It seems however, that Stevens (1964, 1966) was aware of and accounted for several of these sug-gestions. Nelson (1964) gives examples of results from the de-termination of subjective scales, (such as the veg scale by Guilford, 1954), to show that the scales are not fixed. That i s , three different best f i t t i n g equations for the veg scale were determined by three methods: fractionation, constant sum (2 stimuli) and constant sum (5 stimuli). The power ex-ponents were 1.23, 1.06, and 1.17 respectively. Helson (1964) also discusses opposite curvatures found i n ascending and descending orders of judging which resemble the hysteresis loop of Stevens (1963) and therefore must be regarded as order effects. The fact that the hysteresis effect i s present i n some continua and not others does indic-ate a difference i n the form of discrimination, i n supports -24-f o r the two c l a s s e s . There may be merit i n several of the arguments presented. For example, fu r t h e r consideration of the mechan-isms of response l e a r n i n g and response bias i s necessary. However t h i s does not mean that psychophysical studies should be discontinued. Also more accurate s t a t i s t i c a l methods f o r analyzing r e s u l t s and e r r o r components i n c e r t a i n psycho-)"' p h y s i c a l methods should be i n s t i g a t e d . Several of Helson's c r i t i c i s m s such as _ r e f u t i n g the p r o t h e t i c - metathetic d i v -i s i o n can not be substantiated. But the f a c t that varying exponents have been determined by employing d i f f e r e n t methods seems to suggest the need to standardize experimental tech-niques and to eliminate experimental e r r o r as much as p o s s i b l e . Summary S. S. Stevens has formulated the most recent convinc-in g theory explaining the r e l a t i o n s h i p of the psychological continuum to the p h y s i c a l continuum f o r a l l sensory modal-i t i e s . He provides strong evidence i n support of d i f f e r e n t -i a t i n g between a Class I or p r o t h e t i c and Class I I or meta-t h e t i c continuum. The majority of other empirical and physio-l o g i c a l evidence c i t e d supports the d i v i s i o n i n t o two classes, although a few c o n f l i c t i n g views are as yet unexplained. I t appears that not a l l aspects of kinesthesis act s i m i l a r l y , p o s s i b l y suggesting v a r i a t i o n i n performance at the receptor l e v e l . The l i m i t e d but convincing evidence r e -garding k i n e s t h e t i c extent of arm movement suggests that i t i s representative of the metathetic c l a s s . - 2 5 -CHAPTER III METHODS AND PROCEDURES Subjects The subjects were forty right-handed male and female volunteer Physical Education graduate and undergraduate students. The subjects were randomly assigned to one of four groups with the re s t r i c t i o n that equal numbers appear in each group. Apparatus The apparatus consisted of a f l a t piece of white cardboard cut i n an arc of 1 2 0 degrees and attached to the top of a table. Degrees from 0 to 1 2 0 for every half degree were marked on the cardboard. A stationary wood block was used to mark the fixed starting position and a moveable wood block was used as a stopper for each arm movement. Experimental Conditions  Condition I The ratio scaling method of fractionation was used to determine the subjective Kine scale function. Ten physical distance stimuli varying from 2 0 to 1 1 0 degrees represented the independent variables. Each of the ten stimuli were randomly presented five times to each subject. A response was the subject's judgment of one-half the stimulus distance -26-presented. The subject's response distance was recorded to the nearest }. degree. Condition II The category production method was used i n order to produce a subjective category scale. Two stimulus distances, 1 2 $ and 112% degrees, representing categories 1 and 9 re-spectively were presented approximately 5 times i n order for the subjects to "become familiar" with them. After being informed of the nine equally-spaced categories, the subject was asked to reproduce as accurately as possible, the dis-tance representing the category selected. Each category number was presented randomly five times. Therefore each sub-ject i n Condition II was required to make 4-5 responses. The response was recorded to the nearest % degree. Conditions III and IV The ratio scaling method of fractionation (halving) was used i n order to determine whether or not a hysteresis effect was present for kinesthetic extent of movement. The physical distance stimuli varying from 2 0 to 1 1 0 degrees represented the independent variables. Each subject was presented with the total series of physical stimuli i n as-cending order ( 2 0 to 1 1 0 degrees) for Condition III and in descending order ( 1 1 0 to 2 0 degrees) for Condition IV. A response was a subject's judgment of one-half the distance presented. For each condition, the total series was presented five times i n the specified order. The subjective response was recorded to the nearest % degree. -27-Procedure Each subject was blindfolded and did not see the apparatus. Additional cues from clothing were eliminated by-requiring the males to remove shirts and the females to wear a sleeveless top. The subject was seated i n the desired position with his right arm extended over the apparatus, elbow straight, palm down, and hand and fingers extended so that the inside of the subject's right index finger was i n line with the zero degree l i n e , and the stationary wood block, on the outside of the subject's hand, held the starting position constant for each t r i a l . The instructions for Condition I, III, and IV were as follows: Oil the experimenter's command 'move', you w i l l l i f t your hand s l i g h t l y off the table surface and move your straight arm at a relatively constant rate (about 4- seconds for 100 degrees) pa r a l l e l to the top of the table and towards the midline of your-body u n t i l the inside of your index comes i n con-tact with a stopper. Hold the position for approx-imately two seconds and then return i