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An empirical test of a probabilistic model of consumer spatial behaviour Wiginton, John Cameron 1966

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AN EMPIRICAL TEST OF A PROBABILISTIC MODEL OF CONSUMER SPATIAL BEHAVIOUR by John Cameron Wiginton B. A. Sc., University of B r i t i s h Columbia, 1957 i A Thesis i n Commerce Submitted to the Faculty of Graduate Studies In P a r t i a l Fulfilment of the Requirements for the Degree of Master of Business Administration We accept t h i s thesis as conforming to the required standard; THE UNIVERSITY OF BRITISH COLUMBIA August, 1966 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia,, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y aval].able f o r reference and studyo I f u r t h e r agree t h a t p e r m i s s i o n - f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of ^2^2^^ s&foftf/srf&doy?. The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Dat-e i ABSTRACT T h i s t h e s i s i s c o n c e r n e d w i t h making a n e m p i r i c a l t e s t o f a p r o b a b i l i s t i c model o f i n t r a - u r b a n r e t a i l t r a d e i n t e r a c t i o n s . The model, t e r m e d a p r o b a b i l i s t i c model o f consumer s p a t i a l b e h a v i o u r , i s r e l a t e d t o a s e r i e s o f models i n s o c i a l s c i e n c e known a s g r a v i m e t r i c m o d e l s . The p a r t i c u l a r m odel c o n s i d e r e d u s e d s t o r e a r e a a n d d i s t a n c e i n t i m e u n i t s a s I t s m a j o r v a r i a b l e s . I t a l s o i n c l u d e s a n e x p o n e n t i a l p a r a m e t e r , t h e v a l u e o f w h i c h must be e s t i m a t e d f r o m e m p i r i c a l d a t a . The m a j o r h y p o t h e s i s on w h i c h t h e model i s t e s t e d i s b a s e d on t h e b e h a v i o u r o f t h i s p a r a m e t e r . The h y p o t h e s i s s t a t e s t h a t v a l u e s o f t h i s p a r a m e t e r v a r y s i g n i f i c a n t l y , d e p e n d i n g upon t h e t y p e o f s h o p p i n g t r i p b e i n g c o n s i d e r e d . The t y p e o f s h o p p i n g t r i p i s d e t e r m i n e d by t h e p a r t i c u l a r t y p e o f s h o p p i n g goods a p p a r e n t l y s o u g h t by t h e consumer. The h y p o t h e s i s i s t e s t e d by means o f e m p i r i c a l d a t a on consumer p u r c h a s i n g p a t t e r n s g a t h e r e d i n t h e V a n c o u v e r M e t r o p o l i t a n a r e a t h r o u g h t h e u s e o f a n i n t e r v i e w s u r v e y c o n -d u c t e d r a n d o m l y by c e n s u s t r a c t s . The d a t a a r e a n a l y z e d i n a n e s p e c i a l l y w r i t t e n , i t e r a t i v e c omputer programme. S t a t i s -t i c a l t e s t s u s u a l l y a p p l i e d t o s u c h d a t a a r e f o u n d t o be i n a d e q u a t e t o t h e a n a l y s i s o f t h e r e s u l t s . A s p e c i a l t e s t w h i c h i s i n t e n d e d t o show t h e s e n s i t i v i t y o f t h e model t o t h e p a r a m e t e r i s p r e s e n t e d and a p p l i e d . An i n d e p e n d e n t t e s t o f t h e i i r e p r e s e n t a t i v e n e s s o f t h e d a t a i s p r e s e n t e d . The d a t a a r e f o u n d t o be r e p r e s e n t a t i v e , b u t t h e model i s f o u n d t o be i n s e n s i t i v e t o t h e b e h a v i o u r o f t h e p a r a m e t e r . F u r t h e r , m e a s u r e s o f v a r i a t i o n i n o b s e r v e d b e h a v i o u r e x p l a i n e d by t h e model a r e g e n e r a l l y low. I t i s c o n c l u d e d t h a t t h e model i n i t s p r e s e n t f o r m d o e s n o t a p p l y t o V a n c o u v e r . The t h e s i s i s u n a b l e t o c o n c l u d e w h e t h e r c h a n g e s a r e r e q u i r e d i n t h e f a c t o r s o f t h e model o r i n t h e r e l a t i o n s h i p s p e c i f i e d , t h o u g h t h e r e i s e v i d e n c e w h i c h shows t h a t b o t h may r e q u i r e a t t e n t i o n . i i i TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGMENT . v i i Chapter I INTRODUCTION Statement of the Problem 1 Purpose of the Study 2 Methodology of the Study 3 L i m i t a t i o n s of the Study 6 D e f i n i t i o n of Terms 6 Organization of the Wri t t e n Report . . . . 8 I I MODELS IN SOCIAL SCIENCE Par t I : Some Fundamental Notions and L i m i t a t i o n s of Models 10 Types of Models 11 V a r i a b l e s and R e l a t i o n s h i p s . . . . . . . 13 The Problem of Measurement 15 P a r t I I ; Gravimetric Models of Human I n t e r a c t i o n E a r l i e s t Incidences 19 F o r m a l i z a t i o n : S o c i a l Physics 24 Extension of the Concepts 33 Other M o d i f i c a t i o n s and Adaptations . . . 39 Summary 43 I I I SPECIFIC HISTORY OF GRAVIMETRIC MODELS IN MARKETING The Laws of R e t a i l G r a v i t a t i o n 47 E m p i r i c a l Development of the Laws . . . . 51 Tests of the Law of R e t a i l G r a v i t a t i o n . . 57 Summary Iv Chapter p a g e IV A PROBABILISTIC MODEL OF INTRA-URBAN TRADE INTERACTIONS Huff's Approach to Consumer S p a t i a l Behaviour 75 Huff's Concept of U t i l i t y and the Model . 77 Huff's E m p i r i c a l Test and E v a l u a t i o n of the Parameter 85 Results of the E m p i r i c a l Test 90 Summary 96 V EMPIRICAL TEST OF THE PROBABILISTIC MODEL IN METROPOLITAN VANCOUVER The Sample and the Survey 99 Programme XLAMC f o r E v a l u a t i n g the Exponential Parameter 106 A p p l i c a t i o n of Programme XLAMC to Huff's Data 110 A p p l i c a t i o n of Programme XLAMC to Vancouver Data 118 A p p l i c a t i o n of Programme XLAMS to Vancouver Data 133 Concluding Remarks 139 VI CONCLUSIONS AND RECOMMENDATIONS Restatement of the Problem 1^ 3 R e c a p i t u l a t i o n of Purposes i n View of Resu l t s Obtained 1^ 3 The Need f o r a New Model lkk BIBLIOGRAPHY 151 APPENDIX I Huff's Consumer Behavior Questionnaire . . 157 APPENDIX I I Tec h n i c a l D e t a i l of Vancouver Sample . . 158 APPENDIX I I I Questions From Vancouver Survey . . . . l 6 l APPENDIX IV Programme XLAMC : Flow Chart f o r Successive Approximation of Lambda . . . 163 APPENDIX V Programme XLAMC : F o r t r a n 16? V LIST OF TABLES TABLE Page I E v a l u a t i o n of Distance Exponent i n R e i l l y ' s Law 55 I I Reynolds* Test of R e i l l y ' s Law 6 l I I I Reynolds' Test of R e i l l y ' s Law Ev a l u a t i o n of Distance Exponent 62 IV Huff's Sample Data P e r t a i n i n g to the Neighbourhoods Studied 87 V Results of Huff's E m p i r i c a l Test f o r Commodity " C l o t h i n g " , . . . 91 VI R e s u l t s of Huff's E m p i r i c a l Test f o r Commodity " F u r n i t u r e " 92 V I I Composite Re s u l t s of Huff's E m p i r i c a l Test . 93 V I I I D e f i n i t i o n of Composite Commodities . . . . 104 IX Areas of Stores i n M e t r o p o l i t a n Vancouver. . 105 X Comparison of Huff's E v a l u a t i o n s of Para-meter with Programme XLAMC E v a l u a t i o n ( C l o t h i n g ) 112 XI Comparison of Huff's E v a l u a t i o n of Parameter w i t h Programme XLAMC E v a l u a t i o n ( F u r n i t u r e ) 113 X I I Comparison of Huff and XLAMC Parameter Estimates 114 X I I I R e s u l t s of A p p l i c a t i o n of Programme XLAMC f o r Commodity A 121 XIV R e s u l t s of A p p l i c a t i o n of Programme XLAMC f o r Commodity B 123 XV Res u l t s of A p p l i c a t i o n of Programme XLAMC f o r Commodity C 124 XVI Table of Composite Re s u l t s 131 XVII Results of A p p l i c a t i o n of Programme XLAMS to Vancouver Data 135 v i LIST OF FIGURES F i g u r e Page 1 R e i l l y ' s D etermination of the Di s t a n c e Exponent i n R e i l l y ' s Law . . . 56 2 Reynolds's T e s t of R e i l l y ' s Law - R e l a t i v e Frequency 6 3 3 Reynolds' Test of R e i l l y ' s Law - Power of Account 6 4 k Census T r a c t Map of G r e a t e r Vancouver . 100 5 H u f f ' s Data f o r C l o t h i n g 116 6 H u f f ' s Data f o r F u r n i t u r e 117 7 T y p i c a l Curves Produced by Programme XLAMC 120 8 Commodity A: Rmax versus Lambda . . . . 127 9 Commodity B: R ^ x versus Lambda . . . . 128 10 Commodity C: Rmax versus Lambda . . . . 129 v i i ACKNOWLEDGMENT The w r i t e r w i s h e s t o t a k e t h i s o p p o r t u n i t y t o t h a n k D r . L. Moore, D r . F. W e b s t e r , and e s p e c i a l l y D r . S. Oberg, a l l o f t h e F a c u l t y o f Commerce a n d B u s i n e s s A d m i n i s t r a t i o n , f o r t h e i r a s s i s t a n c e and encoura g e m e n t d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . CHAPTER I INTRODUCTION Statement of the Problem The problem to which t h i s t h e s i s i s addressed i s th a t of making an e m p i r i c a l t e s t of a p r o b a b i l i s t i c model of i n t r a - u r b a n r e t a i l trade i n t e r a c t i o n s . This model i s the most recent of a long s e r i e s of models i n s o c i a l science which are c l a s s e d as g r a v i m e t r i c models. In p a r t i c u l a r , the model to be examined i s that put forward by Huff and termed,by him, a p r o b a b i l i s t i c 1 model of consumer s p a t i a l behaviour. The p r a c t i c e of marketing makes many assumptions about the f a c t o r s which i n f l u e n c e consumer s p a t i a l behaviour. G r a v i -metric models are intended to make e x p l i c i t some of these f a c t o r s and t h e i r r e l a t i o n s h i p s . Such models have the general f°rm: R . f(A) f(3) where P - a measure of p o t e n t i a l ; A = a measure of the a t t r a c t i v e f o r c e of an area or p o i n t ; 3 = a measure of the cost i n v o l v e d i n responding to the a t t r a c t i v e f o r c e . The u n i t s used f o r these measures vary w i t h the nature of the problem. The mathematical form of the fu n c t i o n s used f r e q u e n t l y c o n t a i n David L. Huff, Determination of Intra-Urban R e t a i l Trade  Areas (Los Angeles: U n i v e r s i t y of C a l i f o r n i a , Graduate School of Business A d m i n i s t r a t i o n , D i v i s i o n of Research, 1962) pp. 4-5. 2 an exponent or e x p o n e n t i a l parameter. Purpose of the Study The purpose of t h i s study i s to determine whether e m p i r i c a l data gathered i n a p a r t i c u l a r g e o g r a p h i c a l area, namely M e t r o p o l i t a n Vancouver, w i l l tend to support a model which was formulated and t e s t e d i n a p i l o t study elsewhere (the Los Angeles M e t r o p o l i t a n A r e a ) . I f the data tend to support the model then the weight of t h i s a d d i t i o n a l evidence would tend to show that the model might be g e n e r a l l y a p p l i c a b l e to the p r e d i c t i o n of r e t a i l trade i n t e r a c t i o n i n urban a r e a s . On the other hand, i f the data do not support the model, an attempt w i l l be made to determine whether the d e f i c i e n c y l i e s i n the data, or i n the model. I f the data are d e f i c i e n t , then n o t h i n g more can be done with them d i r e c t l y . The task becomes one of determining the manner i n which the data are d e f i c i e n t and of producing a new r e s e a r c h d e s i g n without such l i m i t a t i o n s . I f , however, the model appears to be d e f i c i e n t , t h i s may be the r e s u l t of i n c o r r e c t f a c t o r s or v a r i a b l e s , or i n c o r r e c t r e l a t i o n s h i p s among f a c t o r s , or both. Here, the data, the r e s u l t s of the study, and other sources of i n f o r -mation must be c o n s i d e r e d t o g e t h e r to attempt to d e t e c t the d e f i c i e n c i e s . These may be such as to r e q u i r e a r e d e f i n i t i o n o f f a c t o r s i n c l u d e d or a r e c o n c e p t u a l i z a t i o n of the r e l a t i o n -s h i p s used, or both. In order to prepare f o r t h i s contingency, the development of the present model was examined i n d e t a i l , both f o r i n t e r n a l c o n s i s t e n c y and a g a i n s t a background of 3 p r e v i o u s r e s e a r c h on g r a v i m e t r i c models r e p o r t e d i n d i v e r s e l i t e r a t u r e . M e t h o d o l o g y o f t h e S t u d y The m e t h o d o l o g y o f a s c i e n t i f i c s t u d y must f o l l o w n a t u r a l l y f r o m t h e h y p o t h e s e s t o "be t e s t e d . B e c a u s e t h i s s t u d y s e e k s t o do t h e same t h i n g a s H u f f ' s p i l o t e m p i r i c a l s t u d y , t h e b a s i c h y p o t h e s i s t o be t e s t e d must a l s o be t h e same. H u f f ' s model c o n t a i n s a n e x p o n e n t i a l p a r a m e t e r . The h y p o t h e s i s i s t h a t t h e e m p i r i c a l l y e s t i m a t e d v a l u e o f t h i s p a r a m e t e r i s a f u n c t i o n o f t h e t y p e o f s h o p p i n g t r i p f o r w h i c h i t i s e s t i m a t e d . By t h i s H u f f means t h a t t h e v a l u e d e t e r m i n e d f o r c o n s u m e r s ' e x c u r s i o n s t o buy one t y p e o f s h o p p i n g g ood w i l l be s i g n i f i -c a n t l y d i f f e r e n t f r o m t h a t o b t a i n e d f o r o t h e r s h o p p i n g g o o d s . I n o r d e r t o t e s t h i s h y p o t h e s i s , H u f f u s e d a s a t u r a t i o n s a m p l i n g t e c h n i q u e on a few n e i g h b o u r h o o d s s e l e c t e d t o be homogeneous i n t e r m s o f i n c o m e . The i n d i v i d u a l s t h u s s a m p l e d were a s k e d t o r e p o r t where t h e y l a s t b o u g h t a r a n g e o f s h o p p i n g g oods i t e m s , a n d where t h e y n o r m a l l y b o u g h t t h i s same r a n g e o f i t e m s . The e m p i r i c a l t e s t was made on some o f t h e i n f o r m a t i o n o b t a i n e d i n a nswer t o t h e f i r s t q u e s t i o n . S i n c e t h e p r e s e n t s t u d y was t o be on a b r o a d e r b a s i s , i t c o u l d n o t u s e t h e same s a m p l i n g t e c h n i q u e . N e i t h e r t i m e n o r money p e r m i t t e d t h i s . T h u s , i t was n e c e s s a r y t o u s e a random s a m p l i n g t e c h n i q u e o v e r a l a r g e number o f n e i g h b o u r -h o o d s . F u r t h e r , b e c a u s e t h e d a t a were g a t h e r e d i n a s u r v e y which sought to develop other i n f o r m a t i o n r e g a r d i n g the shopping t r i p as an e n t i t y , the p a r t i c u l a r q u e s t i o n s which were intended t o e l i c i t the e m p i r i c a l data had to be framed d i f f e r e n t l y . The pre s e n t survey focussed on the l a s t shopping t r i p , not on the l a s t purchase. While H u f f ' s q u e s t i o n s normally produced a response f o r every item i n the range, the survey used as a b a s i s f o r t h i s study d i d not o b t a i n a response i n many cases s i n c e many shopping goods are i n f r e q u e n t l y purchased. T h i s reduced the number of items i n the sample, but i t d i d h o l d other f a c -t o r s c o n s t a n t . Given the model and the s e t of e m p i r i c a l o b s e r v a t i o n s , the next task i s to d e v i s e a means of determining whether the theory (model) f i t t e d the f a c t s ( s e t of e m p i r i c a l d a t a ) . T h i s i s the p o i n t a t which the s p e c i f i c h y p o t h e s i s put forward by Huff — t h a t the e x p o n e n t i a l parameter of the model i s a f u n c t i o n of the type of shopping t r i p — i s of c e n t r a l concern. Given a s u i t a b l e h y p o t h e s i s , the t e c h n i c a l d e t a i l of the t e s t must be determined. Huff chose to have the model p r e d i c t what consumer behaviour would be f o r a sequence of va l u e s of the e x p o n e n t i a l parameter. These p r e d i c t i o n s were compared, one by one, wit h the o b s e r v a t i o n s . The g o o d n e s s - o f - f i t of p r e d i c t i o n s with o b s e r v a t i o n s was measured by a t e s t parameter d e f i n e d i n such a way that i t resembled the c o e f f i c i e n t of d e t e r m i n a t i o n used i n r e g r e s s i o n a n a l y s i s . The o b j e c t i v e was to determine the value of the e x p o n e n t i a l parameter which maximized the value of the t e s t parameter. To determine i f the r e s u l t s of t h i s procedure would 5 s u p p o r t t h e h y p o t h e s i s , H u f f p e r f o r m e d some s i m p l e s t a t i s t i c a l t e s t s o f s i g n i f i c a n c e . He c o n c l u d e d t h a t h i s d a t a s u p p o r t e d t h e h y p o t h e s i s . T h i s s t u d y r e - e x a m i n e s H u f f ' s d a t a u s i n g a t e c h n i q u e d e s i g n e d t o show t h e s e n s i t i v i t y o f t h e model t o t h e e x p o n e n t i a l p a r a m e t e r . T h i s t e c h n i q u e e x a mines t h e b e h a v i o u r o f t h e t e s t p a r a m e t e r f o r a r a n g e o f v a l u e s o f t h e e x p o n e n t . When a p p l i e d t o t h e V a n c o u v e r d a t a , t h e t e s t shows t h a t t h e model n e e d s o t h e r f a c t o r s a n d r e l a t i o n s h i p s i n o r d e r t o a c c o u n t f o r consumer b e h a v i o u r i n V a n c o u v e r . The s t u d y t h e n t u r n s t o e x a m i n i n g t h e ways o f a p p r o a c h -i n g t h e t a s k o f d e f i n i n g o t h e r f a c t o r s a n d r e l a t i o n s h i p s . O t h e r r e l e v a n t work i s c o n s i d e r e d f o r i t s i m p a c t on t h i s t a s k , a n d f o r t h e a i d w h i c h i t does o r c a n p r o v i d e . T h r o u g h o u t t h i s s t u d y , t h r e e b a s i c i n t e r - r e l a t e d i d e a s a r e i n t e r w o v e n . F i r s t , t h e r e i s c o n c e r n f o r t h e p r o p e r t i e s a n d p u r p o s e s o f a b s t r a c t m o d e l s . S e c o n d , c o n s i d e r a b l e a t t e n t i o n i s g i v e n t o t h e e m p i r i c a l t e s t s o f g r a v i m e t r i c m o d els u s e d by v a r i o u s r e s e a r c h e r s a t v a r i o u s t i m e s a n d p l a c e s . T h i s i s o f p r i m a r y i n t e r e s t i n d e t e r m i n i n g t h e a p p r o p r i a t e r e s e a r c h d e s i g n f o r t h i s s t u d y . The t h i r d i d e a i s c o n c e r n e d w i t h s u i t a b l e b a s e s on w h i c h t o j u d g e t h e r e s u l t s o f s u c h r e s e a r c h . Too f r e q u e n t l y s i m p l e s t a t i s t i c a l t e s t s mask, o r o v e r l o o k , i n f o r m a t i o n o f f u n d a m e n t a l v a l u e i n t h i s r e g a r d . More d i s c r i m -i n a t i n g t e s t s a r e n e e d e d . 