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On unifying the laws of sensation : an empirical investigation of predictions arising from Norwich's… Davidson, Kelly Patricia 1990

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ON UNIFYING THE LAWS OF SENSATION: AN EMPIRICAL INVESTIGATION OF PREDICTIONS ARISING FROM NORWICH'S ENTROPY THEORY OF PERCEPTION By KELLY PATRICIA DAVIDSON B.A.(Hon), Nipissing University College, Laurentian University, 1982. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PSYCHOLOGY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1990 © Kelly Patricia Davidson, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of p^vjcKolo^y  The University of British Columbia Vancouver, Canada Date / V ^ s f 3 3 , /1l0 DE-6 (2/88) A B S T R A C T i i T h e p r e s e n t thesis const i tutes a n e m p i r i c a l i nves t iga t ion o f the p r e d i c t i o n o f N o r w i c h ' s E n t r o p y T h e o r y o f P e r c e p t i o n that the pos i t i ve e x p o n e n t o f the m a g n i t u d e e s t i m a t i o n p o w e r f u n c t i o n a n d the negat ive exponents o f e q u a t i o n s r e l a t i n g the W e b e r f r a c t i o n a n d s i m p l e r e a c t i o n t i m e to s t imu lus intens i ty s h o u l d , s ince they c a n a l l b e d e r i v e d f r o m the theory 's F u n d a m e n t a l E q u a t i o n , be n u m e r i c a l l y the s a m e . A p i l o t study c o n s i s t i n g o f m a g n i t u d e e s t i m a t i o n a n d r e a c t i o n t i m e e x p e r i m e n t s (us ing p u r e t o n e a u d i t o r y s t i m u l i o f va ry ing in tens i t ies at f i ve f r e q u e n c i e s ) , a n d a " m a i n " study c o m p r i s e d o f m a g n i t u d e e s t i m a t i o n , r e a c t i o n t i m e , a n d W e b e r f r a c t i o n e x p e r i m e n t s a re d e s c r i b e d . T h e resu l ts , w h i l e o f f e r i n g poss ib le c o n f i r m a t i o n o f the p r e d i c t i o n , r e m a i n s o m e w h a t tentat i ve , o w i n g to the pers is tent ly p r o b l e m a t i c t e c h n i q u e o f c u r v e f i t t i n g u p o n w h i c h d e t e r m i n a t i o n o f the r e a c t i o n t i m e a n d W e b e r f r a c t i o n exponents rests. T h e theory , i n l e a d i n g o n e to e v e n a t tempt to c o m p a r e such p r e v i o u s l y u n r e l a t e d m e a s u r e s as m a g n i t u d e e s t i m a t i o n a n d r e a c t i o n t i m e w i t h W e b e r f r a c t i o n s , has y i e l d e d , t h e o r e t i c a l issues as ide , s o m e w o r t h w h i l e e m p i r i c a l resul ts : I have o b t a i n e d measures o n t h r e e d i f f e r e n t p s y c h o p h y s i c a l tasks f r o m the same subjects over (e f fect i ve ly ) the same s t i m u l u s r a n g e fo r e a c h o f those subjects; a n d , m o r e o v e r have , I b e l i e v e fo r the f i rst t i m e , e x p l i c i t l y n o t e d that the W e b e r f r a c t i o n d isp lays the s a m e d e c r e a s e i n exponent w i t h i n c r e a s i n g f requency , f o l l o w e d by a n u p t u r n at the h ighest f r e q u e n c i e s , that c h a r a c t e r i z e s b o t h the e q u a l loudness curves a n d the r e a c t i o n t i m e curves a l a C h o c h o l l e . i i i Sugges t ions are m a d e r e g a r d i n g s u p p l e m e n t a r y curve f i t t ing m e t h o d s by w h i c h to a n a l y z e these d a t a , as w e l l as for fu ture r e s e a r c h i n the p s y c h o p h y s i o l o g i c a l r e a l m w h i c h , i n a d d i t i o n to e x p a n d i n g the scope o f the p r e d i c t i o n that is b e i n g tes ted , m a y p r o v i d e s o m e m u c h n e e d e d ins ight in to the n u m e r i c a l v a l u e s o f the m u l t i p l i c a t i v e a n d a d d i t i v e cons tants that o c c u r i n the equat ions u n d e r c o n s i d e r a t i o n i n th is thes is . iv T A B L E O F C O N T E N T S A b s t r a c t i i T a b l e o f C o n t e n t s iv L i s t o f T a b l e s v i L i s t o f F i g u r e s v i i A c k n o w l e d g e m e n t ix I. I n t r o d u c t i o n 1 II. N o r w i c h ' s E n t r o p y T h e o r y o f P e r c e p t i o n : A s s u m p t i o n s , D e r i v a t i o n s a n d P r e d i c t i o n s 7 III. G e n e r a l B a c k g r o u n d 11 1. M a g n i t u d e E s t i m a t i o n a n d Stevens' P o w e r L a w 11 2 . R e a c t i o n T i m e 13 I V . P i l o t S tudy - M a g n i t u d e E s t i m a t i o n a n d R e a c t i o n T i m e 16 1. M e t h o d 16 1.1 Subjects 16 1.2 S t i m u l i a n d A p p a r a t u s 16 2. P r o c e d u r e 18 2.1 C o n d i t i o n 1: M a g n i t u d e E s t i m a t i o n ( M E ) 18 2.2 C o n d i t i o n 2 : R e a c t i o n T i m e ( R T ) 18 3 . R e s u l t s 2 0 4. R e c o m m e n d a t i o n s 27 V . M a i n S tudy 28 1. R a t i o n a l e 28 2. B a c k g r o u n d o n the W e b e r F r a c t i o n ( W F ) 28 3 . M e t h o d fo r E x p e r i m e n t s M E , R T a n d W F 32 3 .2 Subjects 32 3.3 A p p a r a t u s 32 3 .31 T o n e G e n e r a t o r C h a r a c t e r i s t i c s 33 3.4 S t i m u l i 37 3 .41 S e l e c t i o n o f Intensity V a l u e s 37 3 .411 Pretest : L o w e r A u d i t o r y T h r e s h o l d 37 3 .412 Pretest : U p p e r A u d i t o r y L i m i t 42 4. P r o c e d u r e 43 4.1 O r d e r o f E x p e r i m e n t s 43 4.2 W e b e r F r a c t i o n 44 4.3 R e a c t i o n T i m e 46 4.4 M a g n i t u d e E s t i m a t i o n 46 V V I . P r e l i m i n a r y A n a l y s i s o f R a w D a t a , R e r u n s , a n d P r e p a r a t i o n fo r P r i m a r y A n a l y s i s 48 1. M a g n i t u d e E s t i m a t i o n 48 2 . R e a c t i o n T i m e 55 2 .1 R e r u n s 55 3 . W e b e r F r a c t i o n 62 3 .1 R e r u n s 63 V I I . P r i m a r y D a t a A n a l y s i s a n d R e s u l t s 68 1. M a g n i t u d e E s t i m a t i o n 68 1.1 L i n e a r R e g r e s s i o n A n a l y s i s 68 1.12 R e s u l t s 69 2. T h e N o n - L i n e a r C u r v e F i t t i n g T e c h n i q u e 72 3 . R e a c t i o n T i m e a n d W e b e r F r a c t i o n 75 3.1 T h e F u n c t i o n s 75 3 .2 P r o b l e m s E n c o u n t e r e d i n the C u r v e F i t t i n g T e c h n i q u e 76 4. R e s u l t s 77 V I I I . O v e r a l l R e s u l t s a n d D i s c u s s i o n 87 I X . R e c o m m e n d a t i o n s 91 1. M e t h o d o l o g i c a l 91 2. E m p i r i c a l / T h e o r e t i c a l 92 X . C o n c l u s i o n s 97 X I . B i b l i o g r a p h y 98 X I I . A p p e n d i x 102 1. A - Inst ruct ions to Subjects 102 1.1 M a g n i t u d e E s t i m a t i o n P i l o t S tudy 102 1.2 R e a c t i o n T i m e P i l o t S tudy 103 1.3 Pretest : L o w e r A u d i t o r y T h r e s h o l d 104 1.4 Pretest : U p p e r A u d i t o r y L i m i t 105 1.5 M a i n Study : W e b e r F r a c t i o n 106 1.6 M a i n Study : R e a c t i o n T i m e 107 1.7 M a i n Study : M a g n i t u d e E s t i m a t i o n 108 vi LIST OF TABLES Table 1 Regression method magnitude estimation exponents 70 Table 2 Curve Fitting results for RT and WF - series 4 ^ Table 3 Curve Fitting results for RT and WF - series 17 or v i i L I S T O F F I G U R E S F i g u r e 1 B a s i c shapes o f va r ious p s y c h o p h y s i c a l func t ions 3 F i g u r e 2 R e l a t i o n s h i p s b e t w e e n e q u a t i o n s i n the p resent thesis 6 F i g u r e 3 P o w e r func t ions : i n f l u e n c e o f the m a g n i t u d e o f e x p o n e n t o n 12 s h a p e o f the curve F i g u r e 4 ... . . V a r i a t i o n s i n r e a c t i o n t i m e as a f u n c t i o n o f in tens i ty 14 F i g u r e 5 P i l o t R T curve f i ts : n constrained to rcMEregression values 21 F i g u r e 6 P i l o t R T curve f its: A , /} constrained; Pieron's equation; A = 0.015; fl = 0.150 22 F i g u r e 7 P i l o t R T curve f i ts : A = 6.75; 0 = 6.0 23 F i g u r e 8 P i l o t R T Curve f i ts : (RT-RL) ; R L (min. rt limit) = 150 msec; A = 18; 0= 2.5 24 F i g u r e 9 P i l o t M E a n d R T exponents p l o t t e d together 25 F i g u r e 10 ... W e b e r f r a c t i o n versus in tens i ty fo r 3 5 , 7 0 , 2 0 0 & 1000 H z tones 29 F i g u r e 11 ... W e b e r f r a c t i o n versus in tens i ty fo r 1000, 4 0 0 0 & 10000 H z tones 3 0 F i g u r e 12 ... C o m p i l a t i o n g r a p h o f W e b e r f r a c t i o n studies 31 F i g u r e 13 ... T o n e genera to r character is t ics 34 F i g u r e 14 ... E q u a l loudness contours 36 F i g u r e 15 ... G r a p h e d results o f "raw" d a t a : M E - 7 0 H z 50 F i g u r e 16 ... G r a p h e d results o f "raw" d a t a : M E - 100 H z 51 F i g u r e 17 ... G r a p h e d results o f "raw" d a t a : M E - 2 0 0 H z 52 F i g u r e 18 ... G r a p h e d results o f "raw" d a t a : M E - 1000 H z 53 F i g u r e 19 ... G r a p h e d results o f "raw" d a t a : M E - 10000 H z 54 V l l l Figure 20 ... Graphed results of "raw" data: RT and WF - 70 Hz 55 Figure 21 ... Graphed results of "raw" data: RT and WF - 100 Hz 56 Figure 22 ... Graphed results of "raw" data: RT and WF - 200 Hz 57 Figure 23 ... Graphed results of "raw" data: RT and WF - 1000 Hz 58 Figure 24 ... Graphed results of "raw" data: RT and WF - 10000 Hz 59 Figure 25 ... Weber fraction run records (example) 64 Figure 26 ... Graph of magnitude estimation linear regression results 71 Figure 27 ... Curve fitting time course: escape from a local minimum 73 Figure 28 ... Curve fitting results for RT, WF and M E : series 4 84 Figure 29 ... Curve fitting results for RT and WF: series 17 86 Figure 30 ... Obtained exponents plotted together by subject: nME, nRT & nME versus frequency 89 Figure 31 ... Physiological power functions 93 ix A C K N O W L E D G E M E N T S I w o u l d l i k e to thank m y adv iso r , D r . L a w r e n c e M . W a r d - fo r c o n c e i v i n g o f this e n t i r e p ro jec t a n d t h e n p e r m i t t i n g m e to w o r k o n it , fo r the generos i t y w i t h w h i c h he has g i v e n o f h is t i m e a n d k n o w l e d g e i n a d v i s i n g a n d assist ing m e t h r o u g h o u t e a c h o f its stages, f o r h is so re ly tested p a t i e n c e , a n d fo r the b r o a d s c h o l a r s h i p a n d l i v e l i n e s s h e b r i n g s t o h is w o r k a n d to his l a b . I w o u l d a lso l i k e to t h a n k D r . K . H . N o r w i c h , w h o s e c reat i ve w o r k is p r o v i d i n g us w i t h s u c h a u n i q u e a n d i n t e r e s t i n g e v o l v i n g theory to invest igate ; a n d , p r a c t i c a l l y , f o r m a k i n g a v a i l a b l e to us his S I M P L E X curve f i t t ing p r o g r a m , a n d v a r i o u s o t h e r m a t e r i a l s . S h u j i M o r i has b e e n a great h e l p - i nd i rec t l y t h r o u g h his w o r k o n d e d u c i n g aspects o f c o m p u t e r / e q u i p m e n t in te r faces i n the l a b ; a n d d i rect ly , b o t h i n u n r a v e l l i n g m y i n i t i a l G o r d i a n p r o g r a m m i n g p r o b l e m s , a n d i n his most generous w o r k as a v o l u n t e e r subject i n a l l o f the e x p e r i m e n t s r e p o r t e d i n this thesis . T h a n k y o u , S h u j i . T h a n k s a lso to B i l l T u r k e l fo r w i l l i n g l y shar ing his a w e s o m e k n o w l e d g e o f c o m p u t e r s , a n d f o r b e i n g a subject ; a n d to C a m e r o n K r a u s e for e n d u r i n g m a r a t h o n e x p e r i m e n t a l runs w h e n I was h o p i n g to m e e t a n e a r l i e r d e a d l i n e . F i n a l l y , a n express ion o f a p p r e c i a t i o n to a l l o f the p s y c h o p h y s i o l o g y l a b m e m b e r s w h o s e u n f a i l i n g p leasantness m a d e m y excurs ions in to the i r w o r l d e n j o y a b l e : w i t h s p e c i a l t h a n k s to A d e l l e F o r t h for h e l p i n g m e out w i t h the p o l y g r a p h ; a n d to T i m H a r p u r f o r b e i n g so u n c o m p l a i n i n g as he e x t e n d e d m o r e h e l p w i t h p r e p a r i n g F O R T R A N p r o g r a m s , l e a r n i n g a b o u t the p o l y g r a p h analys is m e t h o d o l o g y , a n d so o n , t h a n h e h a d any r e s p o n s i b i l i t y to g ive o r I any r ight to b e g i ven . I. I N T R O D U C T I O N 1 ~ UNIFICATION of the laws of sensation is the goal ~ Norwich (1990) P s y c h o p h y s i c s i s , as w e k n o w , the b r a n c h o f psycho logy that invest igates the q u a n t i t a t i v e r e l a t i o n s h i p b e t w e e n the p h y s i c a l d i m e n s i o n s o f p e r c e p t u a l ob jects a n d m e a s u r e s o f the h u m a n observer 's p e r c e p t i o n o f those objects . T h e m e a s u r e s c a n b e "consc ious ly " m e d i a t e d , as i n category j u d g m e n t , o r c a n b e u n m e d i a t e d by c o n s c i o u s j u d g m e n t , as i n s i m p l e r e a c t i o n t i m e to s t i m u l u s p r e s e n c e ; they c a n b e "d i rect " e v a l u a t i o n s o f the s t i m u l i , as i n m a g n i t u d e e s t i m a t i o n , o r " ind i rect" , as i n W e b e r f r a c t i o n s u s e d to b u i l d u p a scale o f just n o t i c e a b l e d i f fe rences . W h a t e v e r the m e a s u r e , the g o a l has b e e n to u n c o v e r a l a w f u l r e l a t i o n s h i p b e t w e e n the "subject ive" m e a s u r e a n d the m e a s u r e o f the p h y s i c a l p roper t ies that gave r ise to i t . T h i s is the s a m e t h e o r e t i c a l / s c i e n t i f i c a p p r o a c h that w e a re f a m i l i a r w i t h i n b r a n c h e s o f c l a s s i c a l phys ics . L a w s i n m e c h a n i c s c o n s i d e r the r e l a t i o n s h i p s b e t w e e n fo rces , masses , a n d m o t i o n . I n e l e c t r o m a g n e t i s m , r e l e v a n t v a r i a b l e s i n c l u d e charge , po la r i t y , e tc . ; i n the s tudy o f heat , v o l u m e , pressure , a n d so o n . In o r d e r to expose the r e l a t i o n s h i p s o f in te res t i n systems that m a n i f e s t a l l o f these d i m e n s i o n s , it is h e l p f u l to i d e a l i z e a n d s i m p l i f y those d i m e n s i o n s that a re not o f p r i m a r y interest . T o u n c o v e r the laws o f m e c h a n i c s t h e n , the scient ist chooses to d e a l o n l y w i t h u n c h a r g e d , u n p o l a r i z e d systems; l i k e w i s e , i n the study o f e lect r ic i ty , o n l y systems w i t h n o e las t i c c o m p r e s s i b i l i t y o r o t h e r m e c h a n i c a l a t t r ibutes are b rought u n d e r c o n s i d e r a t i o n . S u c h res t r i c t ions have y i e l d e d , i n the case o f m e c h a n i c s fo r e x a m p l e , N e w t o n ' s L a w (or L a g r a n g e o r H a m i l t o n ' s 2 m o r e s o p h i s t i c a t e d var iants ) , a n d i n the case o f e l e c t r o m a g n e t i s m , the M a x w e l l e q u a t i o n s - laws a n d e q u a t i o n s that a re c a p a b l e o f p r e d i c t i n g s p e c i f i c n u m e r i c a l va lues f o r o b s e r v a b l e quant i t ies r e l e v a n t to the i r u n i q u e d o m a i n s . T h e r m o d y n a m i c s , i n cont rast to the a b o v e examples o f the c l a s s i c a l s c i e n t i f i c p a r a d i g m , is a qu i te d i f f e r e n t k i n d o f theory w h i c h , ra ther t h a n c l a i m i n g p r i m a c y o v e r a p a r t i c u l a r d o m a i n o f systems o r b e i n g c a p a b l e o f spec i f ic n u m e r i c a l p r e d i c t i o n , i ns tead "sets l i m i t s o n p e r m i s s i b l e p h y s i c a l p rocesses a n d establ ishes r e l a t i o n s h i p s a m o n g a p p a r e n t l y u n r e l a t e d p r o p e r t i e s " ( C a l l e n , 1985, p.3). It does so by o p e r a t i n g at a m o r e t h e o r e t i c a l l e v e l that e m b o d i e s a n d ref lects , i n the symmetry o f its f u n d a m e n t a l laws, the m a c r o s c o p i c p r o p e r t i e s o f m a t t e r . In t h e r m o d y n a m i c s , there is a t h e o r e m (Noether ' s , i n C a l l e n , 1985, p .460) that asserts that "every c o n t i n u o u s s y m m e t r y o f the d y n a m i c a l b e h a v i o r o f a s y s t e m i m p l i e s a c o n s e r v a t i o n l a w fo r that system". T h i s is m a n i f e s t e d , fo r e x a m p l e , i n the f i rs t l a w o f t h e r m o d y n a m i c s , w h i c h refers to the s y m m e t r y o f the laws o f phys ics u n d e r s p a c e - t i m e t r a n s l a t i o n a n d r o t a t i o n , a n d the c o n s e q u e n t existence o f c o n s e r v e d energy , m o m e n t u m , a n d a n g u l a r m o m e n t u m f u n c t i o n s . W h a t N o r w i c h is exp l i c i t l y a t t e m p t i n g to d o (see N o r w i c h , 1990) is to d i s c o v e r a c o n s e r v a t i o n law, s i m i l a r to the c o n s e r v a t i o n laws o f t h e r m o d y n a m i c s , at w o r k i n the p s y c h o p h y s i c a l d o m a i n . H i s F u n d a m e n t a l E n t r o p y E q u a t i o n ( to b e e x p l a i n e d b e l o w ) represents the resul t o f this o n g o i n g a t tempt . It is w i t h i n this context that o n e c a n b e g i n to u n d e r s t a n d the u n d e r l y i n g reasons , as o n e e n c o u n t e r s t h e m , for v a r i o u s e l e m e n t s i n N o r w i c h ' s theory : f o r e x a m p l e , i n N o r w i c h ' s r e s t r i c t i o n to steady state s ignals c a n be seen a p a r a l l e l o f the 3 t h e r m o s t a t i s t i c a l c o n v e n t i o n o f rest r ic t ing a t ten t ion to systems that a re m a c r o s c o p i c a l l y s ta t ionary ( i n w h i c h case the m o m e n t u m a n d a n g u l a r m o m e n t u m are a r b i t r a r i l y r e q u i r e d to b e z e r o , l e a v i n g energy to s tand out a m o n g the s e v e n c o n s e r v e d q u a n t i t i e s k n o w n i n m e c h a n i c s - energy , a n d th ree c o m p o n e n t s each o f l i n e a r a n d a n g u l a r m o m e n t u m ) . T h e p r e r e q u i s i t e f o r w o r k i n g out the deta i ls o f a c o n s e r v a t i o n l a w is a d o m a i n that d e m o n s t r a t e s s o m e l a w l i k e re la t ionsh ips that a re n o t exp l i c i t l y l i n k e d . I n psychophys ics , the study o f h o w the intensi ty ( a n d f r e q u e n c y ) o f a s t i m u l u s r e l a t e to v a r i o u s p s y c h o p h y s i c a l m e a s u r e s p rov ides just s u c h a d o m a i n . O n a n e m p i r i c a l l e v e l , it c a n b e o b s e r v e d that as the intensi ty of a n , e.g. a u d i t o r y , s t i m u l u s inc reases , e s t i m a t i o n s o f p e r c e i v e d m a g n i t u d e increase a c c o r d i n g to a p o w e r f u n c t i o n ( e x p o n e n t < 1, cf. K r u e g e r , 1989) u p to s o m e asymptote , as does the m a g n i t u d e o f the e v o k e d p o t e n t i a l ( a p h y s i o l o g i c a l m e a s u r e ) , w h i l e s i m p l e r e a c t i o n t i m e ( R T ) ( C h o c h o l l e , 1945) a n d the W e b e r f r a c t i o n ( W F ) ( R i e s z , 1928) b o t h d i s p l a y e x p o n e n t i a l l y d e c r e a s i n g r e l a t i o n s h i p s to s t i m u l u s intensi ty . INTENSITY INTENSITY FIGURE 1 Basic shape of various psychophysical (and psychophysiological) functions. 4 T h e s i m i l a r i t y o f response curves that c a n be observed b e t w e e n s o m e o f these m e a s u r e s , p a r t i c u l a r l y b e t w e e n m a g n i t u d e e s t i m a t i o n ( M E ) o r s o m e o t h e r m e a s u r e o f sub jec t i ve in tens i ty a n d p s y c h o p h y s i o l o g i c a l curves has, o f course , no t g o n e u n n o t i c e d , a n d t h e r e h a v e b e e n v a r i o u s s tud ies that have exp l i c i t l y t r i e d to d e t e r m i n e w h e t h e r o r n o t sub jec t i ve intens i ty c a n b e p e r f e c t l y p r e d i c t e d by a g i v e n p s y c h o p h y s i o l o g i c a l m e a s u r e (eg. a u d i t o r y e n c e p h a l o g r a p h i c responses vs. M E ( D a v i s et a l , 1 9 6 8 ; M a d e l l & G o l d s t e i n , 1972; D a v i s , 1974; B o t t e et a l . , 1975) , aud i to ry E E G responses vs R T ( M a n s f i e l d , 1970, c i t e d i n Stevens , 1975) , a n d c h o r d a t y m p a n i responses vs gustatory M E s ( B o r g et a l . , 1967)) . W h a t sc ient ists inves t igat ing the h u m a n p e r c e p t u a l systems h a d not b e e n a b l e t o d o , h o w e v e r , p r i o r to N o r w i c h ' s (1977, 1978, 1981, 1984, 1989) f o r m u l a t i o n o f h i s " E n t r o p y T h e o r y o f P e r c e p t i o n " , was to l i n k a l l o f these re la t ionsh ips ( M E , R T , W F , a n d p s y c h o p h y s i o l o g i c a l m e a s u r e s ) , a n d i n fact p r e d i c t t h e m , o n a t h e o r e t i c a l l e v e l . O v e r the past t e n years , N o r w i c h has b e e n e x t e n d i n g his theory to a c c o u n t f o r m o r e a n d m o r e p s y c h o p h y s i c a l laws ( T e g h t s o o n i a n - P o u l t o n l a w ; B l o n d e l - R e y l a w f o r v i s i o n ; H u g h e s ' l a w f o r a u d i t i o n , i n a d d i t i o n to the e a r l i e r w o r k o n p s y h o p h y s i c a l a n d p s y c h o p h y s i o l o g i c a l a d a p t a t i o n curves ( N o r w i c h , 1981a) , a n d o n M i l l e r ' s " M a g i c a l N u m b e r 7" ( N o r w i c h , 1981b) ) . I n the p rocess o f inves t igat ing the ve rac i t y a n d scope o f his theory , w h i c h e v e r p s y c h o p h y s i c a l l a w he has b e e n c o n s i d e r i n g , h is p r o c e d u r e has b e e n to c o n f i r m the p r e d i c t i v e p o w e r o f the e q u a t i o n he has d e r i v e d f r o m his F u n d a m e n t a l E n t r o p y E q u a t i o n by c o m p a r i n g the e q u a t i o n ' s e x p e c t e d results to e m p i r i c a l resu l ts d o c u m e n t e d i n t h e l i t e r a t u r e . 5 G i v e n that N o r w i c h ' s theory is a l ready g r o u n d e d i n ex i s t ing e m p i r i c a l d a t a , o n e m i g h t ask w h y it is necessary to p r o v i d e any "new" d a t a against w h i c h to test h is t h e o r y at a l l . T h e a n s w e r l i es i n the v a r i a b i l i t y o f lab p r o c e d u r e s , e q u i p m e n t , s t i m u l u s charac te r i s t i cs , a n d r a n g e o f s t i m u l i , c o m b i n e d w i t h the use o f d i f f e r e n t subjects that o n e f i n d s i n ex i s t ing s tud ies . It is necessary to e l i m i n a t e these sources o f v a r i a t i o n i n o r d e r t o p r e c l u d e a p o s s i b l y e r r o n e o u s d i s c o n f i r m a t i o n o f N o r w i c h ' s theory r e s u l t i n g f r o m these fac to rs . ' I h a v e s p o k e n o f " test ing" N o r w i c h ' s theory . W h a t p r e c i s e l y a m I g o i n g to test? A l t h o u g h i t m a y s o u n d s o m e w h a t crypt ic at th is p o i n t , the p u r p o s e o f th is thes is is to test the hypothes i s that the exponent n, w h i c h a p p e a r s as the p r i m a r y p a r a m e t e r that d e t e r m i n e s the shape o f m a g n i t u d e e s t i m a t i o n , r e a c t i o n t i m e , a n d W e b e r f r a c t i o n f u n c t i o n s f o r a u d i t o r y in tens i ty (a l l o f w h i c h N o r w i c h is a b l e to d e r i v e by c o n s i d e r i n g his F u n d a m e n t a l E n t r o p y E q u a t i o n i n l ight o f v a r i o u s b o u n d a r y c o n d i t i o n s ) s h o u l d b e q u a n t i t a t i v e l y i d e n t i c a l w h e n subject and s o u n d f r e q u e n c y a re h e l d cons tant . F o r e s h a d o w i n g the a c t u a l der i va t ions , the e q u a t i o n s d e f i n i n g the M E , R T a n d W F s t a n d i n r e l a t i o n to the F u n d a m e n t a l E n t r o p y E q u a t i o n as s h o w n i n F i g u r e 2. E x p r e s s e d m a t h e m a t i c a l l y the hypothes is states that f o r any p a r t i c u l a r sub jec t/ f requency p a i r i n g . 