n a similar manner to the starting position. You w i l l be given two standard movements at each distance and then be asked to reproduce half the standard distance as accurately as you can. Do you have any questions? Each subject was given five random practice t r i a l s to become familiar with the prescribed arm movement. The instructions for Condition II were as follows: On the experimenter's command 'move', you w i l l l i f t your hand s l i g h t l y off the table surface and move your straight arm at a re l a t i v e l y constant rate (approximately 4- seconds for 100 degrees) pa r a l l e l - 2 8 -to the top of the table, and toward the midline of your body u n t i l your index f i n g e r comes i n contact with a stopper. You w i l l be informed that your distance moved i s e i t h e r category 1 or category 9 i n a scale of nine equally-spaced categories. For example, the distance from the s t a r t i n g point to category 1 i s the same distance as between categories 3 and 4-. You w i l l be given several t r i a l s at categories 1 and 9 u n t i l you f e e l f a m i l i a r with them and then be asked to r e -produce any category from 1 through 9 as accurately as p o s s i b l e . Therefore i t i s important that you f e e l f a m i l i a r with categories 1 and 9. Are there any questions? S t a t i s t i c a l Analysis In a l l cases, the p h y s i c a l stimulus i s the independ-ent v a r i a b l e and the subjective response i s the dependent v a r i a b l e . Three of Stevens' f u n c t i o n a l c r i t e r i a (Stevens, 1957) f o r d i f f e r e n t i a t i n g between the p r o t h e t i c and the metathetic continua were employed i n t h i s a n a l y s i s . 1. Category r a t i n g s c a l e . The category scale f o r a prothetic continuum i s concave downward when p l o t t e d against the subjective r a t i o scale whereas f o r a metathetic continuum the category scale p l o t i s l e s s concave and u s u a l l y l i n e a r . 2. Hysteresis e f f e c t . The two functions derived from the subjective judgments of the s t i m u l i presented i n both ascending and descending orders are p l o t t e d on the same graph. Hysteresis occurs i n Class I but not u s u a l l y i n Class I I . 3. Prothetic continua produce a pov/er function when using the method of r a t i o production whereas meta-t h e t i c may not. - 2 9 -Throughout the analysis geometric means and medians were used because, as measures of c e n t r a l tendency, they are l e s s affected by extreme scores than the arithmetic mean, and therefore are t r u e r estimates of the intended response. Condition I - Ratio Production ( F r a c t i o n a t i o n Method) the response values of each subject f o r each stimulus d i s -tance. The median response value was determined f o r the group of ten subjects f o r each stimulus. These median sub- : j e c t i v e values of h a l f the stimulus distance were p l o t t e d on the ordinate against the stimulus distance on the abscissa, on l o g - l o g , l o g - l i n e a r , l i n e a r - l o g , and l i n e a r - l i n e a r paper. The purpose was to obtain the general idea of the best f i t t i n g f u n c t i o n of the data. I f the l o g - l o g co-ordinates give a s t r a i g h t l i n e , a power func t i o n i s the best f i t . I f the l i n e a r - l o g or l o g - l i n e a r give a s t r a i g h t l i n e , the l o g a r i t h -mic function i s the best f i t and i f the l i n e a r - l i n e a r p l o t gives a s t r a i g h t l i n e , a l i n e a r function gives the best f i t . Once the general form was determined, a s p e c i f i c b e s t - f i t t i n g function was derived by the method of l e a s t squares. Also, a c o r r e l a t i o n c o e f f i c i e n t i n d i c a t i n g goodness of f i t of the data to the function was determined and an analysis of var-iance c a r r i e d out. From the derived b e s t - f i t t i n g function, a Kine scale was determined by s u b s t i t u t i n g values where 1 Kine = 10 degrees. (The exact transformation procedure i s i n the appendix A.) The Kine scale transformation i s e s s e n t i a l The geometric mean ( * y i 2 3...n ) was computed f o r - 3 0 -when a comparison to the category production scale i s made and also f o r a standardized comparison with other sensory modalities. These obtained Kine scale values were p l o t t e d against the o r i g i n a l p h y s i c a l stimulus values on four types of co-ordinates: l o g - l o g , l o g - l i n e a r , l i n e a r - l o g , and l i n e a r - l i n e a r i n order to, once again, observe the general form of the best f i t t i n g f unction. Then the s p e c i f i c best-f i t t i n g subjective Kine function was determined by the method of l e a s t squares. An analysis of variance table was derived and a c o r r e l a t i o n c o e f f i c i e n t f o r goodness of f i t c a l c u l a t e d . Condition I I The geometric means f o r each subject f o r each response category were determined. Then the median response f o r each category f o r the group was ca l c u l a t e d . These values are representative of the subjective category scale f o r the group. This subjective category scale i n degrees was p l o t t e d against the nine equally spaced p h y s i c a l categories. To view t h i s r e l a t i o n s h i p , an analysis of variance was computed and the b e s t - f i t t i n g function was derived algebraically. Next, the subjective category scale i n degrees was transformed into Kines using the previously obtained Kine function from Con-d i t i o n I . The nine equal categories were p l o t t e d on the ordinate against the corresponding Kine scale values. This graph was used to i l l u s t r a t e how the category scale corres-ponds to the Kine scale. The b e s t - f i t t i n g function and an analysis of variance table were derived to examine the r e -l a t i o n s h i p a l g e b r a i c l y . I f a l i n e a r r e l a t i o n s h i p f o r the -31-comparison i s found, the metathetic continuum i s indicated, whereas a concave downward curve i s t y p i c a l of a pro t h e t i c continuum. Conditions