6 L i m i t a t i o n s of the Study T h i s study has l i m i t a t i o n s a long three dimensions. F i r s t , the r e s u l t s are v a l i d only f o r the M e t r o p o l i t a n Vancouver area wherein the sample was o b t a i n e d . Second, because the sample was taken c r o s s - s e c t i o n a l l y , the r e s u l t s are only v a l i d f o r that one p o i n t i n time. T h i r d , the study d e a l s only w i t h c e r t a i n c l a s s e s of shopping goods ( c l o t h i n g , home f u r n i s h i n g s , and hardware), and only with c e r t a i n k i n d s of s t o r e s (major department s t o r e s and some planned shopping c e n t r e s ) . I t would be i n c o r r e c t to apply the r e s u l t s of t h i s study beyond these bounds. D e f i n i t i o n of Terms S e v e r a l terms used throughout t h i s study are b a s i c to the understanding of i t . The manner of t h e i r usage should be examined e x p l i c i t l y . 1. Shopping t r i p — I t i s assumed that when a consumer makes an e x c u r s i o n from h i s t r a v e l base p o i n t (assumed to be h i s home) i n quest of shopping goods he a l s o has i n mind the p a r t i c u l a r type of shopping goods he i s seeking. That i s , i t i s assumed t h a t the shopping t r i p i s p u r p o s i v e . I t may, however, e i t h e r be s u c c e s s f u l or u n s u c c e s s f u l i n terms of o b t a i n i n g or not o b t a i n i n g by purchase the d e s i r e d shopping goods. I f the t r i p i s u n s u c c e s s f u l , i t i s more d i f f i c u l t to o b j e c t i v e l y determine the purpose of the t r i p . Thus, f o r t h i s 7 s t u d y , s h o p p i n g t r i p s a r e d e f i n e d t o be o n l y those where s h o p p i n g goods were p u r c h a s e d . W h i l e shopping goods may be o b t a i n e d i n a wide v a r i e t y o f s t o r e s , t h e d e f i n i t i o n o f a s h o p p i n g t r i p , i n t h i s s t u d y , i s r e s t r i c t e d t o i n c l u d e o n l y e x c u r s i o n s t o major department s t o r e s and p l a n n e d s h o p p i n g c e n t r e s . 2. Shopping goods — These a r e the goods f o r which the p r o b a b l e g a i n from making c a r e f u l comparisons i s thought t o be l a r g e r e l a t i v e t o the time and e f f o r t needed t o shop e f f e c t i v e l y . Such goods may be c l a s s e d as homogeneous, which consumers v i e w as e s s e n t i a l l y s i m i l a r , o r h e t e r o g e n e o u s , which a r e non-s t a n d a r d i z e d and may c o n t a i n a s i g n i f i c a n t element o f s t y l e o r f a s h i o n . T h i s s t u d y does n o t e x p l i c i t l y use t h i s c l a s s i f i c a t i o n , b u t does a t t e m p t t o group i t e m s i n t o composite commodities w h i c h a r e c o n s i s t e n t w i t h i t . 3. Major department s t o r e s — These a r e t h e ones which f i t the p a t t e r n o f h a n d l i n g a wide v a r i e t y o f s h o p p i n g goods, d i v i d e d i n t o major departments. With one e x c e p t i o n , they a r e branches o f r e g i o n a l o r n a t i o n a l c o r p o r a t e c h a i n s . 4 . P l a n n e d shopping c e n t r e s — These a r e g e o g r a p h i c a l l y c o n t a i n e d a g g l o m e r a t i o n s o f complementary and competing r e t a i l f i r m s which have been p l a n n e d and d e v e l o p e d as a s e l f - c o n t a i n e d u n i t on a t r a c t of l a n d d e v o t e d e n t i r e l y t o t h a t complex o f a c t i v i t i e s . Such c e n t r e s may be c h a r a c t e r i z e d as neighbourhood, community, o r r e g i o n a l c e n t r e s . T h i s s t u d y i s concerned o n l y w i t h the l a t t e r two t y p e s which a r e l a r g e enough t o have a 8 branch of o n e or more of the regional or national department store chains as a central focus. Organization of the Written Report This report i s divided into chapters, each of which examines a specific aspect of the study. Chapter II i s divided into two parts. The f i r s t part briefly states some basic notions and issues about abstract models and measurement: which are relevant to the discussion of the development of the model and the task of reconceptualization. The second part i s devoted to a survey of the literature of gravimetric models in social science. Particular attention i s paid to the formulation of these models, but some consideration i s also given to empirical tests which have been performed. Chapter III deals with the specific incidence of gravi-metric models in marketing. Previous work is examined for the nature 'iof the processes of conceptualization and formu-lation. Of particular interest, however, i s the nature of empirical research which has been held by some to support, and by others to reject, these models. This aids in the con-sideration of the bases on which evidence can be used to support or reject such models. Chapter IV is devoted to the particular gravimetric model which is the main object of this study. In addition to Huff's original development, two additional approaches are presented. The relative strengths and weaknesses of each are 9 c o n s i d e r e d i n v i e w o f some o f t h e b a s i c i s s u e s o f a b s t r a c t m o d e l s . The c h a p t e r c o n c l u d e s t h a t r e g a r d l e s s o f t h e s e s t r e n g t h s a n d w e a k n e s s e s , i t i s s t i l l w o r t h w h i l e t o c o n d u c t a n e m p i r i c a l t e s t o f t h e m o d e l . C h a p t e r V p r e s e n t s t h e e m p i r i c a l work o f H u f f , a n d o f t h i s s t u d y . The t e c h n i q u e o f e v a l u a t i n g t h e s e n s i t i v i t y o f t h e model t o t h e e x p o n e n t i a l p a r a m e t e r i s p r e s e n t e d . C h a p t e r V I i s d e v o t e d e n t i r e l y t o t h e t a s k o f r e c o n -c e p t u a l i z i n g a n d r e f o r m u l a t i n g t h e m o d e l . The e v i d e n c e d e v e l o p e d i n C h a p t e r V s e r v e s a s t h e i n d i c a t o r o f how t h i s t a s k m i g h t be a p p r o a c h e d . Some v e r y b a s i c i s s u e s , a n d t h e i r p r o b a b l e i m p a c t , i n d e a l i n g w i t h t h e t y p e o f model c o n s i d e r e d i n t h i s s t u d y a r e e x a m i n e d . The n e e d f o r b e t t e r methods o f j u d g i n g t h e r e s u l t s o f s t u d i e s o f t h i s k i n d i s a l s o c o n s i d e r e d . CHAPTER I I MODELS IN SOCIAL SCIENCE P a r t I : Some Fundamental Notions and L i m i t a t i o n s of Models The term model i s being encountered w i t h i n c r e a s i n g frequency i n the l i t e r a t u r e of s o c i a l science. I t i s used by some interchangeably with the term theory, although the term model connotes only the more immediate aspects of conceptual-i z a t i o n and summarization. In a general sense, a model i s a r e p r e s e n t a t i o n of r e a l i t y . In t h i s same general sense i t may be argued that a l l human perceptions of r e a l i t y , and ways of t h i n k i n g about i t , a re models. For example, the number system i s a model by which p h y s i c a l objects can be represented by a b s t r a c t i n g a p a r t i c u l a r common property. There are two b a s i c steps i n v o l v e d i n the use of models: (1) the a b s t r a c t i o n of r e a l o bjects and events i n t o a model; and, (2) the a p p l i c a t i o n of r e s u l t s d e r i v e d from the use of a model. The use of any model may produce a degree of approxima-t i o n i n the r e s u l t s obtained. Consider an example of the use of the number system wherein a housewife has a box of mixed apples with which she wishes to bake p i e s . Using the number system she counts the number of apples but t h i s would not t e l l her e x a c t l y how many p i e s h e l l s to make — the apples are not 11 homogeneous i n s i z e or q u a l i t y . The number system model r e q u i r e s the assumption of complete homogeneity, indeed complete i d e n t i t y , of the objects represented to be complete-l y accurate. Where t h i s assumption i s not f u l f i l l e d i n the processes of a b s t r a c t i o n and s i m p l i f i c a t i o n of the r e a l ( e m p i r i c a l ) s i t u a t i o n , the r e s u l t s of u s i n g the model w i l l be approximate. I t i s important to note i n the example th a t the counting model was a p p l i e d f o r a s p e c i f i c purpose. Models may be developed f o r a number of purposes, but these are g e n e r a l l y acknowledged to i n c l u d e understanding, p r e d i c t i o n , and c o n t r o l of r e a l world events. This i m p l i e s that the type of model that should be used i n a p a r t i c u l a r instance depends on the s p e c i f i c purposes i n v o l v e d . Further, the use of models to accomplish s p e c i f i c purposes has only two bases: (1) determination of r e s u l t s i s f a s t e r , l e s s expensive, or more accurate w i t h the a i d of the model; or (2) o p e r a t i o n on the p h y s i c a l objects and events themselves (experimentation) i s not p o s s i b l e . Types of Models Models are r e p r e s e n t a t i o n s of r e a l i t y , which i m p l i e s a form of correspondence w i t h the r e a l world. Depending upon the nature of the correspondence, models may be c l a s s i f i e d as 1 i c o n i c , analogue, or symbolic. I c o n i c models look l i k e C. West Churchman, R u s s e l l L. A c k o f f , and E. Leonard A r n o f f , I n t r o d u c t i o n to Operations Research (New York: John Wiley and Sons, Inc., 1957),, pp. 158-162. 1 2 r e a l i t y ; analogue models s u b s t i t u t e one p r o p e r t y f o r another; and symbolic models r e p r e s e n t o b j e c t s , events, and processes by symbols of a l o g i c a l and/or mathematical c h a r a c t e r . T h i s study i s concerned with symbolic models, e s p e c i a l l y mathematical models, which are the k i n d of models being a c t i v e l y sought and a p p l i e d i n a l l branches of s o c i a l s c i e n c e , i n c l u d i n g marketing. However, there i s more i n v o l v e d i n the use of mathematical models than j u s t symbolism. Mathematics c o n s i s t s of a s e t of axioms ( b a s i c d e f i n i t i o n s and assumptions) and a s e t of theorems d e r i v e d from the axioms by l e g i t i m a t e processes of l o g i c a l d e d u c t i o n . The t r u t h of a mathematical theorem has n o t h i n g a t a l l to do with whether or not i t corresponds i n any way with any r e a l o b j e c t s or events. A mathematical theorem i s a p u r e l y a b s t r a c t c o n s t r u c t . Because of t h i s b a s i c q u a l i t y of mathematics, i t i s extremely important to note t h a t the use of a mathematical model to r e p r e s e n t r e a l o b j e c t s and events r e s t s s q u a r e l y on the very broad and g e n e r a l assumption that the axioms and theorems of one or more branches of mathematics are l e g i t i m a t e statements when a p p l i e d to some aspect of the r e a l world process which the model p u r p o r t s to r e p r e s e n t . I f t h i s assumption i s not c o r r e c t , then the model and the i n f e r e n c e s drawn from i t are not l e g i t i m a t e . Presumably, t h i s would be d i s c o v e r e d i n the course of experimental or e m p i r i c a l r e s e a r c h 2 when i t i s determined t h a t the model does not f i t the f a c t s . The term " f a c t " i s used here i n the sense of " e m p i r i c a l l y v e r i f i a b l e o b s e r v a t i o n . " See W i l l i a m J . Goode and Paul K. H a t t , Methods i n S o c i a l Research (New York: McGraw-Hill Book Company, Inc., 1 9 5 2 ) , p . 8 . 13 T h e r e f o r e , i t would be b e t t e r to deduce t h i s beforehand to a v o i d w a s t e f u l e m p i r i c a l r e s e a r c h . V a r i a b l e s and R e l a t i o n s h i p s Mathematical models are composed of v a r i a b l e s or f a c t o r s , and the r e l a t i o n s h i p s among them. A v a r i a b l e i s any p r o p e r t y found to have d i f f e r e n t v a l u e s a t d i f f e r e n t times, p l a c e s , or circumstances. The nexus of m o d e l - b u i l d i n g i s the s p e c i f i c a t i o n of the r e l a t i o n s h i p s among v a r i a b l e s , and these may be of s e v e r a l forms. Three forms may be d i s t i n g u i s h e d a s : (1) d e t e r m i n i s t i c c a u s a l i t y ; (2) p r o b a b i l i s t i c c a u s a l i t y ; and ( 3 ) c o r r e l a t i o n . For much of the recorded h i s t o r y of s c i e n t i f i c i n q u i r y , the r e l a t i o n s h i p between v a r i a b l e s has been regarded as that of d e t e r m i n i s t i c c a u s a l i t y , a l s o known as cause and e f f e c t . T h i s i s the c l a s s i c a l viewpoint of p h y s i c a l n e c e s s i t y which dominated s c i e n t i f i c t h i n k i n g up to the n i n e t e e n t h c e n t u r y . More contemporary r e s e a r c h i n mathematical p h y s i c s has shown t h a t the c l a s s i c a l , d e t e r m i n i s t i c concept i s an a r t i f i c i a l one even as a p p l i e d to the s i m p l e s t k i n d s of p h y s i c a l phenomena. The laws of nature are c o n d i t i o n a l . That i s , they are t r u e only i f one a b s t r a c t s from c o n t i n g e n c i e s , and only to the e xtent that these c o n t i n g e n c i e s may be n e g l e c t e d . P r o b a b i l i s t i c c a u s a l i t y has been i n t e r p r e t e d as a r e l a t i o n s h i p i n which one t h i n g i s necessary f o r the occurrence of some other t h i n g , but not s u f f i c i e n t because other f a c t o r s 14 are a l s o involved.-^ The study of such r e l a t i o n s h i p s has been developed i n mathematical s t a t i s t i c s , and i n the study of the nature of s t o c h a s t i c processes. Many of the processes studied i n s o c i a l science are probably s t o c h a s t i c a l and there appears to be a s o r t of m u l t i p l e p r o b a b i l i s t i c c a u s a l i t y o p e r a t i n g . Any one f a c t o r , or s e v e r a l of many d i f f e r e n t f a c t o r s , may cause changes i n a v a r i a b l e . No one causal f a c t o r i s necessary, nor i s any of them s u f f i c i e n t f o r occurrence of the e f f e c t under i n v e s t i g a t i o n , since s t i l l other unknown f a c t o r s may a l s o c o n t r i b u t e to the e f f e c t . This tends to make s t o c h a s t i c process model-building r a t h e r i n t r a c t a b l e . C o r r e l a t i o n i s the weakest form of r e l a t i o n s h i p among v a r i a b l e s . Many types of nonsense r e l a t i o n s h i p s where c o r r e l a -t i o n s are very high have been reported. Such r e l a t i o n s h i p s are 4 fr e q u e n t l y termed spurious c o r r e l a t i o n . Purely s t a t i s t i c a l a s s o c i a t i o n s of t h i s k i n d are f a r l e s s dependable than even p a r t i a l knowledge of a c t u a l causes. On the other hand, i t may be p r e f e r a b l e to base p r e d i c t i o n s on a c o r r e l a t i o n between v a r i a b l e s than on pure guesswork, p r o v i d i n g the inherent dangers R.L. A c k o f f , S c i e n t i f i c Method: Optimizing A p p l i e d Research  Decisions (New York: John Wiley and Sons, Inc., 1962) pp.14-19. 4 The term spurious c o r r e l a t i o n i s used here w i t h a common sense meaning. There i s a l s o a p r e c i s e meaning i n mathematical s t a t i s t i c s r e f e r r i n g to the case where apparent c o r r e l a t i o n i s higher than true c o r r e l a t i o n because of the presence of some t h i r d v a r i a b l e ( e x p l i c i t or i m p l i c i t ) i n common i n the r e l a t i o n -s h i p being s t u d i e d . The e a r l i e s t s u b s t a n t i a l i n v e s t i g a t i o n of t h i s seems to be G. Y u l e , "Why Do We Sometimes Get Nonsense C o r r e l a t i o n s Between Time S e r i e s , " J o u r n a l of the Royal  S t a t i s t i c a l S o c i e t y , LXXXIX (1926), pp. 1-64. 