6 FIGURE 2 Illustrating the relationships between the equations under consideration in this thesis. A l t h o u g h N o r w i c h o f t e n beg ins a d e s c r i p t i o n of his theory (as w i l l I i n the next s e c t i o n ) w i t h a d i scuss ion o f assumpt ions that are o f a m e c h a n i s t i c n a t u r e ( r e c e p t o r ce l ls s a m p l i n g o d o u r a n t s , wave lengths o f l ight o r sound , o r w h a t e v e r ) , i t s h o u l d b e b o r n i n m i n d that th is m e c h a n i s t i c d i m e n s i o n represents a separate e n t e r p r i s e , the "cor rectness" o f w h i c h is no t u n d e r i n v e s t i g a t i o n i n this thesis. I a m focuss ing o n a p r e d i c t i o n o f the theory , a n d the theory i tse l f does not rest u p o n the d iscovery o f the m e c h a n i s m s that w o u l d v a l i d a t e the a s s u m p t i o n s any m o r e t h a n the c o n s e r v a t i o n o f energy l a w d e p e n d s u p o n , f o r e x a m p l e , how, at a m i c r o s c o p i c l e v e l , the w o r k o f s t i r r i n g a l i q u i d is c o n v e r t e d i n t o heat . II. T H E E N T R O P Y T H E O R Y O F P E R C E P T I O N 7 T h e f i rst p u b l i s h e d f o r m u l a t i o n the T h e E n t r o p y T h e o r y o f P e r c e p t i o n ( E T P ) is f o u n d i n a p a p e r e n t i t l e d " O n the i n f o r m a t i o n r e c e i v e d by sensory recepto rs " ( N o r w i c h , 1977) . N o r w i c h beg ins th is p a p e r by h y p o t h e s i z i n g that a sensory n e u r o n f u n c t i o n s as i f i t w e r e a m e t a - i n f o r m a t i o n c h a n n e l , that is , as a c h a n n e l that , r a t h e r t h a n t r a n s m i t t i n g i n f o r m a t i o n a b o u t the s igna l i tself , ins tead t ransmi ts i n f o r m a t i o n a b o u t the c h a n g e i n the l e v e l o f u n c e r t a i n t y e x p e r i e n c e d by the r e c e p t o r as a resu l t o f the p r e s e n c e o f the s i g n a l , r e la t i ve to t h e state o f uncer ta in ty that p r e v a i l e d p r i o r to the p r e s e n c e o f the s i g n a l . T h i s is the f i rst p o s t u l a t e o f the theory . T h e u n c e r t a i n t y o f w h i c h N o r w i c h speaks is v i e w e d as the e r r o r assoc ia ted w i t h the r e c o r d i n g o f a s igna l by a sensory recepto r : " r e c o g n i t i o n o f the (necessary) p r e s e n c e o f a n e r r o r a s s o c i a t e d w i t h the r e c o r d e d intensi ty o f a s t i m u l u s g ives r ise to the u n c e r t a i n t y r e g a r d i n g the 1 r e a l * intensi ty" ( N o r w i c h , 1977, p .454) . T h e s e c o n d p o s t u l a t e o f the theory is that "the g reate r the m a g n i t u d e o f a m e a s u r e d ent i ty , the g reate r , i n g e n e r a l , is the abso lute e r r o r i n v o l v e d i n its m e a s u r e m e n t o r d e t e c t i o n " ( N o r w i c h , 1977, p .454) . T o i l lust rate the p r i n c i p l e , a m o n g o t h e r e x a m p l e s N o r w i c h c o n s i d e r s m e a s u r i n g a tab le versus m e a s u r i n g a r o o m : i n the f o r m e r , the e r r o r m a y b e i n the o r d e r o f 10"2 c m , w h e r e a s i n m e a s u r i n g a r o o m that is about 10 t i m e s l o n g e r t h a n the t a b l e , the e r r o r m a y be i n the o r d e r o f 10' 1 c m . W i t h the a i d o f c e r t a i n s i m p l i f y i n g assumpt ions r e g a r d i n g the r e l a t i o n s h i p s b e t w e e n m a g n i t u d e a n d u n c e r t a i n t i e s i n s ignals detec ted by a sensory n e u r o n , N o r w i c h c o n t i n u e s the e x p l i c a t i o n o f h is theory by const ruc t ing a m a t h e m a t i c a l m o d e l . F i r s t , i n o r d e r to 8 have i m m e d i a t e access to v a r i o u s t h e o r e m s of stat ist ics, he assumes that the s i g n a l in tens i t y is b e i n g " s a m p l e d " by the r e c e p t o r at d isc rete in terva ls . F u r t h e r m o r e , he a s s u m e s that the " rea l " p h y s i c a l s igna l ac tua l l y var ies f r o m m o m e n t to m o m e n t a c c o r d i n g to a n o r m a l ( G a u s s i a n ) d i s t r i b u t i o n w i t h a m e a n , n, a n d v a r i a n c e , a2. H e a l s o assumes that the m e a n a n d v a r i a n c e o f this s igna l d i s t r i b u t i o n a re r e l a t e d b y the e q u a t i o n N e x t , N o r w i c h asks us to suppose that t s a m p l e m e a s u r e m e n t s h a v e b e e n m a d e o f the in tens i t y o f s o m e s igna l d i s t r i b u t e d as above , a n d that the s a m p l e m e a n is m . B y the c e n t r a l l i m i t t h e o r e m of stat ist ics, the s a m p l e m e a n s themse lves w i l l f o r m a n o r m a l d i s t r i b u t i o n w i t h m e a n n a n d v a r i a n c e a 2 / t : O n the bas is o f the f o r e g o i n g assumpt ions , a n d the g r o u n d i n g o f h is E T P i n i n f o r m a t i o n theory , p r o b a b i l i t y theory , statist ics a n d t h e r m o d y n a m i c s ( for m o r e d e t a i l , see e s p e c i a l l y N o r w i c h , 1982) N o r w i c h d e r i v e d his F u n d a m e n t a l E n t r o p y E q u a t i o n b / i n , w h e r e b a n d n a re constants a n d n > 0. (1) (2) H = h In ( l + 0Mn/t) (3) w h e r e H is the e n t r o p y o r uncer ta in ty , e x p e r i e n c e d by a r e c e p t o r s y s t e m af te r t s a m p l e s f r o m a s i g n a l w i t h m e a n /z. 0 is a constant w h o s e deta i l s n e e d n o t c o n c e r n us. 9 F i n a l l y , N o r w i c h must m a k e another a s s u m p t i o n , this t i m e r e g a r d i n g the r e l a t i o n s h i p b e t w e e n u n c e r t a i n t y , o r ent ropy , H a n d the w a y that H is e n c o d e d by the sensory n e u r o n . H e assumes the s imples t poss ib le c o d e : that the n u m b e r o f i m p u l s e s p e r u n i t t i m e p r o d u c e d by the sensory n e u r o n is p r o p o r t i o n a l to H , that is F = k H (4) w h e r e F is the f r e q u e n c y o f n e u r o n a l i m p u l s e s , a n d k is a constant that is g reater t h a n z e r o . H e r e F is c o n s i d e r e d to be the " p e r c e p t u a l " v a r i a b l e ; it represents h o w the r e c e p t o r s y s t e m "sees" the s t imu lus . In later p a p e r s (e.g. N o r w i c h 1984, 1987) the " p e r c e p t u a l v a r i a b l e " was n o longer c o n s i d e r e d to be the f r e q u e n c y o f n e u r o n a l i m p u l s e s , b u t r a t h e r w a s t a k e n as the subject ive m a g n i t u d e o f a s t imu lus : * = k H . (5) A d i s c u s s i o n o f just w h a t the p e r c e p t u a l v a r i a b l e is (or m a y be ) c a n b e f o u n d i n N o r w i c h (1984) b u t n e e d not c o n c e r n us fu r ther i n the p resent o v e r v i e w of the theory . R e t u r n i n g to the 1977 v i e w of the p e r c e p t u a l v a r i a b l e , by i n s e r t i n g E q u a t i o n 4 i n E q u a t i o n 3 , h o l d i n g n constant a n d p l o t t i n g F as a f u n c t i o n o f the n u m b e r o f s a m p l e s o f t ( a f u n c t i o n o f t i m e ) , N o r w i c h is ab le to de r i ve the u s u a l a d a p t a t i o n c u r v e fo r sensory n e u r o n s . I n a d d i t i o n , he is a b l e to der i ve the W e b e r - F e c h n e r l a w by subs t i tu t ing E q u a t i o n 5 i n t o E q u a t i o n 3 a n d c o n s i d e r i n g the resul t o f a p p l y i n g s ignals o f re la t i ve h igh a n d cons tant d u r a t i o n to a sensory receptor to y i e l d the e q u a t i o n $ = n l o g fi + A. 10 (6) A n d f i n a l l y , this p a p e r ( N o r w i c h , 1977) a lso conta ins a d e r i v a t i o n o f the p o w e r l a w of S . S . S tevens . T h e e q u a t i o n l o g H = l o g a + n l o g /x (7) is d e r i v e d f r o m o n e o f the en t ropy equat ions a n d e n a b l e s o n e to o b s e r v e that a s i m p l e p o w e r f u n c t i o n re lates the s t imu lus intensi ty , M> to the e n t r o p y H . A d d i n g the p r e v i o u s l y m e n t i o n e d a s s u m p t i o n that the psychophys ica l v a r i a b l e " p e r c e i v e d in tens i ty " is p r o p o r t i o n a l to en t ropy H ( E q u a t i o n 5) c o m p l e t e s the e m e r g e n c e o f S t e v e n s ' l a w * = aMn. (8) T h i s c o n c l u d e s the b a s i c o u t l i n e o f N o r w i c h ' s E T P . T h e s p e c i f i c d e r i v a t i o n s that h a v e not yet b e e n p r e s e n t e d , but a re re levant to the studies w h i c h w e r e u n d e r t a k e n i n th is thesis , a re d iscussed separate ly b e l o w . III. G E N E R A L B A C K G R O U N D 11 M A G N I T U D E E S T I M A T I O N A N D S T E V E N S ' P O W E R L A W T h e t e c h n i q u e o f m a g n i t u d e e s t i m a t i o n ( M E ) represents Stevens ' s o l u t i o n to the p r o b l e m o f d i r e c t r a t i o s c a l i n g o f sensat ion . It cons is ts o f s i m p l y p r e s e n t i n g s t i m u l i t o o b s e r v e r s a n d a s k i n g t h e m to assign n u m b e r s to t h e m that s e e m e d to c o r r e s p o n d to t h e i r s e n s a t i o n s ( G e s c h e i d e r , 1985). T h e r e is a d i s t i n c t i o n i n M E m e t h o d s to b e m a d e o n the bas is o f the i n s t r u c t i o n s that a r e g i v e n to the subject . In the ra t io M E m e t h o d , o n e s t i m u l u s is p r e s e n t e d as a s t a n d a r d a n d ass igned a n u m e r i c a l v a l u e ( m o d u l u s ) by the e x p e r i m e n t e r . T h e subject assigns n u m b e r s to subsequent s t i m u l i (wh ich s o m e t i m e s i n c l u d e the s t a n d a r d ) so as to r e p r e s e n t t h e i r j u d g e d r a t i o w i t h respect to the s t a n d a r d . T h e subject is f ree to use any n u m b e r s , i n c l u d i n g f rac t ions , fo r responses, a n d n o t h i n g is m e n t i o n e d a b o u t e q u a l i n t e r v a l s b e t w e e n sca le va lues . In a n a l te rnat i ve m e t h o d , r e f e r r e d to as the a b s o l u t e M E m e t h o d , a l t h o u g h there are s t i l l n o l i m i t a t i o n s p l a c e d u p o n the subject as to w h a t n u m b e r s m a y b e u s e d , b u t n o s tandard is p r e s e n t e d a n d the subject assigns any n u m b e r s , as l o n g as they re f lec t j u d g e d rat ios a m o n g s t i m u l i ( B a i r d & N o m a , 1978) . S t e v e n s ' p o w e r l a w is b a s e d u p o n the f i n d i n g that m a g n i t u d e e s t i m a t i o n s f o r a var ie ty o f sensory d i m e n s i o n s i n c r e a s e i n p r o p o r t i o n to the s t imu lus in tens i ty r a i s e d t o a p o w e r (S tevens , 1975) . T h e s ize o f the p o w e r exponent to w h i c h s t i m u l u s in tens i ty is r a i s e d i n o r d e r to p r e d i c t m a g n i t u d e es t imat ions changes d e p e n d i n g o n sensory m o d a l i t y a n d s t i m u l u s c o n d i t i o n s . E x a m p l e s o f p o w e r curves f r o m d i f f e r e n t m o d a l i t i e s , c h a r a c t e r i z e d b y e x p o n e n t less t h a n , e q u a l to , a n d greater t h a n o n e a re d e p i c t e d i n F i g u r e 3 . 12 0 10 20 30 40 50 60 70 80 90 100 Stimulus magnitude FIGURE 3 Power functions: influence of the magnitude of exponent on the shape of the curve. (From Coren & Ward, 1989, p.46) The value of the exponent relating intensity to perceived loudness is given by Stevens (1975) as 0.66 in sound pressure units (0.33 in sound intensity units) although Krueger (1989) suggests that the method of ME may overestimate the value of the exponent by twice that obtained by other methods of determining subjective intensity. The derivation of Stevens' power law from Norwich's Fundamental Entropy Equation has already been presented. To recapitulate, Stevens' law is expressed mathematically as * = a/xn in Norwich's symbols, or, changing /i to I to be consistent with the usual symbol for stimulus intensity in psychophysics and * to ME to reflect Stevens' emphasis on positivistic measures, ME = al\ (9) REACTION TIME Reaction time is defined as the time between the onset of a stimulus and the beginning of an overt response, whereas simple reaction time involves a subject's pressing a key or button immediately upon detection of a stimulus (Norwich, 1989). Simple reaction time is known to grow shorter with increasing stimulus intensity, regardless of modality, asymptoting at a certain minimum value. For example, Figure 4 below shows some classical data relating simple reaction time to tones of differing frequency and intensity (Chocholle, 1945). Pieron (1952; cited in Norwich, 1989) demonstrated that the relationship between reaction time and intensity is a power law t r - t m i n = c r " (io) where tr is reaction time t ^ is minimum observable reaction time, and C and n are constants greater than zero. 14 FIGURE 4 Variations in reaction time (in milliseconds) as a function of intensity (in dBs above threshold) for frequencies indicated on the curves: 50, 250, 1000 and 10000 Hz. (A) Top left - results for subject I. (C) Bottom left - results for subject II. (B) Top right - Chocholle's averaging of results for each frequency from subjects I to HI (subject HI alone not presented). (D) Bottom right - reaction times as a function of the average loudness (in phones above threshold) for the 3 subjects, averaged across the four frequencies. The data for subjects I and II overlap with each other, hence they are represented together by the same curve, while the curve for subject HI appears separately. (Chocholle, 1945, pp.96-97) 15 I n cont ras t to the suggest ions that the p h e n o m e n a d e s c r i b e d by this r e l a t i o n s h i p a re a n e x p r e s s i o n o f energy s u m m a t i o n or i n t e g r a t i o n , a n d i n c o n c e r t w i t h the t e n o r o f h is theory , N o r w i c h p r o p o s e s ins tead that w h a t is b e i n g s u m m a t e d is no t energy b u t i n f o r m a t i o n ( N o r w i c h , 1989) . N o r w i c h (1989) beg ins by r e f e r r i n g to H i c k ' s l a w , w h i c h s h o w e d that f o r c h o i c e r e a c t i o n t i m e , w h e r e a subject has to se lect o n e f r o m a m o n g m c h o i c e s , the m e a n c h o i c e r e a c t i o n t i m e w a s p r o p o r t i o n a l to l o g ( m + 1), w h i c h is a p p r o x i m a t e l y e q u a l to the i n f o r m a t i o n r e q u i r e d to m a k e the co r rec t c h o i c e . It has a lso b e e n s h o w n ( N o r w i c h c i tes H y m a n , 1 9 5 3 ; a n d H e l l y e r , 1963) that the t i m e r e q u i r e d fo r a subject to reac t to a c o m p l e x task is a l i n e a r f u n c t i o n o f the n u m b e r o f bi ts o f i n f o r m a t i o n i n v o l v e d i n the task. N o r w i c h extends this p r i n c i p l e to g o v e r n r e a c t i o n to a s ing le , steady s t i m u l u s a n d succeeds i n d e r i v i n g the f o l l o w i n g e q u a t i o n for s i m p l e R T f r o m his E q u a t i o n 3 ( a g a i n r e p l a c i n g /x w i t h I): 1 RT = (11) A - n'n F u t h e r m o r e , N o r w i c h is a b l e to d e r i v e P i e r o n ' s l a w i tse l f f r o m the E q u a t i o n 3 (see N o r w i c h , 1989) . B e f o r e l e a v i n g this b r i e f d i scuss ion o f r e a c t i o n t i m e , it s h o u l d b e n o t e d that e q u a t i o n (11) is p a r t l y d e f i n e d by a n exponent n. IV. PILOT STUDY - Magnitude Estimation and Reaction Time 16 A pilot study was conducted in order to test the hypothesis that the numerical value of the exponent that is found in the equation derived from ETP to predict magnitude estimation (Equation 9) is quantitatively equal to the exponent found in Norwich's reaction time equation (Equation 11). METHOD Subjects. Five students from the University of British Columbia with no known hearing defects participated in both conditions for pay. Four were males and one was female. All data from one of the male subjects was discarded when it was found that he was unable to consistently detect some of the low frequency-low intensity stimuli. Stimuli and Apparatus. A PDP 11/34A computer was used to control a tone generator to produce the stimuli, to control the timing of stimuli presentation, to provide randomization of stimuli when necessary, and to record responses from subjects. Stimuli consisted of sine-wave signals at 70, 100, 200, 1000 and 10000-Hz. Sound meter limitation precluded the selection of sound intensities below 35-dB for the lower frequencies, while pain thresholds constrained the upper sound intensity limit for the higher frequencies to less than 105-dB. Working within these limitations, and attempting to obtain the maximum range of 10 logarithmically equally spaced intensities for each frequency resulted in use of the following stimulus intensities: 17 Range (dB) 70-Hz: 56 to 101-dB in 5-dB increments; 49 100-Hz: 49 to 103-dB in 6-dB increments; 54 200-Hz: 44 to 98-dB in 6-dB increments; 54 1000-Hz: 35 to 98-dB in 7-dB increments; 63 10000-Hz: 35 to 89-dB in 6-dB increments. 54 The signals were delivered monotically through stereo earphones (Koss Pro-4AAA Plus). The sound intensities were measured at the earphones in a custom made artifical ear with a precision sound-level meter (General Radio). The experimenter exercised control over the presentation of all stimuli for the RT condition, while in the ME condition, the experimenter selected the stimulus frequency and the computer system determined the randomization of the stimulus intensities. In both conditions, the computer system recorded the responses which, in the ME condition, were indicated by subjects on a standard computer keyboard, and in the RT condition, by depression of a telegraph key. In the RT condition, the stimulus was terminated by the computer as soon as the response was registered. For each subject, all trials in one condition were completed before beginning trials in the second condition. The order of condition presentation was alternated and randomly assigned to the subjects. P R O C E D U R E C o n d i t i o n 1: M a g n i t u d e E s t i m a t i o n T h e subjects served i n 10 a p p r o x i m a t e l y 1/2-hr sessions. T h e e x p e r i m e n t a l sessions w e r e p r e c e d e d by a p rac t i ce session o f 100 t r ia ls u s i n g the 1 0 0 0 - H z t o n e at t h e 10 i n t e n s i t i e s o u t l i n e d above . T h e f i rst f i ve e x p e r i m e n t a l sessions p r e s e n t e d t h e f i ve f r e q u e n c i e s (w i th assoc ia ted intensi t ies) i n r a n d o m o r d e r . T h e s e c o n d f i ve sess ions p r e s e n t e d the s a m e f ive f requenc ies i n a d i f fe ren t r a n d o m o r d e r . I n b o t h cases , the r a n d o m o r d e r s d i f f e r e d fo r each subject . T h e m o n a u r a l s t i m u l i w e r e p r e s e n t e d to the s a m e ear as e a c h subject 's p r e f e r r e d h a n d f o r r e s p o n d i n g . S t i m u l i w e r e p r e s e n t e d for 1-sec e a c h . E a c h sess ion c o n s i s t e d o f 3 0 0 t r ia l s , w i t h a n to ta l o f 600 tr ia ls fo r e a c h o f the 5 f r e q u e n c i e s f o r every subject . I n s t r u c t i o n s g i ven to subjects fo r e s t i m a t i n g m a g n i t u d e w e r e a d a p t e d f r o m Z w i s l o c k i a n d G o o d m a n (1980) , (see A p p e n d i x A - l ) . C o n d i t i o n 2 : R e a c t i o n T i m e E a c h subject se rved i n a tota l o f 5 0 a p p r o x i m a t e l y 5 m i n u t e sess ions , g e n e r a l l y c o m p l e t e d i n g roups o f fou r or f ive . A g a i n , the m o n a u r a l s t i m u l i w e r e p r e s e n t e d to the e a r o n t h e s a m e s ide as the p re fe r red response h a n d . T h e s t i m u l u s was t e r m i n a t e d as s o o n as the subject m a d e h is/her response . A p r a c t i c e sess ion o f 5 0 t r ia l s p r e c e d e d e x p e r i m e n t a l sessions o f 110 tr ials e a c h . T h e o r d e r o f the 50 sessions (5 f r e q u e n c i e s at 10 i n t e n s i t i e s ) was q u a s i - r a n d o m l y d e t e r m i n e d a n d d i f f e r e d f o r e a c h subject . T h e f i rst 10 t r ia l s o f e a c h sess ion w e r e des igned as ' w a r m - u p ' t r ia ls a n d w e r e d i s c a r d e d b e f o r e any f u r t h e r ana lys i s o f the d a t a . T h e r e m a i n i n g 100 t r ia ls w e r e g r o u p e d by a c o m p u t e r 19 p r o g r a m i n t o twenty e q u a l s i z e d b ins w h i c h w e r e d e t e r m i n e d by the range o f the R T s . T h e d i s t r i b u t i o n was i n s p e c t e d by the e x p e r i m e n t e r , o b v i o u s o u t l i e r s o n b o t h s ides o f the d i s t r i b u t i o n d i s c a r d e d , a n d the b i n sizes r e c a l c u l a t e d by the c o m p u t e r u s i n g the n e w m a x i m u m a n d m i n i m u m R T va lues . W i t h the d e c r e a s e d b i n s izes , m o r e ou t l i e r s w o u l d o f t e n b e c o m e a p p a r e n t a n d b e d i s c a r d e d , a n d the p rocess w o u l d b e r e p e a t e d (usua l l y 3 o r 4 t i m e s ) u n t i l the m e d i a n o f the d i s t r i b u t i o n a p p r o a c h e d the m e a n as c lose l y as p o s s i b l e . A l l R T s less t h a n 85 w e r e d i s c a r d e d s ince th is was c o n s i d e r e d the p h y s i o l o g i c a l l i m i t f o r R T - t h o u g h i n m a n y cases, if, f o r e x a m p l e , a m i n i m u m v a l u e o f 110 m s e c w a s c lea r l y a n o u t l i e r , i t too w o u l d be d i s c a r d e d . C a r e was t a k e n to r e a c h a c o m p r o m i s e b e t w e e n f i n d i n g a p o i n t at w h i c h the m e d i a n e q u a l l e d the m e a n , a n d not d i s c a r d i n g so m a n y d a t a p o i n t s as to v io la te the rea l i t y o f the o b t a i n e d d i s t r i b u t i o n . T h e f i n a l n u m b e r o f t r ia ls r e t a i n e d i n e a c h c o r r e c t e d d i s t r i b u t i o n was u s u a l l y b e t w e e n 85 a n d 9 5 . T h e m e d i a n o f the c o r r e c t e d d i s t r i b u t i o n was the v a l u e u s e d to rep resent the ent i re d i s t r i b u t i o n i n the f o l l o w i n g resul ts . O u t o f the 2 0 0 sess ions c o m p l e t e d by the 4 subjects , 17 ( d i s t r i b u t e d a m o n g a l l 4) g e n e r a t e d a n o m a l o u s resul ts w h i c h the o r d e r o f p r e s e n t a t i o n suggested m a y have b e e n d u e to fa t igue . T h e s e sessions w e r e r u n a g a i n , a n d the resul ts w h i c h a re r e p o r t e d i n c l u d e the r e - r u n v a l u e s i n p l a c e o f the o r i g i n a l ones . 2 0 R E S U L T S T h e v a l u e s fo r the exponents , o n e at e a c h o f the f ive f r e q u e n c i e s , o b t a i n e d i n the M E e x p e r i m e n t , a v e r a g e d across the fou r f i n a l subjects , a re r e p o r t e d i n the tab les at the b o t t o m o f F i g u r e s 5 - 8 , u n d e r the c o l u m n n M E . T h e s e va lues a lso a p p e a r as the c i rc les i n F i g u r e 9 , w h e r e they c a n be c o m p a r e d w i t h resul ts f r o m a n e a r l i e r M E s tudy ( W a r d , 1990) . C o m p a r i s o n o f the two sets o f resul ts i nd i ca tes that the exponents o b t a i n e d i n the p i l o t s tudy s h o w g o o d agreement fo r the h i g h f requenc ies , but a p p e a r t o b e t o o l o w f o r the l o w e r f r e q u e n c i e s . I expect that this m a y b e d u e to the g e n e r a l u n r e l i a b i l i t y o f resu l ts o b t a i n e d v i a the m e t h o d of m a g n i t u d e e s t i m a t i o n (cf. K r u e g e r , 1989) . R e g a r d i n g the R T e x p e r i m e n t , the resul ts o f v a r i o u s curve f i t t ing p r o c e d u r e s u s e d to e s t i m a t e n R T c a n be f o u n d i n g r a p h f o r m i n F i g u r e s 5 - 8 . T h e curve f i t t i n g w a s a c c o m p l i s h e d u s i n g N o r w i c h ' s (1985, u n p u b l i s h e d ) S i m p l e x C u r v e F i t t i n g P r o g r a m , b a s e d o n the m e t h o d o f N e l d e r & M e a d (1965) . S i n c e , as m e n t i o n e d prev ious ly , R T is d e s c r i b e d i n te rms of a th ree p a r a m e t e r f u n c t i o n : l RT = (11) A - prn a n d t h e r e a re a n i n f i n i t e n u m b e r o f poss ib le so lu t ions to the f u n c t i o n i f a l l t h r e e p a r a m e t e r s a re lef t f ree to vary, it was necessary to e x p e r i m e n t w i t h t h e o r e t i c a l l y j u s t i f i e d p l a c i n g o f const ra in ts u p o n o n e o r m o r e o f the p a r a m e t e r s a n d t h e n f i t t i n g the R T d a t a to E q u a t i o n 11. T h e curves i n F i g u r e 5 w e r e o b t a i n e d by c o n s t r a i n i n g n f o r FIGURE 5 PILOT STUDY RESULTS: Pilot RT curve fits - n constrained to nME regression values 22 FIGURE 6 PILOT STUDY RESULTS: Pilot RT curve fits -A,fi constrained; Pieron's equation; A = 0.015; f3 = 0.150 £n4 t-z o r w c c U f» it ' fe » » ' *> iMTENsrrfU* 1' z ~iS—'—55—'—m iNTtWSITT (tli <. 