I I I and IV - The Hysteresis E f f e c t The geometric means f o r each subjective estimate of h a l f a stimulus distance were obtained f o r each subject f o r both ascending and descending conditions. The median f o r the group f o r each response f o r each condition was then c a l c u l a t e d . The obtained medians of the subjective values were p l o t t e d on the ordinate i n degrees against the ph y s i c a l stimulus values on the abscissa i n degrees. This procedure was c a r r i e d out f o r both ascending and descending orders so that both functions were i l l u s t r a t e d on the same graph. The b e s t - f i t t i n g functions and the c o r r e l a t i o n c o e f f i c i e n t s were calcul a t e d f o r both ascending and descending conditions. Also a trend analysis was used to aid i n comparing the two conditions. The p l o t t e d functions were viewed to determine whether hysteresis was present or not. The hysteresis e f f e c t i s present i f : 1. a l i n e a r function i s obtained f o r the two l i n e s but when p l o t t e d on graph paper the two l i n e s are p a r a l l e l to each other. 2. on l i n e a r coordinates, the two curves are symmetrical, but one concave up and the other concave down. -32-The results from each condition were viewed separ-ately and then together to determine i f the subjective es-timate of extent of arm movement i s subserved by a pro-thetic or metathetic process. -33-CHAPTER IV RESULTS AND DISCUSSION RESULTS Stevens' techniques were applied to the date to de-termine the relationship between the physical stimulus move-ment and the perception of that movement, i . e . to determine whether kinesthetic extent of arm movement i s character-i s t i c of a metathetic or prothetic continuum. Determining the Kine Function In order to test hypothesis one, that the obtained Kine function i s a power function with an exponent of approx-imately one, i t was necessary to determine the best f i t Kine function from the best-fitting function obtained from the median half-responses and their corresponding stimuli. The psychological function i s always a power function for the prothetic continua whereas this i s not always the case for the metathetic continua. The geometric means for each subject's five responses to each of the stimuli for Condition I were determined and the median response for the group for each stimulus was found. (TABLE I) The graphic approach, of plotting the stimulus mag-nitude on the abscissa against the corresponding median re-_ 34-sponses on the ordinate, was carried out on four types of coordinates: log-log, log-linear, linear-log, and linear-linear. From this preliminary examination, the log-log (Figure 1) and the linear-linear (Figure 2) coordinates appeared to i l l u s t r a t e straight l i n e s . TABLE I THE GROUP MEDIAN RESPONSES CORRESPONDING TO EACH STIMULUS IN DEGREES, FOR CONDITION I S R Log S Log R 20 10.150 1.301 1.006 30 15.180 1.477 1.181 40 22.290 1.602 1.348 50 27.460 1.699 1.439 60 34.300 1.778 1.535 70 42.360 1.845 1.627 80 46.930 1.903 1.671 90 54.940 1.954 1.740 100 63.120 2.000 1.800 110 72.000 2.041 1.857 MEAN 65 38.870 1.760 1.520 STANDARD DEVIATION 30.276 20.637 .241 .277 When linear-linear coordinates give a straight line the best f i t function i s a linear function whereas when log--35-log coordinates give a straight line the b e s t - f i t function i s a power function. The only time a power function also f i t s a linear function i s when the power function has an exponent of one. Next, the be s t - f i t t i n g function was determined alge-braically by the method of least squares and the corresponding correlation coefficient found. The equation describing the best - f i t t i n g function was the power function: log R = 1.149 log S - .5014, with a correlation coefficient of r = .999. The equation for 'the best-fitting linear function was: R = .680 S - 5.325, with a correlation coefficient of r = .997. An analysis of variance of the log-log data (TABLE II) and the linear-linear data (TABLE III) i l l u s t r a t e d a signif-icant linear trend i n support of both functions. However a barely significant quadratic term was found for the linear-linear data i n TABLE III. TABLE II ANALYSIS OF VARIANCE TABLE OF LOG S AND LOG R MAGNITUDES Source of Variation Degrees of Mean F P Freedom Squares Ratio Linear term 1 .6890 6890. <.01 Quadratic term 1 .0003 3. >.01 Deviation about Regression 7 .0001 Total 9 -38-TABLE III ANALYSIS OF VARIANCE TABLE OF S AND R MAGNITUDES Source of Variation Degrees of Freedom Mean Squares F Ratio P Linear term 1 3814.461 5387.657 <.01 Quadratic term 1 13.657 19.289 <.01 Deviation about Regression 7 .708 Total 9 The scaling of extent of movement was performed with the function, log R = 1.149 log S - .5014. The unit one Kine was defined as the subjective extent of a movement ten degrees i n length. This value was chosen i n keeping with the defin-i t i o n presented by Hoff (1967), suggesting assigning the value of unity to that stimulus which was judged half of the smallest standard. The values required for scaling subjective extent of movement are given i n TABLE IV. The method of deriving the Kine scale i s described i n the Appendix A. After conversion into Kines, the best f i t equation determined by the method of least squares was: log Kine = 1 . 0 7 5 log S - . 9 9 5 6 . The slope of the line describing the dependence of the psycho-logi c a l extent of movement upon the physical distance i s expressed by: Kine = . 1 0 1 0 S 1 , 0 7 5 -39-TABLE IV THE RELATIONSHIP OP PSYCHOLOGICAL EXTENT OF ARM MOVEMENT IN KINES TO PHYSICAL DISTANCE INDEGREES, USING THE FUNCTION: LOG R = 1.149 LOG S - .5014 Kine Log S S Log Kine 0.250 0.243 1.748 -1.398 0.500 0.647 4.440 -1.699 1.000 1.000 10.000 0.000 2.000 1.307 20.260 0.301 4.000 1.574 37.470 0.602 8.000 1.806 63.950 0.903 16.000 2.008 101.900 1.204 32.000 2.184 152.700 1.505 Mean 7.969 1.346 49.059 .452 Standard Deviation 11.067 .679 54.082 ,737 The intercept 9.889 i s the antilog of .9956. A cor-relation coefficient of .991 was derived i n testing the good-ness of f i t of the data to this l i n e . The analysis of variance (TABLE V) i l l u s t r a t e s the highly significant linear f i t of the log data. The relation of log Kine to log S i s i l l u s t r a t e d i n Figure 3. -AO-TABLE V ANALYSIS OF VARIANCE TABLE LOG KINE OF LOG S VERSUS Source of Variation Degrees of Freedom Mean Squares F Ratio P Linear term 1 8.788 56.387 < .01 Quadratic term 1 .006 < 1.0 Deviation about Regression 5 .156 Total 7 From the exponent 1.075, i t may be seen that the psych-ological variable i s almost li n e a r l y related to the physical variable. Since the linear function would have an exponent of 1 . 