15 a r e r e c o g n i z e d . The P r o b l e m o f Measurement One o f t h e k e y i s s u e s i n t h e d e v e l o p m e n t a n d u s e o f m a t h e m a t i c a l models i n s o c i a l s c i e n c e i s t h a t o f measurement. Measurement may be o f s e v e r a l f o r m s b u t i n common u s a g e i t i m p l i e s some way o f a s s i g n i n g numbers t o p r o p e r t i e s o f o b j e c t s o r e v e n t s . A c k o f f p o i n t s o u t , however, t h a t symbols o t h e r t h a n numbers c a n a l s o be u s e d , a s l o n g a s t h e symbols " a r e r e l a t e d t o e a c h o t h e r i n t h e same way t h a t t h e o b s e r v e d o b j e c t s , e v e n t s , o r p r o p e r t i e s a r e o r c a n b e . " Numbers a r e u s u a l l y d e s i r e d , however, b e c a u s e t h e models i n w h i c h t h e y a r e u s e d c a n be m a n i p u l a t e d m a t h e m a t i c a l l y . E v e n i f measurement i s r e s t r i c t e d t o t h e u s e o f n u m e r i -c a l s y m b o l s , t h e r e a r e s e v e r a l t y p e s o f measurement p o s s i b l e , d e p e n d i n g upon t h e p a r t i c u l a r t y p e o f s c a l e c h o s e n : n o m i n a l , 7 o r d i n a l , o r c a r d i n a l . A n o m i n a l s c a l e i s s i m p l y a measure w h i c h u s e s a number t o name an e n t i t y a n d / o r p r o p e r t y o f an J A u s e f u l d i s c u s s i o n o f t h e u s e o f s t a t i s t i c s , p a r t i c u l a r l y c o r r e l a t i o n , f o r i n f e r e n c e o f c a u s a t i o n i s f o u n d i n H u b e r t M. B l a l o c k , J r . , C a u s a l I n f e r e n c e s i n N o n e x p e r i m e n t a l R e s e a r c h ( C h a p e l H i l l : U n i v e r s i t y o f N o r t h C a r o l i n a P r e s s , 1964.) 6 A c k o f f , op. c i t . , pp. 178-179. 7 Wroe A l d e r s o n a n d P a u l E . G r e e n , P l a n n i n g a n d P r o b l e m  S o l v i n g i n M a r k e t i n g (Homewood, I l l T l R i c h a r d D. I r w i n , I n c . , 1964), pp. 145-148. See a l s o S.S. S t e v e n s , "Measurement, P s y c h o p h y s i c s , a n d U t i l i t y , " Measurement: D e f i n i t i o n s a n d  T h e o r i e s , e d s . C.W. Churchman a n d P h i l b u r n R a t o o s h (New Y o r k : J o h n W i l e y a n d S o n s , I n c . , 1959), PP. 24-26. 16 e n t i t y . I t makes no sense to perform mathematical o p e r a t i o n s on these numbers. Very l i t t l e i n f o r m a t i o n i s conveyed by the use of these numbers other than as a means of i d e n t i f i c a t i o n . An o r d i n a l s c a l e i s a r a n k i n g measure which t e l l s which of any two items i s h i g h e r , lower, or the same on the scale.' While a wide c h o i c e of numbers i s a v a i l a b l e w i t h which to do t h i s , care must be taken t h a t the numbers preserve the o r d e r i n g r e l a t i o n s h i p . Care must a l s o be taken t h a t n o t h i n g i s i n f e r r e d about the numerical d i f f e r e n c e between rank numbers — a s s i g n i n g numerical rankings i m p l i e s n o t h i n g about how f a r a p a r t the items ranked a r e , or how much g r e a t e r a p r o p e r t y i s i n one as compared w i t h another. C a r d i n a l s c a l e s are composed of two s u b c l a s s e s — i n t e r v a l s c a l e s and r a t i o s c a l e s . I n t e r v a l s c a l e s have an a r b i t r a r y zero p o i n t and a constant u n i t of measure. They a r e u s u a l l y d e f i n e d i n an o p e r a t i o n a l way w i t h r e s p e c t to f i x e d r e f e r e n c e p o i n t s . A r a t i o s c a l e has both an a b s o l u t e zero p o i n t and a constant u n i t of measurement. The C e l s i u s and F a h r e n h e i t temperature s c a l e s are examples of i n t e r v a l measures, while the K e l v i n and Rankine s c a l e s are t h e i r r a t i o ( a b s o l u t e ) c o u n t e r p a r t s . Such measures as i n c h e s , pounds, and d o l l a r s are a l s o examples of r a t i o s c a l e s . In the case of i n t e r v a l s c a l e s , d i f f e r e n c e s between s c a l e v a l u e s can be expressed as m u l t i p l e s of each o t h e r , but i n d i v i d u a l v a l u e s cannot. T h i s n o t i o n i s f o r m a l i z e d by s a y i n g t h a t i n t e r v a l s c a l e s are unique up to a p o s i t i v e l i n e a r t r a n s -17 f o r m a t i o n ( s u c h a s t h a t u s e d t o c o n v e r t C e l s i u s t o F a h r e n h e i t m e a s u r e ) . On t h e o t h e r hand, i n t h e c a s e o f r a t i o s c a l e s , i t does make s e n s e t o t a l k a b o u t v a l u e s on one s c a l e b e i n g m u l t i p l e s o f v a l u e s on a n o t h e r . S c a l e s o f t h i s t y p e , h a v i n g a n a b s o l u t e z e r o , a r e u n i q u e up t o a p r o p o r t i o n -a t e t r a n s f o r m a t i o n . A l l t h e m a t h e m a t i c a l o p e r a t i o n s t y p i c a l l y a s s o c i a t e d w i t h numbers c a n be p e r f o r m e d . I t a p p e a r s t h a t c r i t i c s o f m a t h e m a t i c a l m o d els i n s o c i a l s c i e n c e , who c l a i m t h a t m a j o r f a c t o r s c a n n o t be measured, i m p l i c i t l y mean t h a t t h e y c a n n o t be measured on r a t i o s c a l e s . S i n c e t h e y b e l i e v e t h a t t h e s e s c a l e s a r e t h e o n l y ones t h a t c a n be m a n i p u l a t e d m a t h e m a t i c a l l y , t h e y c o n c l u d e t h a t m a t h e m a t i c a l m o d e l - b u i l d i n g i s h o p e l e s s . S u c h c r i t i c s w o u l d do w e l l t o r e c o n s i d e r t h e a s t o u n d i n g p r o g r e s s made i n many b r a n c h e s o f p h y s i c a l s c i e n c e u s i n g o n l y i n t e r v a l s c a l e s f o r measurement. D i f f e r e n c e s c a n be compared r e l a t i v e l y u s i n g i n t e r v a l s c a l e s s i n c e t h e u n i t o f measurement i s c o n s t a n t a n d i t was r e l a t i v e m e a s u r e s , n o t a b s o l u t e m e a s u r e s , w h i c h were s o u g h t by t h e p h y s i c a l s c i e n t i s t s . I n s o c i a l s c i e n c e , a p r o p e r l y c o n s t r u c t e d i n d e x number may be c o n s i d e r e d an a p p r o p r i a t e i n t e r v a l s c a l e . I t w i l l be s e e n t h a t s o c i a l s c i e n c e h a s p a i d c o n s i d e r -a b l e a t t e n t i o n t o t h e r e s u l t s o f p h y s i c a l s c i e n c e i n t e r m s o f m o d e l s o r t h e o r i e s . I n s u f f i c i e n t a t t e n t i o n h as been p a i d t o t h e m e t h o d o l o g y a n d t e c h n i q u e o f p h y s i c a l s c i e n c e , p r o b a b l y b e c a u s e i t i s l a r g e l y e x p e r i m e n t a l , a n d s o c i a l s c i e n c e h a s 1 8 been m o s t l y n o n - e x p e r i m e n t a l . However, t h e i n g e n u i t y a n d c r e a t i v i t y o f e a r l y p h y s i c a l s c i e n t i s t s i n o v e r c o m i n g t h e p r o b l e m s a s s o c i a t e d w i t h o b t a i n i n g s a t i s f a c t o r y measurements s h o u l d p r o v i d e b o t h example and hope t o s o c i a l s c i e n t i s t s . T h i s p o i n t c a n n o t be o v e r - e m p h a s i z e d , s i n c e i t a p p e a r s t o c o n t a i n a v e r y i m p o r t a n t l e s s o n f o r r e s e a r c h i n s o c i a l s c i e n c e s . P a r t I I : G r a v i m e t r i c M o d e l s o f Human I n t e r a c t i o n I n r e c e n t y e a r s i n c r e a s i n g a t t e n t i o n h a s b e e n g i v e n by s o c i a l s c i e n t i s t s t o t h e s o - c a l l e d g r a v i t y a n d p o t e n t i a l c o n -c e p t s o f human i n t e r a c t i o n . T h i s i s a c a s e where, e i t h e r i m p l i c i t l y o r e x p l i c i t l y , a model h a s been d e v e l o p e d by t h e 8 s c i e n t i s t b a s e d on Newton*s U n i v e r s a l Law o f G r a v i t a t i o n . I n g e n e r a l t e r m s , t h e g r a v i t y c o n c e p t o f human i n t e r a c t i o n p o s t u l a t e s t h a t a n a t t r a c t i n g f o r c e o f i n t e r a c t i o n b e t w e e n two a r e a s o f human a c t i v i t y i s c r e a t e d by t h e p o p u l a t i o n masses o f t h e two a r e a s , a n d a f r i c t i o n a g a i n s t i n t e r a c t i o n i s c a u s e d by t h e i n t e r v e n i n g s p a c e o v e r w h i c h t h e i n t e r a c t i o n must t a k e p l a c e . T h a t i s , i n t e r a c t i o n between t h e two c e n t r e s o f popu-T h i s l a w may be e x p r e s s e d a s f o l l o w s : where /W i s a p a r t i c l e o f mass a t p o i n t A , a t a d i s t a n c e ^ f r o m a s e c o n d p a r t i c l e o f mass m a t p o i n t a ; a f o r c e a c t s on e a c h mass, a t t r a c t i n g them t o g e t h e r a l o n g t h e l i n e j o i n i n g them; <2> i s a u n i v e r s a l c o n s t a n t , i n t h i s c a s e t h e g r a v i t a t i o n a l c o n s t a n t . 19 l a t i o n c o n c e n t r a t i o n v a r i e s d i r e c t l y w i t h some f u n c t i o n o f t h e p o p u l a t i o n s i z e o f t h e two c e n t r e s a n d i n v e r s e l y w i t h some f u n c t i o n t h e d i s t a n c e between them. M a t h e m a t i c a l l y t h e r e l a t i o n s h i p may be e x p r e s s e d a s f o l l o w s : Io = f(Pt>Ri) ( i i . l ) f(Du) where Tij = t h e i n t e r a c t i o n between c e n t r e i a n d c e n t r e j ; Pc , Pj = t h e p o p u l a t i o n o f c e n t r e s i a n d j , r e s p e c t i v e l y ; a n d Dij = t h e d i s t a n c e between c e n t r e i a n d c e n t r e j . S t a t e d i n t h i s way, t h e g r a v i m e t r i c h y p o t h e s i s i s b a s e d on t h e p o s t u l a t e s t h a t : (1) t o p r o d u c e i n t e r a c t i o n , i n d i v i d u a l s must be i n c o m m u n i c a t i o n , d i r e c t l y o r i n d i r e c t l y , w i t h one a n o t h e r ; (2) any p a r t i c u l a r i n d i v i d u a l , a s a u n i t o f a l a r g e g r o u p , may be c o n s i d e r e d t o g e n e r a t e t h e same i n f l u e n c e o f i n t e r a c t i o n a s any o t h e r i n d i v i d u a l ; (3) t h e p r o b a b l e f r e q u e n c e o f i n t e r a c t i o n g e n e r a t e d by a n i n d i v i d u a l a t a g i v e n l o c a t i o n i s i n v e r s e l y p r o p o r t i o n a l t o t h e d i f f i c u l t y o f r e a c h -i n g , o r c o m m u n i c a t i n g w i t h , t h a t l o c a t i o n ; and (4) t h e f r i c t i o n a g a i n s t t h i s t r a n s p o r t a t i o n o r c o m m u n i c a t i o n i s d i r e c t l y p r o p o r t i o n a l t o t h e i n t e r v e n i n g p h y s i c a l d i s t a n c e b e tween t h e i n d i v i d u a l a n d t h e g i v e n l o c a t i o n . E a r l i e s t I n c i d e n c e s S t u d e n t s o f m a r k e t i n g a r e aware t h a t g r a v i m e t r i c m o d e l s have been p r o p o s e d f o r some t i m e , b e g i n n i n g w i t h 20 9 R e i l l y ' s Law, w h i c h i n terms o f m a r k e t i n g t h o u g h t i s r e l a -t i v e l y o l d . However, g r a v i m e t r i c m o d e ls a r e a p p a r e n t l y much o l d e r , a s t h e e a r l i e s t known e x p l i c i t f o r m u l a t i o n o f t h e g r a v i t y c o n c e p t o f human i n t e r a c t i o n was made by C a r e y d u r i n g 10 t h e f i r s t h a l f o f t h e n i n e t e e n t h c e n t u r y . C a r e y r e a s o n e d t h a t s o c i a l a n d p h y s i c a l phenomena a r e b a s e d on t h e same f u n d a m e n t a l law, i n a manner n o t u n f a m i l i a r t o d a y . He f o r m u l a t e d h i s c o n c e p t a s f o l l o w s : Man, t h e m o l e c u l e o f s o c i e t y , i s t h e s u b j e c t o f S o c i a l S c i e n c e ... The g r e a t l a w o f M o l e c u l a r G r a v i -t a t i o n i s t h e i n d i s p e n s a b l e c o n d i t i o n o f t h e e x i s t e n c e o f t h e b e i n g known a s man ... The g r e a t e r t h e number c o l l e c t e d i n a g i v e n s p a c e , t h e g r e a t e r i s t h e a t t r a c t i v e f o r c e t h a t i s t h e r e e x e r t e d ... G r a v i t a t i o n i s h e r e , a s e v e r y w h e r e , i n t h e d i r e c t ^ r a t i o o f t h e mass, a n d t h e i n v e r s e one o f d i s t a n c e . A f t e r C a r e y ' s o r i g i n a l s t a t e m e n t o f t h e c o n c e p t , i t was l a r g e l y n e g l e c t e d . I t a p p e a r e d o n l y p a r t l y e x p r e s s e d u n t i l q u i t e r e c e n t l y . I n 1885, R a v e n s t e i n p r o d u c e d an h y p o t h e s i s r e g a r d i n g m i g r a t i o n w h i c h was i m p l i c i t l y b a s e d on o 7 T h i s s o - c a l l e d l a w was f i r s t p u b l i s h e d i n 1931 i n W i l l i a m J . R e i l l y , The Law o f R e t a i l G r a v i t a t i o n (1st e d . ; New Y o r k : W i l l i a m J . R e i l l y Company, 1931). 10 H. C. C a r e y , P r i n c i p l e s o f S o c i a l S c i e n c e ( P h i l a d e l p h i a : J . B. L i p i n c o t t a n d Company, 1858-59). 11 Q u o t e d i n G e r a l d A. P. C a r r o t h e r s , "An H i s t o r i c a l R e v i e w o f t h e G r a v i t y a n d P o t e n t i a l C o n c e p t s o f Human I n t e r a c t i o n , " J o u r n a l o f t h e A m e r i c a n I n s t i t u t e o f P l a n n e r s , X X I I ( S p r i n g , 1956), p. 94. 21 12 the g r a v i m e t r i c concept. Ravenstein was concerned w i t h e x p l a i n i n g m i g r a t i o n and presented e m p i r i c a l evidence suggest-i n g that migratory movement tends to be toward c i t i e s of l a r g e p o p u l a t i o n and that the volume of movement decreased w i t h d i s t a n c e between the source of m i g r a t i o n and the centre of ab s o r p t i o n . The r e l a t i o n s h i p may be expressed mathematically: M n = f (Pi) (II.2 ) where Mij = m i g r a t i o n from source j to centre of absorp-t i o n I ; / ( / / • ' ) = some f u n c t i o n of the p o p u l a t i o n of i ; and £>ij = the distance between source j and centre I . Subsequently, the g r a v i m e t r i c concept does not appear i n the l i t e r a t u r e again f o r some t h i r t y years, u n t i l the l a t e 1 9 2 0 's when Young made a somewhat s i m i l a r attempt to measure 13 m i g r a t i o n . Young hypothesized that the r e l a t i v e volume of mi g r a t i o n to a given d e s t i n a t i o n from each of s e v e r a l source areas v a r i e s d i r e c t l y w i t h the force of a t t r a c t i o n of the d e s t i n a t i o n and i n v e r s e l y w i t h the square of the d i s t a n c e between the source and the d e s t i n a t i o n . That i s : Di 2 ( I I . 3 ) c J 12 I b i d . , p. 9 5 . The o r i g i n a l source i s E.G. Ravenstein, "The Laws of M i g r a t i o n , " J o u r n a l of the Royal S t a t i s t i c a l S o c i e t y , No. 6 ( 1 8 8 5 ) , 167-235 and No. 6 (1889), 241 - 3 0 5 . 13 I b i d . The o r i g i n a l source i s E. C. Young, The Movement  Of Farm Population ( I t h a c a : C o r n e l l A g r i c u l t u r a l Experiment S t a t i o n , B u l l e t i n 4 2 6 , 1 9 2 4 ) . 22 where = t h e f o r c e o f a t t r a c t i o n o f d e s t i n a t i o n i ; and A b o u t t h e same t i m e , R e i l l y p o s t u l a t e d h i s "Law o f 14 R e t a i l G r a v i t a t i o n " w h i c h a p p r o a c h e s t h e g r a v i t y c o n c e p t somewhat d i f f e r e n t l y . W h i l e t h i s c o n c e p t , i t s e m p i r i c a l s u p p o r t , a n d t h e work o f s u b s e q u e n t i n v e s t i g a t o r s w i l l be c o n s i d e r e d i n some d e t a i l i n t h e n e x t c h a p t e r , t h e c o n c e p t i s i n c l u d e d b r i e f l y h e r e f o r c o m p l e t e n e s s o f t h e h i s t o r i c a l s u r v e y . R e i l l y p o s t u l a t e d t h a t a c i t y w i l l a t t r a c t r e t a i l t r a d e f r o m a n i n d i v i d u a l J n i t s s u r r o u n d i n g t e r r i t o r y i n d i r e c t p r o p o r t i o n t o t h e p o p u l a t i o n s i z e o f t h e r e t a i l c e n t r e and i n i n v e r s e p r o p o r t i o n t o t h e s q u a r e o f h i s d i s t a n c e away f r o m t h e c e n t r e . Thus, f o r any two c i t i e s c o m p e t i n g f o r r e t a i l t r a d e , t h e p o i n t o f e q u i l i b r i u m on t h e l i n e j o i n i n g them, where c o m p e t i t i v e i n f l u e n c e i s e q u a l ( t h e b r e a k i n g - p o i n t ) i s d e s c r i b e d by t h e e q u a t i o n : where /?• , Pj = p o p u l a t i o n o f c i t i e s i a n d J , r e s p e c t i v e l y ; X = t h e p o i n t o f e q u i l i b r i u m on t h e l i n e j o i n i n g £ a n d j ; = d i s t a n c e f r o m c i t y i. t o p o i n t x ; = d i s t a n c e f r o m I n a d d i t i o n t o t h e work p r e v i o u s l y c i t e d , t h i s c o n c e p t was e a r l i e r m e n t i o n e d i n W. J . R e i l l y , Methods f o r t h e S t u d y o f  R e t a i l R e l a t i o n s h i p s ( U n i v e r s i t y o f T e x a s , B u r e a u o f B u s i n e s s R e s e a r c h , R e s e a r c h Monograph No. 4, U n i v e r s i t y o f T e x a s B u l l e t i n No. 2994, November, 1929). Jt. = a c o n s t a n t o f p r o p o r t i o n a l i t y . ( I I . 