1 m V I", K 6 -i N T f W S l T T Utt "Si 1 b z K z 1000 ft • 6 {» INTENSITY "Ss % 1 •i , r nutuuer lQOOOMt * ft tb ' K B * ~ INTENSITY U«) "ft) HZ n n SS f — _»£ _ t t jealx lO 70 .46 .99 9.08 100 .44 .73 3.41 200 .41 .50 ' 6.34 1000 .35 .24 18.64 10000 .37 .17 7.72 FIGURE 8 PILOT STUDY RESULTS: Pilot RT curve fits - (RT-RL); RL (reaction time limit) = 150 msec; A = 18; $ = 2.5 z lb ' To ft ' ft INTENSITY <««> -» ft » « sr INTENSITY Utb i n 1 £ S ' I N T I M V T T l i t ) fHfOUENCT 1000 rk u M i* ' ici" m IMTtNsrrf uta £ ""I 1Q000& ~B— ti ' S * «o 5B INTENSITY !••) Hz n n 88 r — _»_. _ i e o U l 0 70 .46 .76 11.05 100 .44 .56 9.77 200 .41 .43 4.52 1000 .35 .25 19.33 10000 .37 .19 7.60 FIGURE 9 PILOT STUDY RESULTS: Pilot ME and RT exponents plotted together log frequency 26 each frequency to the respective n obtained in the ME experiment. Figure 6 shows the "A,B constrained Pieron" condition, in which the RT function to which the data was fitted was Pi6ron's law (RT = AI" + /J). In this condition, A was constrained to 0.015, and to 0.150. In Figure 7 are the results for Equation 11, constraining A to 6.75 and $ to 0.60. Finally, in Figure 8 are the curves obtained by subtracting a constant "motor" response latency (estimated to be 0.150 sec) from the total RT (that is, RT - RL) for all of the RT data, forcing A = 18, f3 = 2.5, and fitting Equation 11. The values obtained for n R T using each of these methods is plotted on the graph in Figure 9, where comparison with the ME exponents suggests that all of the methods yielded RT exponents that meet expectations for the low frequencies, but are too low for the high frequencies - or, in other words, the curves for the higher frequencies are lower than either theory or previous data would lead one to expect. The most likely explanation for this result is that the dynamic range was insufficient for these higher frequencies; more specifically, the lowest intensity values used were not close enough to threshold for these frequencies in order to obtain the longer RTs that are associated with near threshold intensities, and which in turn are responsible for the increase in the absolute value of the exponent. R E C O M M E N D A T I O N S T h e resu l ts o f the p i l o t study w e r e c o n s i d e r e d suf f i c ient l y e n c o u r a g i n g to m e r i t f u r t h e r i n v e s t i g a t i o n . T w o r e c o m m e n d a t i o n s fo r i n c o r p o r a t i o n i n t o the f i n a l p ro jec t w e r e p r o p o s e d : 1) T h a t t h e d y n a m i c ranges b e m a d e as w i d e as p o s s i b l e . T h i s w o u l d m e a n that b o t h the a b s o l u t e a u d i t o r y t h r e s h o l d a n d the u p p e r a u d i t o r y t h r e s h o l d ("highest t o l e r a b l e intensity") w o u l d n e e d to b e e x p e r i m e n t a l l y d e t e r m i n e d f o r e a c h subject at e a c h f requency . 2) T h a t t h e d e g r e e o f r e l i a n c e u p o n results o b t a i n e d v i a the l i n g e r i n g l y c o n t r o v e r s i a l M E m e t h o d be r e d u c e d by i n t r o d u c i n g a n a d d i t i o n a l e x p e r i m e n t - o n e that w o u l d y i e l d W e b e r F r a c t i o n s as the d e p e n d e n t m e a s u r e . N o t o n l y w o u l d this m a k e the s tudy o f the g e n e r a l hypothes is p o s s i b l e , it w o u l d a lso ex tend the r e a c h o f that h y p o t h e s i s . V . M A I N S T U D Y R A T I O N A L E T h e m a i n study was d e s i g n e d to i n c o r p o r a t e the r e c o m m e n d a t i o n s m a d e i n the p i l o t study. It w o u l d cons is t , t h e r e f o r e , o f a r e p l i c a t i o n o f the M E a n d R T s tud ies that this t i m e w o u l d u t i l i z e s t i m u l u s in tens i t ies that s p a n n e d the p a r t i c u l a r subject 's a u d i t o r y range fo r e a c h f r e q u e n c y l e v e l that was b e i n g tested . In a d d i t i o n , it w o u l d i n c l u d e a W e b e r F r a c t i o n e x p e r i m e n t , the b a c k g r o u n d for w h i c h f o l l o w s . B A C K G R O U N D O N T H E W E B E R F R A C T I O N T h e W e b e r f r a c t i o n , a l s o t e r m e d the W e b e r constant , the W e b e r - F e c h n e r r a t i o o r the d i f f e r e n t i a l sensi t iv i ty o f the ear , is d e f i n e d as the ra t io o f the m i n i m u m p e r c e p t i b l e i n c r e m e n t i n s o u n d in tens i ty to the s ta nda rd intensi ty at w h i c h the m i n i m u m p e r c e p t i b l e i n c r e m e n t is m e a s u r e d , a n d is r e p r e s e n t e d as A l / I . A l t h o u g h the W e b e r f r a c t i o n fo r aud i to ry intensi ty h a d b e e n i n v e s t i g a t e d p r e v i o u s l y by o thers (see K n u d s e n , 1923) , the c lass ic p a p e r by R i e s z (1928) w a s the f i rst to use the s a m e subjects to m e a s u r e A l / I over p rac t i ca l l y the ent i re range o f f r e q u e n c i e s a n d in tens i t ies for w h i c h the ea r is c a p a b l e o f sensat ion . C u r v e s r e p r e s e n t i n g the averages across a l l twe lve subjects s h o w e d that at any f requency , A l / I is p r a c t i c a l l y constant fo r in tens i t ies g reater t h a n 10 t imes the th resho ld intensity , b u t that n e a r the a u d i t o r y t h r e s h o l d , A l / I i nc reases . R i e s z f o u n d that W e b e r ' s law, w h i c h states that A l / I = constant , h o l d s a b o v e th is intensi ty , w i t h the v a l u e o f A l / I = cons tant l y i n g b e t w e e n 0.05 a n d 0 .15, d e p e n d i n g o n the f requency . A s a f u n c t i o n o f f r e q u e n c y , A l / I is a m i n i m u m at a b o u t 2 5 0 0 H z (a f r e q u e n c y that co r responds to the r e g i o n o f greatest a b s o l u t e sens i t i v i ty o f the ear ) , the m i n i m u m b e i n g m o r e sharp ly d e f i n e d at l o w in tens i t ies t h a n it is at h i g h . T h e f o l l o w i n g two F i g u r e s (10 & 11) are r e p r o d u c e d f r o m R i e s z ' s 1928 p a p e r , a n d s h o w the average curves o f d i f f e r e n t i a l sens i t iv i ty ( f o r d i f f e r e n t f r e q u e n c i e s ) as a f u n c t i o n o f intensity : — T j t ! ! 1 1— 1 I 1 • i ' I \ I ! \ ! \ i \ 1 ! I"* i < M \ ! 1 Ma ! t j !S I INTENSITY O iDGo \ FIGURE 10 Weber fraction (differential sensitivity) versus intensity (in dBs) for frequencies: 35, 70, 200 and 1000 Hz. ( R i e s z , 1928, p.872) D e s p i t e the u n u s u a l use of the m e t h o d o f beats , w h i c h has b e e n suggested as c o n t r i b u t i n g to the d i f f i cu l ty that e x p e r i m e n t e r s h a v e h a d i n r e p l i c a t i n g his resu l ts , R i e s z ' s s tudy is s t i l l c o n s i d e r e d a n ex t remely i m p o r t a n t c o n t r i b u t i o n , a n d is s t i l l w i d e l y c i t e d , so i t is n o t u n r e a s o n a b l e for N o r w i c h (1987) to have used the d a t a r e p o r t e d i n R i e s z ' s s tudy to test the p red ic t ions o f the o r i g i n a l en t ropy theory e x p r e s s i o n fo r the W e b e r F r a c t i o n that he de r i ved f r o m his E q u a t i o n 3 . 30 3b > § z 1 1 i i i ! ! i 1 1 1 1 ' \ j i i 1 1 i V Vo.ooo VA 7 0 0 0 cp» i I I | 1 1 \\ 1 ; ' i .... i I ' V \\ M M i i = 1 1 i 1 A (V>ft r - - - ^ ^ ^ ^ - - t r 1 i 1 i ; 1 ! [_ ' i l l INTENSITY lOUDOior FIGURE 11 Weber fractions plotted against intensity (in dBs) for the frequencies: 4000, 7000 and 10000. (Riesz, 1928, p.871) N o r w i c h (1984) d e r i v e d h is W F e q u a t i o n d i rect ly f r o m E q u a t i o n 3 to y i e l d the f o l l o w i n g : Al = aH(0' + l' n) . (12) O n c e a g a i n , the r e a d e r ' s a t t e n t i o n is d r a w n to this n e w m a n i f e s t a t i o n o f the exponent n. T h e r e is a great d e a l that c o u l d be discussed w i t h r e g a r d to the W e b e r F r a c t i o n , but m u c h o f i t w o u l d no t b e d i rec t l y re levant to the p u r p o s e o f th is thes is , w h i c h is to e m p i r i c a l l y test a n hypothes is that fo l lows f r o m the E T P . It is no t p o s s i b l e , h o w e v e r , to c o n c l u d e this segment w i t h o u t p o i n t i n g out that a n u m b e r o f s tud ies c o n d u c t e d over the past 10 to 20 years h a v e f a i l e d t o o b t a i n results that a re as b e a u t i f u l l y b e h a v e d as are R i e s z ' s (see fo r e x a m p l e R a b i n o w i t z et a l , 1976; J e s t e a d t et a l , 1977; C a r l y o n & M o o r e , 1984; F l o r e n t i n e et a l , 1 9 8 7 ; a n d V a l t e r , 1988). G r e e n (1976, p .257) p resents a c o m p i l a t i o n g raph (o r i g ina l l y f r o m L u c e & G r e e n , 1974, 31 p. 1559) o f s o m e p rev ious W F study results . T h e g r a p h has b e e n r e p r o d u c e d b e l o w i n F i g u r e 12. 0 20 40 60 80 100 IdBSL FIGURE 12 Summary of several studies of Weber fraction using a signal frequency of 1000 Hz: (•) Riesz (1928); (o) Harris (1963); (a) Harris (1963); (A) McGill and Goldberg (1968); f>; Campbell and Lasky (1967); (m) Luce and Green (1974). (From Luce & Green, 1974, p.1559) O n the bas is o f p rev ious re lat i ve ly recent s tudies t h e n , I s h o u l d be q u i t e p l e a s e d i f t h e W F d a t a to b e o b t a i n e d i n the present study p r o d u c e a n y t h i n g l i k e the s m o o t h n e s s o f R i e s z ' s cu rves . M E T H O D F O R E X P E R I M E N T S M E , R T , A N D W F Subjects . F o u r s tudents f r o m 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 n c l u d i n g the a u t h o r , p a r t i c i p a t e d i n a l l t h r e e e x p e r i m e n t s . T h r e e w e r e m a l e s a n d o n e was f e m a l e . S M , B T , a n d K D w e r e P s y c h o p h y s i c s L a b m e m b e r s . S M was a v o l u n t e e r i n the study, B T a n d K D w e r e a l r e a d y e m p l o y e d by the L a b , a n d C K , f r o m the L a b ' s subject p o o l p a r t i c i p a t e d f o r pay . S M was the most e x p e r i e n c e d subject (over 6 years i n the f i e l d o f psychophys ics as r e s e a r c h e r a n d subject) . B o t h S M a n d K D h a d p a r t i c i p a t e d as subjects i n the p i l o t study. F o r the r e m a i n d e r o f this thesis , w h e r e contex t a l l o w s , B T w i l l be s i m p l y r e f e r r e d to as " B " , C K as C , K D as K , a n d S M as S . A p p a r a t u s . A H e w l e t t - P a c k a r d V e c t r a R S / 1 2 c o m p u t e r was u s e d to c o n t r o l the t i m i n g o f p r e s e n t a t i o n o f s t i m u l i , to p r o v i d e r a n d o m i z a t i o n o f s t i m u l i w h e n necessary , a n d to r e c o r d reponses f r o m subjects . O t h e r e q u i p m e n t was the s a m e as that u s e d i n the p i l o t s tudy : a u d i t o r y s t i m u l i w e r e d e l i v e r e d m o n o t i c a l l y t h r o u g h s te reo h e a d p h o n e s ( K o s s P r o - 4 A A A P l u s ) a n d the s o u n d intensi t ies m e a s u r e d at the e a r p h o n e s , p r i o r to c o n d u c t i n g any o f t h e e x p e r i m e n t s , w i t h a p r e c i s i o n s o u n d - l e v e l m e t e r ( G e n e r a l R a d i o ) ; s ignals w e r e c r e a t e d by the L a b ' s cus tom bu i l t tone g e n e r a t o r , s o m e deta i l s o f w h i c h w i l l be d iscussed b e l o w ; responses w e r e e n t e r e d by the subject o n a s t a n d a r d c o m p u t e r k e y b o a r d i n the M E a n d W F cond i t ions , a n d i n the R T c o n d i t i o n by d e p r e s s i o n o f a t e l e g r a p h key w h i c h i m m e d i a t e l y t e r m i n a t e d the s t i m u l u s . In a l l e x p e r i m e n t s the subject w a s s e a t e d i n a n a c o u s t i c a l c h a m b e r (T racoust ics R E - 1 4 2 C ) . 33 T o n e G e n e r a t o r C h a r a c t e r i s t i c s . T h e o v e r r i d i n g r e q u i r e m e n t f o r the t o n e g e n e r a t o r was that it s h o u l d b e a b l e to p r o d u c e tones over the f u l l range o f s t i m u l i a n d in tens i t i es that m i g h t b e n e e d e d across the s p e c t r u m o f a u d i t o r y psychophys i ca l e x p e r i m e n t s . T h i s it d o e s . A c h i e v i n g this necessary k i n d o f f l ex ib i l i t y w i t h i n c e r t a i n t e c h n i c a l a n d o t h e r cons t ra in ts , t h o u g h , e n t a i l e d a m i n o r t radeof f . T h e resu l t o f this t r a d e o f f is to ta l l y i r r e l e v a n t f o r the intens i ty levels a n d e x p e r i m e n t a l p a r a d i g m s that t yp i ca l l y h a v e b e e n a n d a re b e i n g u s e d i n the L a b . It was o n l y the p e c u l i a r c o n f l u e n c e o f a m e t h o d o l o g y ( a d a p t i v e s ta i rcase W e b e r F r a c t i o n ) a n d a n e x t r e m e s t i m u l u s range ( n e a r t h r e s h o l d ) that that b r o u g h t this m i n o r l i m i t a t i o n i n t o v iew , a n d e v e n t h e n , on ly i n s o m e ins tances . T h e s o u r c e o f the l i m i t a t i o n is th is : O f two e ight b i t registers , a l l e igh t b i ts o f the o n e , r e p r e s e n t i n g 2 " (2048) t h r o u g h 2 3 (8) , a n d th ree bits o f the o t h e r ( 2 2 t o 2°) a re u s e d t o e n c o d e a p o t e n t i a l 2 0 4 8 leve ls o f vo l tage a t t e n u a t i o n . E a c h o f the p o s s i b i l i t i e s is r e p r e s e n t e d by o n e 1 c o m p u t e r n u m b e r 1 (CN). E n t e r i n g , fo r e x a m p l e , t h e m a x i m u m c o m p u t e r n u m b e r o f 2 0 4 8 w o u l d t u r n o n a l l 11 b i ts , a n d there w o u l d b e n o a t t e n u a t i o n o f the v o l t a g e . T h i s m a x i m u m vo l tage w o u l d b e a m p l i f i e d , a n d the m o s t i n t e n s e tone that the g e n e r a t o r is c a p a b l e o f c r e a t i n g w o u l d b e s o u n d e d . C o r r e s p o n d i n g l y , a c o m p u t e r n u m b e r o f 1 w o u l d resul t i n the m a x i m u m a t t e n u a t i o n o f the v o l t a g e to z e r o , a n d n o s o u n d w o u l d b e p r o d u c e d , m u c h less h e a r d by the h u m a n ear . F i g u r e 13, shows h o w the C N s covary w i t h b o t h resu l tant intensi ty l e v e l , i n d B ( e x p o n e n t i a l l y d e c r e a s i n g , t o p curve , le f t v e r t i c a l axis sca le ) , a n d vo l tage (the l o w e r , e x p o n e n t i a l l y i n c r e a s i n g curve , s c a l e d o n the r ight h o r i z o n t a l axis , expressed i n vo l ts ) . 34 | n o i r -I Q CL m f , "co c I I -LL UJ CO CD > o •o c cd CO J£ cc o CO 8 8 -66 -_i 4 4 -22 -• 1 1 1 1 1 1 1 • i 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1111 6 4 . 3 0 0 0 0 7 5 7 . 8 7 0 0 2 - 5 1 . 4 4 0 0 4 4 5 . 0 1 0 0 6 '- 3 8 . 5 8 0 0 8 3 2 . 1 5 0 1 0 r 2 5 . 7 2 0 1 2 7 1 9 2 9 0 1 4 7 1 2 8 6 0 1 6 - 6 .43018 0 I I II I l l ' l I l [ I I I I I I I I I | I I I I I I I I I | I I I I I I I I I | I I I I I M I I | I I I I I I I I I | I I I I I I I I I -50 250 550 850 1150 1450 1750 2050 0 .00020 Computer Numbers (CN) FIGURE 13 Tone generator characteristics. Left and upper curve represents intensity (in dBs); lower, right curve is voltage - both plotted against computer numbers. W h a t o n e n o t i c e s is that very s m a l l changes i n the l o w e n d o f the vo l tage r a n g e m u s t try to rep resent ( p r o d u c e ) the most rap id l y inc reas ing p o r t i o n o f the in tens i ty cu rve , w h i l e at the u p p e r e n d o f the intens i ty curve , there are p len ty o f C N s w i t h w h i c h to p r o d u c e ve ry f ine g r a d a t i o n s o f intens i ty . T h i s s i tuat ion ref lects the l o g a r i t h m i c n a t u r e o f the d B 35 s c a l e : th is is c o n f i r m e d w h e n s o u n d pressure i n d y n e s / c m 2 is p l o t t e d aga ins t vo l tage -a l t h o u g h the vo l tage curve d o e s r ise m o r e rap id l y t h a n the s o u n d p r e s s u r e c u r v e , the two cu rves d i p l a y bas i ca l l y the s a m e shape , a n d are m u c h m o r e s i m i l a r n u m e r i c a l l y t h a n are t h e v o l t a g e a n d d B curves . T h i s , t h e n , is the first l i m i t a t i o n - that the r e g i o n o f greatest in terest , ( n e a r t h r e s h o l d , s i n c e the R T a n d W F curves f l a t t e n out qu i te rap id l y above t h r e s h o l d ) is a l s o the r e g i o n i n w h i c h t h e r e exists the coarsest g r a i n a l o n g the intensi ty p r o d u c t i o n c u r v e . T h e r e s i m p l y a re n o t e n o u g h c o m p u t e r n u m b e r s w i t h w h i c h to represent a l l o f the in tens i ty l eve ls that o n e might l i k e to p r o d u c e . T h e s e c o n d l i m i t a t i o n has to d o w i t h the fact that the vo l tage d o e s not i n c r e a s e w i t h c o m p u t e r n u m b e r at a per fec t l y constant ra te : the in terva ls that s e p a r a t e eve ry m u l t i p l e o f 7 c o m p u t e r n u m b e r s f r o m every m u l t i p l e o f 8 is g reater t h a n a re the i n t e r v e n i n g l e v e l s . T h i s occurs because the shif t f r o m the register w i t h the e ight ac t i ve b i t s to the reg is te r w i t h o n l y the th ree b i ts that a re respons ib le for the va lues 2° t o 2 2 p r o d u c e s s o m e e x t r a a t t e n u a t i o n as the vo l tage a t t e n u a t i o n step s ize is i n c r e a s e d by . 0 0 0 1 2 vo l ts . F o r t u n a t e l y , b o t h o f these l i m i t a t i o n s are m i n i m i z e d at least fo r s t i m u l i at f r e q u e n c i e s that have h igher th resho lds . F i g u r e 14, s h o w i n g e q u a l l o u d n e s s c o n t o u r s a n d t h e h e a r i n g t h r e s h o l d as a f u n c t i o n o f f requency , i l lust rates th is . T h e e x p e c t e d t h r e s h o l d f o r a 7 0 H z t o n e , fo r e x a m p l e , is a b o u t 35 d B , whereas for 1000 H z it is o n l y 5 d B . T h i s m e a n s that i n the case o f the 1000 H z tone , m u c h w i d e r gaps h a v e to b e c o n t e n d e d w i t h t h a n i n the case o f 70 H z . It has b e e n necessary to go in to s o m e d e t a i l o n these po in ts b e c a u s e t h e i r ef fects d o b e c o m e s ign i f i can t la ter , b o t h i n t e r m s o f e f fec t ing the e x p e r i m e n t a l d e s i g n , a n d w i t h r e g a r d to i n t e r p r e t i n g o u r results . 36 FIGURE 14 Equal loudness contours. (From Coren & Ward, 1989, p.197 [from Lindsay & Norman, 1977; data from Robinson & Dadson, 1956].) 37 S t i m u l i . S t i m u l i c o n s i s t e d o f s ine w a v e s ignals at the s a m e f r e q u e n c i e s tes ted i n the p i l o t s tudy : 7 0 , 100, 2 0 0 , 1000 a n d 10000 H z . T h e in tens i t ies o f the s igna ls u s e d a r e d e s c r i b e d i n the s e c t i o n that f o l l o w s i m m e d i a t e l y . S e l e c t i o n o f S t i m u l u s Intensity V a l u e s : T h e i n i t i a l c o n c e p t i o n o f the c h o i c e o f s t imu lus in tens i t i es was s t ra igh t fo rward . E a c h subject 's l o w e r a u d i t o r y t h r e s h o l d w o u l d b e d e t e r m i n e d , as w o u l d the u p p e r aud i to ry l i m i t . T h e u p p e r l i m i t m i n u s t h e l o w e r t h r e s h o l d w o u l d const i tute the range, the range w o u l d be d i v i d e d i n t o 10 e q u a l l y s p a c e d i n t e r v a l s , a n d the re su l t in g s t i m u l i w o u l d b e u s e d i n a l l th ree o f the e x p e r i m e n t s . L o w e r A u d i t o r y T h r e s h o l d : T h e l o w e r a u d i t o r y t h r e s h o l d was d e t e r m i n e d by resul ts o b t a i n e d i n a n a d a p t i v e sta i rcase, t w o - a l t e r n a t i v e f o r c e d - c h o i c e ( 2 A F C ) t e c h n i q u e u s i n g a 1 -up ; 3 - d o w n r u l e (based o n L e v i t t , 1970; see a lso Jesteadt , 1980; K o l l m e i e r et a l . , 1988 ; a n d T a y l o r & C r e e l m a n , 1967). S i n c e this t e c h n i q u e c o n s t i t u t e d the bas is o f the W e b e r f r a c t i o n e x p e r i m e n t , it w i l l be w o r t h w h i l e to go in to m o r e d e t a i l t h a n o n e w o u l d expec t i n d e s c r i b i n g a pretest . T h e a d a p t i v e s ta i rcase m e t h o d , o f w h i c h there are a n i n f i n i t e n u m b e r o f p o s s i b l e v a r i a n t s , is a subset o f a b r o a d e r class o f test ing p r o c e d u r e s k n o w n as s e q u e n t i a l e x p e r i m e n t s , i n w h i c h the s t imu lus l e v e l o n any o n e t r i a l is d e t e r m i n e d by t h e p r e c e e d i n g s t i m u l i a n d responses (Lev i t t , 1971, p .467) . T h e advantages o f t h e m e t h o d ( s ) i n c l u d e h i g h e f f i c iency , robustness , a n d s m a l l - s a m p l e re l iab i l i t y . C h o i c e o f the n u m b e r o f a l t e r n a t i v e s , as w e l l as o f the " ru le" is i n d e p e n d e n t o f the s ta i rcase m e t h o d i tself . T h e 38 2 A F C o p t i o n w a s c o n s i d e r e d suf f ic ient fo r the present study, b u t i n the case o f the r u l e , a l t h o u g h 1 - u p ; 3 - d o w n results i n longer e x p e r i m e n t a l runs t h a n d o e s the 1 -up ; 2 - d o w n r u l e (g i ven that the o t h e r p a r a m e t e r s r e m a i n the s a m e ) , the f o r m e r w a s se lec ted f o r the t h e o r e t i c a l r e a s o n that the p r o b a b i l i t y o f o b t a i n i n g a p o s i t i v e r e s p o n s e at c o n v e r g e n c e is 0 .794 [this m e a n s that the intens i ty o f the c o m p a r i s o n t o n e , r e l a t i v e to the s t a n d a r d , o r b a s e , t o n e w i l l b e ad jus ted to a leve l at w h i c h it is j u d g e d l o u d e r t h a n the s t a n d a r d 8 0 % o f the t i m e a n d sof ter t h a n the s tandard 2 0 % of the t i m e ] , w h e r e a s f o r 1 -up ; 2 - d o w n the p r o b a b i l i t y is 0 .707 ( L e v i t t , 1971, p.471). E m p i r i c a l l y , 1 -up ; 3 - d o w n was se lec ted b e c a u s e p r e v i o u s s tud ies u s i n g the 1-up; 2 - d o w n ru le (eg. F l o r e n t i n e et a l . , 1987) s e e m to p r o d u c e d i s a p p o i n t i n g resul ts (a l though there w e r e c o n t r i b u t i n g factors i n a d d i t i o n to the c h o i c e o f r u l e n o t e d by the authors ) . T h e subject 's task was to l i s ten to two 600 msec in te rva ls ( i n d i c a t e d o n a v i s u a l L E D f e e d b a c k b o x p l a c e d o n a tab le ins ide the a t tenuat ion c h a m b e r ) , a n d to i n d i c a t e i n w h i c h i n t e r v a l h e / s h e h e a r d a tone (see A p p e n d i x A - 3 f o r L o w e r A u d i t o r y T h r e s h o l d Ins t ruc t ions ) . T h e a u d i b l e tone had b e e n assigned r a n d o m l y to o n e o f the two in terva ls . T h r e e c o r r e c t answers c a u s e d the intensity of the tone to b e r e d u c e d by s o m e step s ize , w h i c h i t se l f d e c r e a s e d i n d i rec t response to the subject 's responses as the r u n p rogressed . O n e i n c o r r e c t r e s p o n s e o n the par