0 0 0 , the present exponent of 1 . 0 7 5 i l l u s t r a t e s a very slight more rapid growth of the psychological variable com-pared to the physical variable. The Category Scale In order to test hypothesis two, that i s comparing the subjective category scale to the subjective Kine scale, i t was f i r s t necessary to determine the subjective category scale of the category production task. The geometric means of each subject's nine response categories in Condition II were computed and the nine median response categories for the group were determined. -4-2-TABLE VI represents the median results of the category pro-duction procedure. TABLE VI THE MEDIAN RESPONSES OP THE GROUP FOR CATEGORY PRODUCTION TASK THE Category Physical Category (Degrees) Subjective Category 1 12.5 18.870 2 25.0 36.930 3 37.5 4-6.500 4- 50.0 60.790 5 62.5 71.010 6 75.0 81.050 7 87.5 91.760 8 100.0 98.74-0 9 112.5 111.590 Mean 5 62.5 68.580 Figure 4- i l l u s t r a t e s the linear function, y = ,889x + 12.99, derived from the median response magnitudes (Y) corresponding to the nine equally-spaced category mag-nitudes (x) i n degrees. A correlation coefficient of .995 was derived. The analysis of variance (TABLE VII), showing the linear trend to be significant, i l l u s t r a t e s that the subjective cate-gory scale i s linea r l y related to the physical category scale. - D A -TABLE VII ANALYSIS OP VARIANCE TABLE OP STIMULUS CATEGORIES VERSUS MEDIAN RESPONSE CATEGORIES Source of Variation Degrees of Freedom Mean Squares F Ratio P Linear term 1 7416.793 2030.639 <.01 Quadratic term 1 40.504 11.080 >.01 Deviation about Regression 6 3.653 Total 8 The Category Scale Plotted Against the Subjective Kine Scale In accordance with hypothesis two, the subjective cat-egory scale was compared to the subjective Kine scale. A concave downward function i s characteristic of the prothetic continuum whereas a linear function i s characteristic of the metathetic continuum. That i s , on a metathetic continuum the scale derived from the category production method and the scale derived from the ratio production method are related linearly, whereas, on a prothetic continuum the scales are not li n e a r l y related. The subjective category medians were transformed into Kines using the previously obtained Kine function, log Kine = 1.075 log S - .9956. This transformation of the subjective categories into Kines was necessary for the com-parison of the two scales to be made in subjective Kine units. -45-The corresponding Kine values are l i s t e d in TABLE VIII and the results of the category production procedure are plotted against the scale of subjective movements i n Kines in Figure; 5. TABLE VIII A TABLE OF THE SUBJECTIVE CATEGORY SCALE WITH THE CORRESPONDING KINE VALUE Category Subjective Response Subjective Response in Degrees i n Kines 1 18.870 2.370 2 36.930 4.880 3 46.500 6.266 4 60.790 8.356 5 71.010 9.870 6 81.050 11.380 7 91.760 13.020 8 98.740 14.075 9 111.590 16.020 Mean 5 68.580 9.582 The best - f i t t i n g function, y = 1.645x + 1.355, derived from the category scale (x) plotted against the scale of subjective movement in Kines (y), i s linear with a correlation coefficient of .997. The analysis of variance(TABLE IX) i l -lustrating a highly significant linear function indicates that discriminal s e n s i t i v i t y i s comparatively uniform over the lengths of the scale. -47-TABLE IX ANALYSIS OP VARIANCE TABLE OP THE CATEGORY SCALE VERSUS THE KINE SCALE Source of Variation Degrees of Freedom Mean Squares F Ratio P Linear tern- 1 162.416 2588.476 <.01 Quadratic term 1 .567 8.353 >.01 Deviation about Regression 6 .068 Total 8 The Hysteresis Effect In order to test hypothesis three, that i s , whether or not a hysteresis effect i s present, i t was necessary to deter-mine the bes t - f i t function for the responses of both ascending (Condition III) and descending (Condition IV) conditions, and then to examine the relationship between the two functions. Hysteresis i s present i f : 1. a linear function i s obtained for both lines but when plotted on graph paper the two lines are p a r a l l e l , or 2. on linear coordinates, the two curves are symmetrical, but one concave up and the other concave down. Hysteresis i s a characteristic of prothetic continua but not metathetic continua. The geometric means were determined for the five re-sponses of each subject for each of the ten stimuli i n Condition III (presented i n ascending order) and Conditiion IV (presented i n descending order). The median responses for both the -48-ascending and descending groups were computed for each stimulus and the results are presented i n TABLE X. TABLE X MEDIAN RESPONSES FOR GROUP III (ASCENDING) AND GROUP IV (DESCENDING) CORRESPONDING TO THE STIMULI IN DEGREES S III (R) IV (R) 20 10.300 11.260 30 15.620 15.520 40 21.340 21.420 50 28.560 26.710 60 36.950 34.240 70 42.990 39.820 80 50.980 45.120 90 56.880 52.460 100 64.970 59.260 110 73.970 64.150 Mean 65 40.256 36.996 Standard Deviation 30.276 21.489 18.336 Figure 6 i s a graph of the medians for both conditions on linear coordinates. Algebraically., the best f i t t i n g lines were found to be l i n -ear. The function for Condition III i s : y = .605* - 2.327 with a correlation coefficient of .998 and the condition IV i s : y = ,709x - 5.814 with a correlation coefficient of .999. - 5 0 -I t appeared that the ascending and descending conditions were very similar. However to examine i f an interaction occurred between the two conditions over the