4 ) 2 3 c i t y j t o p o i n t x ; a n d Dcj = afyi + . I n t h e 1 9 3 0 * s , a n e a r l y s o c i o l o g i c a l a p p l i c a t i o n o f t h e c o n c e p t was made by B o s s a r d who e x amined t h e f u n c t i o n o f 1 6 d i s t a n c e a s a f a c t o r i n m a r r i a g e s e l e c t i o n . B o s s a r d f o u n d , A c t u a l l y , R e i l l y e x p r e s s e d h i s c o n c e p t somewhat d i f f e r e n t l y . I n R e i l l y , The Law o f R e t a i l G r a v i t a t i o n , p . 9 » i s f o u n d : Two c i t i e s a t t r a c t r e t a i l t r a d e f r o m any i n t e r m e d i a t e c i t y o r town i n t h e v i c i n i t y o f t h e b r e a k i n g p o i n t a p p r o x i m a t e l y i n d i r e c t p r o p o r t i o n t o t h e p o p u l a t i o n o f t h e two c i t i e s a n d i n i n v e r s e p r o p o r t i o n t o t h e s q u a r e o f t h e d i s t a n c e s f r o m t h e s e two c i t i e s t o t h e i n t e r m e d i a t e town. M a t h e m a t i c a l l y : &6 Pb Oa. * where 3* , 3& - p r o p o r t i o n s o f r e t a i l t r a d e f r o m t h e i n t e r -m e d i a t e town a t t r a c t e d by c i t i e s A and & , r e s p e c t i v e l y ; Pa > P& - p o p u l a t i o n s o f c i t i e s A and 3 , r e s p e c t i v e l y ; Da » Os - d i s t a n c e s f r o m t h e i n t e r m e d i a t e town t o c i t i e s /J a n d 3 , r e s p e c t i v e l y . E q u a t i o n ( I I . 4 ) above c a n be d e r i v e d f r o m R e i l l y ' s f o r m u l a , s i n c e a t t h e b r e a k i n g p o i n t , where c o m p e t i t i v e i n f l u e n c e s a r e e q u a l , &a = 3s , o r t h e i n t e r m e d i a t e community d i v i d e s i t s t r a d e e q u a l l y between t h e two c i t i e s . T h u s , &a/B6 = 1 , a n d : P6 Da* o r , ^ £ L - EL w h i c h i s E q u a t i o n ( I I . 4 ) . 1 6 James H.S. B o s s a r d , " R e s i d e n t i a l P r o p i n q u i t y a s a F a c t o r i n M a r r i a g e S e l e c t i o n , " A m e r i c a n J o u r n a l o f S o c i o l o g y , X X X V I I I ( S e p t e m b e r , 1 9 3 2 ) , 2 1 9 - 2 4 4 . 24 from e m p i r i c a l t e s t s i n P h i l a d e l p h i a , that the number of marriages decreased as the distance between the premarriage residences of the p r i n c i p a l s i n c r e a s e d . F o r m a l i z a t i o n : S o c i a l Physics Beginning i n the e a r l y 1 9 4 0 ' s , the g r a v i t y concept of 17 human i n t e r a c t i o n was g e n e r a l i z e d by Stewart and Z i p f . By r e f e r r i n g to the r e l e v a n t p h y s i c a l laws (models), equations Published works by John Q. Stewart t r a c i n g the development of h i s t h i n k i n g i n t h i s area are: "An Inverse Distance V a r i -a t i o n f o r C e r t a i n S o c i a l I n f l u e n c e s . " Science, X C I I I , (January, 1 9 4 1 ) , 8 9 - 9 0 ; " E m p i r i c a l Mathematical Rules Concerning the D i s t r i b u t i o n and E q u i l i b r i u m of P o p u l a t i o n , " Geographical Review. XXXVII, ( J u l y , 1 9 4 7 ) , 4 6 1 - 4 8 5 ; "Suggested P r i n c i p l e s of S o c i a l P h y s i c s , " Science, CVI, (August, 1 9 4 7 ) , 1 7 9-180; "Demographic G r a v i t a t i o n : Evidence and A p p l i c a t i o n s , " Sociometry, X I , (February-May, 1948), 3 1 - 5 8 ; "Concerning S o c i a l P h y s i c s " S c i e n t i f i c American, CLXXVIII (May, 1 9 4 8 ) , 2 0 - 2 3 ; "The Development of S o c i a l P h y s i c s , " American J o u r n a l  of P h y s i c s , X V I I I , (May, 1 9 5 0 ) , 2 3 9 - 2 5 3 ; " P o t e n t i a l of Popu-l a t i o n and i t s R e l a t i o n s h i p to Marketing," Theory i n Marketing, eds. Reavis Cox and Wroe Alderson (Chicago: Richard D. I r w i n , Inc., 1 9 5 0 ) , pp. 1 9-40; "A Basis f o r S o c i a l P h y s i c s , " Impact of Science on S o c i e t y , I I I (Summer 1 9 5 2 ) , 1 1 0 - 1 3 3 . Published works by George K. Z i p f t r a c i n g the development of h i s t h i n k i n g i n t h i s area are: N a t i o n a l Unity and D i s u n i t y (Bloomington: P r i n c i p i a Press, 1 9 4 1 ) ; "The Unity of Nature, L e a s t - A c t i o n and N a t u r a l S o c i a l Science," Sociometry, V, (February, 1 9 4 2 ) 48 - 6 2 . ; "The P 1P 2/D Hypothesis: On I n t e r c i t y Movement of Persons," American S o c i o l o g i c a l Review, X I , (December, 1 9 4 6 ) , 6 7 7 - 6 8 6 ; "Some Determinants of the C i r c u l a t i o n of Information," American J o u r n a l of Psychology, LIX ( J u l y , 1 9 4 6 ) , 401-421; "The Pj_P?/D Hypothesis: the Case of Railway Express," J o u r n a l of Psychology, XX ( J u l y , 1 9 4 6 ) , 3 - 8 ; "The Hypothesis of the •Minimum Equation* as a U n i f y i n g S o c i a l P r i n c i p l e : w i t h Attempted Synthesis," American  S o c i o l o g i c a l Review, X I I (December, 1 9 4 7 ) , 6 2 7 - 6 5 0 ; Human  Behavior and the P r i n c i p l e of Least E f f o r t (Cambridge: Addison-Wesley Press, 1 9 4 9 ) . 25 d e s c r i b i n g demographic f o r c e , energy and p o t e n t i a l have been developed. In f a c t , Stewart, an a s t r o p h y s i c i s t , claims that t h i s g e n e r a l i z a t i o n i s the ba s i s of a new science, which he c a l l s " S o c i a l Physics' 1. According to Stewart: In i t s a p p l i c a t i o n , s o c i a l physics r e l a t e s d i r e c t l y to the economic theory of l o c a t i o n and to general marketing theory — f o r example, to the c o n s t r u c t i o n of maps of t r a d i n g centers ... The general p o s s i b i l i t i e s of i t s s o c i a l physics usefulness extend over the e n t i r e f i e l d of s o c i a l s t a t i s t i c s wherever averages r a t h e r than i n d i v i d u a l human c h a r a c t e r i s t i c s are st u d i e d . Emphasis added. ^ y S o c i a l physics i s defined by Stewart as i n c l u d i n g a l l s t u d i e s of a mathematical type r e l a t i n g to human behaviour. S o c i a l p h y s i c s , he s t a t e s : ... analyses demographic, economic, p o l i t i c a l and s o c i o l o g i c a l s i t u a t i o n s i n terms of the purely p h y s i c a l f a c t o r s of time, d i s t a n c e , mass of m a t e r i a l , and number of people w i t h recourse a l s o to s o c i a l f a c t o r s which can be shown to operate i n a s i m i l a r way to two other p h y s i c a l agents, namely temperature and e l e c t r i c a l charge. 19 He w r i t e s that temperature as a s o c i a l f a c t o r r e f e r s to the l e v e l of a c t i v i t y of people as shown i n bank deposits per c a p i t a , mileage of r a i l w a y s and highways per square mile of area, and percentage of workers i n manufacturing i n d u s t r i e s , 20 f o r example. The s o c i a l q u a n t i t y t h a t Stewart s u b s t i t u t e s f o r e l e c t r i c charge ( p o s i t i v e and negative) i s d e s i r e . He 18 Stewart, Theory i n Marketing, p. 1 9 . 19 7 Stewart, Impact of Science on S o c i e t y , I I I , p. 110. 2 0 I b i d . , pp. 120-121. 2 6 w r i t e s : S i n c e d e s i r e i n t h e s e n s e o f s i m p l e h u n g e r i s n e u t r a l i z e d by i t s a p p r o p r i a t e s a t i s f a c t i o n , i t i s i n t h i s r e s p e c t l i k e a n e g a t i v e e l e c t r i c c h a r g e , w h i l e t h e d e s i r a b l e o b j e c t p o s s e s s e s t h e n e u t r a l i z i n g p o s i t i v e c h a r g e . 2 1 T h i s k i n d o f framework i s t e r m e d d i m e n s i o n a l a n a l y s i s by S t e w a r t . As a p p l i e d t o s o c i a l p h y s i c s , a p r i n c i p a l o b j e c t i v e o f d i m e n s i o n a l a n a l y s i s i s t o d e s i g n a t e t h e l e a d i n g f a c t o r s by r e f e r e n c e t o w h i c h s o c i a l p h y s i c s a s a whole c a n be summarized. The i m m e d i a t e o b j e c t i v e o f t h e s o c i a l p h y s i c i s t , a c c o r d i n g t o S t e w a r t , i s t o d i s c o v e r u n i f o r m i t i e s i n s o c i a l b e h a v i o u r w h i c h c a n be e x p r e s s e d i n m a t h e m a t i c a l f o r m more o r l e s s c o r r e s p o n d i n g t o t h e known p a t t e r n s o f p h y s i c a l s c i e n c e . He i s o f t h e o p i n i o n t h a t enough s u c h r e g u l a r i t i e s e x i s t t o j u s t i f y t h e c o n c l u s i o n t h a t c e r t a i n t y p e s o f human r e l a t i o n s , on t h e a v e r a g e a n d o n l y on t h e a v e r a g e , c o n f o r m t o m a t h e m a t i c a l f o r m u l a s r e s e m b l i n g t h e p r i m i t i v e l a w s o f p h y s i c s . A t t h e hands o f S t e w a r t a n d Z i p f , a r e t u r n was made t o t h e o r i g i n a l f o r m u l a t i o n i n t e r m s o f N e w t o n i a n p h y s i c s , a s f i r s t s e t f o r t h by C a r e y , namely t h a t t h e f o r c e o f i n t e r a c t i o n b e t w e e n two c o n c e n t r a t i o n s o f p o p u l a t i o n , a c t i n g a l o n g a l i n e j o i n i n g t h e i r c e n t r e s , i s d i r e c t l y p r o p o r t i o n a l t o t h e p r o d u c t o f t h e p o p u l a t i o n s o f t h e two c e n t r e s a n d i n v e r s e l y p r o p o r t i o n a l I b i d . , p . 1 2 3 . 2 7 t o t h e s q u a r e o f t h e d i s t a n c e between them. T h a t i s , m a t h e m a t i c a l l y : rij = Q.py d i .5) where Pij = t h e f o r c e o f i n t e r a c t i o n between c o n c e n t r a t i o n s * a n d j . F o l l o w i n g t h e a n a l o g y f r o m p h y s i c s , t h e e n e r g y o f i n t e r a c t i o n between t h e two c e n t r e s w h i c h r e s u l t s f r o m t h i s f o r c e w o u l d be: £cj -  p£Pj ( I I . 6 ) where &i) = e n e r g y o f i n t e r a c t i o n between i a n d J ; and & = a c o n s t a n t o f p r o p o r t i o n a l i t y e q u i v a l e n t t o t h e g r a v i -2 2 t a t i o n a l c o n s t a n t o f p h y s i c s . T h u s , t h e e n e r g y o f i n t e r -a c t i o n between any two c e n t r e s o f p o p u l a t i o n i n c r e a s e s a s t h e p r o d u c t o f t h e two p o p u l a t i o n s i n c r e a s e s , a n d f a l l s o f f a s t h e d i s t a n c e between t h e two c e n t r e s i n c r e a s e s . The t o t a l 22 The c o n c e p t o f d e m o g r a p h i c e n e r g y i s b a s e d on Newton's e q u a t i o n f o r t h e m u t u a l e n e r g y o f two masses i n t h e g r a v i -t a t i o n a l f i e l d : m where <S = t h e g r a v i t a t i o n a l c o n s t a n t ; M = mass a t a p o i n t ; r# = mass a t a n o t h e r p o i n t ; a n d a = d i s t a n c e b etween t h e two p o i n t s . M u t u a l e n e r g y r e f e r s t o t h e f o r c e e a c h mass e x e r t s on t h e o t h e r , o p e r a t i n g t o b r i n g t h e two masses t o g e t h e r . 2 8 energy of i n t e r a c t i o n of a given r e g i o n c would be the sum of the energy of i n t e r a c t i o n of i w i t h each of the r> other regions i n t o which a given universe may be d i v i d e d . Formally: ( I I . 7 ) where £ i = t o t a l energy of i n t e r a c t i o n of re g i o n I . Z i p f , 2 3 Stewart, and others have t e s t e d and a p p l i e d t h i s f o r m u l a t i o n of the g r a v i t y concept e m p i r i c a l l y , measuring the energy of i n t e r a c t i o n between p a i r s of c i t i e s by a v a r i e t y of character-i s t i c s such as telephone c a l l s , bus passenger movements, newspaper c i r c u l a t i o n , a i r l i n e t r i p s , and s i m i l a r a c t s of communication. Stewart has extended the p h y s i c a l analogy to in c l u d e the concept of p o t e n t i a l of p o p u l a t i o n , which may be thought of as a measure i n d i c a t i n g the i n t e n s i t y of the p o s s i b i l i t y of i n t e r a c t i o n . At a given l o c a t i o n i , the p o t e n t i a l i n f l u e n c e , or p o s s i b i l i t y of i n t e r a c t i o n , w i t h respect to an 2 3 These others i n c l u d e : Theodore R. Anderson, " I n t e r -m etropolitan M i g r a t i o n : A Comparison of the Hypotheses of Z i p f and S t a u f f e r , " American S o c i o l o g i c a l Review, XX (June, 1 9 5 5 ) , 2 8 7 - 2 9 1 ; John P. Boland, "On the Numbsr and S i z e s of Radio S t a t i o n s i n R e l a t i o n to the Populations of t h e i r C i t i e s , " Sociometry, X I , (February-May, 1 9 4 8 ) , 1 1 1 - 1 1 6 ; Joseph A. Cavanaugh, "Formulation, A n a l y s i s , and Testing of the I n t e r -actance Hypothesis," American S o c i o l o g i c a l Review, XV (December, 1 9 5 0 ) , 7 6 3 - 7 6 6 ; John F o l g e r , "Some Aspects of M i g r a t i o n i n the Tennessee V a l l e y , " American S o c i o l o g i c a l Review, XVIII (June, 1 9 5 3 ) , 2 5 3 - 2 6 0 ; Fred Strodtbeck, "P o p u l a t i o n , Distance, and M i g r a t i o n from Kentucky," Sociometry, X I I I (May, 1 9 5 0 ) , 1 2 3 - 1 3 0 . 29 i n d i v i d u a l a t i , w h i c h i s g e n e r a t e d by t h e p o p u l a t i o n o f a n y g i v e n a r e a j , w i l l be g r e a t e r a s t h e p o p u l a t i o n o f j i s l a r g e r a n d w i l l be l e s s a s t h e d i s t a n c e between c a n d j i n c r e a s e s . M a t h e m a t i c a l l y : V(j _- I (II.8 ) where Vij = p o t e n t i a l a t i o f t h e p o p u l a t i o n o f a r e a J . The t o t a l p o s s i b i l i t y o f i n t e r a c t i o n between a n i n d i v i d u a l a t / a n d t h e p o p u l a t i o n o f a l l o t h e r a r e a s i n t h e p a r t i c u l a r u n i v e r s e u n d e r c o n s i d e r a t i o n , t h a t i s , t h e t o t a l p o p u l a t i o n p o t e n t i a l a t / , w o u l d be: p. ( I I . 9 ) where Vi = t o t a l p o p u l a t i o n p o t e n t i a l a t i . A l t h o u g h S t e w a r t d i d n o t c a r r y o u t t h i s n e x t s t e p , i t c o u l d be a r g u e d t h a t i f Vij r e p r e s e n t s t h e p o s s i b i l i t y o f i n t e r a c t i o n f r o m i w i t h a p a r t i c u l a r a r e a j , a n d V/ r e p r e s e n t s t h e t o t a l p o s s i b i l i t y o f i n t e r a c t i o n f r o m i w i t h a l l r e l e v a n t j , t h e n i n a r e l a t i v e f r e q u e n c y s e n s e t h e p r o b a b i l i t y o f i n t e r a c t i o n f r o m i w i t h a p a r t i c u l a r /' i s : £ Pj , Vi An e q u a t i o n o f t h i s f o r m w i l l be e n c o u n t e r e d l a t e r . The i n t e r p r e t a t i o n t o be g i v e n t o t h e c o n c e p t o f d e m o g r a p h i c p o t e n t i a l i s n o t e n t i r e l y c l e a r . S t e w a r t w r i t e s o f 30 p o p u l a t i o n p o t e n t i a l of a p o i n t as a measure of the p r o x i -mity of people to t h a t p o i n t , as a measure of aggregate a c c e s s i b i l i t y , and more simply as a measure of i n f l u e n c e of people a t a d i s t a n c e . To p o i n t up the s i g n i f i c a n c e of t h i s concept, Stewart and h i s a s s o c i a t e s have conducted a number of e m p i r i c a l s t u d i e s . He r e p o r t s h i g h c o r r e l a t i o n w i t h i n the U n i t e d S t a t e s of the s p a t i a l v a r i a t i o n of p o p u l a t i o n p o t e n t i a l w i t h s p a t i a l v a r i a t i o n i n a wide v a r i e t y of s o c i o l o g i c a l phenomena. Among these are r u r a l p o p u l a t i o n d e n s i t y , r u r a l nonfarm p o p u l a t i o n d e n s i t y , r u r a l nonfarm r e n t s , farmland v a l u e s , m i l e s of r a i l r o a d t r a c k per square m i l e , m i l e s of r u r a l f r e e d e l i v e r y routes per square m i l e , d e n s i t y of r u r a l 2 4 wage earners i n manufacturing, and r u r a l death r a t e s . In a more r e c e n t a r t i c l e , Stewart and Warntz compare c i t i e s i n the U n i t e d S t a t e s and Great B r i t a i n equal i n p o p u l a t i o n but l o c a t e d i n areas of v a r y i n g p o p u l a t i o n poten-t i a l . They f i n d t h a t c i t i e s i n areas of low p o t e n t i a l tend to have l a r g e r a r e a s , lower taxes, and a g r e a t e r excess of b i r t h s over deaths than c i t i e s of equal p o p u l a t i o n i n areas 2 5 of high p o t e n t i a l . In the h i s t o r i c a l development of the g r a v i t y model, the works of Z i p f have a l s o been i n f l u e n t i a l . Z i p f w r i t e s v a r i o u s l y of the PX/Q and P* Pi I Q f a c t o r s which, u s i n g the same 2 4 Stewart, Sociometry, XI, 31-58• 25 Stewart and W i l l i a m Warntz, "Macrogeography and S o c i a l S c i e n c e , " G e o g r a p h i c a l Review, XLVIII ( A p r i l 1 9 5 8 ) . 