t o f the subject, h o w e v e r , c a u s e d a n i n c r e a s e i n the in tens i ty o f the t o n e to b e detec ted . T h e ac tua l l ength o f the r u n d e p e n d e d u p o n the subject 's r e s p o n s e s , t h o u g h g e n e r a l constra ints w e r e p l a c e d u p o n the s t i m u l u s p r e s e n t a t i o n t h r o u g h the s e l e c t i o n , by the e x p e r i m e n t e r , o f v a r i o u s p a r a m e t e r s such as s ta r t ing a m p l i t u d e v a l u e s , i n i t i a l step size, va lue o f the step s i ze d e c r e m e n t m u l t i p l i e r , n u m b e r o f d i r e c t i o n changes w i t h i n any g i ven reversa l , n u m b e r o f reversa ls b e f o r e a 39 c h a n g e i n step s ize , a n d so o n ) . T h e va lues set fo r these p a r a m e t e r s w e r e g u i d e d by a l t e r n a t i v e s suggested i n the l i te ra tu re , a n d by e x p e r i e n c e g a i n e d by the a u t h o r as p r e -s tudy s u b j e c t / e x p e r i m e n t e r / p r o g r a m m e r t h r o u g h e x p l o r i n g these a l t e r n a t i v e s i n h u n d r e d s o f r u n s i n the sta i rcase p r o g r a m d e v e l o p m e n t stage, a n d la ter , i n a n e x p l i c i t a t t e m p t to sett le u p o n p a r a m e t e r s that w o u l d be m a x i m a l l y e f f i c ient . M a x i m u m e f f i c i e n c y w a s i m p o r t a n t f o r a m e t h o d that w o u l d be used for 5 0 runs p e r subject i n the W F e x p e r i m e n t . T h e f i n a l n u m e r i c a l v a l u e that was used to represent the a u d i t o r y t h r e s h o l d that this t e c h n i q u e is d e s i g n e d to converge u p o n was c a l c u l a t e d by a v e r a g i n g the r e v e r s a l po in ts o b t a i n e d at the smal les t m i n i m u m step s ize . T h e a b o v e m e n t i o n e d p a r a m e t e r s i n c l u d e d : 1) s t a r t i n g a m p l i t u d e va lues - f o l l o w i n g L e v i t t (1971) a n d J e s t e a d t (1980) e a c h r u n w a s b e g u n by f i n d i n g a c o m p u t e r n u m b e r that c o r r e s p o n d e d to a s t a r t i n g a m p l i t u d e ( in d B s ) , s e l e c t e d m a n u a l l y w i t h re fe rence to the lab's c a l i b r a t i o n r e c o r d s , that e x c e e d e d the s tep s i ze t w o o r th ree f o l d . T h i s a l l o w e d fo r re lat ive ly easy j u d g m e n t s i n i t i a l l y that w e r e e n c o u r a g i n g to the subject , a n d m o r e o v e r a l l o w e d s o m e t i m e f o r the a u d i t o r y sys tem to "sett le d o w n " ; 2) i n i t i a l step s ize - the a m o u n t by w h i c h the s t i m u l u s is d e c r e a s e d f o l l o w i n g th ree c o r r e c t j u d g m e n t s o r i n c r e a s e d f o l l o w i n g a n incor rec t o n e - set to 4 c o m p u t e r n u m b e r s , w h i c h i n the n e a r t h r e s h o l d range a m o u n t s to qu i te la rge steps i n d B s ; 3 ) s tep s ize d e c r e m e n t m u l t i p l i e r - a f ter 6 reversals at a g i v e n s tep s i ze t h e step s ize w a s r e d u c e d by s o m e m u l t i p l e - i n this pretest , the m u l t i p l e was 0 .5 , w h i c h m e a n t that f o r the f i rst 6 reversa ls the step s ize was 4 c o m p u t e r n u m b e r s , f o r the next 6 reversa ls it 40 was 2, and for the final reversal it was 1; 4) number of direction changes within any given reversal - a direction change is simply the point at which an incorrect judgment has caused an increase in the variable tone, or at which three consecutive correct judgments have resulted in a decrease in that tone. 5) number of reversal changes - a "reversal" is a collection of a prespecified number of changes in the direction of the variable tone for which the step size is the same. In this pretest, each reversal consisted of 4 changes of direction - this means that for each step size, 4, then 2, then 1, there will be 4 direction changes. If 5 reversals were requested, but there were only a maximum of 3 step size decrements possible before reaching the minimum step size, then the first reversal would be at step size = 4, the next 2, and the final three reversals at a step size of 1. This enables one to obtain more estimates of the reversal points, relative to the total number of reversals, over which to average in the calculation of a final threshold value. Despite careful selection of these starting parameters, it was clear even to the subjects who had no knowledge of the tone generator characteristics discussed above, that there were problems with some of the experimental runs. Particularly in the case of the 1000 Hz tone, it turned out that the intensity that was just discernable must have fallen in the "gap" between computer numbers 7 (12.6 dB) and 8 (19.8 dB) because the discriminations were either too easy (when the CN was 8), or impossible (when it was 7). The length of the run for 1000 Hz clearly substantiated the subjective reports: once subjects were in the CN 7-8 range, they alternated perfectly between 3 correct responses to 8, and 79% of the time an incorrect response to 7, making these runs maximally 41 e f f i c i e n t i n t e r m s of t i m e , but s ingu lar l y u n i n f o r m a t i v e about the p r e c i s e v a l u e o f the h e a r i n g t h r e s h o l d . T h i s was the o n e case i n w h i c h n o t h i n g c o u l d b e d o n e to r e m e d y the s i t u a t i o n : f o r the o ther f r e q u e n c i e s , w h e r e s i m i l a r though less severe g a p p r o b l e m s w e r e s u s p e c t e d , t h e r e was s o m e r o o m to take th is in to account w h e n s e t t l i n g o n t h e lowest in tens i t y l e v e l . T h i s " r o o m " was p r o v i d e d by the fact that it was k n o w n at t h e outset that t h e a c t u a l v a l u e o b t a i n e d i n this pretest w o u l d not be used as the f i rst in tens i t y l e v e l b e c a u s e , i n the case o f the W e b e r f r a c t i o n study this c o u l d i n t r o d u c e the c o n f o u n d o f o n e o f the in tens i t ies r e p r e s e n t i n g a d e t e c t i o n task w h i l e the o t h e r n i n e w o u l d be d i s c r i m i n a t i o n tasks ; fo r the r e a c t i o n t i m e e x p e r i m e n t , the w e l l k n o w n s p o n t a n e o u s f l u c t u a t i o n o f the aud i to ry t h r e s h o l d w o u l d m e a n that at t imes the subjects w o u l d s i m p l y never h e a r a t o n e o n s o m e t r ia ls , a n d that w o u l d m a k e the t r i a l m e a n i n g l e s s ; a n d i n the case o f m a g n i t u d e e s t i m a t i o n , b e c a u s e r e q u i r i n g a subject to j u d g e the m a g n i t u d e o f a t o n e that c a n n o t a lways be h e a r d presents a n ev ident p r o b l e m . I n s t e a d , the resul ts o f the p re tes t w e r e to be u s e d to d e t e r m i n e the t h r e s h o l d so that a lowest in tens i t y v a l u e c o u l d be s e l e c t e d that was as c lose to the t h r e s h o l d as poss ib le w h i l e s t i l l b e i n g r e l i a b l y a u d i b l e . T o th is e n d , 2 d B w e r e a d d e d to the o b t a i n e d t h r e s h o l d , a n d w h e r e there was a s u s p e c t e d g a p p r o b l e m , the pretest r u n i n f o r m a t i o n was used to e s t i m a t e w h e r e the " t rue" t h r e s h o l d was w i t h i n the gap . A s m e n t i o n e d prev ious ly , i n the case o f 1000 H z , the r e v e r s a l p o i n t s o n e i the r s ide o f the gap w e r e so evenly d i s t r i b u t e d that i t w a s i m p o s s i b l e to extract any i n f o r m a t i o n f r o m t h e m , so he re a d d i n g the 2 d B w a s bypassed , a n d the u p p e r s ide o f the gap was left to s t a n d as the f irst in tens i ty v a l u e that c o u l d b e r e l i a b l y d i s c e r n e d . 4 2 U p p e r A u d i t o r y L i m i t : It was expected that subjects ' u p p e r " t o l e r a t i o n " t h r e s h o l d w o u l d p r o b a b l y e x c e e d the m a x i m u m intensity o f 110 d B that c a n b e p r o d u c e d by the t o n e g e n e r a t o r ; n e v e r t h e l e s s , I tested to see that this l e v e l was " t o l e r a b l e " f o r the subjects by h a v i n g t h e m l i s t e n to a ser ies o f tones that w e r e i n c r e m e n t e d i n d i sc re te steps. T h e subject w a s i n s t r u c t e d that at any t i m e , s h o u l d the in tens i t y o f the tones b e c o m e i n t o l e r a b l e , h e / s h e c o u l d t e r m i n a t e the series by p r e s s i n g a s p e c i f i e d k e y b o a r d b u t t o n ( for U p p e r A u d i t o r y L i m i t Instruct ions, see A p p e n d i x A - 4 ) . T h e in tens i ty p r i o r to the o n e that m e r i t e d t e r m i n a t i o n o f the t r ia l was t h e n c o n s i d e r e d the u p p e r a u d i t o r y l i m i t fo r that subject , f o r that f requency . M y expectat ions w e r e c o n f i r m e d : n o n e o f the subjects t e r m i n a t e d any o f the runs. H a v i n g o b t a i n e d the i n d i v i d u a l aud i to ry ranges, the s t i m u l u s set f o r e a c h subject w a s c a l c u l a t e d by d i v i d i n g the range b e t w e e n the l o w e r " t h r e s h o l d + 2 d B " a n d the u p p e r t h r e s h o l d i n t o 10 a p p r o x i m a t e l y equa l l y spaced in terva ls ( b a s e d o n d B s ) . P R O C E D U R E O r d e r o f E x p e r i m e n t s . T h e e x p e c t a t i o n o f gap p r o b l e m s a n d the r e a l i z a t i o n o f t h e m i n the l o w e r t h r e s h o l d p re tes t c o m p e l l e d m e to fo rego w h a t w o u l d n o r m a l l y have b e e n a fu l l y e x p e c t e d , c o m p l e t e l y r a n d o m i z e d "order o f e x p e r i m e n t s " p r e s e n t a t i o n . Ins tead , i t w a s t h o u g h t a d v i s a b l e t o r u n a l l subjects o n the W F e x p e r i m e n t f irst , because i f i t w a s necessary to " w o r k a r o u n d " any o f the e q u i p m e n t d i scont inu i t i es by, fo r e x a m p l e , i n c r e a s i n g the target c o m p u t e r n u m b e r by 1, a s l ight ly d i f fe ren t s t imu lus w o u l d be b e i n g p r e s e n t e d to the subject i f that subject h a d a l ready c o m p l e t e d s o m e of the o ther e x p e r i m e n t s u s i n g the c o m p u t e r n u m b e r c o r r e s p o n d i n g to the "des i red" d B that was o r i g i n a l l y c a l c u l a t e d . B y r u n n i n g a l l subjects i n the W F e x p e r i m e n t f i rst , i t c o u l d b e e n s u r e d that the s t i m u l u s sets u s e d f o r any g i v e n subject w o u l d i n fact be i d e n t i c a l across a l l e x p e r i m e n t s , w h i c h w a s o n e o f the m a i n r e q u i r e m e n t s o f the ent i re pro ject . In the course o f l o o k i n g at the resu l ts f r o m the W F e x p e r i m e n t , h o w e v e r , a n d i n t h i n k i n g about the w h o l e s tudy i n g e n e r a l , I c a m e to accept that it was m o r e i m p o r t a n t to try to get the best p o s s i b l e d a t a p o i n t s ( those that w e r e least c o n f o u n d e d by gap p r o b l e m s ) f r o m w h i c h to cons t ruc t the cu rves a n d to e s t i m a t e the i r exponents , t h a n it was to r ig id ly s t ick to e i t h e r the i d e a o f p r e c i s e l y e q u a l in tens i ty in terva ls across the range , o r o f exact c o r r e s p o n d e n c e o f the s t i m u l u s sets across the e x p e r i m e n t s . S o , a l t h o u g h , as p l a n n e d , the W F e x p e r i m e n t was r u n f i rst f o r a l l subjects a n d the o r d e r o f the subsequent R T a n d M E e x p e r i m e n t s r a n d o m i z e d , the s t imu lus va lues i n the s e c o n d two w e r e not f o r c e a b l y c a r r i e d t h r o u g h f r o m the W F e x p e r i m e n t . 44 W e b e r F r a c t i o n T h e c o m p u t e r p r o g r a m that c o n t r o l l e d the p r e s e n t a t i o n o f the s t i m u l i i n the W F e x p e r i m e n t w a s b a s e d u p o n the same pr inc ip les o f a d a p t i v e s ta i rcase t e c h n i q u e as those e m p l o y e d i n the l o w e r a u d i t o r y th resho ld ( L A T ) p re tes t p r o g r a m , the o n l y two d i f f e r e n c e s b e i n g : f i rst , that i n the L A T case, on ly o n e o f the t w o in te rva ls c o n t a i n e d a n o n - n u l l t o n e , w h e r e a s i n the W F case, each i n t e r v a l c o n t a i n e d a t o n e (one at the intens i ty f o r w h i c h the W F was b e i n g m e a s u r e d , r e f e r r e d to as the "base tone" o r the "s tandard tone" ; the o t h e r b e i n g a tone c a l l e d the "var iab le" , o r " c o m p a r i s o n " tone that d i f f e r e d i n in tens i ty f r o m the base tone by vary ing a m o u n t s ) a n d the subject 's task was to i d e n t i f y the i n t e r v a l that c o n t a i n e d the louder o f the two t o n e s ; a n d s e c o n d , that W F runs w e r e c o m p o s e d o f t w o r a n d o m l y in ter leaved sequences , e a c h o f w h i c h was r e q u i r e d to r e a c h a c e r t a i n n u m b e r o f d i r e c t i o n changes a n d reversa ls , w h i l e the L A T cons i s ted o f o n l y o n e such s e q u e n c e . T h e purpose of the two sequences ( r e c o m m e n d e d by L e v i t t (1971) a n d others ) w a s to p revent , as m u c h as poss ib le , the d e v e l o p m e n t o f expectancy ef fects , p a r t i c u l a r l y those that c o u l d be re la ted to t o o e v i d e n t e x p o s u r e o f the 1-up; 3 - d o w n r u l e . E a c h subject s e r v e d i n 5 0 (5 f requenc ies x 10 in tens i t ies ) W e b e r f r a c t i o n runs . T h e o r d e r o f the runs w a s r a n d o m i z e d independent l y fo r e a c h subject . R u n s las ted f r o m 5 to 14 m i n u t e s e a c h . T h e l e n g t h o f the r u n re f lected a host o f p o s s i b l e factors r a n g i n g f r o m fat igue c a u s i n g e r r o r s a n d thus p r o l o n g i n g the r u n , to the ease o r d i f f i cu l t y o f the j u d g m e n t s ( r e c a l l the 5 m i n u t e runs for the 1000 H z L A T case ) , t h r o u g h to the i n f l u e n c e o f the e x p e r i m e n t e r ' s s e l e c t i o n o f i n i t i a l s tart ing a m p l i t u d e v a l u e s , s tep s izes , m i n i m u m step s ize , a n d so o n . R u n lengths w e r e not dependent u p o n e i t h e r the n u m b e r o f 45 reversa ls o r the n u m b e r o f d i r e c t i o n changes w i t h i n a reve rsa l s i n c e these p a r a m e t e r s w e r e h e l d cons tant at 4 a n d 4. A l t h o u g h not h e l d constant across r u n s as the p r e c e d i n g t w o p a r a m e t e r s w e r e , e a c h r u n i n c o r p o r a t e d step s izes , step s ize m u l t i p l e s , a n d m i n i m u m step s izes that , together , e n s u r e d that there w e r e 2 c o m p l e t e reversa ls a t t h e m i n i m u m step s i ze , w h i c h i n t u r n m e a n t that there w e r e a t o t a l o f 16 r e v e r s a l p o i n t s o v e r w h i c h to ave rage to o b t a i n a v a l u e fo r the u p p e r b o u n d o f the j n d (1 + AI ) - 4 d i r e c t i o n changes x 2 reversa ls x 2 r a n d o m l y i n t e r l e a v e d sequences ( c a l l e d A a n d B ) . T h e e x p e r i m e n t a l sessions w e r e p r e c e e d e d by a d i scuss ion w i t h the sub ject o f the i n s t r u c t i o n s (see A p p e n d i x A - 5 ) a n d a p rac t i ce sess ion . In a t y p i c a l e x p e r i m e n t a l sess ion , a subject w o u l d c o m p l e t e f r o m 5 to 12 runs (w i th b r e a k s ) , b u t t h e n u m b e r o f r u n s i n any g i v e n sess ion v a r i e d , d e p e n d i n g large ly o n the l e n g t h o f t h e p r e v i o u s runs . T h e longest st retches o f runs , w i t h b r e a k s b e t w e e n every 5 o r so , w e r e l o g g e d by C K , w h o c o m p l e t e d 22 i n o n e e x p e r i m e n t a l sess ion . R e c o r d s w e r e k e p t o f the r u n w i t h w h i c h a sess ion was b e g a n , w h e r e the b r e a k s o c c u r r e d , a n d o n any sub jec t i ve i m p r e s s i o n s the subject h a d b e e n e n c o u r a g e d to o f fe r (eg. that r u n w a s t o o easy ; I was t i r e d a n d lost m y c o n c e n t r a t i o n m i d w a y t h r o u g h that o n e ; I don ' t t h i n k that I c o u l d h e a r s o m e o f the tones ; a n d the l i k e ) . T h e s e records w e r e la te r r e f e r r e d to w h e n i n s p e c t i n g the r a w d a t a . Subjects w e r e not r e q u i r e d to p r e - a d a p t i n the s o u n d c h a m b e r u n l e s s they w e r e a w a r e o f h a v i n g b e e n recent l y e x p o s e d to u n u s u a l n o i s e leve ls , o r u n l e s s they h a p p e n e d t o b e b e g i n n i n g a sess ion o r r e t u r n i n g f r o m a n out o f l a b b r e a k w i t h o n e o f the " t h r e s h o l d + 2 d B " in tens i ty leve ls . W h i l e subjects d i d e m e r g e f r o m t h e c h a m b e r b e t w e e n runs f o r the c o u p l e o f m i n u t e s n e e d e d to set the c o m p u t e r u p f o r the next runs , 46 i t w a s t h o u g h t that this w o u l d not be p a r t i c u l a r l y d i s r u p t i v e to the sens i t i v i ty o f the a u d i t o r y s y s t e m s ince the l a b tends to be qu iet , a n d , o n c e b a c k i n the b o o t h , the subject w o u l d b e r e s e n s i t i z e d (as d iscussed i n the Pretest : L A T sec t ion ) by the g r a d u a l r e d u c t i o n i n w h a t w e r e i n i t i a l l y la rge step sizes, m a k i n g c o m p a r a t i v e l y easy j u d g m e n t s o v e r t h e f i rst r e v e r s a l at least , a n d a l l o w i n g the aud i to ry s y s t e m to re -ad jus t to t h e b o o t h ' s r e d u c e d n o i s e l e v e l i n the m e a n t i m e . R e a c t i o n T i m e T h e p r o c e d u r e f o r the r e a c t i o n t i m e c o m p o n e n t o f the m a i n study w a s i d e n t i c a l to that a l r e a d y d e s c r i b e d a n d c a r r i e d out i n the r e a c t i o n t i m e p i l o t study. T h e o n l y d i f f e r e n c e i n v o l v e d subst i tu t ing for the gener ic s t i m u l u s set o f the p i l o t study the set c u s t o m d e s i g n e d t h r o u g h the m a i n study pretest p r o c e d u r e s . M a g n i t u d e E s t i m a t i o n A s i n t h e case o f r e a c t i o n t i m e , the m a i n study m a g n i t u d e e s t i m a t i o n e x p e r i m e n t u t i l i z e d t h e n e w s t i m u l u s set a n d essent ia l ly the s a m e p r o c e d u r e d e s c r i b e d i n its r e s p e c t i v e p i l o t study. T h e on ly d i f fe rence to be f o u n d w a s i n the ins t ruc t ions (see A p p e n d i x A - 7 ) . I n the m a i n study M E e x p e r i m e n t , the subject was i n s t r u c t e d to ass ign a v a l u e o f 5 0 to t h e f i rst t o n e (the 5th intensity l e v e l u n i q u e to subject a n d f r e q u e n c y ) h e / s h e h e a r d . T o e n s u r e that the subject h a d u n d e r s t o o d a n d was c o m f o r t a b l e w i t h ass ign ing n u m b e r s i n a c c o r d a n c e w i t h this p r o c e d u r e , a n d w i t h the n o t i o n o f a m o d u l u s , the e x p e r i m e n t e r c o n d u c t e d a short , i n f o r m a l test i n w h i c h the subject was a s k e d to e s t i m a t e 47 l i n e lengths d r a w n o n a b l a c k b o a r d w i t h re ference to a m o d u l u s o f 100. N o n e o f the subjects h a d any d i f f i cu l t y w i t h this task (cf Stevens, 1975) . V I . P R E L I M I N A R Y A N A L Y S I S O F R A W D A T A . R E R U N S . A N D P R E P A R A T I O N F O R F I N A L A N A Y S I S M A G N I T U D E E S T I M A T I O N R a w d a t a f i l e s w e r e i n s p e c t e d f o r m i s s i n g v a l u e s , w h i c h w e r e t h e n s u p p l i e d b y a v e r a g i n g t h e t w o p r i o r a n d the two f o l l o w i n g responses g i v e n to the s a m e s t i m u l u s . M i s s i n g v a l u e s " a p p e a r " w h e n a subject p r e m a t u r e l y , o r o the rw ise e r r o n e o u s l y , presses the k e y b o a r d ' s < S P A C E B A R >, thereby i n i t i a t i n g the next t r i a l a n d b y p a s s i n g the o p p o r t u n i t y to e n t e r a response to the p r e v i o u s t r i a l ; i n this study there w e r e very f e w m i s s i n g v a l u e s ( u n d e r 1 % ) , r e f l e c t i n g the e x p e r i e n c e a n d care fu lness o f the subjects . T h e r a w d a t a f i les w e r e a lso in spec ted f o r the o c c u r r e n c e o f changes i n r e s p o n s e m a d e by t h e subject d u r i n g the e x p e r i m e n t w h i c h a p p e a r e d as, f o r e x a m p l e , 100c90 i n the r e s p o n s e r e c o r d . I n such cases, the r e s p o n s e was r e e n t e r e d to re f lec t o n l y the subject 's f i n a l i n t e n d e d response (here , 90) . S i n c e t h e r e h a d b e e n two 3 0 0 t r i a l sessions r u n p e r subject fo r e a c h f r e q u e n c y , these o r i g i n a l " h a l f f i l es " w e r e c o p i e d , a n d the c o p i e s c o n c a t e n a t e d i n t o s ing le 6 0 0 t r i a l " f u l l f i les" . H a l f f i l e s w e r e r e t a i n e d i n o r d e r to b e a b l e to eva lua te the s tab i l i t y o v e r t i m e o f the subject 's u s e o f n u m b e r , w h i c h has a lways b e e n o n e o f the m a j o r sources o f d e b a t e r e g a r d i n g m a g n i t u d e e s t i m a t i o n . T h e r e s u l t i n g t h r e e f i les fo r e a c h subject at e a c h f r e q u e n c y w e r e t h e n p r o c e s s e d in to d i f fe ren t f o r m s fo r f u r t h e r analys is by b o t h the l i n e a r r e g r e s s i o n m e t h o d , a n d the curve f i t t i n g m e t h o d . R a t h e r t h a n p l o t t i n g a l l 600 o f the d a t a p o i n t s c o n t a i n e d i n the f u l l f i l es , the t ru l y r a w d a t a h a v e b e e n r e p r e s e n t e d i n F i g u r e s 15 -19 by p l o t t i n g the a r i t h m e t i c m e a n o f the 49 60 responses g i v e n to e a c h o f the 10 s t i m u l i intensi ty leve ls . H e r e , as i n o t h e r g raphs e n c o u n t e r e d i n th is thes is , the g r a p h i n g p r i n c i p l e o f s e l e c t i n g sca le v a l u e s i n o r d e r to m a x i m i z e d i s c r i m i n a b i l i t y a m o n g the d a t a po in ts w h i l e m a i n t a i n i n g the a b i l i t y to r e p r e s e n t a l l o f the d a t a w i t h i n the g r a p h b o u n d a r i e s has b e e n a b a n d o n e d i n favour o f u n c h a n g i n g sca le ranges , the la t te r o f w h i c h it is b e l i e v e d f a c i l i t a t e s c o m p a r i s o n a m o n g the g r a p h s . P l e a s e n o t e that i n the cases o f H z . , S 1 0 0o H » a n d S 1 0 0 0 0 H z , the p o i n t c o r r e s p o n d i n g to the h ighest in tens i ty l e v e l exceeds the m a x i m u m sca le v a l u e a n d has b e e n e n t e r e d by h a n d , r a t h e r t h a n i n c l u d i n g it w i t h i n the g r a p h b o u n d a r i e s ( w h i c h w o u l d h a v e r e q u i r e d a f u r t h e r c o m p r e s s i o n o f o ther a l ready re la t i ve l y d i m i n u t i v e p lo ts ) . 50 FIGURE 15 - Main Study "Raw" Data Points - Magnitude Estimation: 70 Hz - (each point = arithmetic mean of sixty responses per intensity level) Subject: B - Freer. 