stimuli, a trend analysis of the geometric mean responses of the ten subjects for the la s t nine stimuli was carried out. The results of the trend analysis are presented i n TABLE XI. TABLE XI ANALYSIS OP VARIANCE TABLE FOR TWO CONDITIONS, TEN SUBJECTS, AND NINE STIMULI OF CONDITIONS III AND IV Source of Variation Degrees of Freedom Mean Squares F Ratio P Conditions 1 112.78 Subjects (Conditions) 18 610 .69 < 1 Stimuli 8 9535.15 110.4-4- «<r.oi Stimuli (Linear) 1 69701.4-38 76.312 <.01 Stimuli (Quadratic) 1 4-27.952 < 1 Conditions x Stimuli 8 823 .03 <1 Subjects (Conditions) x Stimuli 14-4-179 913 .37 Neither the variation between conditions nor between subjects was found to be significant. The only significant variable was between stimuli. Also i t was shown that the functions previously derived were significantly linear. Therefore there was found to be no significant d i f -ference between the functions obtained from the results when the stimuli are presented i n ascending or descending order. -52-DISCUSSION The differentiation between the prothetic and meta-thetic continua by Stevens (1957) i s an attempt to explain how various sense modalities function. Although the specific terms, prothetic and metathetic are not clearly defined, a set of characteristics of each class i s employed i n providing an adequate distinction. I t i s thought that a clear separ-ation of prothetic and metathetic sensory modalities may be found at the physiological l e v e l . However, since direct physiological studies of such processes are very d i f f i c u l t , i f not impossible to conduct at this time, psychological methods such as psychophysical scaling are used in an attempt to c l a s s i f y various sensory modalities. From these studies, implications are inferred to the physiological l e v e l . Hoff (1967) has attempted to derive psychophysical scales for five attributes of kinesthesis, and Ronco (1963) has suggested that further study on kinesthetic extent of movement may prove i t to be representative of a metathetic continuum. The present study has viewed kinesthetic extent of movement on the basis of three distinct c r i t e r i a , which as a whole, were thought to be substantial indicators of the respective continuum. The Kine Function The ratio production method of fractionation by halv-ing was chosen to develop a Kine function to meet the f i r s t c r i t e r i o n . In determining the b e s t - f i t t i n g function from -53-the median response data, both the l o g - l o g and l i n e a r - l i n e a r functions were shown to provide a good f i t , with c o r r e l a t i o n c o e f f i c i e n t s of .999 and .997 r e s p e c t i v e l y . However the l i n e a r - l i n e a r analysis of variance i l l u s t r a t e d a very small but s i g n i f i c a n t quadratic term. This quadratic trend was not evident f o r the l o g - l o g function and therefore i t was employed to determine the Kine function. These l o g - l o g data gave a power function of the form: l o g R = 1.14-9 lo g S - .5014-. The Kine function, l o g Kine = 1.075 l o g S - .9956 or Kine = .1010 S 1 ' 0 ^ , was transformed from the preceding power function using as a base, one Kine equals ten degrees. Being a power function may suggest that k i n e s t h e t i c extent of arm movement could f i t e i t h e r the metathetic or p r o t h e t i c c l a s s i f i c a t i o n . However the f a c t that the exponent, 1.075, i s very close to 1.000 which ind i c a t e s a l i n e a r function, suggests a very uniform s c a l i n g function. The uniformity i s i l l u s t r a t e d i n Figure 3. The psychological functions, de-termined by r a t i o techniques, of sensory modalities believed to belong to the metathetic c l a s s , have u s u a l l y been des-cribed as e s s e n t i a l l y l i n e a r . This r e f l e c t s a constant ac-curacy of descrimination over the e n t i r e sensory continuum. Examples of such sense modalities are p i t c h (Ekman, 1961), v i s u a l p o s i t i o n ( M i l l , 1961) and l i n e i n c l i n a t i o n (Stevens, 1961) which have a l l been described as belonging to the meta-t h e t i c c l a s s and e x h i b i t i n g e s s e n t i a l l y l i n e a r r a t i o scales of subjective judgment. The greater the deviation of a power function from l t000, the l e s s uniform i s the subjective d i s c r i m i n a t i o n s c a l e . Also the more the exponent deviates -54-from 1.000, the less linear i s the relationship between the subjective and the physical scales. The fact that the median stimulus and response mag-nitudes f i t t e d a linear function (Figure 2 and TABLE III) almost as well as the power function also strongly suggests an equality of the accuracy of discrimination of kinesthetic extent of arm movement over the range of i t s scale. The greater increase of the psychological scale than the physical scale with an exponent of 1.075, i s extremely minimal and would not be noticed as a deviation from the physical u n t i l an extremely large distance which would be outside the range of possible arm movement. The present results support the finding of Ronco (1963) and Hoff (1967) who, although using a different task for ex-tent of movement found exponents of 1.05 and 1.21 respect-ive l y . Also, the present task being one involving position discrimination, supports S. S. Stevens' (1961) report that judgments based on position provide some of the clearest ex-amples of the Class II continua. The uniformity of the perception of movement over the range as indicated by the exponent close to one and the l i n -earity of the median results related to the stimulus values supports the general hypothesis that kinesthetic extent of movement i s representative of the metathetic class. The Category Scale The category production method was used to develop an interval scale for kinesthetic extent of arm movement, - 5 5 -referred to as the category scale. The subjective category scale plotted against the physical category scale, i l l u s -trated in Figure 4, i s a linear function, y = ,889x + 12 . 9 9 , with a correlation coefficient of . 