31 s y m b o l s a s a b o v e , a r e t h e ^J'/Q^J a n d Pi P j / f a c t o r s , r e s p e c t i v e l y . T h e s e f a c t o r s c l o s e l y r e s e m b l e S t e w a r t ' s c o n -c e p t s o f p o t e n t i a l and d e m o g r a p h i c e n e r g y a n d have b e e n i n t e r p r e t e d a s i d e n t i c a l w i t h t h e s e c o n c e p t s on a number o f o c c a s i o n s . However, i n e m p i r i c a l a n a l y s i s , Z i p f ' s u s e o f h i s r e l a t i o n s h i p s o r f a c t o r s d i f f e r s f r o m S t e w a r t ' s u s e o f h i s ( S t e w a r t ' s ) d e m o g r a p h i c c o n c e p t s . E s s e n t i a l l y , Z i p f e x a m i n e s , f o r p a i r s o f c i t i e s , i n t e r a c t i o n phenomena an d t h e PJ/DIJ f a c t o r where t h e e n t i r e f a c t o r i s r a i s e d t o some e x p o n e n t . F o r example, f o r e a c h p a i r s e l e c t e d f r o m a s e t o f c i t i e s f o r w h i c h m a t e r i a l s a r e a v a i l a b l e he p l o t s a l o g - l o g g r a p h o f t h e f r e q u e n c y o f a p a r t i c u l a r phenomena "and t h e i r Pi Pj IOij f a c t o r . When t h e d a t a f o r a l l t h e p o s s i b l e p a i r s o f c i t i e s a r e p l o t t e d , Z i p f f i n d s a s t r a i g h t - l i n e r e l a t i o n s h i p . Z i p f f i n d s s t r a i g h t - l i n e r e l a t i o n s h i p s between t h e Pi Pj I Dij f a c t o r a n d many i n t e r a c t i o n phenomena be t w e e n p a i r s o f c i t i e s , s u c h a s bus p a s s e n g e r t r i p s , r a i l w a y p a s s e n g e r t r i p s , a i r l i n e p a s s e n g e r t r i p s , t e l e p h o n e c a l l s , a n d t o n n a g e o f r a i l w a y e x p r e s s s h i p m e n t s . However, i t must be remembered t h a t i n e a c h c a s e t h e s l o p e o f t h e s t r a i g h t l i n e i s t h e e x p o n e n t o f t h e e n t i r e Pi'P/'/^j f a c t o r , a n d n o t t h e e x p o n e n t o f Dij a l o n e a s i n S t e w a r t ' s c o n c e p t s . T h u s , Z i p f ' s f i n d i n g s 26 a r e n o t a d i r e c t t e s t o f t h e v a l i d i t y o f S t e w a r t ' s c o n c e p t s . I n a p p l i c a t i o n , t h e g r a v i t y a n d p o t e n t i a l m o d e ls l e n d ^ D However, i n t h e n o n - t y p i c a l c a s e , when t h e s l o p e o f Z i p f ' s s t r a i g h t l i n e i s u n i t y , Z i p f ' s u s e o f h i s PIP^/D r e l a t i o n -s h i p becomes i d e n t i c a l w i t h S t e w a r t ' s u s e o f d e m o g r a p h i c e n e r g y . 3 2 themselves to being mapped M t o p o g r a p h i c a l l y " to show contours of equal p o t e n t i a l . From such maps, areas of d i f f e r e n t poten-t i a l are r e a d i l y d i s c e r n i b l e , and i n t e r r e l a t i o n s between areas are e a s i l y v i s u a l i z e d . Stewart has p l o t t e d such maps f o r the United S t a t e s . 2 7 C a r r o l l describes a technique by which a somewhat 28 d i f f e r e n t aspect of the concepts i s mapped. C a r r o l l was concerned with d e s c r i b i n g the area over which urban centres have i n f l u e n c e . As measures of i n f l u e n c e , he used numbers of telephone c a l l s and volume of highway t r a f f i c . S t a r t i n g w i t h assumptions of (1) a f l a t t e r r a i n , (2) urban i n f l u e n c e p r o p o r t i o n a l to c i t y s i z e , and (3) constant r a t e of change i n the d e c l i n e of i n f l u e n c e w i t h d i s t a n c e , C a r r o l l a r b i t r a r i l y assigned the p o p u l a t i o n s i z e as the measure of maximum i n f l u -ence of each centre. Since i n f l u e n c e f a l l s away i n a l l d i r e c t i o n s , he generated a surface (or r a t h e r contours on t h i s surface below the maximum point ) from the equation: Un = *• _ £ L _ ( 1 1 . 1 1 ) where = urban i n f l u e n c e of centre c upon any p o i n t j ; and a. = a constant exponent. The l i n e s of i n t e r s e c t i o n of the surfaces f o r the various adjacent centres c o n s t i t u t e a 2 ? Stewart i n c l u d e d such maps i n h i s a r t i c l e s i n : Geographical  Review, XXXVII, 4 6 1 - 4 8 5 ; American J o u r n a l of P h y s i c s , X V I I I , 2 3 9 - 2 5 3 ; and Theory i n Marketing, 19-40T 28 Douglas C a r r o l l , " S p a t i a l I n t e r a c t i o n and the Urban-M e t r o p o l i t a n D e s c r i p t i o n , " T r a f f i c Q u a r t e r l y , IX, ( A p r i l , 1 9 5 5 ) » 1 4 9 - 1 6 1 . 33 map of the areas of i n f l u e n c e of the centres. C a r r o l l c a r r i e d out t h i s process f o r 21 major c i t i e s i n southern Michigan, and determined a value of 2.8 f o r the exponent of distance from e m p i r i c a l t e s t s of telephone messages and i n t e r - c i t y t r a v e l . Extension of the Concepts Many researchers have c o l l e c t e d e m p i r i c a l evidence i n an e f f o r t to t e s t hypotheses developed from the general permutations of the concepts of demographic f o r c e , energy, and p o t e n t i a l discussed above. This has l e d some to concentrate t h e i r a t t e n t i o n on the distance f a c t o r , and others on the pop u l a t i o n f a c t o r . Such r e s u l t s , and the m o d i f i c a t i o n s to the models developed therefrom are of relevance i n c o n s i d e r i n g the f a c t o r s i n v o l v e d i n the model which i s the main object of t h i s t h e s i s . The work i s reviewed under the two f a c t o r s — dist a n c e and po p u l a t i o n . A. The Distance Factor 29 E m p i r i c a l evidence developed by P r i c e and I k l e among oth e r s , as w e l l as C a r r o l l , suggests that the impact of dis t a n c e i s not uniform, and that i t s r e l a t i o n s h i p i n the ba s i c Equations ( I I . 6 ) and (II.8) i s not a simple i n v e r s e one, but one i n which distance i s r a i s e d to some power other than 7 D.O. P r i c e , "Distance and D i r e c t i o n as Vectors of I n t e r n a l M i g r a t i o n , 1 9 3 5 - 4 0 , " S o c i a l Forces, XXVII (October, 1948), 4 8 - 5 3 ; Fred C. I k l e , " S o c i o l o g i c a l R e l a t i o n s h i p of T r a f f i c to Pop u l a t i o n and Distance," T r a f f i c Q u a r t e r l y , V I I I (December, 1 9 5 4 ) , 1 2 3 - 1 3 6 ; and C a r l Hammer and F.C. I k l e , " I n t e r c i t y Telephone and A i r l i n e T r a f f i c Related to Distance and the •Propensity to I n t e r a c t ' , " Sociometry, XX, (December, 1 9 5 7 ) » 3 0 6 - 3 1 6 . 3 4 one or two. Various exponents have been used as a r e s u l t of e m p i r i c a l t e s t i n g , ranging from one-half to over t h r e e . I s a r d and Peck have deri v e d a value of 1.7 e m p i r i c a l l y . - ^ 0 C a r r o l l 31 r e p o r t s determined exponents of over three. Anderson has suggested t h a t the exponent i t s e l f i s a 32 v a r i a b l e , i n v e r s e l y r e l a t e d to the s i z e of the p o p u l a t i o n . This i m p l i e s , t a k i n g the p o t e n t i a l r e l a t i o n s h i p f o r example, t h a t : Vij - 1 i L _ (11.12) A 7 * where x = f (*/Pj) On the other hand, Carrothers argues that the evidence may be i n t e r p r e t e d d i f f e r e n t l y ; namely, that the exponent may be a v a r i a b l e f u n c t i o n r e l a t e d i n v e r s e l y to dis t a n c e i t s e l f , 3 3 r a t h e r than to po p u l a t i o n . I n t h i s case, the i n t e r p r e t a t i o n would be that f r i c t i o n per u n i t of dis t a n c e a g a i n s t i n t e r a c t i o n caused by short distances i s d i s p r o p o r t i o n a t e l y g r e a t e r than f r i c t i o n per u n i t of di s t a n c e caused by longer d i s t a n c e s . For i n s t a n c e , f r i c t i o n a g a i n s t movement w i t h i n an urban area i s g e n e r a l l y g r e a t e r than t h a t caused by an equal d i s t a n c e i n the l e s s densely developed space between two such areas. Or, again, an e x t r a u n i t of distance added to a long movement i s of 3° Walter I s a r d and Merton J . Peck, "Location Theory and Inter-n a t i o n a l and I n t e r r e g i o n a l Trade Theory," Quarterly J o u r n a l of  Economics, LXVIII (February, 1 9 5 4 ) 97-U 1*. 31 C a r r o l l , op. c i t . Anderson, op. c i t . , p. 287. 33 J J Carrothers, op. c i t . , p. 97, 35 l e s s importance than an e x t r a u n i t added to a short movement.— i t i s obvious that the r e l a t i v e e f f e c t i s l e s s . I t i s i n t e r e s t i n g to note that many f r e i g h t and passenger t r a n s p o r t a -t i o n r a t e s t r u c t u r e s a l s o r e f l e c t t h i s p o i n t . In any event, the exponent of distance i n Equation (11.12) becomes x = /(l/c>ij) I f , however, distance were measured i n terms of t r a v e l time, and increments i n r e l a t i v e (say, percentage) terms, r a t h e r than absolute terms, t h i s argument of p o p u l a t i o n versus d i s t a n c e i n the exponent f u n c t i o n tends to v a n i s h . This i s a case where the type of measure (s c a l e ) used may be h i d i n g the r e a l i s s u e . In r e p o r t i n g i n v e s t i g a t i o n s of the r e l a t i o n of distance to m i g r a t i o n , P r i c e r e p o r t s another c o m p l i c a t i o n . The e m p i r i c a l evidence suggests that the f u n c t i o n of d i s t a n c e i s a f f e c t e d by the p a r t i c u l a r d i r e c t i o n from the d e s t i n a t i o n i n which the measures are made, and that the distance f a c t o r of 3 4 Equation ( I I . 8 ) v a r i e s a c c o r d i n g l y . This i m p l i e s that d i s t a n c e should be t r e a t e d as a v e c t o r , and not as a s c a l a r . B . The P o p u l a t i o n Factor In c a l c u l a t i n g p o p u l a t i o n p o t e n t i a l s and energy of i n t e r a c t i o n f o r v a r i o u s c o u n t r i e s , and f o r v a r i o u s kinds of a c t i v i t y w i t h i n a country, Stewart found that f r e q u e n t l y an area would have a p u l l , or i n f l u e n c e , e i t h e r greater or l e s s 3^ P r i c e , op. c i t . 3 6 t h a n w o u l d be e x p e c t e d f r o m t h e s i m p l e f o r m u l a t i o n s o f t h e 35 c o n c e p t s . He c o n c l u d e d t h a t p o p u l a t i o n u n d e r one s e t o f c i r c u m s t a n c e s i s n o t n e c e s s a r i l y o f t h e same i m p o r t a n c e a s u n d e r o t h e r c i r c u m s t a n c e s , t h a t i s , t h e p o p u l a t i o n s a r e n o t homogeneous. T h i s i s c o n t r a r y t o t h e a s s u m p t i o n s i m p l i e d i n h i s o r i g i n a l f o r m u l a t i o n where a v a l u e o f u n i t y was i m p l i c i t l y 36 a s s i g n e d a s a w e i g h t t o t h e p o p u l a t i o n e l e m e n t . By a s s i g n -i n g v a l u e s o t h e r t h a n one t o t h e s e w e i g h t s , he s o u g h t t o a c c o u n t f o r d i f f e r e n c e s i n d e g r e e o f i n f l u e n c e w h i c h w o u l d r e s u l t f r o m d i f f e r e n t c h a r a c t e r i s t i c s o f t h e p o p u l a t i o n s . When t h i s i s done, t h e b a s i c e n e r g y e q u a t i o n becomes: £ L j = k mL Pj '.nnjPj ( 1 1 . 1 3 ) Dcj where m, , mj = m o l e c u l a r w e i g h t s o f a n i n d i v i d u a l i n * , and a n i n d i v i d u a l i n / ' , r e s p e c t i v e l y . S t e w a r t i n t e r p r e t s m o l e -c u l a r w e i g h t i n t h i s c o n t e x t a s a measure o f t h e i n d i v i d u a l ' s c a p a c i t y f o r s o c i o l o g i c a l i n t e r a c t i o n . H a v i n g a d m i t t e d t h e s e m o l e c u l a r w e i g h t s , however, i t may be a r g u e d t h a t a d j u s t m e n t h a s been made f o r c u l t u r a l d e t e r m i n a n t s o f human b e h a v i o u r , and t h a t t h e r e l a t i o n s h i p s a r e no l o n g e r t h e p u r e l y o b j e c t i v e , d e m o g r a p h i c s p e c i e s u pon w h i c h S t e w a r t f o u n d e d h i s whole c o n c e p t o f s o c i a l p h y s i c s . S t e w a r t , T h e o r y i n M a r k e t i n g , pp. 3 2 - 3 6 . T h i s w e i g h t i s p h y s i c a l a n a l o g y . i s e q u i v a l e n t t o m o l e c u l a r w e i g h t i n t h e 3 7 T h i s f o r m u l a t i o n i s s i m i l a r t o Dodd's i n t e r a c t a n c e h y p o t h e s i s i n w h i c h he i n t r o d u c e s v a r i a b l e s o t h e r t h a n t h o s e o f p o p u l a t i o n numbers a n d d i s t a n c e i n t o t h e o r i g i n a l f o r m u l a t i o n , by making them m u l t i p l i e r s o f t h e b a s i c v a r i a b l e s , i n o r d e r t o a c c o u n t f o r d i f f e r e n t i a l s i n s e x , income, e d u c a t i o n , a n d o t h e r 37 ( d e m o g r a p h i c ) c h a r a c t e r i s t i c s . The b a s i c e n e r g y e q u a t i o n t h u s becomes: Pj (11.14) whereE(j6,- , E 0 y = weighting f a c t o r s f o r p o p u l a t i o n P} and Pj , r e s p e c t i v e l y . But the a p p l i c a t i o n of simple indexes to the population f a c t o r i n the equation may not be enough to account f o r observed d i f f e r e n c e s i n the i n f l u e n c e of p o p u l a t i o n i n d i f f e r e n t circumstances. For in s t a n c e , a l a r g e r p o p u l a t i o n i n one area than i n another may of i t s e l f r e s u l t i n an i n f l u e n c e f o r the f i r s t area l a r g e r p r o p o r t i o n a t e l y than can be accounted f o r by the m o d i f i c a t i o n of po p u l a t i o n s i z e by a simple m u l t i p l i e r . This may be the case where agglomeration economies are present, f o r example. Anderson suggests the p o s s i b i l i t y of r a i s i n g the numerator of the ba s i c equation to some power other than u n i t y ( i m p l i c i t above). D But t h i s s t i l l i m p l i e s t h a t populations of 3 ? S t u a r t C. Dodd, "The Interactance Hypothesis: A G r a v i t y Model F i t t i n g P h y s i c a l Masses and Human Groups," American  S o c i o l o g i c a l Review, XV ( A p r i l , 1 9 5 0 ) 2 4 5 - 2 5 6 . O Q Anderson, op. c i t . ; and " P o t e n t i a l Models and S p a t i a l D i s t r i b u t i o n of P o p u l a t i o n , " Papers and Proceedings of the  Regional Science A s s o c i a t i o n , I I ( 1 9 5 6 ) . 3 8 d i f f e r e n t k i n d s have e q u a l i n f l u e n c e . T h e r e f o r e , i t w o u l d seem more l o g i c a l t o r a i s e t h e i n d i v i d u a l p o p u l a t i o n e l e m e n t s t o some power o t h e r t h a n u n i t y , a n d n o t n e c e s s a r i l y t h e same power ( n o r e v e n n e c e s s a r i l y a c o n s t a n t p o w e r ) . p o p u l a t i o n f a c t o r s t h a t have been d i s c u s s e d a r e a d d e d t o t h e b a s i c e n e r g y e q u a t i o n , i t becomes: C l e a r l y , t h e t a s k o f e v a l u a t i n g t h i s model a g a i n s t a p a r t i c u l a r s e t o f e m p i r i c a l d a t a seems v e r y l a r g e i n d e e d . F o r e v e r y p a i r o f p o s s i b l e w e i g h t i n g f a c t o r s , t h e t h r e e e x p o n e n t s must be e v a l u a t e d s i m u l t a n e o u s l y . E x p e r i e n c e w i t h q u a d r a t i c programming p r o b l e m s w o u l d i n d i c a t e t h a t t h i s i s l i k e l y t o be v e r y t i m e -c o n s u m i n g , e v e n on a h i g h - s p e e d d i g i t a l c o m p u t e r . F u r t h e r , t h e r e s u l t s o f s u c h an e v a l u a t i o n f o r a v e r y much s i m p l e r m o d e l , t o be r e p o r t e d l a t e r , r a i s e t h e d i s t i n c t p o s s i b i l i t y t h a t s u c h a n e v a l u a t i o n may p r o v e t o be i n c o n c l u s i v e . I t may t h e n be p r o p e r t o q u e s t i o n t h e p r o b a b l e r e t u r n t o t h e e f f o r t e x p e n d e d i n s u c h a c t i v i t y . More c a r e f u l a n a l y s i s o f t h e f a c t o r s , a n d t h e measures o f t h e f a c t o r s , seems t o o f f e r a g r e a t e r p o t e n t i a l r e t u r n . The a d j u s t m e n t s o f t h e b a s i c c o n c e p t s o f g r a v i t y a n d p o t e n t i a l m o d e ls w h i c h h ave b e e n v a r i o u s l y p r o p o s e d , a n d d e s c r i b e d a b o v e , open t h e way f o r i n t r o d u c i n g d i f f e r e n t k i n d s When t h e v a r i o u s m o d i f i c a t i o n s o f t h e d i s t a n c e a n d ( 1 1 . 1 5 ) 39 o f k e y v a r i a b l e s i n p l a c e o f p o p u l a t i o n a n d d i s t a n c e . D i f f e r e n t k e y v a r i a b l e s a l s o i m p l y new i n t e r p r e t a t i o n s t o be g i v e n t o t h e g r a v i t y a n d p o t e n t i a l r e l a t i o n s h i p s d e v e l o p e d . O t h e r M o d i f i c a t i o n s a n d A d a p t a t i o n s S t o u f f e r h a s s u g g e s t e d a m o d i f i c a t i o n , o f i n t e r e s t i n 39 c o n n e c t i o n w i t h m e a s u r i n g p o p u l a t i o n m o b i l i t y . He s u g g e s t s t h a t t h e r e i s no n e c e s s a r y r e l a t i o n s h i p between d i s t a n c e a n d m o b i l i t y , b u t t h a t t h e number o f p e r s o n s g o i n g a g i v e n d i s t a n c e i s d i r e c t l y p r o p o r t i o n a l t o t h e number o f o p p o r t u n i t i e s a t t h a t d i s t a n c e a n d i n v e r s e l y p r o p o r t i o n a l t o t h e number o f i n t e r v e n -i n g o p p o r t u n i t i e s . T h a t i s , t h e f u n c t i o n o f d i s t a n c e i s n o t n e c e s s a r i l y c o n t i n u o u s i n t h e f o r m u l a t i o n . M a t h e m a t i c a l l y , t h e r e l a t i o n may be e x p r e s s e d : A tf s a • 4x (11.16) /Is x. • /is where A</ = number o f p e r s o n s m oving f r o m a p o i n t o f o r i g i n t o a c i r c u l a r b a n d o f w i d t h /?s ; X = c u m u l a t e d number o f o p p o r t u n i t i e s between t h e o r i g i n a n d d e s t i n a t i o n 5 ; 4x = number o f o p p o r t u n i t i e s w i t h i n t h e band o f w i d t h /Is ; a n d ^ = a c o n s t a n t . I n t e s t i n g t h e h y p o t h e s i s , S t o u f f e r e n c o u n t e r e d 39 Samuel S t o u f f e r , " I n t e r v e n i n g O p p o r t u n i t i e s : A T h e o r y R e l a t i n g M o b i l i t y a n d D i s t a n c e , " A m e r i c a n S o c i o l o g i c a l Review, V (December 1940), 845-857. 