70 Hz Subject C - Free- 70 Hz 700 6 16 25 36 46 NTEN6fTY (dvn98/cm-2) S 16 26 36 INTENSITY (dynee/enra 46 Subject S - Freq 70 Hz Subject: K - Freq: 70 Hz 6 16 25 35 N T B d T Y (d>OGa/cm -2 ) 46 700 600 600 400 300 200 -100 -0 -I I l _ 6 16 25 36 45 NTENSfTY (dyn8a/cm-2) FIGURE 16 - Main Study "Raw" Data Points - Magnitude Estmation: 100 Hz - (each point = arithmetic mean of sixty responses per intensity level) Sublect: B - Freq 100 Hz 6 16 26 36 NTENSTTY (dyrea/cm'^ ) Subject: C - Freq: 100 Hz 6 16 26 36 INTENSITY Wynea/onT2> 46 Subject: S - Freq: 100 Hz 700 600 600 400 300 -200 -100 0 - i 1 1 r - i i i u 6 16 25 36 46 NTENBTTY (dynea/cm~2) Subject K - Freq: 100 Hz 6 16 26 36 46 INTENSfTY (dyrWom-2) FIGURE 17 - Main Study "Raw" Data Points - Magnitude Estimation: 200 Hz - (each point = arithmetic mean of sixty responses per intensity level) Subject: B - Freq 200 Hz 6 16 25 30 NTENSfTY (dyn8e/cm*2> 46 Subject: C - Freq: 200 Hz 6 16 26 36 INTENSITY (<}yT)96/crrr2) 46 Subject: S - Freq: 200 Hz 5 16 26 36 NTENBTTY (dynoe/cm-2) 46 Subject K - Freq: 200 Hz 6 16 26 36 INTENSITY (dynea/cm~2) 46 FIGURE 18 - Main Study "Raw" Data Points - Magnitude Estimation: 1000 Hz - (each point = arithmetic mean of sixty responses per intensity level) Subject: B - Freq 1000 Hz Subject C - Freq 1000 Hz 6 16 2 5 3 6 KTEN6TTY (oyB8/cm~2) 6 16 26 36 NTEN9TY <Ofi*a/air2) Subject; S * « - (22,975) Freq 1000 Hz Subject K - Freq 1000 Hz 6 16 2 6 3 5 4 6 NTENBTTY (dyoBa/om-2) 6 16 2 6 3 6 INTENSITY toyn98/om-2) FIGURE 19 - Main Study "Raw" Data Points - Magnitude Estimation: 10000 Hz - (each point = arithmetic mean of sixty responses per intensity level) Subject: B - Freq: 10000 Hz Subject C - Freq 10000 Hz 6 16 26 36 NTEN6TTY ( o y w « / a n * 2 ) - 6 6 15 2 6 3 6 Subject: S - Freq: 10000 Hz Subject K - Freq 10000 Hz • « - (28,720) 700 6 16 25 36 NTENBfTY (oyi9a/om-2) 46 6 16 26 36 INTENBfTY (Oroa/cm~Z) 55 R E A C T I O N T I M E R e a c t i o n t i m e d a t a was s u b m i t t e d to the same p r e l i m i n a r y analys is as it was f o r the p i l o t study, the p u r p o s e o f w h i c h was to exc lude o b v i o u s o u t l i e r s ( o n b o t h the shor t a n d the l o n g s ides ) , a n d t h e n to a r r i ve at a s ingle v a l u e that a p p r o a c h e d b o t h the m e a n a n d the m e d i a n o f the r e m a i n i n g d i s t r i b u t i o n . P l e a s e r e f e r b a c k to s e c t i o n I V . P r o c e d u r e fo r f u r t h e r de ta i l s . R e r u n s . T h e r e s u l t i n g R T v a l u e s w e r e p l o t t e d against in tens i t y fo r e a c h subject at e a c h f r e q u e n c y . W h e n these g raphs were e x a m i n e d a n d p o i n t s f o u n d that s e e m e d to c l e a r l y " l ie o f f the curve" , the r e c o r d s o f the p r e l i m i n a r y analys is w e r e d o u b l e c h e c k e d f o r the a p p r o p r i a t e n e s s w i t h w h i c h out l ie rs w e r e d i s c a r d e d , o r fo r any o t h e r a n o m a l i e s , a n d i f t h e r e s e e m e d to b e n o e v i d e n c e o f o ther factors b e i n g at w o r k , the r u n w a s r e r u n a n d a n e w p o i n t c lose by o n the intensi ty scale was a lso r u n . I f the r e r u n a n d the n e w r u n y i e l d e d s i m i l a r v a l u e s , the o r i g i n a l ques t ionab le r u n resul t w a s d i s c a r d e d a n d the r u n r e p l a c e d w i t h the r e r u n v a l u e ; i f there was a sp read a m o n g the o l d r u n , the r e r u n , a n d the n e w r u n , a n i n t e r m e d i a t e R T va lue was e s t i m a t e d . A n o t h e r s i t u a t i o n cons i s ted o f o n e i n w h i c h t h e r e w a s a g r o u p o f R T vs intensi ty p o i n t s that m a d e it i m p o s s i b l e to d e t e r m i n e w h e t h e r any , a n d i f so w h i c h , might be "faulty" . In th is case , runs o f i n t e r m e d i a t e in tens i ty v a l u e s w e r e c o n d u c t e d a n d i f the resul ts f r o m these runs c o u l d b e s e e n to l i e a l o n g a c u r v e , the po in ts that w e r e not o n that c u r v e w e r e c o n s i d e r e d suspect a n d w e r e sub jec ted to the s a m e r e r u n p r o c e d u r e a p p l i e d to i n d i v i d u a l p o i n t s that d i d not s e e m to f i t as d e s c r i b e d a b o v e . A l t h o u g h the n u m b e r o f re runs p e r subject w i l l s e e m l a rge ( B T = 1 9 , C K = 2 3 , K D = 2 0 , S M = 18) it is i m p o r t a n t to note that the a b s o l u t e v a l u e i n m i l l i s e c o n d s that c o u l d denote a p o i n t that lay o f f the curve was i n the o r d e r o f 2 0 -that is , a 2 0 m i l l i s e c o n d e r r o r i n , o n average , a 2 0 0 to 3 0 0 m i l l i s e c o n d r e a c t i o n t i m e . T h e s e R T r e r u n s w e r e p r o m p t e d by m u c h less ser ious concerns o v e r the p o i n t s ' v a l i d i t y t h a n i n t h e W F case. T h e f i n a l d a t a p o i n t s ( f o l l o w i n g the r e r u n p r o c e d u r e ) a re p l o t t e d i n F i g u r e s 2 0 - 2 4 . T h e s e f i ve f igures show the R T d a t a p o i n t s fo r e a c h subject f r e q u e n c i e s 7 0 , 100, 2 0 0 , 1000 a n d 10000 H z respect ive ly . A l o n g w i t h the R T d a t a , e a c h f i gu re a lso d isp lays the " raw" W F d a t a p o i n t s , to b e d iscussed next . FIGURE 20 - Main Study "Raw" Data Points -RTandWF: 70 Hz Subject: B - Free? 70 Hz Subject C - Freq: 70 Hz 1.0 -0 6 oo 1 I " - 1 1 8 o 0 0 1 0 1 i . . 1 6 16 25 35 NTENSTTY ( o y » e / c m - 2 ) 46 - 6 6 16 25 36 46 INTENSITY WyneB/cm-2) Subject S - Freq 70 Hz Subject: K - Freq 70 Hz - 6 6 15 25 35 46 NTENBTTY (dynea/cnT2> 6 15 26 36 INTENSITY fcVnea/oni~2) 45 RT(88C) - . : WEBER FRACTION - o 8 S § RTCsec) - . : WEBER FRACTION - o § b 8 RTCsec) - . ; WEBER FRACTON - o 8 8 S CD I T I CD £> 8 N So o 5 b ft RTteec) - . : WEBER FRACTION - o 8 8 <3 1 r r • • 0 oo • o o o CO & •ar o O i - n I 00 FIGURE 22 - Main Study "Raw" Data Points -RTandWF: 200 Hz Subject: B - Freq 200 Hz Subject: C - Freq 200 Hz 6 16 26 36 NTENSfTY (dyr*w/cm"2) - 6 6 16 26 36 NTENSTY « y n e e / c n T 2 ) Subject: S - Freq: 200 Hz Subject K - Freq 200 Hz ~i 1 1 r-1X3 06 00 6 16 26 36 NTENBTTY (dyfwa/cm-2) 46 - 5 6 16 26 36 46 INTENBTTY <dynoa/cm~2) FIGURE 23 - Main Study "Raw" Data Points -RTandWF: 1000 Hz Subject: B - Freq 1000 Hz Subject: C - Freq 1000 Hz 1.0 0J5 1 oo .oo o - 5 6 16 25 35 NTENSTTY (dynee/cm*2) 46 INTENSITY (dynee/cm*2> Subject S - Freq: 1000 Hz Subject K - Freq 1000 Hz 10 0 6 0 0 -i r" o o 8 ft :o i 5 16 25 35 NTENBTTY (dynes/cm-2) 46 6 16 26 36 INTENSITY (dynns/'cnr2) 46 FIGURE 24 - Main Study "Raw" Data Points -RTandWF: 10000 Hz Subject: B - Freq: 10000 Hz Subject C - Free- 10000 Hz 6 16 25 36 NTENSTTY (dyree/cnY^ 46 6 16 26 35 INTENSfTY toynee/cnT2) 45 Subject: S - Freq: 10000 Hz Subject K - Freq 10000 Hz 5 16 26 36 NTENBITY (dyr »a/cm-2) 46 6 16 26 36 INTENBTY <ctyn8a/air2) 46 W E B E R F R A C T I O N T h e f i rst l e v e l o f d a t a analys is r e q u i r e d i n this e x p e r i m e n t w a s c a l c u l a t i o n o f the averages o f the reve rsa l p o i n t s at the m i n i m u m step s ize v a l u e (as d e s c r i b e d a b o v e ) to o b t a i n a n e s t i m a t e o f (I + Al), the intens i ty l e v e l (here expressed i n d B s ) that is just n o t i c e a b l y d i f f e r e n t f r o m the s t a n d a r d t o n e . Al is t h e n o f course jus t (I + Al) m i n u s I. T h e s e v a l u e s p r o v i d e d suf f ic ient i n f o r m a t i o n to c a l c u l a t e , u s i n g the f o l l o w i n g e q u a t i o n , t h e u n i t l e s s m e a s u r e k n o w n as the W e b e r f r a c t i o n . WF = A l 10 IdB + AldB 10 10 IdB 10 10 IdB 10 T h e a b o v e ca lcu la t ions w e r e c o m p l e t e d by the c o m p u t e r that was c o n t r o l l i n g the e x p e r i m e n t a l r u n at the t i m e o f the r u n . I h a v e a l r e a d y o u t l i n e d i n s o m e d e t a i l the d i f f i cu l t ies a n t i c i p a t e d i n d e t e r m i n i n g the l o w e r in tens i t y W e b e r f rac t ions , so a l t h o u g h at the e n d o f each e x p e r i m e n t a l r u n a W e b e r f r a c t i o n va lue h a d b e e n o b t a i n e d , i n s t e a d o f g o i n g straight to S Y G R A P H , p l o t t i n g the W F s , a n d l o o k i n g fo r ou t l i e r s , the next step i n the ana lys is w a s r a t h e r to assess the v a l i d i t y o f that v a l u e before p l o t t i n g it , o therw ise it w o u l d h a v e b e e n c o n c e i v a b l e to have b e e n l o o k i n g at "good" resul ts f r o m u n i n t e r p r e t a b l e d a t a . 63 R e r u n s . In o r d e r to b e a b l e to rec reate , fo r la ter analys is , the e v o l u t i o n o f a n a d a p t i v e s ta i rcase m e t h o d r u n , m o r e " raw data" i n f o r m a t i o n m u s t b e s t o r e d t h a n i n m o s t p s y c h o p h y s i c a l e x p e r i m e n t s . T h i s is espec ia l l y t rue w h e n o n e is d e a l i n g w i t h runs c o n s i s t i n g o f t w o r a n d o m l y i n t e r l e a v e d sequences - i n a way , t w o e x p e r i m e n t s a re g o i n g o n at the s a m e t i m e ; e a c h o n e , m e t h o d o l o g i c a l l y sens i t i ve to o n l y its o w n h istory , yet , t a k e n toge ther , "psycho log ica l l y " sensit ive to b o t h . A n a l y s i s p r o g r a m s w e r e w r i t t e n that w o u l d extact i n f o r m a t i o n f r o m these r a w d a t a f i les a n d a r r a n g e that i n f o r m a t i o n i n v a r i o u s f o r m s d e s i g n e d to e n a b l e the e x p e r i m e n t e r to look at the r u n . T h e output o f the p r o g r a m s u s e d i n this s e c o n d l e v e l o f d a t a analys is w e r e c a l l e d r u n r e c o r d s . R e p r o d u c e d i n F i g u r e 10 is a t y p i c a l r u n r e c o r d i n w h i c h a gap p r o b l e m exits . L o o k i n g at the g r a p h i c a l r e p r e s e n t a t i o n ( l a b e l l e d E ) , one c a n see that the "truly" just n o t i c e a b l e d i f f e r e n c e l i es s o m e w h e r e b e t w e e n 61.87 ( = C N 127) a n d 62 .5 ( = C N 128) d B s , m e a n i n g , ( r e f e r r i n g to the c o l u m n " A s s o c . d W F s " ) , that the W F was b e t w e e n .296 a n d .486. 64 rfeqaency ( t a r g e t ) : 100 I n i t i a l step s i z e (CN): i change d i r e c t i o n s : 2 isi: 10 pwint: SO Date: 07-20-1986 A a p i i t u d e lor Standard Tone (CN): 119 Value (or 'c' used: .5 I r e v e r s a l s : 6 l u l t i p l i e r (I of r e v e r s a l s s t d u r : 600 i t i : 200 t a r d u r : £00 T u e : 12:59:33 P r o g r a i n a i e : «E8'64.bas Total t i t e taken (or KD's run ( s e s s i o n 25a): 309 (sec) —=> a p p r o i . 5 t i n u t e s . ! • d e l t a I (dB): 61.92122 Averaged I • d e l t a I: 127.75 CN Standard (dB): 60.74437 Standard tone : 119 CN d e l t a I (dB): 1.176853 Veber F r a c t i o n ( V a l t e r Eqn.): .3112494 RUNNING I t d e l t a Nuiber u i u l t y R S 1 15 2 15 3 15 4 15 ! and UEBER CALCULATIONS » » » » u i u l t y R I I+delta I Running Ueber — * 1 0 0 16 126.5 61.83582 .2857149 16 127.125 61.87862 .2984514 16 127.5833 61.90938 .3078303 FIGURE 10 (A) - PARAMETER VALUES at s i a l l e s t s s ) : 4 (B) - NUMERICAL RESULTS OF THE RUN (C) - "RUNNING WEBER" - A cummulative running calculation of the Weber fraction - that is, a time series, with a "picture" of the WF taken at the end of each minimum step size reversal. "Number" = first, second....Iast reversal at minumum step size. » » » » » » » » » » » » » » » » » » » » » » » » » » » » » » ) » » » ) » » » » » » oc d e value NaxAN HinAN BHaiAN BKinAN 54 1 2 126 0 0 126 0 59 ; 3 126 0 0 0 126 67 1 4 127 0 0 127 0 69 1 5 127 0 0 0 127 76 2 5 1.27 0 127 0 0 77 2 6 128 0 0 128 0 84 2 7 127 0 0 0 127 89 3 7 129 129 0 0 0 91 4 7 129 0 129 0 0 94 4 8 128 0 0 128 0 96 5 8 129 129 0 0 0 39 5 9 128 0 0 0 128 102 6 9 128 0 128 0 0 105 7 9 128 128 0 0 0 112 8 9 126 0 126 0 0 121 9 9 131 131 0 0 0 graph of C:\qbasic\weber\KD25a.dat's run: P r o g r a i Naie: HEB'64.8AS «ip(dB!) CN TRIALS r60.74437 119 61.80141 126 B b 61.87008 127 B b a = (ti«>=================> Assoc.d NT's 0 a .27557 .2958978 I • I 62.46478 128 B B • 62.53237 129 A a 62.59945 130 62.66601 131 S u n a r y values for t h i s r u n . . . . SAapN! (dB): 60.74437 SAipN(CN): 119 TAtpN' (dB): 61.92122 TAipNCCN): 127,75 d e l t a l t d ! ) : b a A .4860773 .5093B68 .5328786 A .5565531 Heber: .3112494 d e l t a ! (dB): 1.176853 (D) REVERSAL POINTS (in computer numbers-CN) "OC" = occurrence ( = trial number) \alue" = CN " MaxAN" = maximum amplitude number, sequence A " MinAN" = minimum amplitude number, sequence A "BMaxAN" - maximum amplitude number, sequence B "BMinAN" = minimum amplitude number, sequence B "d" & "c" encodes the hitting of a res'ersal point, whether a maximum or a minimum value from sequence A or B respectively. (E) - GRAPHICAL REPRESENTATION of the final portion of the run upon which the summary statistics are based. Plotted are the reversal points that are averaged to yield the WF. A - maximum, sequence A a - minimum, sequence A B - maximum, sequence B b - minimum, sequence b Doited line represents a "gap" • the intensity le\>el between CN 127 and 128 (1.41 dB) is approximately 7 times that of the other inten'als. Standard tone ( = SAmpN) is the first line of the plot. The estimate of I + delta I ( = TAmpN) is the average CN over the 16 reversal points. "Assoc.d WFs" calculates what the WF would be if the amplitude on which it was based was in fact the I + delta I value. This allows one to see how wide the gap is in WF terms and assists in the process of assessing the validity of the WF ("Weber") summary statistic. 65 O n e c o u l d i m a g i n e a var ie ty o f c o m p l i c a t e d a l g o r i t h m s fo r t r y ing to i n t e r p o l a t e w i t h i n that r a n g e w h e r e the "true" v a l u e fa l ls f r o m the p a t t e r n o f r e v e r s a l p o i n t s o n e i t h e r s i d e o f the g a p , b u t it r e m a i n s that w i t h o u t v e r i f i c a t i o n o f those a l g o r i t h m s t h e m s e l v e s , i t is d i f f i c u l t to c o n f i r m a s ingle v a l u e . O n e o p t i o n f o r d e a l i n g w i t h th is w o u l d b e to a l l o w the edges o f the gap to represent the e r r o r o f m e a s u r e m e n t ( the m a g n i t u d e o f w h i c h w o u l d m a k e it i m p o s s i b l e to cu rve f i t w i t h any c o n f i d e n c e ) , b u t i t is r e a l l y n o t as b a d as a l l that . A s m e n t i o n e d a b o v e , the o p t i o n that I t o o k was to c o n d u c t a " n e a r - r e r u n " that u t i l i z e d a s l ight ly d i f fe rent in tens i ty f o r the s t a n d a r d t o n e a n d w o u l d , o n e h o p e d , resu l t i n the (I + AI) v a l u e f i n d i n g i tse l f i n a r e g i o n w e l l p o p u l a t e d by c o m p u t e r n u m b e r s . S e l e c t i n g those s tandard t o n e va lues f o r the re runs , as it h a d b e e n fo r the i n i t i a l runs , w a s a p a i n s t a k i n g , m a n u a l , h a l f i n tu i t i ve ( i n the sense o f at least s e e m i n g u n f o r m a l i z a b l e w i t h o u t t e m p t i n g B o n i n i ' s p a r a d o x ) task, o w i n g to the fact that e a c h o f t h e 2 0 4 8 c o m p u t e r n u m b e r s stands i n a u n i q u e r e l a t i o n s h i p to the w i d t h o f the in te rva ls b e t w e e n i t se l f a n d its ne ighbours , to the a b s o l u t e in tens i ty l e v e l that it rep resents , a n d t o the gaps w h i c h themselves are u n i q u e i n p r e c i s e l y the s a m e t e r m s ; a n d by the fact that the n u m b e r o f p e r m u t a t i o n s i n v o l v e d i n these c o m p l e x i t i e s is d w a r f e d b y the p r o b l e m i n t r o d u c e d by the d i f fe rences i n i n d i v i d u a l subjects a n d , by d e f i n i t i o n , the l a c k o f a p r i o r i k n o w l e d g e about the AI va lues that a re u n d e r i n v e s t i g a t i o n . T h e p o i n t he re is that h o w e v e r m u c h I w o u l d l i k e to have b e e n a b l e to d i v o r c e m y s e l f as t h e e x p e r i m e n t e r f r o m the poss ib i l i t y o f i n f l u e n c i n g the d a t a t h r o u g h m a n u a l s e l e c t i o n o f the s tar t ing p a r a m e t e r s by c r e a t i n g a n a l g o r i t h m to select those p a r a m e t e r s f o r m e - a n a l g o r i t h m that I c o u l d p resent a n d w h i c h w o u l d b e o p e n to e x a m i n a t i o n - h o w e v e r m u c h I w o u l d l i k e to have a c c o m p l i s h e d that , I d i d n o t s u c c e e d . 66 M o s t n e a r - r e r u n s p r o d u c e d i n t e r p r e t a b l e W F s , s o m e r e q u i r e d t w o o r t h r e e m o r e n e a r - r e r u n s , a n d there w e r e a few fo r w h i c h , e v e n af ter g o i n g t h r o u g h t h e p rocess a f e w m o r e t i m e s u n t i l there w e r e n o m o r e s t a n d a r d t o n e C N s that h a d n o t b e e n tes ted i n a g i v e n in tens i t y r e g i o n , there was n o a l te rnat i ve lef t but to i n t e r p o l a t e as b e s t I c o u l d u s i n g w h a t e v e r i n f o r m a t i o n was c o n t a i n e d i n a l l o f the n e a r - r e r u n s , o r , i n t h e very wors t s c e n a r i o , to d r o p that intensi ty l e v e l W F m e a s u r e . A f t e r the r e q u i r e d n e a r - r e r u n s u n c o v e r e d by the above analys is w e r e c o m p l e t e d the W F s w e r e p l o t t e d against the intens i ty a n d po in ts i d e n t i f i e d that l o o k e d l i k e out l ie rs , I w e n t b a c k to the r u n records to see w h e t h e r the W F that h a d b e e n c a l c u l a t e d l o o k e d r e p r e s e n t a t i v e o f the r u n . If a n y t h i n g s e e m e d u n u s u a l - if, f o r e x a m p l e , o n e o f the r e v e r s a l p o i n t s was so d i f fe rent f r o m the others that it was p u l l i n g t h e a v e r a g e o f those p o i n t s i n o n e d i r e c t i o n , a n d w i t h it , the W e b e r f r a c t i o n , o r i f a l l o f o n e s e q u e n c e h a d r e a c h e d its d i r e c t i o n change a n d n u m b e r o f reversa ls c r i t e r i a b e f o r e t h e o t h e r h a d e v e n b e g u n to r e c o r d reversa l po in ts at the smal les t step s ize , the resu l t w a s c o n s i d e r e d q u e s t i o n a b l e . A g a i n , there was a n e l e m e n t o f j u d g m e n t i n v o l v e d i n th is stage. It h a d to b e d e c i d e d w h e t h e r the W F c o u l d b e e s t i m a t e d by f i n d i n g a c e n t r a l t e n d e n c y i n the r e v e r s a l p o i n t s est imates a n d u s i n g the assoc ia ted W F fo r that r u n ; w h e t h e r the d i s t r i b u t i o n o f the reve rsa l p o i n t s was so i n d e f i n i t e that the r u n s h o u l d b e r e r u n , a n d i f so w h e t h e r w i t h a n i d e n t i c a l s t a n d a r d tone , o r w i t h o n e that w a s a l i t t l e d i f f e r e n t ; a n d so o n . T h e n , af ter o b t a i n i n g the r e r u n va lues , there was m o r e j u d g m e n t o f t h e var ie ty m e n t i o n e d i n the a n a l o g o u s phase o f the R T analys is - w h a t to d o w i t h t h e o l d runs , the r e r u n s , the n e a r - r e r u n s , the n e w runs ( in s o m e cases a d d e d to g ive a p o i n t o n the W F b e t w e e n t w o ex is t ing p o i n t s i n the l o w intens i ty r e g i o n , w h i c h is w h e r e t h e m a x i m u m 67 influence of the exponent is observed): whether to delete some in favour of others; to average the WF obtained across some or all; to force the number of points on the curve to stay at 10, as intended in the initial design or to allow a couple more points in that seemed to really constribute to the definition of the curve, or whether to allow in these more informative points and keep the number of points at 10 by dropping a point or two in the flatter region of the curve (an option never taken); et cetera. At the end of this entire process the final data consisted of 9 to 12 WF vs Intensity pairings in each of the 20 (4 Subj x 5 Freq) conditions. These data were plotted, and appear, as previously mentioned, in Figures 20-24. V I I . P R I M A R Y D A T A A N A L Y S I S A N D R E S U L T S 68 L e t us r e c a l l that the p u r p o s e o f a l l o f the p r i m a r y d a t a analyses is to d e t e r m i n e the n u m e r i c a l v a l u e o f the exponent f o u n d i n the e q u a t i o n , re levant to the p s y c h o p h y s i c a l task , that is thought to e m b o d y the r e l a t i o n s h i p b e t w e e n s t imu lus ( h e r e , in tens i t y ) a n d r e s p o n s e . M A G N I T U D E E S T I M A T I O N L i n e a r R e g r e s s i o n A n a l y s i s . I e m p l o y e d the m e t h o d that is t yp ica l l y u s e d to d e t e r m i n e the e x p o n e n t f o r M E d a t a i n the e q u a t i o n M E = kl° : l i near reg ress ion analys is . F r o m a l g e b r a it is k n o w n that a n e x p o n e n t c a n be c o n v e r t e d i n t o a m u l t i p l i c a t i v e c o n s t a n t b y t a k i n g the l o g a r i t h m o f the e q u a t i o n . T h u s , the e q u a t i o n M E = k i " c a n a l s o b e w r i t t e n l o g M E = n l o g l + logk . T h i s e q u a t i o n is o f the s a m e f o r m as that o f a s t ra ight l i n e : Y = m X + b, w h e r e m represents the s lope o f the l i n e . I f l o g M E d a t a a r e s u b m i t t e d as va lues o f the d e p e n d e n t v a r i a b l e , a n d l o g l d a t a as v a l u e s o f the i n d e p e n d e n t v a r i a b l e to a stat is t ica l p r o g r a m (I u s e d the M u l t i v a r i a t e G e n e r a l L i n e a r H y p o t h e s i s ( M G L H ) m o d u l e o f the S Y S T A T stat is t ica l p a c k a g e ) , the p r o g r a m w i l l f i n d t h e r e g r e s s i o n l i n e that m i n i m i z e s the s u m o f the s q u a r e d d istances b e t w e e n the d a t a p o i n t s a n d the h y p o t h e s i z e d l i n e , a n d the r e s u l t i n g s l o p e o f that l i n e w i l l b e the e x p o n e n t . I c a r r i e d out this p r o c e d u r e for b o t h o f the h a l f f i les , a n d the f u l l f i l e f o r a l l subject a n d f r e q u e n c y c o m b i n a t i o n s . 69 R e s u l t s . T h e n u m e r i c a l resul ts o f the l i n e a r reg ress ion analys is as p r e s e n t e d i n T a b l e 1 a n d are g r a p h e d i n F i g u r e 26. I f o n e f i rst focuses o n the curve w i t h the la rge , f i l l e d p e n t a g o n a l s y m b o l s that r e p r e s e n t the average o f the i n d i v i d u a l subjects ' exponents at e a c h f r e q u e n c y , o n e observes a d e c l i n e i n e x p o n e n t w i t h i n c r e a s e d f requency . T h i s t r e n d c o n t i n u e s u p to a n d i n c l u d i n g 1000 H z . B y 10000 H z t h o u g h , the exponent has i n c r e a s e d b a c k u p to the l e v e l o b t a i n e d at 200 H z . A l l o f the i n d i v i d u a l subjects re f lec t th is b a s i c p a t t e r n w i t h t w o except ions : S at 200 H z shows a n i n c r e a s e i n e x p o n e n t ; a n d B at 1000 H z shows a p r e m a t u r e u p t u r n i n the v a l u e o f the exponent ; O v e r a l l t h e n , t w o p o i n t s c a n b e n o t e d w i t h regard to these d a t a . F i r s t , i t is c lear that the e x p o n e n t s s h o w a decrease i n m a g n i t u d e w i t h i n c r e a s i n g f r e q u e n c y f o r a l l subjects, f o l l o w e d by a s l ight i n c r e a s e i n exponent at the h ighest f r e q u e n c y that was tested (10 K H z ) . T h e A N O V A tes t ing fo r this ef fect was s ign i f i cant at the 0.001 l e v e l [ F 4 1 2 = 31.558]. Sub ject S shows this effect m o s t m a r k e d l y , a l t h o u g h the " b u m p " at 200 H z («a» HZ = -902) m a k e s the H z to nm m d r o p , f r o m 1.335 to .882, s o m e w h a t less d r a m a t i c . O n e c o u l d r e c a l l that S w a s the m o s t e x p e r i n c e d subject w h o , o n e m i g h t specu la te , feels m o r e at ease u s i n g a n expans ive range o f n u m b e r s i n a m a g n i t u d e e s t i m a t i o n task. S e c o n d , w h i l e th is d o m i n a n t p a t t e r n across f r e q u e n c i e s is b a s i c a l l y p r e s e r v e d fo r a l l subjects , t h e r e a p p e a r to b e s ign i f i cant d i f fe rences b e t w e e n the a b s o l u t e v a l u e s o f the e x p o n e n t s o b t a i n e d f o r d i f fe ren t subjects. T h i s was c o n f i r m e d by a n A N O V A [ F 3 1 2 = 30.205, p < 0.001]. P a r t i c u l a r l y s t r i k ing , a g a i n , a re the resu l ts f o r sub ject S , a l l o f whose M E exponents a re c o n s i d e r a b l y h igher t h a n those o f the o t h e r t h r e e subjects , espec ia l l y at the l o w e r f r e q u e n c i e s . REGRESSION METHOD: Magnitude Estimation Exponents SUBJECTS B K Half F i l e s : HF l ( t r i a l s 5-300) HF 2 ( t r i a l s 305-600) Avg F u l l F i l e : 70 Hz 100 Hz 200 Hz 1000 Hz 10000 Hz Avg n 0.603 0.324 0.461 0.630 0.720 0.677 0.544 0.517 0.530 1.335 1.327 1.335 0.778 0.722 0.751 0.454 0.268 0.361 0.517 0.629 0.576 0.468 0.406 0.439 0.928 0.834 0.882 0.592 0.534 0.550 0.213 0.280 0.246 0.495 0.516 0.507 0.364 0.332 0.350 0.690 1.111 0.902 0.441 0.560 0.501 0.351 0.194 0.272 0.333 0.411 0.370 0.232 0.241 0.236 0.609 0.611 0.661 0.381 0.364 0.372 0.375 0.225 0.300 0.376 0.575 0.530 0.277 0.353 0.316 0.586 0.782 0.684 0.404 0.484 0.458 0.399 0.258 0.328 0.470 0.570 0.532 0.377 0. 