9 9 5 . The linear function means that subjects had the a b i l i t y to divide the movement continuum into nine approximately equal sized categories. According to Stevens ( 1 9 5 7 ) , only when sensitivity i s uniform over the continuum can linear results from category ratings be expected. Although categories 1 and 9 were given as reference points for the beginning and end of the scale, these end points did not appear to distort the linear trend. The tendency, according to Brown, Knaft, and Rosenbaum (1948),to overshoot shorter distances and undershoot longer distances may have occurred but a compensation over the total range to keep the categories relatively equal must also have occurred. Overshooting the shorter distances particularly was i l l u s -trated by the median response category of 18.87 degrees corresponding to the f i r s t physical category of 12 .5 degrees. The a b i l i t y to retain relative equality over the nine categories as evidenced by the linear relation support the metathetic hypothesis. This compensation effect i s an example of the purpose of employing subjective units. A subject's perception, although uniform, may not equate s p e c i f i c a l l y with physical units. The Category Scale Versus the Kine Scale It i s the perception of subjective units which i s the basis of psychophysical scaling. Probably due to a physio--56-l o g i c a l difference, the subjective category scale derived from an interval method and the subrjective scale derived from a ratio method are l i n e a r l y related for the metathetic continua but non-linear, usually concave downward, for the prothetic. This non-linear trend i n the prothetic continua i s due to the fact that a subject's sensitivity varies. Usually at the lower end of the scale discrimination i s good, categories are narrow, and the slope of the function i s steep; but further along the scale, stimuli are less easy to detect, categories broaden and the slope declines. In Class II or metathetic continua, where sensitivity measured i n sub-jective units remains rel a t i v e l y constant, i t i s possible to produce category scales that are lin e a r l y related to subjective ratio scales. In the present study when the subjective categories i n degrees were changed into subjective Kine units, by means of the Kine function, the plot of the nine equal physical categories against the subjective Kine categories was found to be linear. The relationship, y = 1.64-5x + 1.355 with a correlation coefficient of .997, i s i l l u s t r a t e d i n Figure 5. This highly significant linear function supports the hypothesis that a linear function would result from the plot and suggests that kinesthetic extent of arm movement i s representative of the metathetic class. These findings support the results of Ronco (1963), who also found a linear relationship when the category scale was plotted against the Kine scale. He suggested that although -57-the evidence strongly indicated that extent of movement represented the metathetic class, a further cr i t e r i o n must be investigated to make definite conclusions. The Hysteresis Effect The hysteresis or lagging behind effect was the third criterion tested i n the present study. Hysteresis i s an effect found most clearly i n bisection or equipartition experiments. It i s present i f the position of the divisions or the bisection point i s higher or lower depending upon whether the stimuli are presented i n ascending or descending order. This phenomenon i s exhibited by prothetic continua, due probably to different subjective zeros (Eisler and Ottander, 1965), but not by metathetic continua due to the fact that sen s i t i v i t y i s more uniform over the subjective scale, regard-less of the starting point. No hysteresis effect was exhibited i n the present experiment for kinesthetic extent of arm move-ment as shown graphically by Figure 6. In fact the two con-ditions appear to be very similar. Algebraically the two functions, y = ,605x - 2.527 and y = ,709x - 5.814 with cor-relation coefficients of .998 and .999 respectively, were found to be linear. Trend analysis also showed that there was no difference i n the trend of the functions over the last nine stimuli. These results clearly suggest that no hysteresis effect was present. Such a result supports the hypothesis that hysteresis i s not present and implies that kinesthetic extent -58-of arm movement represents a metathetic continuum. Thus, i t would seem to make l i t t l e d i fference to extent of arm movement perception i f the movement was increasing or de-creasing i n length. A Synthesis of the I n d i v i d u a l Results Each of the tested conditions suggested a r e s u l t cor-responding to the p a r t i c u l a r hypothesis concerned. F i r s t l y , the derived subjective Kine function, Kine = .1010S^ , ( " ) ' ' 7 ' ' , was a power function with an exponent very close to one, supporting the metathetic c l a s s . Secondly, the subjective category scale was found to be l i n e a r when p l o t t e d against the p h y s i c a l category scale. Also, when the i n t e r v a l category scale was p l o t t e d against the subjective Kine scale, a l i n e a r function was derived. Such a f i n d i n g supports the metathetic hypothesis. T h i r d l y , the hysteresis e f f e c t was not present when ascending subjective values were p l o t t e d against corresponding descending subjective values. This f i n d i n g also provides evidence to place k i n e s t h e t i c extent of arm movement i n the metathetic c l a s s . Therefore, when viewed together, the conclusion from .the present psychophysical experiment must be that the sub-j e c t i v e estimate of extent of arm movement i s subserved by a metathetic process. I t has been determined that subjects have a very uniform perception of extent of movement over at l e a s t a 110 degree arc of the shoulder j o i n t . The findings suggest that sub-j e c t i v e perception of any length of movement within the range -59-w i l l not necessarily coincide with the physical scale i n degrees but w i l l be essentially li n e a r l y related. Unlike Class I or prothetic continuum