4 0 d i f f i c u l t p r o b l e m s o f m e a s u r i n g o p p o r t u n i t i e s . One a p p r o a c h u s e d by him was t o c o n s i d e r t o t a l i n - m i g r a n t s a s s u c h a m easure, b u t , a s A n d e r s o n h a s p o i n t e d o u t , t h e r e i s a n 4 0 e l e m e n t o f c i r c u l a r i t y i n v o l v e d . O t h e r t e s t s o f S t o u f f e r ' s h y p o t h e s i s have been c a r r i e d o u t by B r i g h t a n d Thomas, P o l g e r , I s b e l l , a n d S t r o d t b e c k . A l l have e x p e r i e n c e d s i m i l a r d i f f i -4 1 c u l t y w i t h t h e measure o f o p p o r t u n i t i e s . F o r p u r p o s e s o f d e v e l o p i n g models f o r p r o j e c t i n g n a t i o n a l and r e g i o n a l p r o d u c t , I s a r d a n d F r e u t e l c o n s i d e r a g g r e g a t e income t o be a c r i t i c a l v a r i a b l e a n d s u g g e s t t h e u s e o f a n income p o t e n t i a l measure, i n w h i c h r e g i o n a l ( o r n a t i o n a l ) 4 2 income i s s u b s t i t u t e d f o r t h e p o p u l a t i o n f a c t o r . They a l s o s u g g e s t t h a t t h e f r i c t i o n a g a i n s t i n t e r a c t i o n c a u s e d by d i s t -a n c e i s n o t so much a f u n c t i o n o f t h e i n t e r v e n i n g p h y s i c a l s p a c e , b u t r a t h e r a f u n c t i o n o f t h e c o s t o f t r a v e r s i n g t h i s s p a c e . They t h e r e f o r e u t i l i z e a measure o f e f f e c t i v e o r e c o n o m i c d i s t a n c e , i n w h i c h p h y s i c a l d i s t a n c e i s m o d i f i e d by 4 0 A n d e r s o n , A m e r i c a n S o c i o l o g i c a l Review, XX, 289. 4 1 M a r g a r e t B r i g h t a n d D o r o t h y S. Thomas, " I n t e r s t a t e M i g r a t i o n and I n t e r v e n i n g O p p o r t u n i t i e s , " A m e r i c a n S o c i o -l o g i c a l Review, VI ( December, 1941), 773-WTi F o l g e r , "Some A s p e c t s o f M i g r a t i o n ... i n t h e T e n n e s s e e V a l l e y , " A m e r i c a n S o c i o l o g i c a l Review, X V I I I , 253-260; E l e a n o r G. I s b e l l , " I n t e r n a l M i g r a t i o n i n Sweden and I n t e r v e n i n g O p p o r t u n i t i e s , " A m e r i c a n S o c i o l o g i c a l Review, IX, (December, 1944), 627-639; F r e d - S t r o d t b e c k , " E q u a l O p p o r t u n i t y I n t e r -v a l s : A C o n t r i b u t i o n t o t h e Method o f I n t e r v e n i n g O p p o r t u n i t y A n a l y s i s , " A m e r i c a n S o c i o l o g i c a l Review, XIV, ( A u g u s t , 1949), 490-497. 4 2 W a l t e r I s a r d and Guy F r e u t e l , " R e g i o n a l a n d N a t i o n a l P r o d u c t P r o j e c t i o n s a n d t h e i r I n t e r r e l a t i o n s , " Long-Range E c o n o m i c  P r o j e c t i o n s , S t u d i e s i n Income and W e a l t h , V o l . XVI by t h e C o n f e r e n c e on R e s e a r c h i n Income a n d W e a l t h ( P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s , f o r t h e N a t i o n a l B u r e a u o f E c o n o m i c R e s e a r c h , 1954). 41 t r a n s p o r t c o s t . T h u s , u n d e r t h i s h y p o t h e s i s , t h e b a s i c p o t e n t i a l e q u a t i o n becomes: K£f Yy ( 1 1 . 1 7 ) where W; = income p o t e n t i a l a t i ; = income o f r e g i o n j ; a = a c o n s t a n t e x p o n e n t ; a n d = a p a r a m e t e r w h i c h d i f f e r s f r o m one p a i r o f r e g i o n s t o a n o t h e r a n d w h i c h i s some f u n c t i o n o f t r a n s p o r t c o s t between e a c h p a i r o f r e g i o n s , b u t n o t e t h a t t h i s seems a l s o t o be u l t i m a t e l y d e p e n d e n t on d i s t a n c e . C a r r o l l a n d A n d e r s o n have s u g g e s t e d t h a t a t i m e - c o s t 43 measure o f d i s t a n c e would be a p p r o p r i a t e i n many c i r c u m s t a n c e s . F o r i n s t a n c e , i t i s a r g u e d t h a t t h i s m i g h t be p a r t i c u l a r l y p e r t i n e n t w i t h r e s p e c t t o i n t r a - m e t r o p o l i t a n i n t e r a c t i o n when t h e t i m e i n v o l v e d i n c o m m u n i c a t i v e a c t i v i t i e s a p p e a r s t o be a c r i t i c a l f a c t o r . I n a p p l y i n g t h e g r a v i t a t i o n p r i n c i p l e t o t r a f f i c a n a l y s i s , V o o r h e e s , S h a r p e , a n d S t e g m a i e r u s e t i m e o f t r a v e l a s a measure o f d i s t a n c e , c l a s s i f y i n g movement by mode o f 44 t r a v e l t o a c c o u n t f o r d i f f e r e n c e s i n r a t e s . A t t h e same t i m e , t h e y s u g g e s t c l a s s i f y i n g t r i p s by t h e n a t u r e o f t h e d e s t i n a t i o n , s u c h a s a s h o p p i n g a r e a , a n d m e a s u r i n g t h e s i z e ^ 3 C a r r o l l , T r a f f i c Q u a r t e r l y , IX, 1 4 9 - 1 6 1 ; A n d e r s o n , P a p e r s a n d P r o c e e d i n g s o f t h e R e g i o n a l S c i e n c e A s s o c i a t i o n , I I , ( 1 9 5 6 ) . 4 4 A l a n M. V o o r h e e s , G.B. S h a r p e , a n d J . T . S t e g m a i e r , S h o p p i n g  H a b i t s and T r a v e l P a t t e r n s ( W a s h i n g t o n : Highway R e s e a r c h B o a r d , S p e c i a l R e p o r t 1 1-B, 1 9 5 5 ) . 42 o f t h e a t t r a c t i n g i n f l u e n c e o f t h e d e s t i n a t i o n i n terms s u c h a s f l o o r a r e a d e v o t e d t o t h e s a l e o f a p a r t i c u l a r commodity o f i n t e r e s t . H a r r i s h a s e x t e n d e d t h e p o t e n t i a l c o n c e p t t o i n c l u d e a 45 measure o f m a r k e t p o t e n t i a l . J He measures t h e m a r k e t s i n t e r m s o f r e t a i l s a l e s and d i s t a n c e i n terms o f t r a n s p o r t c o s t s , t h e n sums o v e r a l l m a r k e t s . Thus, t h e m a r k e t p o t e n t i a l i s : rt #i =^ Sj_ (11.18) where Ri = market p o t e n t i a l a t / ; Sj = volume o f r e t a i l s a l e s i n r e g i o n j ; a n d C<j - t r a n s p o r t c o s t f r o m * t o y . N o t e t h a t t h i s r e l a t i o n s h i p i m p l i c i t l y c o n s t i t u t e s r a i s i n g t h e d i s t a n c e f a c t o r t o a v a r i a b l e e x p o n e n t s i n c e t r a n s p o r t c o s t s a r e a p r o d u c t o f d i s t a n c e a n d r a t e , and r a t e i s a f u n c t i o n o f 46 d i s t a n c e . A n d e r s o n has g e n e r a l i z e d t h e p o t e n t i a l c o n c e p t i n s u c h a way t h a t t h e n u m e r a t o r may be i d e n t i f i e d a s any g i v e n r e s o u r c e ( i n c l u d i n g p o p u l a t i o n ) whose d i s t r i b u t i o n may be 45 C D . H a r r i s , "The M a r k e t a s a F a c t o r i n t h e L o c a l i z a t i o n o f I n d u s t r y i n t h e U n i t e d S t a t e s , " A n n a l s o f t h e A s s o c i a t i o n o f  A m e r i c a n G e o g r a p h e r s , XLIV (December, 1954), 315-348. 46 The t e r m f u n c t i o n i s u s e d h e r e i n a s e n s e s l i g h t l y d i f f e r e n t f r o m m a t h e m a t i c s , i n t h a t w h i l e d i s t a n c e i s a f a c t o r , many t r a n s p o r t r a t e s a r e d e t e r m i n e d i n a somewhat a r b i t r a r y f a s h i o n w i t h a v i e w t o c o m p e t i t i o n , s u b s i d i e s , and s i m i l a r i n f l u e n c e s . 4 3 u s e f u l I n d e s c r i b i n g t h e v a r i a t i o n s i n i n t e n s i t y o f p o t e n t i a l 4 7 o f i n t e r a c t i o n w h i c h may be f o u n d among a r e a s . I n t h e e x p o n e n t t o w h i c h he r a i s e s t h e d i s t a n c e f a c t o r i s i n c l u d e d a measure o f t h e i m p a c t o f t e c h n o l o g i c a l i n n o v a t i o n w h i c h w i l l , i n g e n e r a l , t e n d t o d e c r e a s e t h e e n e r g y consumed i n t r a v e r s i n g d i s t a n c e . T h i s g e n e r a l i z e d f o r m u l a t i o n may be e x p r e s s e d a s : Ni - ( 1 1 . 1 9 ) j.-t Dcja where f-fi - p o t e n t i a l a t i c r e a t e d by r e s o u r c e X i n r e g i o n j j = t h e measure o f a g i v e n r e s o u r c e i n r e g i o n j ; a n d a. = a c o n s t a n t e x p o n e n t . Summary As h a s been p o i n t e d o u t , t h e g r a v i t y and p o t e n t i a l c o n c e p t s o f human i n t e r a c t i o n were d e v e l o p e d o r i g i n a l l y f r o m 4 8 a n a l o g y t o N e w t o n i a n p h y s i c s o f m a t t e r . The b e h a v i o u r o f i n d i v i d u a l m o l e c u l e s o f m a t t e r i s n o t n o r m a l l y p r e d i c t a b l e , b u t i n l a r g e numbers m o l e c u l a r b e h a v i o u r i s p r e d i c t a b l e on t h e b a s i s o f m a t h e m a t i c a l p r o b a b i l i t y . S i m i l a r l y , w h i l e i t i s n o t y e t p o s s i b l e t o d e s c r i b e t h e a c t i o n s a n d r e a c t i o n s o f t h e i n d i v i d u a l human i n m a t h e m a t i c a l t e r m s , i t i s t h e i m p l i c i t h y p o t h e s i s o f a l l t h e a b o v e - m e n t i o n e d a u t h o r s — a s w e l l a s ^ A n d e r s o n , P a p e r s a n d P r o c e e d i n g s o f t h e R e g i o n a l S c i e n c e  A s s o c i a t i o n , I I (1956). ^ I t i s i n t e r e s t i n g t o n o t e , p a r e n t h e t i c a l l y , t h a t t h e r e h a v e b e e n no r e p o r t e d a t t e m p t s t o a n a l y z e human i n t e r a c t i o n u s i n g t h e r e l a t e d c o n c e p t s d e v e l o p e d by E i n s t e i n i n h i s v a r i o u s r e l a t i v i t y t h e o r i e s . 4 4 e c o n o m e t r i c i a n s , a n d t h o s e i n v o l v ^ i n s o c i o m e t r y a n d p s y c h r o -m e t r i c s t h a t i n t e r a c t i o n s o f g r o u p s o f p e o p l e may be d e s c r i b e d m a t h e m a t i c a l l y . N o t e , however, t h a t none o f t h e a u t h o r s c i t e d u s e d a p r o b a b i l i s t i c a p p r o a c h , a n d where p r o b a b i l i t y was m e n t i o n e d , i t was a d d e d t o t h e d i s c u s s i o n f o u n d i n t h e l i t e r a t u r e . The p o s s i b i l i t y t h a t i n t e r a c t i o n may be d e s c r i b e d m a t h e m a t i c a l l y i s s u g g e s t e d by t h e phenomenon o b s e r v e d i n a l l s o c i a l s c i e n c e s t h a t p e o p l e do, i n f a c t , b ehave d i f f e r e n t l y i n g r o u p s t h a n t h e y do a s i n d i v i d u a l s . A g r o u p , however, i s n o t t h e same t h i n g a s an a g g r e g a t i o n o f i n d i v i d u a l s a c t i n g i n d e p e n d e n t l y , a n d p e r h a p s a l s o w i t h o u t k nowledge o f t h e 4 9 a c t i o n o f o t h e r s . I t seems i m p o r t a n t t o k e e p i n mind, w h i l e e v a l u a t i n g t h e v a r i o u s works d i s c u s s e d a b o v e , t h a t a l t h o u g h t h e u s e o f a n a l o g y i n d e v e l o p i n g a c o n c e p t may be a t t r a c t i v e , f l e x i b i l i t y must be m a i n t a i n e d . O t h e r w i s e , s c i e n c e i s r e p l a c e d by dogma. T h e r e may be a p o i n t a t w h i c h r e f e r r i n g b a c k t o t h e o r i g i n a l a n a l o g y c a n be d y s f u n c t i o n a l i n t h a t i t may r e t a r d f o r w a r d p r o g r e s s . A t t h i s p o i n t , t h e o r i g i n a l a n a l o g y , a n d p e r h a p s e v e n much o f t h e r e s u l t s g e n e r a t e d f r o m i t , s h o u l d be a b a n -d o n e d i n f a v o u r o f some o t h e r , more f r u i t f u l a p p r o a c h . I n t h e c a s e o f g r a v i t y and p o t e n t i a l m o d e l s , a f u n d a m e n t a l d i f f i c u l t y a r i s e s f r o m t h e d i f f e r e n t n a t u r e o f t h e ^9 P a u l A. S a m u e l s o n d i s c u s s e s t h e " f a l l a c y o f c o m p o s i t i o n " , w h i c h i s a r e l a t e d n o t i o n . See h i s E c o n o m i c s : An I n t r o d u c t o r y  A n a l y s i s , 3rd e d . (New Y o r k : M c G r a w - H i l l Book Company, I n c . , 1955), P P . 9-19. . 45 two b a s i c e n t i t i e s i n v o l v e d i n t h e o r i g i n a l p h y s i c a l model a n d t h e human i n t e r a c t i o n m o d e l s : t h e i n d i v i d u a l human b e i n g c a n make d e c i s i o n s w i t h r e s p e c t t o h i s a c t i o n s , w h i l e t h e i n d i v i d u a l m o l e c u l e ( p r e s u m a b l y ) c a n n o t . T h i s d o e s n o t n e c e s s a r i l y i m p l y t h a t t h e i n t e r a c t i o n o f humans i n l a r g e numbers c a n n o t be d e s c r i b e d m a t h e m a t i c a l l y , b u t i t d o e s c a s t some d o u b t r e g a r d i n g t h e v a l i d i t y o f a g g r e g a t i v e p r o c e d u r e s u s e d i n e m p i r i c a l l y t e s t i n g t h e m a t h e m a t i c a l r e l a t i o n s h i p s e v o l v e d . I t do e s a p p e a r t h a t t h e r e e x i s t s some f o r m o f t h r e s h o l d where t h e power o f i n d i v i d u a l d e c i s i o n - m a k i n g c r i t i c a l l y a f f e c t s t h e r e s u l t s o b t a i n e d . T h i s t h r e s h o l d may v a r y among i n t e r a c t i o n phenomena a l o n g many d i f f e r e n t d i men-s i o n s . I t w o u l d seem d e s i r a b l e t o i n v e s t i g a t e somehow t h i s n o t i o n o f t h r e s h o l d s b e f o r e c o n c e p t s b a s e d on a g g r e g a t i o n a r e b r o a d l y a p p l i e d . T h i s i s t a n t a m o u n t t o i n v e s t i g a t i n g w h e t h e r t h e b a s i c m a t h e m a t i c a l o p e r a t i o n s a r e l e g i t i m a t e p r o c e d u r e s i n m o d e l s o f human b e h a v i o u r . CHAPTER I I I S P E C I F I C HISTORY OF GRAVIMETRIC MODELS IN MARKETING A l l t h e s o - c a l l e d l a w s o f r e t a i l g r a v i t a t i o n now a p p e a r i n g i n t h e l i t e r a t u r e o f m a r k e t i n g , e x c e p t t h e one t o be d i s c u s s e d i n C h a p t e r IV o f t h i s p a p e r , a r e b a s e d on t h e a n a l y s i s o f r e t a i l t r a d e begun by R e i l l y i n 1927.^ T h i s work was p r e v i o u s l y m e n t i o n e d i n C h a p t e r I I i n o r d e r t o p l a c e i t i n i t s p r o p e r h i s t o r i c a l p e r s p e c t i v e w i t h r e s p e c t t o g r a v i -m e t r i c models i n o t h e r b r a n c h e s o f s o c i a l s c i e n c e . R e i l l y ' s s t u d i e s were c a r r i e d o u t o v e r a p e r i o d o f more t h a n t h r e e y e a r s , a n d had a s t h e i r o b j e c t i v e t h e d i s c o v e r y o f some method f o r m e a s u r i n g t h e r e t a i l t r a d e i n f l u e n c e o f a 2 c i t y . C o n v e r s e h a s p e r f o r m e d c o n s i d e r a b l e work i n e x t e n d i n g 3 a n d r e f i n i n g t h e r e s u l t s o b t a i n e d by R e i l l y . A c c o r d i n g t o S c h w a r t z , t h e n e t r e s u l t o f t h e combined e f f o r t s o f R e i l l y a n d C o n v e r s e c a n be summarized by s i x W i l l i a m J . R e i l l y , The Law o f R e t a i l G r a v i t a t i o n (1st e d . ; New Y o r k : W i l l i a m J . R e i l l y Company, 1931). 2 R e i l l y d e f i n e s " r e t a i l t r a d e i n f l u e n c e o f a c i t y " a s " t h e amount o f t r a d e a c i t y draws f r o m i t s s u r r o u n d i n g a r e a . " ( I b i d . , p . 56). 3 P a u l D. C o n v e r s e , A S t u d y o f R e t a i l T r a d e A r e a s i n E a s t  C e n t r a l I l l i n o i s ( U r b a n a : U n i v e r s i t y o f I l l i n o i s P r e s s , 19^3); R e t a i l T r a d e A r e a s i n I l l i n o i s ( U r b a n a : U n i v e r s i t y o f I l l i n o i s P r e s s , 1946); "New Laws o f R e t a i l G r a v i t a t i o n , " J o u r n a l o f  M a r k e t i n g , XIV ( O c t o b e r , 1 9^9). 47 equations which are the laws of r e t a i l g r a v i t a t i o n . I t i s the purpose of t h i s chapter to survey these equations i n the order i n which they were developed, to d i s c u s s b r i e f l y some as p e c t s of the methodology by which e m p i r i c a l support was drawn, and to look a t some more r e c e n t work p e r t a i n i n g to them. The Laws of R e t a i l G r a v i t a t i o n The o r i g i n a l law, with which i t a l l began, was s t a t e d by R e i l l y : Two c i t i e s a t t r a c t r e t a i l trade-* from any i n t e r m e d i a t e c i t y o r town i n the v i c i n i t y of the b r e a k i n g p o i n t " approximately i n d i r e c t p r o p o r t i o n to the p o p u l a t i o n of the two c i t i e s and i n Inverse p r o p o r t i o n to the square of the d i s t a n c e s from these two c i t i e s t o the i n t e r m e d i a t e town.7 M a t h e m a t i c a l l y , t h i s statement has the f o l l o w i n g form: = fk x £±1 ( 1 1 1 . D 3b Pb Oa x George Schwartz, Development of Marketing Theory ( C i n c i n a t t i : South-Western P u b l i s h i n g Co., 1 9 6 3 ) , p. 11. ^ The term " r e t a i l t r a d e " i n R e i l l y ' s work i s a p p a r e n t l y meant to i n c l u d e purchases of shopping goods, s p e c i a l t y goods, and convenience goods ( R e i l l y , op. c i t . ) . On the other hand, Converse r e f e r s to purchases of shopping goods, e s p e c i a l l y f a s h i o n goods (Converse, R e t a i l Trade Areas i n I l l i n o i s ) . ^ The term "breaking p o i n t " i s d e f i n e d by R e i l l y to be "the p o i n t up to which one c i t y shows domination of r e t a i l t r a d e , and beyond which the other c i t y dominates" ( R e i l l y , op. c i t . p. 64). I b i d . , p. 9 . 