370 0. 374 0.830 0.933 0.882 0.519 0.533 0.526 TABLE 1 Linear regression method magnitude estimation exponents. 71 LINEAR REGRESSION: Magnitude Estimation FIGURE 26 Linear regression method magnitude estimation exponents plotted. 72 T H E N O N L I N E A R C U R V E F I T T I N G T E C H N I Q U E T h e a c t u a l curve f i t t ing a l g o r i t h m u s e d was the s a m e as that u s e d i n the p i l o t study -a S I M P L E X o p t i m i z a t i o n r o u t i n e , s u p p l i e d i n w o r k i n g p r o g r a m f o r m by D r . N o r w i c h . T h i s p r o g r a m searches t h r o u g h its N d i m e n s i o n a l ( N equa ls the n u m b e r o f p a r a m e t e r s i n the f u n c t i o n that is b e i n g f i t ) s o l u t i o n space i n search o f the m i n i m a l s u m s o f squares r e s i d u a l . T h e S S ^ is , as o n e w o u l d expect , the s u m of the d is tance , s q u a r e d , b e t w e e n e a c h m e a s u r e d d a t a p o i n t o f c o o r d i n a t e s [Intensity^ m e a s u r e d R T J , a n d t h e p o i n t [ Intensity^ c a l c u l a t e d R T J that has b e e n c a l c u l a t e d . Success ive c o n d e n s a t i o n s a n d e x p a n s i o n s a re the processes by w h i c h the r o u t i n e m o v e s t h r o u g h the s p a c e : expans ions , w h i l e not p r e v e n t i n g the p r o g r a m f r o m f a l l i n g in to a va l ley , d o p r o v i d e a m e c h a n i s m that p revents that v a l l e y b e i n g o n e f r o m w h o s e b o u r n n o t rave l le r re tu rns ( t h o u g h s o m e m a y n o t ) ; w h i l e c o n d e n s a t i o n s a l l o w f o r inc reas ing ly f ine l y g r a i n e d e x p l o r a t i o n o f va l leys that m a y , i n fact , t u r n out to be g l o b a l m i n i n a . F i g u r e 2 7 , t a k e n essent ia l l y at r a n d o m f r o m a m o n g the 40 sets o f d a t a that w e r e f i t to curves , shows a n escape f r o m a l o c a l m i n i m u m . 0 50 100 NUMBER OF ITERATIONS (Data: RS70a.SYS) FIGURE 27 Time course of a typical curve fitting nm - escape from a local minimum. 74 The upper three graphs depict a zoom-in on the sum of squares residual (note the changes in scale). The progress of the curve fitting run is represented on the horizontal, with the number of iterations (attempted solutions) increasing from left to right. One notices in the third graph from the top a sudden drop in the S S ^ at about iteration 28. This indicates the existence of a local minimum. The program obviously manages to expand its way back out of the hole, ultimately finding a radically different (note the reversal in relative positions between the multiplicative constant A and the exponent n that occurs in the bottom graph in the vicinity of 50 to 65: iterations) and more stable solution. What cannot be prevented however, is the possibility of falling into a genuine tradeoff situation between some parameter values, and this is precisely the problem that was encountered in much of the curve fitting in this project. A glimpse of this can be caught in the previous example: between about iteration 50 through to about 75, the multiplicative parameter is observed moving in opposition to the value of the exponent -that is, they are trading off. This issue leads me into the next section, which begins with a brief detour, taken to ensure that when I talk about parameters, the meaning will be clear. 75 REACTION TIME AND WEBER FRACTION CURVE FITTING The approaches that were taken to the curve fitting were applied in parallel to both the RT and the WF data. Even the functions that were fitted were the same. This requires some explanation. The Functions In the initial presentation of the equations derived by Norwich from his Fundamental Entropy Equation (Equation 3 ) , it will be recalled that the resulting RT equation was: 1 RT = (11) A + 0I" n and for WF was: WF = aH (/?' + I"n) (12) Using a different derivation from Equation 3 , however, Norwich was also able to approximate Pieron's RT equation (a power function with a multiplicative constant, an additive constant, and a negative exponent, RT = k i " + b), so one is as justified in using Pieron's RT equation as one is in using Norwich's RT formulation. Turning back to the WF equation, expanding Equation 12 results in: WF = aHI" n + aH/T. Since aH is a constant, and aH/T, too, is a constant, the WF, like the RT, can be treated as a power function of the form WF = k i " + b. This is the rationale behind fitting both sets of data with the same equation. 76 A w o r d n o w about the constants . T h e m u l t i p l i c a t i v e constant is r e f e r r e d to i n th is thes is as " A " o r "k", as i n the e q u a t i o n s d i rec t l y a b o v e , o r as a r g u m e n t , o r " a r g ( l ) " , i n the c u r v e f i t t i n g p r i n t o u t s . A s a multiplicative constant , i t assumes a s c a l i n g r o l e , i n t e r a c t i n g w i t h the e x p o n e n t to d e t e r m i n e the cu rva tu re o f the f i t ted curve . T h e r e w e r e n o a p r i o r i e x p e c t a t i o n s as to w h a t the n u m e r i c a l va lue (s ) the p a r a m e t e r A s h o u l d t a k e . I n the case o f the add i t i ve constant "b", ("arg(2)" i n the p r i n t o u t s ) h o w e v e r , i t s e e m e d r e a s o n a b l e that b s h o u l d be the asymptote o f the curve b e i n g suggested by the d a t a p o i n t s . R e c a l l i n g the f i n a l "raw" d a t a p r e s e n t e d i n F i g u r e s 2 0 - 2 4 , o n e c a n see that this a s y m p t o t e is eas i l y b e i n g c lose ly a p p r o a c h e d by the u p p e r in tens i ty l e v e l p o i n t s . I n m y s u b s e q u e n t c u r v e f i t t ing , I t yp ica l l y c o n s t r a i n e d th is b p a r a m e t e r to a v a l u e a v e r a g e d o v e r the d e p e n d e n t measures o b t a i n e d i n response to the 3 h ighest in tens i t y leve ls . I n t e r m s o f i n t e r p r e t a t i o n , o n e c o u l d c o n s i d e r the asymptote i n the case o f R T to b e s o m e cons tant , m i n i m u m a m o u n t o f t i m e (pe rhaps w h o l l y a t t r i b u t a b l e to the l e n g t h o f t i m e t a k e to i n i t i a t e a n d carry out the m o t o r response , p lus s o m e cons tant m i n i m u m a m o u n t o f t i m e r e q u i r e d for r e c e p t i o n o f i n f o r m a t i o n ) ; i n te rms o f W F , as r e p r e s e n t i n g s o m e r e s o l u t i o n l i m i t o f the aud i to ry system's o n intens i ty d i s c r i m i n a t i o n . P r o b l e m s E n c o u n t e r e d i n the C u r v e F i t t i n g T e c h n i q u e N o w , a r e t u r n to the p r o b l e m o f curve f i t t ing . A s i n the p i l o t study, " f ree" f i ts , that i s , w i t h n o const ra in ts p l a c e d o n any o f the p a r a m e t e r s - a l l p a r a m e t e r s a r e f ree to t a k e o n a n y v a l u e s i n the process o f o p t i m i z a t i o n - p r o d u c e d h igh ly u n p r e d i c t a b l e resu l ts . S i n c e there ex is ted a r a t i o n a l e f o r d o i n g so (as d e s c r i b e d a b o v e ) , b w a s the f i rs t p a r a m e t e r that I d e c i d e d to c o n s t r a i n cons is tent ly . 7 7 L e a v i n g u n c o n s t r a i n e d b o t h A a n d n, I q u i c k l y d i s c o v e r e d that t h e r e w a s a t r a d e o f f b e t w e e n theses t w o p a r a m e t e r s : the S S « , c o u l d be m i n i m i z e d by i n c r e a s i n g A a n d l e t t i n g n f a l l , o r b y the reverse . W h a t was n e e d e d , t h e n , a g a i n as i n the case o f the p i l o t study, w a s s o m e m e a n s by w h i c h I c o u l d l e g i t i m a t e l y c o n s t r a i n o n e o r the o t h e r o f these p a r a m e t e r s a n d t h e r e b y o v e r c o m e the i n c o n c l u s i v e n e s s g e n e r a t e d by the t r a d e o f f s i t u a t i o n . S i n c e the e x p o n e n t was the m e a s u r e I was a t t e m p t i n g to u n c o v e r , i t was e v i d e n t that w h a t w a s n e e d e d was a const ra in t o n A i n p a r t i c u l a r . M a n y t e c h n i q u e s w e r e a t t e m p t e d , o f d i f f e r i n g c o m p l e x i t y , to so lve th is p r o b l e m . D e t a i l s o f the m e t h o d s that w e r e unsuccessfu l w i l l n o t b e r e p o r t e d , a l t h o u g h I t h i n k i t w o u l d b e f a i r to say that i n e a c h case, I h a d a r e a s o n a b l y g o o d i d e a as to w h y they w e r e i n a d e q u a t e . R E S U L T S O n l y t w o m e t h o d s w i l l b e r e p o r t e d - the f i rst , to i l l us t ra te the inve rse p r o b l e m o f u n p r e d i c t a b l e resu l ts a n d t radeof fs , i n w h i c h I d e m o n s t r a t e that , g i v e n e v e n p e r f e c t l y sys temat ic c o n s t r a i n t s on A, i t is poss ib le to f o r c e the e x p o n e n t s f r o m R T a n d W F to b e n u m e r i c a l l y e q u i v a l e n t ; the s e c o n d to r e p o r t the resul ts f r o m w h a t I c o n s i d e r a l e g i t i m a t e c o n s t r a i n t m e t h o d . In a l l that f o l l o w s , w h e n I re fe r to "a cu rve fit", I a m t a l k i n g a b o u t the m e t h o d u s e d to o b t a i n a s ing le e x p o n e n t , (or a n es t imate o f A w h i c h w o u l d a l l o w us, t h r o u g h s u b s e q u e n t c u r v e f i ts , to o b t a i n a n v a l u e fo r the e x p o n e n t ) f o r o n e s ing le s u b j e c t / f r e q u e n c y p a i r . A " run" h e r e , t h e n m e a n s , f o r e x a m p l e , the p rocess o f f i t t ing the 10 d a t a p o i n t s f o r subject B T at a f requency o f 70 H z . O b t a i n i n g the exponents fo r a n 78 e n t i r e e x p e r i m e n t ( R T o r W F ) t h e r e f o r e consists o f curve f i t t i n g a t o t a l o f 2 0 runs (4 S u b j x 5 F r e q ) . I n the f i rst m e t h o d , the v a l u e o f A was o b t a i n e d by a v e r a g i n g t h e v a l u e s o f A o v e r f i ve c o n s t r a i n t runs that w e r e themse lves o b t a i n e d by c o n s t r a i n i n g n t o the r e g r e s s i o n n M E p r e v i o u s l y o b t a i n e d f o r e a c h subject ( i n c l u d i n g the "average subject" ) that c o r r e s p o n d e d to the f r e q u e n c y u n d e r c o n s i d e r a t i o n (that is , f o r B T , 7 0 H z , cons t ra in t 1 g e n e r a t e d a v a l u e ( A l ) re su l t in g f r o m c o n s t r a i n i n g n to reg ress ion n M E f o r B T 7 0 ; c o n s t r a i n t 2 g e n e r a t e d A 2 by c o n s t r a i n i n g n to regress ion n M E C K 7 0 , a n d so o n , u p to A 5 . A l to A 5 w e r e t h e n a v e r a g e d to y i e l d A a v g ) . Nex t , n i n e steps o f A , e a c h o n e c o v e r i n g a w i d t h that was e q u a l to o n e - t e n t h the m a g n i t u d e o f the a v e r a g e d A ( A a v g ) v a l u e , w e r e t h e n cons t ruc ted , 4 o n e a c h s ide o f A a v g . C u r v e f i t resu l ts u s i n g these c o n s t r a i n e d " A " s w e r e o b t a i n e d , a n d the e x p o n e n t r e p o r t e d ( i n T a b l e 2 ; i n g r a p h i c f o r m , F i g u r e 2 8 , th is m e t h o d r e f e r r e d to as p r o d u c i n g results "Ser ies 4") w a s the v a l u e f o r the 5 t h l e v e l o f A (that is , A a v g ) . H e r e , i t is ev ident that the exponents i n b o t h the W F a n d the R T cases c o u l d b e m a d e e q u a l just by u s i n g the r e g r e s s i o n n M E to a l l o w the e s t i m a t i o n o f a v a l u e f o r A w h i c h c o u l d t h e n be c o n s t r a i n e d w h i l e the exponent was f ree to vary . T h i s m e t h o d is a n a l o g o u s to the n - c o n s t r a i n e d m e t h o d u s e d i n the p i l o t study. T h e s e c o n d m e t h o d that w i l l b e r e p o r t e d is c o n s i d e r e d m o r e l e g i t i m a t e b e c a u s e it r e d u c e s the degrees o f f r e e d o m a v a i l a b l e to the e s t i m a t e d / c o n s t r a i n e d v a l u e s o f A . H e r e , the e s t i m a t e d va lues fo r A w e r e those o b t a i n e d by a v e r a g i n g A o v e r f i ts f r o m the f i ve f r e q u e n c i e s , e a c h o n e c o n s t r a i n e d to that p a r t i c u l a r subject 's , that p a r t i c u l a r run 's a s s o c i a t e d r e g r e s s i o n n M E . T h i s a v e r a g e d v a l u e o f A was t h e n u s e d as a c o n s t r a i n t i tse l f 79 f o r a l l f r e q u e n c i e s fo r that subject. T h u s , a s ing le es t imate o f A p e c u l i a r to e a c h subject ( a n d to e a c h e x p e r i m e n t ; R T vs. W F ) was u s e d to f i t a l l f ive f r e q u e n c i e s . T h e resu l ts o f th is ana lys is a r e i d e n t i f i e d as "Ser ies 17", a n d a re p r e s e n t e d i n T a b l e 3 , a n d i n g r a p h i c f o r m i n F i g u r e 29 ( the M E regress ion resul ts a re a l s o r e p r o d u c e d i n T a b l e 3 a n d F i g u r e 29 to f a c i l i t a t e ease o f c o m p a r i s o n ) . A s i n t h e M E resul ts sec t ion , the a p p r o a c h to F i g u r e 14 c a n b e g i n by l o o k i n g , i n e a c h case ( R T a n d W F ) , at the curve j o i n i n g the la rge , f i l l e d p e n t a g o n a l s y m b o l s that r e p r e s e n t the average o f the exponents across subjects at e a c h o f the f ive f r e q u e n c i e s . L o o k i n g at the R T g r a p h , o n e sees a b e a u t i f u l d e c r e a s e i n the e x p o n e n t f r o m .765 at 7 0 H z , d o w n t o .330 at 1000 H z , f o l l o w e d by a n u p t u r n i n the 10000 H z case to a n e x p o n e n t o f . 4 7 1 . T h i s bas ic pat te rn , w h i c h was a lso o b s e r v e d as a n o u t c o m e o f the M E r e g r e s s i o n analys is , is r e p e a t e d here w i t h v a r y i n g degrees o f f ide l i t y by the curves d e p i c t i n g the e x p o n e n t s o b t a i n e d for the i n d i v i d u a l subjects : S , w h i l e d e m o n s t r a t i n g cons is tent l y h i g h e r exponents at a l l f requenc ies t h a n any o f the o t h e r subjects , best p reserves th is r e l a t i o n s h i p - i n d e e d , o n e m i g h t expect this to be the case, s ince the g rea te r m a g n i t u d e o f this subject's exponents exerts a n u n d u e i n f l u e n c e o n the o u t c o m e o f the a v e r a g i n g across subjects ; C shows a b s o l u t e e x p o n e n t va lues c lose r to the r a n g e o f the r e m a i n i n g two subjects w h i l e r i v a l l i n g S i n m a i n t a i n i n g the p a t t e r n ; subject K exh ib i t s b o t h a f a i l u r e o f the exponent at 2 0 0 H z to d e c e n d m u c h b e l o w the 100 H z e x p o n e n t , a n d a m i s s i n g out o f the u p t u r n at 10000 H z ; w h i l e B r e a c h e s h is m i n i m u m e x p o n e n t at 2 0 0 H z a n d m a i n t a i n s this v a l u e at 1000 H z . E v e n a c k n o w l e d g i n g these i n d i v i d u a l p e c u l i a r i t i e s , however , o v e r a l l these R T d a t a r e v e a l a r e m a r k a b l e cons is tency that is e s p e c i a l l y e n c o u r a g i n g i n l ight o f the r a t h e r i n d i r e c t m e t h o d o f ana lys is . 80 T u r n i n g to the W e b e r f r a c t i o n g r a p h o f F i g u r e 29, o n e f inds a s o m e w h a t m e s s i e r p i c t u r e , b u t not a l a r m i n g l y so . T h e curve o f the average W F e x p o n e n t s p a r a l l e l s r e a s o n a b l y w e l l , i n b o t h s h a p e a n d abso lu te va lue , the average R T a n d M E exponents . N o t e that t h e 7 0 H z v a l u e s a re n u m e r i c a l l y qu i te c lose ( .698 f o r W F ; .765 f o r R T ; a n d .751 f o r M E ) , a n d that the o v e r a l l d r o p i n exponent f r o m 7 0 to 2 0 0 H z is 4 4 % f o r the W F , 4 2 % f o r R T , a n d 3 3 % f o r M E . T h e m a j o r d e p a r t u r e o c c u r s pos t 2 0 0 H z . F i r s t , the 1000 H z W F exponent is h i g h e r t h a n the 200 H z W F e x p o n e n t , w h i c h is c l e a r l y not the case f o r R T , a n d g e n e r a l l y no t the case fo r M E (subject B b e i n g the e x c e p t i o n ) ; a n d s e c o n d , the 10000 H z W F e x p o n e n t is very h igh (.647) - a l m o s t e q u a l to t h e v a l u e o b t a i n e d at 7 0 H z ( .698) . F o r R T , the exponent at 10000 H z ( .471) is c l o s e r to the e x p o n e n t at 2 0 0 H z ( .431) t h a n i t is to the exponent o f any o f the o t h e r f r e q u e n c i e s . A s m e n t i o n e d p rev ious l y , th is is a l s o the case fo r M E . In p a s s i n g , it c o u l d b e n o t e d that at least i n the e q u a l - l o u d n e s s c o n t o u r s s h o w n i n F i g u r e 9 , tones at 10000 H z a re j u d g e d to b e m o r e s i m i l a r i n p e r c i e v e d l o u d n e s s to a t o n e o f e q u a l s o u n d p ressure at a f r e q u e n c y o f 200 H z t h a n they a re to a 7 0 H z tone . W a r d (1990) , however , notes p r o b l e m s w i t h re l y ing o n the v a l u e s s u p p l i e d i n any g i v e n set o f e q u a l - l o u d n e s s c o n t o u r s w h e n the c o n d i t i o n s (acous t i ca l a n d m e t h o d o l o g i c a l ) u n d e r w h i c h the contours w e r e o b t a i n e d d o not c o r r e s p o n d to the c o n d i t i o n s p r e v a i l i n g o v e r the a c q u i s i t i o n o f the m e a s u r e s that a re i n s o m e way b e i n g c o m p a r e d to the c o n t o u r s . H e r e , this m e a n s that , w i t h o u t k n o w i n g that the c o n d i t i o n s i n w h i c h the p r e s e n t s tud ies t o o k p l a c e w e r e c o m p a r a b l e to those i n w h i c h t h e d a t a i n F i g u r e 9 w e r e o b t a i n e d , these e q u a l - l o u d n e s s curves c a n n o t necessar i ly b e u s e d to p r o v i d e any ins ight i n t o w h e t h e r , a n d i f so w h i c h , set o f 10000 H z data , the W F , the R T , o r the M E , m a y be 81 i n s o m e w a y q u e s t i o n a b l e . T h i s is doub l y t rue s ince it is at the h igher e n d s o f the in tens i ty a n d f r e q u e n c y d i m e n s i o n s that the v a r i a b i l i t y o f e q u a l - c o n t o u r resu l t s b e c o m e s m o s t p r o n o u n c e d ( W a r d , 1990). I f t h e r e a r e n o d e f i n i t i v e t h e o r e t i c a l g rounds u p o n w h i c h to base a r e a s o n e d guess a b o u t th is s t range i n c o n g r u i t y i n the 10000 H z resul ts , it c a n at least b e n o t e d that i t was the i n the W F e x p e r i m e n t that subject ive repor ts i n d i c a t e d that there m a y h a v e b e e n s o m e t h i n g u n u s u a l a b o u t the s o u n d qua l i t y o f d i f f e r e n t in tens i t ies at 10000 H z . W h e t h e r th is m a y h a v e b e e n d u e to the subjects' p e r c e p t u a l appara tus , o r to the e x p e r i m e n t a l e q u i p m e n t a p p a r a t u s is n o t c lear . A l t h o u g h I have n o e v i d e n c e , not e v e n a n y t h i n g o t h e r t h a n a f e w subject ive repor ts to l e a d m e to p a r t i c u l a r l y suspect t h e m , it is n e v e r t h e l e s s c o n c e i v a b l e that at s o m e p o i n t i n the s igna l p r o d u c t i o n a n d t r a n s m i s s i o n s y s t e m u s e d i n th is study, t h e r e w e r e s o m e d is tor t ions at h i g h f r e q u e n c i e s a n d h igh in tens i t ies . I f so , at best I m i g h t specu la te that such d is tor t ions c o u l d va ry i n the i r i n f luence o n s u c h d i f f e r e n t tasks as s igna l d e t e c t i o n ( R T ) a n d s igna l d i s c r i m i n a t i o n ( W F ) . A l t e r n a t e l y , p o s s i b l y - i n d e p e n d e n t o f the l i n k w i t h e q u a l - l o u d n e s s c o n t o u r s p e r c e , this h i g h W e b e r f r a c t i o n at h i g h f requenc ies ( m e a n i n g that intens i ty d i s c r i m i n a t i o n at very h i g h f r e q u e n c i e s , eg . 10000 H z is m o r e d i f f i cu l t t h a n it is at m i d r a n g e f r e q u e n c i e s s u c h as 1000 H z ) m a y be a g e n u i n e p roper ty o f the h u m a n a u d i t o r y system, a r i s i n g f r o m c o c h l e a r b i o m e c h a n i c s a n d a n a t o m y . W e r e this to b e the case, I might h a v e less r e a s o n to p a u s e h e r e to q u e s t i o n the qual i ty o f the 10000 H z W F d a t a , a n d m o r e r e a s o n to try to a c c o u n t f o r th is resul t i n l ight o f N o r w i c h ' s theory . L e a v i n g o f f s p e c u l a t i o n , a l t h o u g h it w i l l c o m e as n o surpr ise after v i e w i n g the resul ts o f the R T a n d W F curve f i t t ing results , I s h o u l d r e p o r t he re that , as i n the case o f M E , the ef fect o f f r e q u e n c y o n exponent was stat is t ica l ly s ign i f i cant ( fo r R T , F 4 1 J = 3 6 0 , p . 0 0 1 ; f o r W F , F 4 1 2 = 2 0 5 , p < .001 . T h e ef fect o f subject , o n the o t h e r h a n d , was not s ign i f i can t i n e i the r case ( R T , F 3 1 2 = . 8 9 1 , p = .474 ; W F , F 3 1 2 = 1.311, p = .316) . 83 CURVE FITTING: Series 4 (Constraint 65; Est(A) =avg(A c o n s t r j | n t s 1-5) ; Est(B)=measYmin) Reaction Time Exponents: SUBJECTS B C K S Avg 70 Hz 0. 683 0. 662 0. 677 0. 693 0.679 100 HZ 0. 538 0-534 0. 517 0. 529 0.53C 200 Hz 0. 428 0. 448 0. 438 0. 454 0.442 1000 Hz 0. 348 0. 350 0. 347 0. 355 0.350 10000 Hz 0. 450 0. 469 0. 424 0. 428 0.443 Avg n 0. 489 0. 493 0. 479 0. 492 0.489 Weber Fraction Exponents: SUBJECTS B C K S Avg 70 Hz 100 Hz 200 Hz 1000 Hz 10000 Hz Avg n Magnitude Estimation Exponents: SUBJECTS B C K S Avg 70 Hz 0.461 0. 677 0. 530 1.335 0.751 100 Hz 0.361 0. 576 0. 439 0.882 0.550 200 Hz 0.246 0. 507 0. 350 0.902 0.501 1000 Hz 0.272 0. 370 0. 236 0.661 0.372 10000 Hz 0.300 0. 530 0. 316 0.684 0.458 Avg n 0.328 0. 532 0. 374 0.882 0.526 TABLE 2 Curve fitting results for RT and WF - series 4. (With reproduced ME regression results). 0.696 0.659 0.679 0.674 0.677 0.527 0.531 0.508 0.536 0.526 0.437 0.448 0.444 0.458 0.447 0.364 0.346 0.357 0.349 | 0.354 0.481 0.458 0.412 0.424 | 0.444 0.501 0.488 0.480 0.488 | 0.490 CURVE FITTING: Reaction Time - Series 4 CURVE FITTING Weber Fraction - Seres 4 LINEAR REGRESSION: Magnitude Estimation FIGURE 28 Curve fitting results for RT and WF, plotted - series 4. (With reproduced ME regression results). 85 CURVE FITTING: Series 17 (Est(A)/Subj=avg over freqs(Est(A)using n=nMEreg); Est(B)=measYmin) Reaction Time Exponents: SUBJECTS B C K S Avg 70 Hz 0.500 0. 735 0. 624 1.200 0.765 100 Hz 0.390 0. 595 0. 450 0.970 0.601 200 Hz 0.250 0. 372 0. 438 0.662 0.431 1000 Hz 0.250 0. 316 0. 263 0.500 0.330 10000 Hz 0.400 0. 490 0. 295 0.700 0.471 Avg n 0.358 0. 502 0. 