examples such as loudness or electric shock for which discrimination i s best at one point and then decreases, the extent of arm move-ment can be discriminated consistently over the range. As was suggested previously, once the outward character-i s t i c s of the sensory modality have been determined, implic-ations to the underlying mechanisms may be made. It may be suggested, therefore, that the sense receptors from kines-thetic extent of movement act substitutively or qualitatively. That i s , a different population of receptors would be activ-ated in order to perceive a change i n the angle of the shoulder joint rather than an increase i n the number. In keeping with Skoglund's (1956) c l a s s i f i c a t i o n of three types of sensory receptors of the joint, i t would f o l -low that the receptors i n the shoulder joint, responsible for the perception of extent of movement, are slow-adapting with Golgi-type endings, signalling the exact position of the joint, and considered to act substitutively. Since extent of movement i s only one attribute of kin-esthesis, i t i s necessary for several c r i t e r i a of the psycho-physical approach to be used to test the additional attributes of kinesthesis. Once a general picture of man's kinesthetic sense i s understood from the psychophysical approach, not only may implications for the underlying mechanisms of kines-thesis be made but also implications may possibly be made toward applications such as the role of perception of move--60-ment i n the acquisition of movement s k i l l s . However, as S. S. Stevens states, i t must be realized that an attempt to impose simple order on the complex system of perceptions, runs the risk of oversimplifying what i s not simple. -61-CHAPTER V SUMMARY AND CONCLUSIONS The purpose of this study was to determine the re-lationship between a physical stimulus continuum and the interpreted psychological continuum for one attribute of the kinesthetic sense: extent of arm movement. Forty right-handed male and female volunteer physical education students were randomly assigned to one of four conditions with the re-s t r i c t i o n that equal numbers appear i n each group. The arm movement task i n each condition involved a rotation of the shoulder joint, with a straight arm, palm down movement i n a horizontal position, towards the midline of the body. The conditions were attempting to provide tests of. three function-al c r i t e r i a , outlined by S. S. Stevens (1957) for d i f f e r -entiating between two classes of continua, prothetic and metathetic, into which the majority of sensory modalities naturally f a l l . In Condition I, the psychophysical ratio scaling method of fractionation was employed. The subjects were presented with ten random stimuli varying from 20 to 110 degrees and asked to reproduce half the distance. In Condition II, the psychophysical interval scaling method of category production was employed. The subjects were required to produce their subjective estimate of an equally spaced nine category scale. - 6 2 -I n C o n d i t i o n I I I and I V , t h e p s y c h o p h y s i c a l method o f f r a c t i o n a t i o n was u s e d . Ten s t i m u l i d i s t a n c e s ( f r o m 20 t o 110 d e g r e e s ) were p r e s e n t e d i n a s c e n d i n g o r d e r f o r C o n d i t i o n 111 and i n d e s c e n d i n g o r d e r f o r C o n d i t i o n I V , and t h e s u b -j e c t s were r e q u i r e d t o r e p r o d u c e h a l f t h e s t i m u l u s d i s t a n c e . A p s y c h o p h y s i c a l K i n e f u n c t i o n , K i n e = . l O l O S " 5 " * 0 ^ , was d e r i v e d f r om C o n d i t i o n I f o r e x t e n t o f arm movement. A l i n e a r f u n c t i o n r e s u l t e d f r om C o n d i t i o n I I when t h e i n t e r v a l c a t e g o r y s c a l e was p l o t t e d a g a i n s t t h e K i n e s c a l e . From C o n -d i t i o n s I I I and I V , i t was d e t e r m i n e d t h a t t h e h y s t e r e s i s e f f e c t was n o t p r e s e n t f o r k i n e s t h e t i c e x t e n t o f arm movement. The s p e c i f i c c o n c l u s i o n s w e r e : 1. T h a t t h e K i n e f u n c t i o n , K i n e = . 1 0 1 0 S 1 , 0 7 5 , was c h a r -a c t e r i s t i c o f t h e m e t a t h e t i c c l a s s . 2 . T h a t t h e f u n c t i o n d e r i v e d f r om t h e p l o t o f t h e c a t e -g o r y s c a l e a g a i n s t t h e s u b j e c t i v e K i n e s c a l e was l i n e a r and , t h e r e f o r e , r e p r e s e n t s t h e m e t a t h e t i c c l a s s . 3. Tha t t h e h y s t e r e s i s e f f e c t was n o t p r e s e n t when t h e a s c e n d i n g s u b j e c t i v e v a l u e s were p l o t t e d a g a i n s t t h e c o r r e s p o n d i n g d e s c e n d i n g s u b j e c t i v e v a l u e s . The a b -sence o f t h e h y s t e r e s i s e f f e c t i s c h a r a c t e r i s t i c o f t h e m e t a t h e t i c c l a s s . The g e n e r a l c o n c l u s i o n , d e t e r m i n e d f r om a s y n t h e s i s o f t h e t h r e e t e s t e d f u n c t i o n a l c r i t e r i a , was t h a t one a t t r i b -u t e o f k i n e s t h e s i s , e x t e n t o f arm movement, i s s u b s e r v e d by a m e t a t h e t i c p r o c e s s . T h e r e f o r e t h e s u b j e c t i v e p e r c e p t i o n o f e x t e n t o f movement i s e s s e n t i a l l y u n i f o r m i n i t s d i s --63-crimination over the range of movement studied. Recommendations 1. Further study using psychophysical techniques, to test a series of functional c r i t e r i a for the remain-ing attributes of kinesthesis. 2. Further study of kinesthetic extent of movement on additional joints. -64-REFERENCES Adams, J . A., Response Feedback and Learning, Psychological B u l l e t i n , 70:486-504, 1968. 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T., Kinesthesis: A Model f o r Movement Feedback, New Perspectives of Man i n Action, Brown, R. C. and-Cratty, R. J . , - e d i t o r s , Englewood C l i f f s : P r e n tice-H a l l , 1969. 