48 where 3# ,3b = the proportions of r e t a i l trade from the i n t e r -mediate town a t t r a c t e d by c i t i e s A and £> , r e s p e c t i v e l y ; Pa , = the populations of c i t i e s A) a n d & , r e s p e c t i v e l y ; and Da , Db = the distances from the intermediate town to c i t i e s A? a n d 3 , r e s p e c t i v e l y . Converse and h i s a s s o c i a t e s developed a second equation, known as the breaking p o i n t formula, used to measure the movement of shopping goods tra d e , and w r i t t e n : O/, = DA * Db ( I I I . 2 ) ^ Pb In t h i s equation, the breaking p o i n t between any two c i t i e s i s the intermediate community which d i v i d e s i t s shopping goods trade e q u a l l y between the two c i t i e s . According to Converse, Equation ( I I I . 2 ) can be used to determine the boundaries of a town's normal t r a d i n g area without performing any f i e l d - w o r k , p r o v i d i n g the d i f f e r e n c e s i n populations of the towns and c i t i e s being s t u d i e d are r e l a t i v e l y s m a l l . Distances used are those measured along improved automobile highways. Another law of r e t a i l g r a v i t a t i o n developed by Converse s t a t e s : A t r a d i n g center and a town i n or near i t s trade area d i v i d e the trade of the town approximately i n d i r e c t p r o p o r t i o n to the populations of the two towns and i n v e r s e l y as the squares of the distance f a c t o r s , u s i n g 4 as the distance f a c t o r of the home town.8 Converse, J o u r n a l of Marketing, XIV, p. 382. 49 M a t h e m a t i c a l l y , t h i s s t a t e m e n t has t h e e q u a t i o n : - 3L x ( I I I . 3 ) 3i yh where sa = t h e p r o p o r t i o n o f t r a d e g o i n g t o t h e o u t s i d e town; £>h= t h e p r o p o r t i o n o f t r a d e r e t a i n e d "by t h e home town; ^§ = t h e p o p u l a t i o n o f t h e o u t s i d e town; = t h e p o p u l a t i o n o f t h e home town; ^ = t h e d i s t a n c e t o t h e o u t s i d e town; a n d 4- = t h e i n e r t i a f a c t o r . The i n e r t i a - d i s t a n c e f a c t o r , a c c o r d i n g t o C o n v e r s e , r e f l e c t s t h e e f f o r t r e q u i r e d t o overcome t h e i n e r t i a o f t r a v e l l i n g t o a s t o r e c l o s e a t hand. C o n v e r s e s p e c u l a t e s t h a t t h e p u r c h a s e s n o t made i n t h e n e a r b y l a r g e c i t y b e c a u s e o f t h i s i n e r t i a f a c t o r a r e made e i t h e r i n t h e home town o r i n 9 s h o p p i n g c e n t r e s i n t h e s u b u r b s o f t h e l a r g e c i t y . E q u a t i o n ( I I I . 3 ) d e a l s o n l y w i t h one s m a l l town a n d one l a r g e town, w h i c h i s n o t a l w a y s t h e a c t u a l a r r a n g e m e n t . I f a s m a l l town l o s e s a c o n s i d e r a b l e amount o f t r a d e t o two o r more l a r g e r towns, t h e n , a c c o r d i n g t o C o n v e r s e , t h e p r o p o r t i o n l o s t t o t h e s e towns i s d e t e r m i n e d by u s i n g m u l t i p l e s o f 4 t o o b t a i n 10 a t o t a l i n e r t i a f a c t o r . C o n v e r s e s t a t e s t h a t he h a s e x p e r i -mented w i t h t h i s method, and he has c o n c l u d e d t h a t i t seems t o 11 work s a t i s f a c t o r i l y . Two a d d i t i o n a l l a w s o f r e t a i l g r a v i t a t i o n d e v e l o p e d by I b i d . I o I f home town B l o s e s t r a d e t o two l a r g e r towns, i n e r t i a f a c t o r = 2 x 4 o r 8, a n d so f o r t h . I I I b i d . 50 Converse a r e b a s i c a l l y E q u a t i o n s ( I I I . l ) and ( I I I . 2 ) , b u t m o d i f i e d t o i n c r e a s e p r e d i c t i v e a c c u r a c y i n those cases where a t r a d i n g c e n t r e i s more than twenty t i m e s t h e s i z e of the i n t e r m e d i a t e town. The r e v i s e d e q u a t i o n s a r e : Converse r e g a r d s E q u a t i o n s ( I I I . 4 ) and ( I I I . 5 ) as t e n t a t i v e , because "we do n o t have enough d a t a as y e t t o measure the 12 a c c u r a c y o f t h i s a d j u s t m e n t . " The f i n a l e q u a t i o n d e v e l o p e d by Converse i s a m o d i f i -c a t i o n o f E q u a t i o n ( I I I . 3 ) , and was t e s t e d o n l y i n Chicago and v i c i n i t y . Because o f urban c o n g e s t i o n , n e i g h b o u r i n g s m a l l towns were found t o r e t a i n a l a r g e r p r o p o r t i o n o f t h e i r f a s h i o n goods t r a d e than E q u a t i o n ( I I I . 3 ) p r e d i c t s . U s i n g s u i t a b l e d a t a , i t was d e t e r m i n e d t h a t the i n e r t i a - d i s t a n c e f a c t o r s h o u l d be 1.5 f o r towns i n the v i c i n i t y o f C h i c a g o , and t h u s the e q u a t i o n i s : ( I I I . 4 ) ( H i . 5 ) &a = Pa y J-3Z ( I I I . 6 ) 3h Ph *2 1 2 I b i d . , p. 383. 5 1 E m p i r i c a l D e v e l o p m e n t o f t h e Laws S c h w a r t z s t a t e s t h a t " t h e method u s e d t o d e v e l o p R e i l l y ' s l a w i s an example o f t h e u s e o f r e g r e s s i o n a n a l y s i s t o d i s c o v e r 13 a s c i e n t i f i c l a w . " J T h i s s t a t e m e n t may n o t be a n e n t i r e l y a c c u r a t e c h a r a c t e r i z a t i o n o f what R e i l l y a c t u a l l y d i d , b u t t h i s w i l l become more a p p a r e n t f r o m t h e f o l l o w i n g d i s c u s s i o n . R e i l l y c h o s e t o i n v e s t i g a t e t h e phenomenon t h a t he c a l l e d " t h e d i v i s i o n o f r e t a i l t r a d e between two c i t i e s " t o d e t e r m i n e what f a c t o r s w o u l d e x p l a i n t h i s phenomenon ( i n a c a u s a l s e n s e ) , a n d what r e l a t i o n s h i p t h e f a c t o r s b o r e t o e a c h o t h e r and t h e phenomenon. P r i m a r i l y , i t a p p e a r s , by u s i n g i n t r o s p e c t i o n ( a n d p e r h a p s a l s o i m p l i c i t r e f e r e n c e t o Newton's Law o f G r a v i t y ) R e i l l y a s c e r t a i n e d t h a t t h e s p e c i f i c f a c t o r s o f p o p u -l a t i o n o f t h e c i t i e s a n d d i s t a n c e f r o m t h e c i t i e s must be t h e d e t e r m i n i n g f a c t o r s . R e i l l y e x p l i c i t l y s t a t e s : I t i s so r e a d i l y a c c e p t a b l e t h a t t h e amount o f o u t s i d e t r a d e w h i c h a c i t y e n j o y s i n any s u r r o u n d i n g town i s a d i r e c t f u n c t i o n o f t h e p o p u l a t i o n o f t h a t c i t y a n d an i n v e r s e f u n c t i o n o f t h e d i s t a n c e o f t h e c i t y f r o m t h a t town, t h a t t h e g e n e r a l l a w n e e d s no s u p p o r t . 1 ^ R e i l l y u s e s d i s t a n c e i n a n i n v e r s e r e l a t i o n s h i p b e c a u s e he i m p l i c i t l y seems t o assume t h e image o f an e c o n o m i c man, a n d S c h w a r t z , op. c i t . , p. 1 6 . I k R e i l l y , op. c i t . , p. 7 1 . The " g e n e r a l law" i s g i v e n b e l o w a s E q u a t i o n ( I I I . 7 ) . 5 2 r e a s o n s t h a t i s a p r i o r i i n c o n v e n i e n t a n d c o s t l y f o r p e o p l e t o t r a v e l t o s h o p . On t h e o t h e r hand, R e i l l y i s n o t n e a r l y so c e r t a i n i n h i s c h o i c e o f p o p u l a t i o n i n a d i r e c t r e l a t i o n s h i p . He a s s e r t s t h a t i t i s n o t m e r e l y t h e f a c t o f a l a r g e c l u s t e r o f p e o p l e a l o n e w h i c h c a u s e s o t h e r p e o p l e t o t r a v e l t o a l a r g e c i t y t o shop, b u t r a t h e r i t i s t h e e x i s t e n c e o f s u c h a t t r a c t i o n s a s l a r g e r e t a i l s t o r e s w i t h a wide v a r i e t y o f g o o d s , s o c i a l a n d amusement a t t r a c t i o n s , a n d a d v e r t i s i n g media i n t h e l a r g e 1 5 c i t y w h i c h c a u s e p e o p l e t o buy t h e r e . ^ B u t R e i l l y a r g u e s , d e d u c t i v e l y , t h a t t h e e x i s t e n c e o f t h e a t t r a c t i o n s w o u l d n o t be e c o n o m i c a l l y f e a s i b l e i n t h e a b s e n c e o f l a r g e r p o p u l a t i o n , e r g o t h e r e must be a d i r e c t r e l a t i o n s h i p between t h e f a c t o r s w h i c h i n d u c e p e o p l e t o t r a v e l t o a c i t y t o shop and t h e s i z e o f 1 6 t h a t c i t y ' s p o p u l a t i o n . R e i l l y c o n c l u d e s t h a t , i n v i e w o f t h i s r e l a t i o n s h i p , i t i s r e a s o n a b l e t o u s e p o p u l a t i o n as a 1 7 " p r o x y " v a r i a b l e f o r t h e s e o t h e r f a c t o r s . I t i s i n t e r e s t i n g t o n o t e t h a t t h e i m p l i c i t model o f t h e s h o p p e r w h i c h R e i l l y R e i l l y , op. c i t . , pp. 7 3 - 7 5 * v e n t u r e s a v e r y l o n g l i s t o f f a c t o r s w h i c h may i n f l u e n c e d i v i s i o n o f r e t a i l t r a d e , and p o p u l a t i o n a n d d i s t a n c e a r e o n l y two among t h e many. 1 6 S u b s e q u e n t s t u d i e s have u s e d o t h e r m e a s u r e s . I n F r a n k S t r o h k a r k a n d K a t h e r i n e P h e l p s , "The M e c h a n i c s o f C o n s t r u c t i n g a M a r k e t A r e a Map." J o u r n a l o f M a r k e t i n g , X I I ( A p r i l , 1 9 k 8 ) , 4 9 5 - 4 9 6 , i t i s c o n c l u d e d t h a t s u b s t i t u t i o n o f " s h o p p i n g l i n e s a l e s " i n p l a c e o f p o p u l a t i o n y i e l d e d more a c c u r a t e r e s u l t s . T h e r e i s a s e r i o u s q u e s t i o n h e r e as t o w h e t h e r t h e r e s u l t i n g s t a t e m e n t i s l a r g e l y t a u t o l o g i c a l , s i n c e i n a c a u s a l s e n s e i t w o u l d s t a t e t h a t s h o p p i n g l i n e s a l e s c a u s e s h o p p i n g l i n e s a l e s . 1 7 R e i l l y , op. c i t . , pp. 3 0 - 3 2 . 53 seems t o u s e i n r e a c h i n g t h i s c o n c l u s i o n i s r a t h e r d i f f e r e n t f r o m t h e e conomic man model o f t h e s h o p p e r who i s i n f l u e n c e d by d i s t a n c e . How t h e s e d i f f e r e n t images o f t h e s h o p p e r a f f e c t t h e o p e r a t i o n o f t h e model i s n o t e v i d e n t . R e i l l y has summed up t h e b a s i s f o r h i s u s e o f p o p u l a t i o n a n d d i s t a n c e a s v a r i a b l e s i n h i s law a s f o l l o w s : ... E v i d e n c e s e c u r e d ... shows t h a t t h e p o p u l a t i o n o f a c i t y a n d t h e " d i s t a n c e f r o m t h a t c i t y t o a n o t h e r c o m p a r a b l e c i t y a r e t h e p r i m a r y f a c t o r s t h a t c o n d i t i o n t h e r e t a i l t r a d e i n f l u e n c e o f t h a t c i t y ; t h a t p o p u -l a t i o n and d i s t a n c e a r e r e l i a b l e i n d e x e s o f t h e b e h a v i o r o f o t h e r f a c t o r s ; t h a t o t h e r f a c t o r s a r e e i t h e r so c l o s e l y r e l a t e d t o , o r so d i r e c t l y d e p e n d e n t upon, t h e s e two p r i m a r y f a c t o r s t h a t t h e e f f e c t s o f t h e d e p e n d e n t f a c t o r s t e n d t o b a l a n c e o u t when c i t i e s a r e compared on t h e b a s i s o f p o p u l a t i o n a n d d i s t a n c e . ^ ' H a v i n g d e d u c e d and s u p p o r t e d h i s p o s i t i o n t h a t t h e s e a r e t h e f a c t o r s a n d t h e d i r e c t i o n o f t h e i r i n f l u e n c e , R e i l l y p u t s t h e r e l a t i o n s h i p i n g e n e r a l f o r m : &a Rz Db" ( I I I . 7 ) 3b Ph~ D*" The e m p i r i c a l phase o f h i s r e s e a r c h t h e n became t h e p r o b l e m o f e v a l u a t i n g t h e e x p o n e n t s N a n d n . R e i l l y d e t e r m i n e d t h a t t h e v a l u e o f s h o u l d be one on t h e b a s i s o f s t u d i e s w h i c h he does n o t p r e s e n t . He w r i t e s : 1 0 I b i d . , pp. 31-32. The " e v i d e n c e s e c u r e d " i s n o t p r e s e n t e d . 5 ^ ... By p e r s o n a l i n v e s t i g a t i o n we have a p p r o x i m a t e d t h e r a t e a t w h i c h o u t s i d e t r a d e drawn, by a c i t y i n c r e a s e s w i t h t h e p o p u l a t i o n o f t h a t c i t y , a n d we have e v i d e n c e t o s u p p o r t the^ u s e o f t h e f i r s t power o f t h e p o p u l a t i o n v a r i a b l e 1 9 R e i l l y t h e n s o l v e d h i s g e n e r a l e q u a t i o n f o r / 7 , a n d s u b s t i t u t e d a p p r o p r i a t e v a l u e s f o r t h e o t h e r v a r i a b l e s . P o p u l a t i o n d a t a were r e l a t i v e l y e a s i l y o b t a i n e d . R e i l l y c h o s e t o e v a l u a t e n o n l y f o r t h o s e i n t e r m e d i a t e towns w h i c h were b r e a k i n g p o i n t s between any two p a i r o f t r a d i n g c e n t r e s . T h u s , Ba /E>b = 1 , and R e i l l y n e e d e d o n l y t o d e t e r m i n e w h i c h towns were b r e a k i n g p o i n t s . T h i s w o u l d a l s o g i v e h i m t h e n e c e s s a r y d i s t a n c e s . R e i l l y ' s d i s c u s s i o n o f how t h i s was h a n d l e d i s i n t e r e s t i n g : We d r o v e a l o n g t h e main a u t o m o b i l e h i g h w a y s c o n n e c t i n g l a r g e r c i t i e s a n d c a l l e d u pon t h e s e c r e t a r y o f t h e R e t a i l C r e d i t Men's A s s o c i a t i o n i n i n t e r m e d i a t e c i t i e s a n d towns. On t h e b a s i s o f t h e r e c o r d s o f c r e d i t i n q u i r i e s k e p t i n t h e s e o f f i c e s , we were a b l e t o f i n d t h a t p o i n t a t w h i c h t h e p r e p o n d e r a n c e o f r e t a i l t r a d e c e a s e d t o f l o w i n t h e d i r e c t i o n o f t h e c i t y we had l e f t a n d began t o f l o w i n t h e d i r e c t i o n o f t h e c i t y we were a p p r o a c h i n g . 2 0 When t h e b r e a k i n g p o i n t town was d e t e r m i n e d i n t h i s manner, R e i l l y c o u l d a s c e r t a i n t h e d i s t a n c e s by c o n s u l t i n g a highway map. I n t h i s manner, / i was e v a l u a t e d i n 2 2 5 c a s e s , a n d t h e f r e q u e n c y d i s t r i b u t i o n o f t h e r e s u l t s i s p r e s e n t e d i n T a b l e I a n d F i g u r e 1 . From t h i s d i s t r i b u t i o n R e i l l y c o n c l u d e d : 1 9 I b i d . , pp. 7 1 - 7 2 . 2 0 I b i d . , p. 64. 55 TABLE I E v a l u a t i o n o f D i s t a n c e E x p o n e n t  i n R e i l l y ' s Law V a l u e o f 'No. o f C a s e s # -of T o t a l 0.00 - 1.5 45 20.0 1.51 - 2.5 87 38.8 2.51 - 3.5 35 15.6 4.51 - 5.5 15 6.7 5.51 - 6.5 14 6.2 6.51 - 7.5 6 2.7 7.51 - 8.5 5 2.2 8.51 - 9 .5 12 .. 5.3. 9.51 -10.5 5 2.2 10.51 -11.5 3 1.3 11.51 -12.5 4 1.8 T o t a l 225 - These d a t a computed f o r p l o t t i n g p u r p o s e s : See F i g u r e 1. S o u r c e : R e i l l y , Methods f o r t h e S t u d y o f R e t a i l R e l a t i o n s h i p s ( A u s t i n : U n i v e r s i t y o f Texas P r e s s , 1929) , P- 4 9 . 57 ... a c l e a r mode o c c u r s i n t h e r a n g e o f 1.51 - 2.50 w h i c h shows t h a t t h e e x p o n e n t o f d i s t a n c e i s n e a r e r t o t h e s e c o n d power t h a n t o any o t h e r e v e n p o w e r . 2 ^ O v e r t h e i n t e r v e n i n g y e a r s t h e f a c t t h a t t h e r e was a d i s t r i b u t i o n i n t h e v a l u e s o f t h e e x p o n e n t , an d t h a t t h e v a l u e c h o s e n was a modal o r maximum l i k e l i h o o d e s t i m a t e , h a s been l o s t f r o m s i g h t . The r e s u l t i s t h a t s t u d e n t s who e n c o u n t e r R e i l l y ' s law — o f t e n i n a f i r s t c o u r s e i n m a r k e t i n g — g a i n t h e i m p r e s s i o n t h a t i t i s much more p r e c i s e a n d r i g o r -ous t h a n has b e e n shown. T h i s i s u n f o r t u n a t e , s i n c e many s t u d e n t s t h e n c a n n o t u n d e r s t a n d why o t h e r s i m i l a r l y w e l l -d e f i n e d r e l a t i o n s h i p s h a v e n o t b e en u n c o v e r e d i n m a r k e t i n g . T e s t s o f t h e Law o f R e t a i l G r a v i t a t i o n R e i l l y u n d e r t o o k t o t e s t h i s l a w by c o m p u t i n g t h e l o c a t i o n o f t h e b r e a k i n g p o i n t between e a c h p a i r o f c i t i e s i n a s e t o f t h i r t y p a i r s o f t r a d i n g c e n t r e s by means o f h i s e q u a t i o n . He t h e n c o n d u c t e d f i e l d s t u d i e s t o d e t e r m i n e t h e a c t u a l l o c a t i o n o f t h e b r e a k i n g p o i n t towns, an d compared t h e r e s u l t s . The c o m p a r i s o n w i l l n o t be p r e s e n t e d h e r e b e c a u s e o f i t s l e n g t h , b u t R e i l l y c o n c l u d e d t h a t h i s l a w was 22 " s t a r t l i n g l y " a c c u r a t e . T h e r e i s no d o u b t i n e x a m i n i n g t h e c o m p a r i s o n t h a t t h e a g r e e m e n t between t h e p r e d i c t e d a n d a c t u a l r e s u l t s i s v e r y c l o s e . I t w o u l d have been i n t e r e s t i n g W i l l i a m J . R e i l l y , Methods f o r t h e S t u d y o f R e t a i l R e l a t i o n -s h i p s ( A u s t i n : U n i v e r s i t y o f T e x a s P r e s s , 1929), p. 50. 2 2 R e i l l y , The Law o f R e t a i l G r a v i t a t i o n , pp. 25-29. 