414 0.806 0.520 Weber Fraction Exponents: SUBJECTS B C K S Avg 70 Hz 0.550 0.560 0.520 1.161 0.698 100 Hz 0.390 0.510 0.450 0.757 0.527 200 Hz 0.246 0.412 0.250 0.720 0.407 1000 Hz 0.285 0.365 0.390 0.660 0.425 10000 Hz 0.720 0.700 0.436 0.730 0.647 Avg n 0.438 0.509 0.409 0.806 0.541 B SUBJECTS C K S Avg 70 Hz 0.461 0.677 0.530 1.335 0.751 100 Hz 0.361 0.576 0.439 0.882 0.550 200 Hz 0.246 0.507 0.350 0.902 0.501 1000 Hz 0.272 0.370 0.236 0.661 0.372 10000 Hz 0.300 0.530 0.316 0.684 0.458 Avg n 0.328 0.532 0.374 0.882 0.526 TABLE 3 Curve fitting results for RT and WF - series 17. (With reproduced ME regression results). CURVE FITTING: Reaction Time - Series 17 CURVE FITTING: Weber Fraction - Series 17 FREQUENCY OotfHz)) FREQUENCY Oog(Hz)) LINEAR REGRESSION: Magnitude Estimation Figure 29 Curve fitting results for RT and WF - series 17. (With reproduced ME regression results). VI I I . O V E R A L L R E S U L T S A N D D I S C U S S I O N 87 T h e e m p i r i c a l hypothes is that underscores th is thesis is that the e x p o n e n t s i n m a g n i t u d e e s t i m a t i o n , r e a c t i o n t i m e , a n d W e b e r f r a c t i o n studies s h o u l d a l l b e n u m e r i c a l l y t h e s a m e f o r any g iven subject. T h e i d e a is that if a c o n s e r v a t i o n p r i n c i p l e is at w o r k i n t h e p s y c h o p h y s i c a l r e a l m of in tens i ty p e r c e p t i o n , as N o r w i c h pos tu la tes it is , the " s t u f f that is b e i n g conserved is i n f o r m a t i o n ( i n the str ict i n f o r m a t i o n theory sense ) ; that is , w h a t e v e r i n f o r m a t i o n is r e c e i v e d by a sensory r e c e p t o r w i l l b e , M U S T b e t r a n s m i t t e d to the " b r a i n " w i t h o u t loss ( N o r w i c h , 1990, u n p u b l i s h e d ) . I h a r p o n w h a t m a y s e e m l i k e a n o u t - o f - p l a c e t h e o r e t i c a l i n t e r j e c t i o n h e r e f o r a r e a s o n . N o r w i c h is , w i t h the conserva t ion p r i n c i p l e F = k H a l o n e , m a k i n g n o a s s u m p t i o n s a b o u t how much i n f o r m a t i o n is b e i n g r e c e i v e d by the r e c e p t o r s . L e t us a s s u m e that t h e r e a re i n d i v i d u a l d i f ferences i n the sensory receptors - i n the o p e r a t i n g charac te r i s t i cs o f s ing le receptors , i n the d i s t r i b u t i o n o f recepto rs , i n the d e g r e e o f d a m a g e to any o r m a n y receptor (s ) , or whatever . W h a t the c o n s e r v a t i o n p r i n c i p l e is t r y i n g to t e l l us is that the receptors w i l l be a b l e to g a i n w h a t e v e r i n f o r m a t i o n the i r o w n c a p a c i t i e s a n d l i m i t a t i o n s e n a b l e t h e m to, but o n c e they have that i n f o r m a t i o n , it w i l l be c o n s e r v e d w i t h i n the sys tem (a l l ow ing for k, the m u l t i p l i c a t i v e constant , w h i c h is no t necessar i l y asser ted to b e 1). I nd i v idua l d i f fe rences t h e n , w h i c h is w h e r e I s ta r ted th is s e c t i o n , b e c o m e very in te res t ing sources of p e r i p h e r a l in te r - sub jec t v a r i a b i l i t y that o n e w o u l d h o p e , i f the c o n s e r v a t i o n p r i n c i p l e is co r rec t ( its i m p l i e d asser t ion o f p e r f e c t l i n e a r i t y past the r e c e p t o r l e v e l b e i n g put to the s ide ) , w o u l d be r e f l e c t e d i n m i n i m i z e d i n t r a - s u b j e c t v a r i a b i l i t y across the d o m a i n . 88 H o w d o the o v e r a l l resul ts o f this study c o m p a r e w i t h this e x p e c t a t i o n ? I h a v e l o o k e d at the v a l u e s o b t a i n e d fo r these exponents i n e a c h o f the e x p e r i m e n t a l c o n d i t i o n s s e p a r a t e l y : w h e n these a re p u t together , the results p l o t t e d i n F i g u r e 3 0 c a n b e o b s e r v e d . I n these g raphs the M E exponents o b t a i n e d t h r o u g h the l i n e a r r e g r e s s i o n m e t h o d , a n d the cu rve f i t t i n g resul ts f o r R T a n d W F (ser ies 17) a r e p l o t t e d against f r e q u e n c y f o r e a c h subject . H e r e , as i n p r e v i o u s graphs , the scales h a v e b e e n h e l d cons tant . F i r s t , as w a s n o t e d w h e n c o n s i d e r i n g the results o f the th ree e x p e r i m e n t s separate ly , i t c a n b e o b s e r v e d that a l m o s t a l l o f the curves o f the exponents i n a l l th ree o f the e x p e r i m e n t s , a n d f o r e a c h subject , s h o w the s a m e b a s i c p a t t e r n : h i g h at the lowest f r e q u e n c y , d r o p p i n g i n the f r e q u e n c y m i d r a n g e , a n d r i s i n g a g a i n at the h ighest f r e q u e n c y l e v e l . Sub jec ts B a n d C s h o w the largest u p t u r n . I n d e e d , the re a re s o m e q u i t e e x t r a o r d i n a r y u p t u r n s that a p p e a r - espec ia l l y fo r the W F c o n d i t i o n s f o r subject B , a n d to a lesser extent fo r C . A s d iscussed prev ious ly , these e x t r e m e v a l u e s m a y have b e e n a t t r i b u t a b l e to u n i n t e n d e d s o u n d p r o p e r t i e s o f the 10000 H z tone that s o m e subjects s e e m e d to detect i n the h i g h e r in tens i ty 10000 H z tones . S e c o n d , o n e s h o u l d n o t e that the d r o p i n e x p o n e n t w i t h f r e q u e n c y d i f fe rs a m o n g subjects : S , fo r e x a m p l e , shows by fa r the s teepest d e c r e a s e i n exponent , w h i l e K ' s curves are f lat by c o m p a r i s o n . W h a t is o f m o s t in teres t h e r e t h o u g h , i n l ight o f the i m m e d i a t e l y f o r e g o i n g d i s c u s s i o n , is the r e l a t i o n s h i p b e t w e e n the va lues o f the exponents o n the th ree e x p e r i m e n t s w i t h i n , as o p p o s e d to across subjects. H e r e , it i t w o u l d a p p e a r that the e x p o n e n t s a r e m o r e c lose l y r e l a t e d to the sub ject/ f requency p a i r i n g they represent , t h a n they a r e to the e x p e r i m e n t fac to r . 89 Subj: B - EXPONENTS (RT.WF.MEreg) Sub): C - EXPONENTS (RT.WF.MEreg) 1 7 5 2 2 2 2 6 9 3 .16 3 6 3 4 1 0 FREQUENCY OogtHz)) 1.75 2 2 2 2 6 9 3 1 6 3 6 3 4 .10 FREQUENCY OogCHz)) Subj: K - EXPONENTS (RT.WF.MEreg) Sub; S - EXPONENTS (RT.WFJvEreg) 1.75 222 259 3.16 363 4>10 FREQUENCY Oog(Hz» 1.76 222 269 3.16 3.63 4.10 FREQUENCY CtogCHz)) FIGURE 30 Obtained exponents plotted together: nKIE(regiession), nR1{series 17) and nlir(serics 17) versus frequency. Results shown for each subject. 90 A b e t w e e n / w i t h i n A N O V A c o n f i r m s this : b o t h the f r e q u e n c y ef fect ( ¥ A A 5 = 4 .05, p < .01) a n d the subject ef fect ( F 3 ^ [ I „ W i m . s , E i S s F i ^ S l l E x F 1 = 6 4 . 6 1 5 4 , p < .001) w e r e s ign i f i cant , w h i l e the e x p e r i m e n t effect was not s ign i f i cant ( F ^ = 0 .04 , p > .96) . T h i s m e a n s that , desp i te s ign i f i cant d i f fe rences b e t w e e n the e x p o n e n t s o b t a i n e d f o r b o t h d i f f e r e n t subjects a n d f o r d i f fe ren t f requenc ies , the e x p o n e n t s d o n o t d i f f e r s ign i f i cant l y across e x p e r i m e n t s . O n e sees h e r e that i n d i v i d u a l d i f fe rences , as i n o t h e r b r a n c h e s o f psycho logy , m a y b e m o r e o f a b l e s s i n g t h a n a c o n f o u n d , but the r e a s o n i n th is case is to ta l l y d i f f e r e n t t h a n the ones u s u a l l y c o n s i d e r e d (e.g. u n d e r s t a n d i n g w h a t is " n o r m a l " f o r a s y s t e m by b e i n g a b l e to study it i n v a r i o u s stages o f a b n o r m a l i t y ; i n a m o r e e x i s t e n t i a l i s t / p s y c h o l o g i c a l v e i n , l o o k i n g t o w a r d the e x c e p t i o n as a way o f l e a d i n g o thers a n d o n e s e l f to a r i c h e r a p p r e c i a t i o n o f h u m a n poss ib i l i t y , et cetera . ) - fo r h e r e it suggests, o r re f lec ts the ex is tence o f a c o n s e r v a t i o n p r i n c i p l e such as N o r w i c h ' s theory is p r o p o s i n g . I X . R E C O M M E N D A T I O N S 91 M E T H O D O L O G I C A L T h e c u r v e f i t t i n g t e c h n i q u e r e m a i n s p r o b l e m a t i c , no t just f o r the p r e s e n t p ro jec t , b u t i n a b r o a d c o m p u t a t i o n a l sense. In "the o l d days", b e f o r e a u t o m a t e d c o m p u t a t i o n was a v a i l a b l e o r a f f o r d a b l e , d a t a such as ours w o u l d have b e e n "fit b y h a n d " - that is , a curve d r a w n by h a n d w o u l d b e the t h e o r e t i c a l cu rve , S S r a w o u l d b e c a l c u l a t e d by h a n d , a n d so o n . It m i g h t b e i n t e r e s t i n g to l o o k at the resul ts o f such a n analys is o n the p r e s e n t d a t a . T o a t t a c k the t h e o r e t i c a l p r o b l e m o f curve f i t t ing , as w e l l as to a n s w e r the e m p i r i c a l q u e s t i o n s r a i s e d i n this thesis, it might b e h e l p f u l to c o n d u c t a m o r e f i n e l y g r a i n e d e x p l o r a t i o n o f the A-n t radeoff . F o r ins tance , a l t h o u g h it is t rue that i n o n e o f the u n r e p o r t e d c u r v e f i t t i n g m e t h o d s I l o o k e d at the ef fect o n n o f c o n s t r a i n i n g A f r o m the r a t h e r a r b i t r a r y v a l u e s o f .2 to 1.2, i n .2 u n i t steps, th is m e t h o d c o n s t i t u t e d a q u i t e c o a r s e ana lys i s that m a y have b e e n at o n c e t o o rest r ic t ive i n its range , a n d t o o b r o a d i n its s tep s i ze . C o n s i d e r a b l y m o r e inves t iga t ion i n this d i r e c t i o n m i g h t p r o v e f r u i t f u l . A f i n a l p o s s i b i l i t y w o u l d be to we ight the d a t a po in ts that are b e i n g f i t i n s u c h a way as to c a p t u r e the m o s t " impor tant" sect ions o f the curves , a n d to f o r c e the t h e o r e t i c a l c u r v e t o c o n f o r m to these sect ions. T h i s , o n e w o u l d h o p e , w o u l d p r e v e n t those s i t u a t i o n s i n w h i c h the S S r a c a n be m i n i m i z e d by a f u n c t i o n that fa i l s to c a p t u r e the extent o f the c u r v a t u r e i n the data . T h e p r o b l e m w i t h this poss ib i l i t y , o f c o u r s e , l ies i n t r y ing to d e c i d e exact ly w h a t the w e i g h t i n g f u n c t i o n s h o u l d be , b u t it is c o n c e i v a b l e that s o m e w e i g h t i n g , n o m a t t e r h o w in i t ia l l y c r u d e , m i g h t w e l l g ive m o r e r e p r e s e n t a t i v e curve f i t t i n g resu l ts t h a n I h a v e o b t a i n e d to date . E M P I R I C A L / T H E O R E T I C A L A t t i m e s , i n n e r psychophys ics , at least , does s e e m l i k e a ve ry b l a c k box . C l a r k L . H u l l , s t i l l a b e h a v i o u r i s t , b u t t e n d i n g t o w a r d the n e o , u s e d to t a l k a b o u t " o r g a n i s m i c v a r i a b l e s " - that is , he r e c o g n i z e d that the d i m e n s i o n s o f the b o x w e r e r e l a t i v e , d e p e n d i n g u p o n w h i c h d e p e n d e n t measures were u n d e r i n v e s t i g a t i o n . If o n e w e r e to b r i n g u n d e r c o n s i d e r a t i o n c e r t a i n p s y c h o p h y s i o l o g i c a l m e a s u r e s , t h e n , p e r h a p s o n e c o u l d a lso m a n a g e to s h r i n k the box a l i t t le , a n d i n d o i n g so , r e v e a l i n f o r m a t i o n that m a y b e o f use i n e v a l u a t i n g b o t h the m e c h a n i s t i c , a n d the c o n s e r v a t i o n a l s ides o f N o r w i c h ' s en te rp r i se . T h o u g h the n a t u r e o f the theory that p r o v o k e s such a d i r e c t i o n o f s tudy m a y b e d i f fe ren t , e m p i r i c a l l y this w o u l d , o f course , not be a f i rst : n u m e r o u s a t t e m p t s have b e e n m a d e to re la te p s y c h o p h y s i c a l results to p h y s i o l o g i c a l m e a s u r e s . In t e r m s o f r e l a t i n g the P o w e r L a w o b t a i n e d i n M E e x p e r i m e n t s to n e u r a l events, S tevens has t e n d e d to m a r s h a l l e v i d e n c e i n suppor t o f the n o t i o n that b o t h subject ive e x p e r i e n c e o f intens i ty a n d n e u r a l f i r i n g rates c a n be d e s c r i b e d by a p o w e r f u n c t i o n u n i q u e to the g i v e n m o d a l i t y . H e c i tes (Stevens , 1970), fo r e x a m p l e , the c lass ic e x p e r i m e n t o f A d r i a n & M a t t h e w s (1927) o n the eye of the e e l , w h i c h showed that n e u r a l o u t p u t w a s a l o g a r i t h m i c f u n c t i o n o f s t imu lus intensi ty . Stevens r e p l o t t e d th is d a t a a n d f o u n d that a p o w e r f u n c t i o n w i t h a n exponent o f 0.32 c o u l d a lso fit the d a t a w e l l (Stevens , 1970) . T h e c o r r e s p o n d i n g psychophys i ca l f u n c t i o n for the p e r c e p t i o n o f b r igh tness by h u m a n subjects has a n e x p o n e n t o f 0.33 (Stevens, 1975). Stevens (1970) a lso s h o w e d that H a r t l i n e & G r a h a m ' s (1932) study o n a s ingle f ib re o f the o p t i c n e r v e r e v e a l e d a p o w e r f u n c t i o n w i t h a n e x p o n e n t o f 0.29 (see F i g u r e 31 b e l o w , f r o m Stevens , 1975, p. 2 0 9 ) . 93 Power function responses in toe eye of the horseshoe crab Limulia. The stars represent frequencies of nerve impulses recorded by Hanline and Graham in 1932 from a single fiber of the optic nerve. Fre-quency was measured 3.5 seconds after the onset of the light. The lower three functions represent responses to 0.02-second flashes of light, recorded with intracellular electrodes in single ommatidia. The plotted points represent the areas under oscillographs published by Fuortes and Hodgkin in 1964. The straight lines in the log-log coordinates are power functions, whose slopes (ex-ponents) are 0.29 for the stars and 0.32 for the other three functions. Both coordinates give relative values only. Eyes were dark adapted. (From Stevens 1970, Scienct. 170, 1043-1050. Copyright 1970 by the American Association for the Advancement of Science. Reprinted by permission.) FIGURE 31 Physiological power functions. (From Stevens, 1975, p.209) Stevens a l s o c i tes M o u n t c a s t l e et a l . , 1962 a n d M o u n t c a s t l e et a l . , 1966, w h o s e w o r k has s h o w n that f o r the pass ive m o v e m e n t o f the m o n k e y l i m b the r e l a t i o n b e t w e e n j o i n t p o s i t i o n a n d n e u r a l o u t p u t c a n be d e s c r i b e d by a p o w e r f u n c t i o n , a n d that the n e u r a l r e s p o n s e to m e c h a n i c a l p ressure o n the g l a b r o u s s k i n o f the m o n k e y ' s h a n d c a n b e d e s c r i b e d by a p o w e r f u n c t i o n w i t h a n e x p o n e n t that c lose ly matches that w h i c h has b e e n f o u n d w h e n h u m a n subjects j u d g e d the in tens i ty o f pressure a p p l i e d to the p a l m o f t h e h a n d (Stevens , 1975: J . C . Stevens & M a c k , 1959) . F i n a l l y , Stevens (1975) re fers to the w o r k o f B o r g et a l (1967) , w h i c h d e m o n s t r a t e d that the intensi ty charac te r i s t i cs o f n e r v e f i b r e s e n d i n g o n gustatory receptors c o u l d b e successfu l ly a p p r o x i m a t e d by p o w e r f u n c t i o n s . R e g a r d i n g the h y p o t h e s i z e d p o w e r f u n c t i o n r e l a t i n g in tens i ty to e v o k e d p o t e n t i a l s , S tevens (1970) presents s u p p o r t i n the w o r k o f B o u d r e a u (1965) , w h o m a n a g e d to -i 1 1 1 1 ; O O 0I 1 1 1 1 '— 0 10 20 JO 4 0 50 Relative light intensity in decibel* 94 r e c o r d , i n o n e out o f his 25 cat subjects, potent ia ls a r i s i n g f r o m the s u p e r i o u r o l i v a r y c o m p l e x that f o l l o w e d a p o w e r f u n c t i o n over a range o f 60 d e c i b e l s , K e i d e l a n d S p r e n g (1965) , w h o m e a s u r e d the a m p l i t u d e o f the N I c o m p o n e n t o f the ve r tex p o t e n t i a l i n r e s p o n s e to tones , e lec t r i c cu r rent , a n d v i b r a t i o n , a n d D a v i s et a l (1968) w h o o b t a i n e d p o w e r f u n c t i o n s f i t ted to ve r tex p o t e n t i a l d a t a fo r f i ve h u m a n subjects i n r e s p o n s e to r e p e a t e d bursts o f tones . O n the bas is o f these a n d o the r s tudies t h o u g h , Stevens was u l t i m a t e l y c o m p e l l e d to c o n c l u d e that e l e c t r o p h y s i o l o g i c a l exponents w e r e s m a l l e r t h a n p s y c h o p h y s i c a l exponents (Stevens , 1975). T h e r e is o the r e v i d e n c e , too , i n d i c a t i n g that the exponents m a y not b e i d e n t i c a l (see K n i b e s t o l & V a l l b o , 1976, V a l l b o & H a g b a r t h , 1968, B o t t e e l a l , 1975) . Y e t , e v e n h a v i n g s a i d th is , it c a n s t i l l b e a r g u e d that there is su f f i c ient s u p p o r t i n g e v i d e n c e to m e r i t f u r t h e r study o n e m p i r i c a l g rounds a l o n e , a n d there is a d e f i n i t e n e e d i n t e r m s o f N o r w i c h ' s theory to invest igate the va l id i t y o f the F = k H a s s u m p t i o n ( w h e n F is i n t e r p r e t e d as rate o f n e u r a l i m p u l s e s ) . T h e r e is n o exp l i c i t d e r i v a t i o n o f a "magn i tude o f e v o k e d p o t e n t i a l " e q u a t i o n to b e f o u n d i n N o r w i c h ' s w o r k . T h e i n c l u s i o n o f this m e a s u r e w o u l d b e j u s t i f i e d , h o w e v e r , by S tevens ' ( a n d others ' - see F l e t c h e r & M u n s o n , 1933; Stevens & D a v i s , 1938; Z w i s l o c k i , 1969) suggest ion that the l o u d n e s s o f sounds c o u l d b e d e t e r m i n e d by the m a g n i t u d e o f n e u r a l act iv i ty a l o n g the a u d i t o r y t ract . In the case o f Stevens i n p a r t i c u l a r , h is "sensory t r a n s d u c e r theory" , w h i c h m a i n t a i n s that a l l o f the n o n - l i n e a r i t y b e t w e e n i n p u t [ a p p l i c a t i o n o f the s t imulus] a n d ou tput [sensation] occurs at the r e c e p t o r l e v e l ( M c K e n n a , 1985, b rackets m i n e ) , appears to share w i t h N o r w i c h a n e m p h a s i s u p o n the sensory r e c e p t o r i tse l f as a c c o m p l i s h i n g a l l o f the necessary e n c o d i n g - i n S tevens ' te rms , 95 a l l o f the " c o m p r e s s i o n " o f a s t imu lus range that is p o t e n t i a l l y m a n y o r d e r s o f m a g n i t u d e g r e a t e r t h a n is the range o f the n e u r a l f i r i n g rate that m u s t u l t i m a t e l y ca r r y the s i g n a l ; i n N o r w i c h ' s v i e w , a l l o f the " s a m p l i n g " o f the d isc re te "uni ts" ( t e r m m i n e ) o f sensory i n p u t . I n t h e s a m e w a y that fo r Stevens, i f a l l subsequent t r a n s f o r m a t i o n s o f the s i g n a l a re l i n e a r t h e e l e c t r o p h y s i o l o g i c a l a n d psychophys i ca l exponents s h o u l d b e i d e n t i c a l , so s h o u l d they b e f o r N o r w i c h , i f F = k H is t rue . I n s t u d y i n g th is , the a m p l i t u d e a n d la tency o f the 5 t h c o m p o n e n t o f the a u d i t o r y e v o k e d p o t e n t i a l ( the N I wave ) c o u l d b e u s e d as the d e p e n d e n t v a r i a b l e s that w o u l d be r e l a t e d to s t i m u l u s intensi ty . T h e task w o u l d b e i d e n t i c a l i n m e t h o d to the psychophys i cs R T e x p e r i m e n t , except that b o t h p s y c h o p h y s i c a l a n d p s y c h o p h y s i o l o g i c a l d a t a w o u l d b e r e c o r d e d . T h e p l a n w o u l d be to f irst conduct a s m a l l "p i lo t " study to d e t e r m i n e w h e t h e r i t w a s p o s s i b l e to o b t a i n c lea r e n o u g h measures o f changes i n a m p l i t u d e a n d / o r l a t e n c y as a f u n c t i o n o f s t i m u l u s intens i ty to ca l cu la te a v a l u e fo r the p s y c h o p h y s i o l o g i c a l p o w e r f u n c t i o n e x p o n e n t . F o u r d e p e n d e n t measures , t h r e e o f a m p l i t u d e a n d o n e o f la tency , that h a v e b e e n r e p o r t e d i n the l i te ra tu re as p o t e n t i a l c a n d i d a t e s c o u l d b e t e s t e d : 1) d i f f e r e n c e b e t w e e n the largest pos i t i ve a n d negat i ve d e f l e c t i o n s f r o m the b a s e l i n e ( f o l l o w i n g F r a n z e n & O f f e n l o c h , 1969), 2 ) largest negat i ve t rough to base l ine d e f l e c t i o n (based o n R i e t v e l d & T o r d o i r , 1965) , 3 ) ' pos i t i ve to b a s e l i n e ' ( K e i d e l & S p r e n g , 1965) , a n d 4 ) la tency o f a pos i t i ve c o m p o n e n t ( P 3 0 0 ? ) (af ter V a u g h n & H u l l , 1965) . R e s u l t s f r o m the a b o v e studies d o show s o m e d i s c r e p a n c i e s , b u t suggest that o n e m i g h t expec t t h e t h i r d m e a s u r e to y i e l d exponents c losest to those o b t a i n e d by the 96 psychophysics measures. This was the conclusion reached by Keidel & Spreng (1965) who utilized the first three measures and were able to fit all of them using power functions (though each required a different exponent). Assuming success in finding an adequate electrophysiological measure, one could hope to extend the domain over which the present thesis' hypothesis could be evaluated, as well as to further investigate both some of the constants and some of the variables that appear in Norwich's derived equations. X . C O N C L U S I O N S 97 H o w e v e r in tu i t i ve l y in t r i gu ing are m a n y o f the c o m m o n a l i t i e s , the extent to w h i c h N o r w i c h ' s a t t e m p t to un i f y the laws o f s e n s a t i o n t h r o u g h his f u n d a m e n t a l e q u a t i o n strictly p a r a l l e l s t h e c o n s e r v a t i o n o f energy l a w of t h e r m o d y n a m i c s , o r the d e g r e e to w h i c h h is i d e a s a b o u t sensory recepto r s a m p l i n g o f i n f o r m a t i o n p r o v i d e s a r i g o r o u s a n a l o g u e t o the p r i n c i p l e s o f thermostat is t ics is not yet c lea r . Y e t the re is n o r e a s o n w h y this s h o u l d b e te r r ib l y t roub l ing at this p o i n t s ince , h o w e v e r p l e a s i n g it m a y b e to the P y t h a g o r i a n sens ib i l i t ies , strict p a r a l l e l i s m is not the i m m e d i a t e g o a l . 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N e w Y o r k : S p r i n g e r - V e r l a g . R a b i n o w i t z , W . M . , L i m , J .S . , B r a i d a , L . D . & D u r l a c h , N. I . (1976) Intens i ty p e r c e p t i o n : V I . S u m m a r y o f recent d a t a o n dev ia t ions f r o m W e b e r ' s l a w f o r 1 0 0 0 - H z t o n e pu lses . J o u r n a l o f the A c o u s t i c a l Soc ie ty o f A m e r i c a . 59, 1506 -1509 . 101 R i e t v e l d , W . J . & T o r d o i r , W . E . M . (1965) T h e i n f l u e n c e o f f l a s h in tens i ty u p o n the v i s u a l e v o k e d r e s p o n s e i n the h u m a n cortex . A c t a P h y s i o l o g i c a et P h a r m a c o l o g i c a  N e e r l a n d i c a . 13, 160 -170 . R i e s z , R . R . (1928) D i f f e r e n t i a l intensi ty sensi t iv i ty o f the e a r f o r p u r e tones . P h y s i c a l  R e v i e w . 31, 8 6 7 - 8 7 5 . S t e i n , R . B . ( 1 9 6 7 b ) T h e i n f o r m a t i o n capac i t y o f ne rve ce l l s u s i n g a f r e q u e n c y c o d e . B i o p h y s i c a l J o u r n a l . 7, 7 9 7 - 8 2 6 . S tevens , S .S . (1970) N e u r a l events a n d the p s y c h o p h y s i c a l l a w . S c i e n c e . 170(3962), 1 0 4 3 - 1 0 5 0 . S tevens , S .S . (1975) Psychophys ics : I n t r o d u c t i o n to its p e r c e p t u a l , n e u r a l , a n d s o c i a l  p rospec ts . N e w Y o r k : W i l e y . S tevens , S .S . & D a v i s , H . (1938) H e a r i n g , its psycho logy a n d phys io logy . N e w Y o r k : W i l e y . S tevens , S .S . & M a c k , J . D . (1959) Sca les o f a p p a r e n t f o r c e . J o u r n a l o f E x p e r i m e n t a l  P s y c h o l o g y . 