31-47. -66-Stevens, J . C. and Cain, W. S., E f f o r t i n Muscular Con-t r a c t i o n s Related to Force, Level, and Duration, Perception and Psychophysics, 8:240-244, 1970. Stevens, J . C. and Rubin, L. L., Psychophysical Scales of Apparent Heaviness and the Size-Weight I l l u s i o n , Perception and Psychophysics, 8:225-230, 1970. Stevens, S. S., On the Psychophysical Law, Psychological Review, 64:153-182, 1957. Stevens, S. S. and Galanter, E. H., Ratio Scales and Category Scales f o r a Dozen Perceptual Continua, Journal of  Experimental Psychology, 54:377-411, 19571 Stevens, S. S., Cross Modality V a l i d a t i o n of Subjective Scales f o r Loudness, V i b r a t i o n , and E l e c t r i c Shock, Journal  of Experimental Psychology, 57:201^209, 1959.U. Stevens, S. S., Mathematics, Measurement, and Psychophysics, Handbook of Experimental Psychology, S; S. Stevens, e d i t o r , New York: John Wiley and Sons, 1959.(b) Stevens, S. S., The Qu a n t i f i c a t i o n of Sensation, Daedabes Journal of American Academy of Arts and Science, 1959. Cc) Stevens, S. S., The Psychophysics of Sensory Function, W. A. Rosenblith, e d i t o r , Sensory Communication, New York: M.I.T. Press and Wiley, 1961.(a) Stevens, S. S., To Honour Fechner and Repeal His Law, Science, 33:80-86, 1961.(b) Stevens, S. S., The Surp r i s i n g S i m p l i c i t y of Sensory Metrics, American Psychologist, 17:29-39, 1962. Stevens, S. S., Concerning the Psychophysical Law, Quarterly  Journal of Experimental Psychology, 26: part IV, 1964. Stevens, S. S., Matching Functions between Loudness and Ten Other Continua, Perception and Psychophysics, 1:5-8, 1966. Williams, H. G., Neurological Concepts and Perceptual Motor Behaviour, New Perspectives of Man i n Action, R. C. Brown arid -B. J. Cratty, e d i t o r s , Englev/ood C l i f f s : P r e n t i c e - H a l l , 1969. Wood, H., Psychophysics of Active Kinesthesis, Journal of Experimental Psychology, 79:480-485, 1969. APPENDIX A CALCULATION OF THE K INE SCALE Calculation of the Kine Scale from the Function Log E = 1.14-9 Log S - .5014-, obtained by the Halving Method of Ratio Production. Using as a base: 1 Kine = 10 degrees or Log S = 1.000 log R = 1.14-9 (1) - .5014-log R = .646 R = 4-.4-40 corresponding to .5 Kines .5 Kines •-•= 4-.4-40 or Log S - .648 log R = .24-3 R = 1.750 corresponding to .25 Kines Transforming the preceding function to: = Log R + .5014-— r r m — = 1.000 (1 Kine) = 1.04- + .5014-1.149 = 1.307 = 20.26 corresponding to 2 Kines = 1.307 (2 Kines) = 1.307 + .5014 — r r m — = 1.574-= 37.47 corresponding to 4 Kines = 1.574 (4 Kines) = 1.57^ + .5014-I7I7+9 = 1.806 = 63.95 corresponding to 8 Kines = 1.806 (8 Kines) = 1.806 + .5014 I7Ff9 = 2.008 = 101.90 corresponding to 16 Kines = 2.008 (16 Kines) = 2.008 + .5014-1717+9 = 2.184 = 152.70 corresponding to 32 Kines log s log R log S log S S log R log S log S S log R log S log S S log R log S log S s log R log S log s s APPENDIX B INDIVIDUAL GEOMETRIC MEAN RESPONSES STIMULI 2 0 3 0 40 50 60 7 0 80 9 0 1 0 0 1 1 0 1 9 . 3 2 1 3 . 2 7 1 7 . 7 6 26 .38 32.82 44.84 45.41 55.64 6 5 . 1 3 82 . 9 7 2 9.24 1 3 . 3 9 16.64 23.28 2 9 . 6 9 3 2 . 9 3 40 . 0 5 47.84 5 6 . 0 2 60.08 3 1 1 . 9 6 1 9 . 3 6 23.90 27.62 3 6 . 4 7 4 3 . 8 7 50.18 57.46 62 . 5 0 75.62 4 1 1 . 9 6 1 5 . 7 9 24 .01 28 .39 3 7 . 8 8 41 . 4 9 5 1 . 9 6 58.81 6 5 . 5 8 6 7 . 7 8 5 1 0 . 1 0 1 5 . 8 8 2 2 . 5 7 2 7 . 6 5 3 5 . 5 0 4 3 . 2 3 52.41 5 6 . 0 2 6 6 . 3 6 74.81 6 1 1 . 0 7 16 . 3 5 2 3 . 9 8 2 9 . 6 8 3 1 . 4 3 41 . 2 0 4 5 . 7 5 5 0 . 0 0 5 6 . 2 0 5 7 . 2 7 Subjects 7 10.16 1 3 . 9 5 2 2 . 0 0 2 7 . 2 9 3 5 . 4 7 4 3 . 9 5 46 . 8 3 54.24 54.90 61.48 8 1 0 . 1 3 14 .57 2 3 . 0 3 28.24 3 5 . 5 8 4 4 . 9 3 5 0 . 8 8 61.18 6 6 . 4 4 7 8 . 6 8 9 1 0 . 5 7 16.28 2 0 . 9 7 2 5 . 6 7 3 3 . 1 3 41 .38 4 7 . 0 3 5 2 . 1 5 6 3 . 7 4 73.26 1 0 7 . 5 2 1 1 . 5 0 16.06 2 2 . 5 6 2 9 . 6 9 3 5 . 3 5 40 . 12 5 1 . 8 6 61 .89 7 0 . 7 4 CONDITION I - Ten Subjects' Geometric Mean Responses for Ten Stimuli Distances. Subjects CATEGORIES 4 5 6 7 8 1 21.55 33.49 42.78 56.82 60.49 75.05 87.28 98.74 107.38 2 18.09 39.98 52.04 61.89 74.95 87.55 91.76 97.78 111.59 3 18.87 31.49 42.60 55.45 71.01 68.41 83.72 95.82 116.38 4 21.32 36.93 55.54 72.77 83.66 89 .32 98.62 108.26 116.54 5 17.44 34.87 47.25 60.79 71.01 86.86 97.17 104.69 120.00 6 24.31 41.03 52.14 60.88 78.51 90.26 9 5 . 9 2 105.65 118.09 7 16.43 26.65 37.59 68.09 78.78 84 .95 92.21 98.99 111.29 8 44.68 46.50 56.78 68.99 71.61 79.64 95.83 105.97 115.13 9 25.32 49.92 46.50 56.20 65.81 81 .05 85 .59 92.78 104.49 10 15.60 25.83 33.57 46.58 60.08 70.94 81 .09 92.97 105.94 CONDITION I I - Ten Subjects' Geometric Mean Responses Corresponding to the Nine Categories. STIMULI Subjects 20 30 40 50 60 70 80 90 100 110 1 10.87 15.99 21.87 28.76 37.09 42.99 49.39 56.97 65.57 77.44 2 8.83 12.60 19.95 28.58 36.81 49.56 52.04 56.97 64.19 73.46 3 10.96 16.72 22.33 28.53 40.50 44.51 50.80 59.00 65.05 71.65 4 8.67 13.46 20.80 24.92 31.65 38.19 46.00 55.08 60.99 69.21 5 9.89 15.11 20.08 27.44 32.23 39.44 46.50 52.87 66.80 74.47 6 10.25 15.24- 20.57 27.21 35.33 41.66 51.16 56.78 66.07 75.16 7 9.20 13.93 20.23 25.55 31.17 38.13 42.04 48.70 51.60 57.73 8 11.19 16.31 22.89 30.27 37.79 45.66 53.42 57.46 64.37 76.55 9 10.34- 17.68 23.31 31.09 38.01 42.99 52.69 54.61 65.64 69.83 10 10.76 17.12 25.16 31.52 38.27 50.18 57.35 66.30 69.97 80.88 CONDITION I I I - Ten Subjects' Geometric Mean Responses f o r Ten S t i m u l i D i s t a n c e s . STIMULI Subjects 20 30 40 50 60 70 80 90 100 110 1 9.35 14.61 18.40 24.79 30.51 35.96 40.20 52.41 65.08 74.49 2 10.35 14.94 20.98 27.28 31.04 39.44 45.58 . 54.16 61.70 70.26 3 5.57 8.42 15.13 19.53 24.68 30.33 33.32 38.77 45.08 58.03 4 10.48 16.43 21.86 28.39 36.71 42.75 52.32 55.27 63.65 67.40 3 10.28 15.67 20.56 26.14 35.69 40.20 46.00 50.62 56.97 64.04 6 10.38 14.47 20.69 25.78 31.84 36.72 42.92 51.96 53.87 62.00 7 12.17 19.17 24.92 30.67 37.46 46.15 48.70 55.83 60.19 64.25 8 11.24 18.12 22.96 28.87 35.67 41.42 45.08 56.12 65.76 76.94 9 13.04 17.53 21.84 29.98 35.68 41.11 45.16 52.50 54.43 60.36 10 10.00 15.36 20.53 25.96 32.81 38.48 44.68 52.04 58.32 63.09 CONDITION IV - Ten Subjects' Geometric Mean Responses for Ten Stimuli Distances 

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