58, 4 0 5 - 4 1 3 . T a y l o r , M . M . & C r e e l m a n , C D . (1967) P E S T : E f f i c i e n t es t imates o n p r o b a b i l i t y f u n c t i o n s . J o u r n a l o f the A c o u s t i c a l Soc ie t y o f A m e r i c a . 41(4), 7 8 2 - 7 8 7 . V a l t e r , K . M . (1988) T h e ent ropy theory o f p e r c e p t i o n as a p p l i e d to in tens i ty  d i s c r i m i n a t i o n o f aud i to ry p u r e tones . U n i v e r s i t y o f T o r o n t o : M . A . T h e s i s . V a u g h n , H . G . & H u l l , R . C (1965) F u n c t i o n a l r e l a t i o n b e t w e e n s t i m u l u s in tens i ty a n d p h o t i c a l l y e v o k e d c e r e b r a l responses i n m a n . N a t u r e . 206, 7 2 0 - 7 2 2 . W a r d , L . M . (1990) C r i t i c a l bands a n d m i x e d - f r e q u e n c y s c a l i n g : s e q u e n t i a l d e p e n d e n c i e s , e q u a l - l o u d n e s s contours , and p o w e r f u n c t i o n exponents . P e r c e p t i o n & Psychophys i cs . 47(6), 5 5 1 - 5 6 2 . Z w i s l o c k i , J . (1960) T h e o r y o f t e m p o r a l a u d i t o r y s u m m a t i o n . J o u r n a l o f the A c o u s t i c a l  Soc ie t y o f A m e r i c a . 32(8), 1046-1060. Z w i s l o c k i , J . (1969) T e m p o r a l s u m m a t i o n o f l oudness : A n ana lys is . J o u r n a l o f the  A c o u s t i c a l S o c i e t y o f A m e r i c a . 46, 4 3 1 - 4 4 1 . Z w i s l o c k i , J . J . & G o o d m a n , D . A . (1980) A b s o l u t e s c a l i n g a n d sensory m a g n i t u d e s : A v a l i d a t i o n . P e r c e p t i o n & Psychophys ics . 28, 2 8 - 3 8 . XII. APPENDIX A - INSTRUCTIONS TO SUBJECTS 102 A-l : Magnitude estimation pilot study Psychophysica Experiment; MONMXF ABSOLUTE MAGNITUDE ESTIMATION Supervising Professort Dr. Lawrence M. Ward Research Assistant t Kelly Davidson Instructions; In this experiment we would like to find out how loud various intensities of sound appear to you. For this purpose, you wi l l hear a series of tones, one at a time, through one side of the earphones. Tour task w i l l be to assign a number to every tone in such a way that your impressjon of how large the number is matches your impression of how loud the sound i s . Tou may use any positive numbers that appear appropriate to you - whole numbers or decimals (convert fractions to decimals). Do not use zero. Do not think of physical units of measurement, such as decibels, and do not worry about running out of numbers - there w i l l always be a smaller number-than the smallest you use. Do not worry about numbers you assigned to pre-ceding sounds. You w i l l notice that the pitch of the tones within each session w i l l a l l be the same, but between sessions, the pitch may vary. There i s no need to worry about this. Consider the sessions to be unrelated to each other, and just follow the foregoing instructions for each. Procedure; You w i l l be seated inside a ventilated sound attenuation chamber with a response keyboard. Tones w i l l be presented through one side of the headphones to the ear on the same side as that of the preferred hand i n a telegraph key task. It is important that you not begin your response U n t i l after the tone has ended. Responses are made by pressing the top row of numbered keys on the keyboard. Each numerical response must be followed by a press of the termina-tion key, marked located on the upper right hand corner of the keyboard. This w i l l send your response to the computer and trigger the next stimulus. Because the computer i s programmed to accept only the f i r s t response i t receives, i t i s not possible to correct mistakes. If you hit a wrong key accidentally, just repeat the correct response, press the termination key as usual, and con-tinue with the experiment. Please remember that we are concerned with the accuracy of your responses. It is important that you respond as carefully and as conscientiously as possible at a l l times. Please press the keyboard keys firmly, l i f t i n g your finger from the key completely with every response. There w i l l be five (5) approximately one hour sessions consisting of two, three-hundred (300) tone runs each, for which you w i l l be paid $4.00 per hour. Payment occurs after a l l of the sessions have been completed. Thank you for your interest and participation. 103 A-2: R e a c t i o n t i m e p i l o t study Psychophysics Experiment: RTFRBQ Supervis ing Professor : Dr. Lawrence M . Ward Research A s s i s t a n t : K e l l y Davidson I n s t r u c t i o n s : In t h i s experiment you w i l l be presented w i t h tones of a p a r t i c u l a r frequency (pi tch) and i n t e n s i t y ( loudness) . Tour task w i l l be to respond to the onset of the tone by press ing down a telegraph key as q u i c k l y as p o s s i b l e . The p i t c h and loudness of the tones w i l l vary across sessions but your task w i l l remain the sane. Procedure: Tou w i l l be seated ins ide a v e n t i l a t e d sound attenuation chamber w i t h a te legraph key placed on the s ide of your preferred response hand. Tones w i l l be presented through one s ide of the headphones to the ear on the same s ide as that of the preferred hand. Responses are made by press ing down the te legraph key as q u i c k l y as poss ib le i n response to the onset of the tone. P r e s s i n g the key causes the computer to terminate the tone, record the reac -t i o n t ime , and prepare for the next tone. Tou w i l l not ice that the time i t takes f o r the next tone to be presented v a r i e s from t r i a l to t r i a l . This i s done f o r experimental reasons, and does not i n d i c a t e any mechanical malfunc-t i o n . Tou w i l l always know whether you have pressed the telegraph key prop-e r l y or not because i f you have n o t , the tone w i l l cont inue. I t Is important that you respond as q u i c k l y as poss ib le at a l l t imes. A f t e r an i n i t i a l pract ice s e s s i o n , there w i l l be f i f t y (50) approximately ten (10) minute sessions c o n s i s t i n g of one-hundred and ten (110) t r i a l s each, f o r which you w i l l be paid $4.00 per hour . Payment occurs a f t e r a l l of the sess ions have been completed. A p r i s e of $10.00 w i l l be paid to the p a r t i c i -pant w i t h the lowest s ing le sess ion average r e a c t i o n t ime. Thank you f o r your in teres t and p a r t i c i p a t i o n . 104 A-3: Pretest: Lower auditory threshold PSYCHOPHYSICS EXPERIMENT A u d i t o r y Range - AUDRANDM.BAS Instructions! This experiment can be considered a pretest which i s designed to determine your lower auditory range for i n t e n s i t y for tones of f i v e d i f f e r e n t frequencies. Your task w i l l be to judge i n which of two i n t e r v a l s a tone -appears, and to enter a response representing your judgment into the keyboard. E n t e r a " 1 " i f you heard t h e tone in the f i r s t i n t e r v a l , and a "2" i f i n t h e second. There i s no need to be disconcerted i f you f i n d i t d i f f i c u l t to make a decision i n some c a s e s - p l e a s e j u s t c o n t i n u e t o be as careful and a t t e n t i v e as you can. Pr oc e d u r e i You w i l l be s e a t e d i n a sound-proof booth w i t h earphones on and a r e s p o n s e keyboard and d i g i t a l d i s p l a y box on the t a b l e b e f o r e you. A l l of t h e t o n e s t h a t you hear w i l l be p r e s e n t e d t o the ear t h a t i s on t h e same s i d e as your r e f e r r e d r e s p o n s e hand. The p r o c e d u r e f o r each t r i a l 15 as f o l l o w s . You w i l l n ote t h a t p r i o r t o b e g i n n i n g , t h e d i g i t a l d i s p l a y box i s showing a COOl. Once you a r e s e a t e d c o m f o r t a b l y i n the booth and a r e ready t o b e g i n , you 11.ay p r e s s t h e spacebar on the keyboard t o s t a r t t h e s e s s i o n . A f t e r a b r i e f pause, t he d i s p l a y box w i l l change t o C993, which i s a "warning" s i g n a l , a l e r t i n g you t h e f a c t t h a t t h e f i r s t i n t e r v a l w i l l commence i n about h a l f of a second. The s t a r t o f t h e f i r s t i n t e r v a l w i l l be i n d i c a t e d by C101. T h i s d i s p l a y w i l l c o n t i n u e for t h e d u r a t i o n of t h e f i r s t i n t e r v a l . The d i s p l a y w i l l t h e n change t o C021 f o r t h e d u r a t i o n of t h e second i n t e r v a l . The m i d d l e 600 msec w i t h i n each of t h e s e two 1000 msec i n t e r v a l s w i l l c o n t a i n e i t h e r t he a u d i t o r y t o n e , or t h e "dummy t o n e " ( t h a t i s , the s i l e n t i n t e r v a l ) . As mentioned p r e v i o u s l y , you a r e t o judge which i n t e r v a l c o n t a i n e d t h e tone. A f t e r t he second i n t e r v a l i s o v e r , t h e [02] w i l l be r e p l a c e d w i t h C001, i n d i c a t i n g t h a t you can now make your keyboard r e s p o n s e . E n t e r i n g your r e s p o n s e a l s o c a uses t h e i n t i a t i o n of t h e next t r i a l . Thus t h e s t r u c t u r e of each t r i a l i s : tone I or (tone COO 3 [991 CI03 C021 [001 n e u t r a l w arning i n t e r v a l 1 i n t e r v a l 2 P l e a s e do not enter a r e s p o n s e b e f o r e t h e C00] appears because the computer cannot accept a response b e f o r e t h e second i n t e r v a l i s c o m p l e t e l y o v e r , and re s p o n d i n g e a r l y w i l l cause t h a t p a r t i c u l a r t r i a l t o be re p e a t e d from t h e b e g i n n i n g . There w i l l be f i v e r u n s i n t h i s p r o c e d u r e , one at each of f i v e d i f f e r e n t f r e q u e n c i e s . Each run s h o u l d l a s t no more than a p p r o x i m a t e l y 10 mi n u t e s . The r a t e of r e n u m e r a t i o n i s • 5 . 0 0 per hour, which w i l l p a i d f o l l o w i n g c o m p l e t i o n of the end of t h e e n t i r e s e r i e s of t e s t s w i t h i n t h i s m u l t i l e v e l e x p e r i m e n t s . Thank you f o r your i n t e r e s t and p a r t i c i p a t i o n . Dr. Lawrence l i . Ward P r o f e s s o r of P s y c h o l o g y K e l l y Davidson Research S t a f f 105 A-4: Pretest: Upper auditory limit PSYCHOPHYSICS EXPERIMENT Upper Auditory Range - UPAUDM.BAS Ins t ruct ions ! The purpose of t h i s b r i e f experiment i s to determine the upper 1imit of your auditory threshold for i n t e n s i ty for tones of f i v e d i f f e r e n t frequencies . Your task w i l l be to l i s t e n to a s e r i e s of 600 msec tones of ascending i n t e n s i t y , (presented one at a t ime) , and to terminate the s e r i e s at the point at which the i n t e n s i t y of the tone seems d e f i n i t e l y i n t o l e r a b l e . No more tones w i l l be presented in that run after you s ignal termination, and the experiment w i l l move on to the next run. Procedure: You w i l l be seated in a sound-proof booth with earphones on and a response keyboard and d i g i t a l d i s p l a y box on the table before you. A l l of the tones that you hear w i l l be presented to the ear that i s on the same s^de as your refer red response hand. The procedure for each t r i a l i s as fo l lows . You w i l l note that pr ior to beginning, the d i g i t a l d i s p l a y box i s showing a COOl. Once you are seated comfortably i n the bu-.-th and are ready to begin, you may press the spacebar on the keyboard to s tar t the sess ion . After a b r i e f pause, the d i s p l a y box w i l l change to C 11 3 , and at the same time you wi11 hear a tone which wi11 last for 600 msec. This i s the maximum length of any tone to which you w i l l be exposed in any of the three other experiments that make up the present study. Once you have l i s t e n e d to t h i s tone, you are to judge whether or not you are able to to lera te a louder tone than you just heard. If the answer i s " d e f i n i t e l y not" , then press "S" <for s top) , and the run w i l l end. If you are not sure, then press any key except " S " , and the same tone w i l l be presented again for you to judge. Eve?n i f you are e n t i r e l y sure that an increase in i n t e n s i t y w i l l not be problematic , the f i r s t tone at the given i n t e n s i t y w i l l be presented a second time. As in a l l cases, just pr ess any key except "S" to cont i nue. Entering your response also causes the i n t i a t i o n of the next t r i a l . Please do not enter a response before the tone i s over because the computer cannot accept a response before t h i s time, and responding early w i l l cause that p a r t i c u l a r t r i a l to be repeated from the beginning. It i s important for you to know that there i s an upper l i m i t upon the i n t e n s i t y level which our tone generator i s capable of producing, and each s c r i e s has a d e f i n i t e spontaneous end p o i n t . The chances are very good that you wi11 never have to "S"top any run yourself and that i t w i l l therefore be the instrument 1 i mi t at i oris f wi 11 determine the maximum i n t e n s i t y level which w i l l be used in the subsequent parts of the study. There w i l l be f i v e runs in t h i s procedure, one at each of f ive d i f f e r e n t frequencies . Each run should las t no more than approximately 5 minutes. The rate of renumeration i s *5.00 per hour, which w i l l paid fol lowing completion of the end of the e n t i r e s e r i e s of test s within t h i s muIti level e^peri ment. Thank you for your i nter est and par t i c i pat i on. Dr. Lawrence M. Ward Professor of Psychology K e l l y Davidson Research S t a f f 106 A-5: M a i n study: W e b e r f r a c t i o n PSYCHOPHYSICS EXPERIMENT Weber Frac t ion Using Adaptive Staircase Technique •'. 1 up - 3 down; 2AFC) Inst ruct ions : In t h i s experiment, your task w i l l be to attend to the i n t e n s i t y of many p a i r s of separate tones, to Judge, i n each case, whether the more intense (the louder) tone was presented dur i ng the f i r st or the second inter v a l , and to enter a response representing your judgment i n t o the keyboard. Enter a "1" i f you heard the louder tone in the f i r s t i n t e r v a l , and a "2" i f in the second. There i s no need to be disconcerted i f you find i t d i f f i c u l t to make a dec i s i on i n some cases - pi ease just c ont i nue to be as c ar eful and att en t i ve as you can. Procedure: You w i l l be seated in a sound-proof booth with earphones on and a response keyboard and d i g i t a l d i s p l a y box on the table be for e you. A l l of the tones t hat you hear wi11 be presented to the ear that i s on the same s ide as your preferred response hand. The procedure for each t r i a l i s as f o i l o w s . You w i l l note that pr ior to beginning, the d i g i t a l d i s p l a y box i s showing a C003. Once you are seated comfortably i n the booth and are ready to begin, you may press the spacebar on the keyboard to s tar t the sess ion . After a b r i e f pause, the d i s p l a y box w i l l change to t"9'33, which i s a "warning" s i g n a l , a l e r t i n g you the fact that the f i r s t i n t e r v a l w i l l commence in about•half of a second. The star t of the f i r s t i n t e r v a l w i l l be indicated by C101. This d i s p l a y w i l l continue for the duration of the f i r s t i n t e r v a l . The display w i l l then change to C021 for the durat ion of the second i n t e r v a l . The middle £00 msec within each of these two 1000 msec lnter vals w i l l contain an audi tory tone. As ment i oned pr ev i ous 1y, you are to judge which i n t e r v a l contained the louder tone. After the second i n t e r v a l i s over, the C02 3 w i l l be r ep1ac ed with C 00 D, i n d i c a t i n g that you c an now make your keyboard response. Ent er i ng your re$>onse also causes the i n t i a t i o n of the next t r i a l . Thus the s t ructure of each t r i a l i s : tone!I Itone2 [coo: C-391 C 101 C021 C001 neutral war n i ny in terval 1 i n t e r v a l 2 Please do not enter a response before the C00D appears because the computer cannot accept a response before the second interval i s completely over, and responding ear ly w i l l cause thai p a r t i c u l a r t r i a l to be repeated from the beginning. There w i l l be f i f t y runs in t h i s experiment, ten d i f fer ent runs at each of f ive d i f f e r ent f r equenc i es . Eac h run contains 300 t r i a l s , and should last no more than approximately 10 minutes. In a d d i t i o n , the experiment w i l l i nc1ude one shor t pr act i ce run consi st i ny of 50 t r i a l s . The r at e of r enumer at i on i s *5.00 per hour , wh ich w i l l paid at the end of the e n t i r e exper iment. Thank you for your interest and p a r t i c i p a t i o n . Dr. Lawrence M. Ward Professor of Psychology Kel1y Davidson Research Staf f 107 A-6: M a i n s tudy : R e a c t i o n t i m e P s y c h o p h y s i c s Experiment: RTPREQ S u p e r v i s i n g P r o f e s s o r t Dr. Lawrence M. Ward R e s e a r c h A s s i s t a n t J K e l l y Davidson I n s t r u c t i o n s : I n t h i s experiment you w i l l be p r e s e n t e d w i t h tones o f a p a r t i c u l a r f r e q u e n c y ( p i t c h ) and i n t e n s i t y ( l o u d n e s s ) . Tour t a s k w i l l be t o respond to t h e onset o f the tone by p r e s s i n g down a t e l e g r a p h key as q u i c k l y as p o s s i b l e . The p i t c h and loudness o f t h e tones w i l l v a r y a c r o s s s e s s i o n s but y o u r t a s k w i l l remain the same. P r o c e d u r e : Tou w i l l be seated i n s i d e a v e n t i l a t e d sound a t t e n u a t i o n chamber w i t h a t e l e g r a p h key p l a c e d on the s i d e of y o u r p r e f e r r e d response hand. Tones w i l l be p r e s e n t e d through one s i d e of t h e headphones t o the e a r on the same s i d e as t h a t o f t h e p r e f e r r e d hand. Responses a r e made by p r e s s i n g down t h e t e l e g r a p h key as q u i c k l y as p o s s i b l e i n response t o the onset o f the tone. P r e s s i n g t h e key causes the computer t o t e r m i n a t e the t o n e , r e c o r d the r e a c -t i o n t i m e , and p r e p a r e f o r the next t o n e . Tou w i l l n o t i c e t h a t the time i t t a k e s f o r the n e x t tone t o be p r e s e n t e d v a r i e s from t r i a l t o t r i a l . T h i s i s done f o r e x p e r i m e n t a l reasons, and does n o t i n d i c a t e any mechanical m a l f u n c -t i o n . Tou w i l l always know whether you have p r e s s e d t h e t e l e g r a p h key p r o p -e r l y o r not because i f you have n o t , t h e tone w i l l c o n t i n u e . I t i s Important t h a t you respond as q u i c k l y as p o s s i b l e a t a l l t i m e s . A f t e r an i n i t i a l p r a c t i c e s e s s i o n , t h e r e w i l l be f i f t y (50) a p p r o x i m a t e l y t e n (10) minute s e s s i o n s c o n s i s t i n g o f one-hundred and t e n (110) t r i a l s e a c h , f o r w h i c h you w i l l be p a i d $5.00 p e r h o u r . Payment o c c u r s a f t e r a l l o f the s e s s i o n * have been completed. A p r i s e o f $10.00 w i l l be p a i d t o the p a r t i c i -p ant w i t h the l o w e s t s i n g l e s e s s i o n average r e a c t i o n t i me. Thank you f o r your i n t e r e s t and p a r t i c i p a t i o n . A-7: Main study: Magnitude estimation PSYCHOPHYSICS EXPERIMENT Magnitude Estimation Instruct ions: In this experiment we would like to find out how loud various intensities of sound appear to you. For this purpose, you will hear a series of tones, one at a time, through one side of the earphones. Your task will be to assign a number to every tone in such a way that your impression of how large the number is matches your impression of how loud the sound i s . The task will begin with the presentation of a tone ("modulus tone") to which you are to mentally assign the number 50. All numbers assigned to subsequent tones are to be Judged relative to this modulus tone: that is i f , for example, a tone is perceived as being twice as loud as the modulus tone, the number assigned to the non-modulus tone should be 100; similarly, i f the tone is half as loud as the modulus tone, the number assigned should be 25, and so on. You may use any positive numbers that appear appropriate to you - whole numbers or decimals (convert fractions to decimals). Do not use zero. Do not think of physical units of measurement, such as decibels, and do not worry about running out of numbers -there will always be a smaller number than the smallest you • use. Do not worry about numbers you assigned to preceding sounds. You will notice that the pitch of the tones within each session will a l l be the same, but between sessions, the pitch may vary. There is no need to worry about this. Consider the sessions as being unrelated to each other, and just follow the foregoing instructions for each. Procedure: You will be seated inside a ventilated sound attenuation chamber with a response keyboard. Tones will be presented through one side of the headphones to the ear on the same side as that of the preferred hand in a telegraph key task. When you are ready to begin the experiment, press the spacebar on the keyboard. The first tone that you hear will be the modulus tone. You do not need to respond to this tone - just mentally assign a "50" to the loudness that you perceive when listening to i t . After a short pause the experiment will automatically present the first tone to be judged. Afts*- the tone has completely ended and you have A-7 (cont.): Main study: Magnitude estimation judged the loudness, enter your numerical response by pressing the top row of numbered keys on the keyboard. Because the computer is programmed to accept only the first response i t receives, it is not possible to actually correct mistakes, but you can change your response by typing a "c" (for change or correct) and then entering your corrected response, e.g. 100c90. Do not try to type any spaces in such a correction because the spacebar is reserved throughout the experiment as the key that you will press to advance to the next tone. Throughout the experiment, after every ten tones requiring your judgment the modulus tone w i l l be presented again i n order t o refresh your memory of i t . There i s no need for you to count the number of tones i n order t o know when i t i s the modulus tone that i s being presented; instead, you will be alerted by a slightly longer pause after you have pressed the spacebar. This signals you to look at the visual feedback display box. A C333 will appear on the box, indicating that the next tone that you hear will be the modulus tone. After the tone has been presented, the visual feedback box will return to C003, and you are free to press the spacebar in order to advance the experiment to the next set of tones that will need to be judged. Please remember that we are concerned with the accuracy of your responses. It is important that you respond as carefully and as conscientiously as possible at a l l times. Please press the keyboard keys firmly, l i f t i n g your finger from the key completely with every response. There will be a 100 tr ial practice session in which to familiarize yourself with this task. The experiment i tself will consist of a total of 10 approximately one-half hour sessions composed of 300 tr ials each. Payment for participation in this experiment is *5.00 per hour and will be made after the sessions in al l parts (Weber Fraction, Reaction Time, Magnitude Estimation, Magnitude of Evoked Potentials) of this thesis study have been completed by the "subject". Thank you for your interest and participation. Dr. Lawrence M. Ward Professor of Psychology Kelly Davidson Research Staff 

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