58 i f R e i l l y had also worked backward from his empirical r e s u l t s and calculated the value of n for these t h i r t y pairs of trading centres. Undoubtedly the values would be close to two. In 19 k 3 , Converse published a study i n which he reported the results of his e f f o r t s to check the accuracy of 23 R e i l l y ' s law. His study compared the predictions of R e i l l y ' s law with the movement of r e t a i l trade ascertained through consumer surveys wherein he determined how many families i n the towns studied made th e i r shopping goods purchases i n nearby primary trading centres. Using R e i l l y ' s law i n i t s o r i g i n a l form, Converse predicted the r e l a t i v e d i v i s i o n of r e t a i l trade for thirteen intermediate towns l y i n g between Champaign-Urbana and f i v e other primary trading centres. From the survey data he computed the percentages of consumers who bought shopping goods i n Champaign-Urbana and the competing trading centres. He then calculated the c o e f f i c i e n t of co r r e l a t i o n between the predictions and the empirical r e s u l t s (least squares, r = 0 . 8 8 , r 2 = 0 . ? 7 ) , and concluded that, "on the whole, i t r "I 1 2 4 R e i l l y ' s j l a w j works with a r e l a t i v e l y high degree of accuracy. Converse l a t e r reported another study i n the same geo-graphical area involving the d i v i s i o n of fashion goods purchases. In t h i s study, the d o l l a r amounts spent on fashion goods by the inhabitants of each town i n the nearby trading J Converse, A Study of R e t a i l Trade Areas i n East Central I l l i n o i s , pp. 2 3 - 5 k . 2 4 Ibid., p. 48. 59 centres were estimated. Prom these, the r e l a t i v e d i v i s i o n was c a l c u l a t e d and compared wi t h p r e d i c t i o n s based on R e i l l y ' s law. In t h i s study, the agreement was c l o s e r than before ( l e a s t squares, r = 0 . 9 3 , r 2 = 0 . 8 6 ) . 2 ^ Even so, Converse f e l t i t necessary to q u a l i f y and e x p l a i n the high degree of agree-ment i n a note of c a u t i o n : I t should not, however, be concluded that the law w i l l measure the movement of trade w i t h such accuracy i n a l l t e r r i t o r i e s . The towns i n c l u d e d i n the computations used here are i n a t e r r i t o r y i n which the primary t r a d i n g centers are consider-ably l a r g e r than the intermediate towns. In areas where there i s l e s s d i f f e r e n c e i n s i z e between the primary and secondary t r a d i n g centers or between the t r a d i n g centers and the towns from which they draw trade, the law of r e t a i l g r a v i t a t i o n may perhaps not p r e d i c t the movement of trade with the accuracy found i n the t e r r i t o r y here studied.2° More r e c e n t l y , another t e s t of the law of r e t a i l g r a v i -27 t a t i o n was conducted by Reynolds. Reynolds used the same equation as R e i l l y d i d to determine the value o f n , the exponent of the di s t a n c e f a c t o r . While R e i l l y had concluded that t h i s exponent should be equal to two on the b a s i s of h i s e m p i r i c a l data, Reynolds sought to f i n d out i f other e m p i r i c a l data would support R e i l l y ' s c o n c l u s i o n . Reynolds used data c o l l e c t e d i n 1935 f ° r ninety-one Iowa c o u n t i e s , and data 2 5 Converse, R e t a i l Trade Areas i n I l l i n o i s , pp. 13-18. 2 6 I b i d . , p. 18. 2 7 Robert B. Reynolds, "A Test of the Law of R e t a i l G r a v i -t a t i o n , " J o u r n a l of Marketing, XVII (January, 1953"), 273-277. 60 c o l l e c t e d i n 1949 f o r southwest Iowa. From these, he computed r e g r e s s i o n equations f o r g r o c e r i e s , eggs and p o u l t r y , movies, farm machinery, lumber and cement, p h y s i c i a n s 1 s e r v i c e s , women's coats and dresses, men's good shoes, and men's s u i t s . The data which Reynolds used described the t r a d i n g areas of s p e c i f i c t r a d i n g centres f o r each of the products mentioned. The study i t s e l f i s described as f o l l o w s : From ... [the ... p o i n t s , where t r a d i n g area boundaries crossed roads, highway distances to the nearest mile ... were measured to the two t r a d i n g centers i n question and the populations of the centers ... were recorded to the nearest hundred. Two values each, D ( D i / D 2 and D 2 / D 1 ) and P ( P 1 / P 2 and P 2 / P 1 ) then were computed f o r every breaking p o i n t . ... The problem r e s o l v e d i t s e l f to f i n d i n g the value of the constant, b, i n the f o l l o w i n g l i n e a r equation, ... : l o g D = b l o g P. This b i s the exponent of P i n Converse's formula, where i t i s . 5 0 , or the square root.2° In a d d i t i o n to c a l c u l a t i n g a b value f o r each of the commodities he s t u d i e d , Reynolds a l s o computed c o e f f i c i e n t s of determination f o r these products. The r e s u l t s are shown i n Table I I , which i n c l u d e s a l s o the conversion of b values back to t h e i r equivalent n values. These n values are cumulated i n Table I I I , and the r e l a t i v e frequency i s p l o t t e d i n Figure 2 . These can be compared with Table I I I and Figure 3 . I t i s granted that the number i n the sample i s very s m a l l , Note that only some of these commodities are shopping goods; namely, women's coats and dresses, men's good shoes, and men's s u i t s . 29 Reynolds, op. c i t . , p. 275 • 61 TABLE I I R e y n o l d s ' T e s t o f R e i l l y ' s Law P r o d u c t 6 1935 6 1949 a G r o c e r i e s • 31 3 . 2 4 .68 . 4 6 2.17 • 71 Eggs and P o u l t r y , M a r k e t i n g o f .32 3.12 .68 . 1 7 b 5.39 . 1 2 b M o v i e s • 31 3.24 .59 .38 2.67 . 4 8 Lumber and Cement .22 4.54 .58 n. s. - n. s. Farm M a c h i n e r y - n. s. - n . s. • 51 1.96 .63 P h y s i c i a n .22 4 .54 .43 .27 3.70 . 2 4 Women's C o a t s and D r e s s e s .44 2.27 • 79 . 5 1 ° 1.96 .90° Men's S u i t s .45 2.22 .76 n.n. - n. s. Men's Good Shoes n. s. — n. s. • 50 2.00 • 75 S o u t h w e s t Iowa o n l y b - Eggs o n l y c - Women's C o a t s o n l y n . s . n o t s t u d i e d S o u r c e : R o b e r t B. R e y n o l d s , "A T e s t o f t h e Law o f R e t a i l G r a v i t a t i o n , " J o u r n a l o f M a r k e t i n g V o l . 17 ( J a n u a r y , 1953), pp. 275-276. TABLE I I I Reynolds' Test of R e i l l y ' s Law - E v a l u a t i o n of Distance Exponent Value of No. of Cases % of T o t a l 0.00 - 1.5 0 0 1 . 5 1 - 2 . 5 6 42.8 2 .51 - 3.5 4 28 .6 3.51 - 4 . 5 . 1 7-2 4 .51 - 5.5 2 14 . 3 5.51 - 6 .5 1 7.2 To t a l 14 63 64 ... i. •vi-i ! * 1 ! • f t - !" I j . 1 . , . .1 . . ! . i i ( -i I 1 -I 1 •f i * —*-i \~\ [ f l f -I 1.1 1 - r - i ~ | - 4 }! - r - - | >--t— •1 -t r r i r 1--T 0. •! 1 ft-T : i- 1 i -I •!-- 4 -I 11: ' T 'T i -i i- 1 • 111 t ! t -•i , -i-1 !'! 1 r !• •r i r ••H i -i-;~t - V 1 4 4 4 -CO CD III IN': J i • 4 ^ j 5 I-.) ;s xx... x.c*:. l-i-i-» -t--i i f ? .iii] ! h f t - -1 4 * • 1 1 -:. \J \? , :o •t r .! ! O G o ° ! ! • t t I i ! 7 8. 7 H T I I I ! t t - ! - t -I t * ! ' ! -'bi.i: . I'l.'l-- ( - I -, . . . . 11 t- - • mi • i r - (• i 1 ! i •!:F 1 i -f r © i 11 > i ; . O i l !... i : i.; j ; i : i.i i I H I I . . . . . . I ! I I ! f -I i ; f -t i i ! . O.. r -i Ti I i 4 t i-r-f '! t I.I. t i r UXJ.. i I: -J-J-I 4 • ] i ' 1 1 1 i r l ! • 'I T tl 1 i ! -f r •f I I T •| CO; 110-1 ,.i„ 4-.... f .-4 I'!-.1.. ! H • f i ::l:Li 111 0 T c ] 8-1-:i4 i j .{... 4 f-i t . 4 ! - p ~ t - t -r -I -I l-t' J - I... .14,.. •! I ••4 +-Vl I I .1j . | „ .i j . . i n '(".r"f"f' I : r i -II l-t hi : , 1 . , . r i n : l . l . . m < i i r i-i M 4-: o: ' J J "t. 44-i ; i 7 . 1 -! J ! 1 f - i -i i i ! I i l' J J.!. 44' "i r r -r ... n I . f j -i- ! ! 1 i 1 t ! 1 L H - 1 -hi-; ; 4 i 4 I •t -i -• - i • \ 1 ; • 'i i • 'i \ri~ t . i _ !..| J !-I -i i-, i . .1 I I • ! .! " I i i - } H H-• t ;- r i :• 1 I- ; 1- I - i . •j I • i .j-: r r f i i i ; ; t i ' i ' , U - i ~ - j ~ . i n T 7 r r • t 4 i •I; I J -j 14 i i i X i X i -» 1 14 i " 11. r; i t ' r X J n Ti'TT' II- i i j - i i I I:ji7 4 i 4 1? I • JT^'TT ! I ' 4 4 ! 111!' .1., :. 1 }•;•: i J t X 1 " .nr. i -: 4 l.r.i; i •p 65 b u t e v e n s o , t h e r e i s a d i s t i n c t r e s e m b l a n c e t o R e i l l y ' s d a t a shown i n t h i s way. Shown t h e s e r e s u l t s , R e i l l y w o u l d u n d o u b t e d -l y s u g g e s t t h a t t h e y s u p p o r t h i s o r i g i n a l c o n t e n t i o n . R e y n o l d s t r e a t e d t h e r e s u l t s d i f f e r e n t l y , a n d t e s t e d t h e s i g n i f i c a n c e o f t h e d i f f e r e n c e between e a c h b v a l u e and 0 . 5 0 . He f o u n d t h a t f o r 1935 t h e d i f f e r e n c e s were s i g n i f i c a n t a n d c o n c l u d e d t h a t t h e s e r e s u l t s were e v i d e n c e w a r r a n t i n g r e j e c t i o n o f t h e b r e a k i n g p o i n t e q u a t i o n ( E q u a t i o n I I I . 2 ) . S i m i l a r t e s t s p e r f o r m e d f o r t h e 19^9 r e s u l t s p r o d u c e d s i g n i f i c a n t d i f f e r e n c e s f o r t h e p r o d u c t s s t u d i e d , e x c e p t f o r t h e s h o p p i n g goods i t e m s . S i n c e t h e model was o r i g i n a l l y f o r m u l a t e d f o r s h o p p i n g g o o d s , i t w o u l d n o t seem t h a t R e y n o l d s ' i n v e s t i g a t i o n p r o d u c e d any e v i d e n c e on w h i c h i t c o u l d be d e c i d e d t h a t t h e model s h o u l d be r e j e c t e d . R e y n o l d s c a r r i e d h i s i n v e s t i g a t i o n a l i t t l e f u r t h e r , a n d computed c o e f f i c i e n t s o f d e t e r m i n a t i o n i n o r d e r t o f i n d t h e e x t e n t t o w h i c h t h e v a r i a t i o n i n t h e l o c a t i o n o f t h e b r e a k i n g p o i n t towns between t r a d i n g c e n t r e s was e x p l a i n e d by t h e m o d e l . T h e s e c o e f f i c i e n t s a r e shown i n T a b l e I I . F o r t h e s h o p p i n g g oods s t u d i e d , t h e s e c o e f f i c i e n t s f a l l w i t h i n t h e r a n g e 0.75 t o 0 . 9 0 . However, o v e r t h e whole sample t h e r a n g e i s c o n s i d e r a b l y w i d e r , a n d t h e r e seemed t o be some r e l a t i o n s h i p b etween t h e e x p o n e n t n and t h e c o e f f i c i e n t F o r t h i s r e a s o n , a s c a t t e r d i a g r a m has b een p l o t t e d , a n d i s g i v e n a s F i g u r e 3. W h i l e no a t t e m p t h a s b e e n made t o f i t a c u r v e t o t h e s e d a t a , i t c a n be s e e n by i n s p e c t i o n t h a t t h e r e i s a d e f i n i t e t e n d e n c y t o r a p i d l y d e c l i n i n g r-* v a l u e s w i t h i n c r e a s -66 i n g v a l u e s . I n words, t h i s means t h a t t h e power o f a c c o u n t o f t h e model — measured by t h e c o e f f i c i e n t o f d e t e r m i n a t i o n — d e c r e a s e s w i t h i n c r e a s i n g v a l u e s o f t h e e x p o n e n t . Whether t h i s i s s i g n i f i c a n t , o r l a r g e l y t a u t o l o g i c a l , i s d i f f i c u l t t o d e t e r m i n e . F u r t h e r , s i n c e R e y n o l d s ' d a t a a r e n o t d i r e c t l y c o m p a r a b l e t o R e i l l y ' s o r C o n v e r s e ' s , i t seems t h a t t h e r e s u l t s , a t b e s t , a r e i n c o n c l u s i v e i n t e s t i n g t h e b r e a k i n g p o i n t r e l a t i o n s h i p . A f t e r p u b l i c a t i o n o f R e y n o l d s ' s t u d y , t h e r e was a n e xchange o f v i e w s w i t h C o n v e r s e w h i c h u n f o r t u n a t e l y g e n e r a t e d 30 much h e a t a n d l i t t l e l i g h t . Thus, t h e s i t u a t i o n r e m a i n e d u n c h a n g e d . R e l a t i v e l y r e c e n t l y , J u n g p r e s e n t e d some d a t a w h i c h , he 31 m a i n t a i n s , c o n t r a d i c t R e i l l y ' s l a w . J u n g w r o t e t h a t a s t u d y o f b u y i n g h a b i t s o f t h e r e s i d e n t s o f C o l u m b i a , M i s s o u r i showed no p r e f e r e n c e f o r b u y i n g a t s t o r e s i n S t . L o u i s r a t h e r t h a n K a n s a s C i t y . A c c o r d i n g t o R e i l l y ' s law, however, i t w o u l d be e x p e c t e d t h a t C o l u m b i a r e s i d e n t s w o u l d have made a s u b s t a n -t i a l l y l a r g e r p r o p o r t i o n o f t h e i r p u r c h a s e s i n S t . L o u i s . R e i l l y h a s c o u n t e r e d t h i s r e s u l t by a r g u i n g t h a t J u n g h a s n o t i n v a l i d a t e d h i s g e n e r a l l a w . The most t h a t h a s b een done, a c c o r d i n g t o R e i l l y , i s t o d i s p u t e t h e judgment t h a t D P.D. C o n v e r s e , "Comment on Movement o f R e t a i l T r a d e i n Iowa," an d R.B. R e y n o l d s , " R e j o i n d e r t o C o n v e r s e ' s Comment on R e y n o l d s ' T e s t o f t h e Law o f R e t a i l G r a v i t a t i o n , " J o u r n a l  o f M a r k e t i n g , X V I I I ( O c t o b e r , 1953). 31 A . F . J u n g , " I s R e i l l y ' s Law o f R e t a i l G r a v i t a t i o n A l w a y s T r u e ? " J o u r n a l o f M a r k e t i n g , XXIV ( O c t o b e r , 1959), 6 2 - 6 3 . 67 N = 1 and n = 2. This immediately r a i s e s the question of what e x a c t l y i s R e i l l y ' s law? Common usage, and decreasing frequency of reference to the o r i g i n a l work, has produced the b e l i e f that the law i s the p a r t i c u l a r equation i n which N = 1 and n = 2. I t would appear that Jung might not have consulted the o r i g i n a l e m p i r i c a l work of R e i l l y , nor noted the q u a l i f i -c a t i o n s which Converse a p p l i e d , both of which have been discussed above. Summary The s p e c i f i c h i s t o r y of g r a v i m e t r i c models i n marketing evolves from the bas i c law of r e t a i l g r a v i t a t i o n s t a t e d by R e i l l y . Although R e i l l y , h i m s e l f , made no mention of Newton's Law of G r a v i t a t i o n , nevertheless the s i m i l a r i t y e x i s t s . As was mentioned e a r l i e r ( i n Chapter I I ) , Stewart has w r i t t e n t h a t R e i l l y ' s law can be derived from Newton's Law, and f u r t h e r t h a t i t represents the f i r s t r e c o g n i t i o n of demographic g r a v i -33 t a t i o n . In h i s s e v e r a l s t u d i e s , R e i l l y attempted to f i n d methods f o r measuring the r e t a i l trade i n f l u e n c e of a c i t y . - Proceeding i n a d e d u c t i v e l y a n a l y t i c a l manner, he concluded t h a t distance and population (as a proxy v a r i a b l e f o r a range of a t t r a c t i v e Schwartz, op. c i t . , p. 28, rep o r t s that R e i l l y s t a t e d t h i s p o s i t i o n at the I l l i n o i s Symposium i n October, 1959• 33 John Q. Stewart, "Demographic G r a v i t a t i o n : Evidence and A p p l i c a t i o n , " Sociometry, XI (February-May, 1 9 k8), 35. The survey given i n Chapter I I shows that Stewart was not s t r i c t l y c o r r e c t i n h i s statement. 68 i n f l u e n c e s ) were t h e m a j o r d e t e r m i n a n t s . He t h e n , i n e f f e c t , h y p o t h e s i z e d t h e g e n e r a l f o r m o f t h e r e l a t i o n s h i p i n v o l v i n g t h e v a r i a b l e s , a n d p r o c e e d e d t o e v a l u a t e h i s p a r a m e t e r s e m p i r i c a l l y . On t h i s b a s i s , he c o n c l u d e d i t s h o u l d have t h e f o r m o f E q u a t i o n ( I I I . l ) a b o v e . R e i l l y t h e n t e s t e d t h e r e s u l t i n g f o r m a g a i n s t a n o t h e r , s m a l l e r sample a n d f o u n d 34 r e m a r k a b l y good a g r e e m e n t . As h a s been d i s c u s s e d , o t h e r s h a v e d e v e l o p e d e v i d e n c e w h i c h d o e s n o t a g r e e so c l o s e l y , b u t i n v i e w o f t h e o r i g i n a l method o f d e v e l o p i n g t h e law, t h i s c a n n o t be t a k e n a s s u f f i c i e n t e v i d e n c e w a r r a n t i n g r e j e c t i o n . C o n v e r s e ' s r e f o r m u l a t i o n s a r e a n o t h e r i s s u e , a s t h e law was r e w o r k e d f o r a p p l i c a t i o n i n r a t h e r d i f f e r e n t c i r c u m s t a n c e s . J S c h w a r t z , op. c i t . , p . 31, h a s computed t h e c o e f f i c i e n t o f c o r r e l a t i o n between p r e d i c t e d a n d a c t u a l d i s t a n c e s i n t h i s s ample t o be + 0.98, b u t p o i n t s o u t a v e r y s l i g h t upward b i a s . CHAPTER IV A PROBABILISTIC MODEL OF INTRA-URBAN TRADE INTERACTIONS The g r a v i m e t r i c models d i s c u s s e d i n t h e p r e c e d i n g c h a p t e r s have r e v e a l e d one i m p o r t a n t c h a r a c t e r i s t i c i n common; t h e y a r e a l l m a c r o - m o d e l s . The human i n t e r a c t i o n s s t u d i e d a r e t r e a t e d i n a h i g h l y a g g r e g a t e d b a s i s , a t t h e n a t i o n a l , r e g i o n a l , o r i n t e r - u r b a n l e v e l . M a r k e t i n g , however, a l s o o p e r a t e s a t t h e m i c r o l e v e l , t h e l e v e l o f t h e f i r m s and consumers where d e c i s i o n s a r e made. S i n c e t h e t i m e o f t h e i n t r o d u c t i o n o f R e i l l y ' s l a w i n t o t h e m a r k e t i n g l i t e r a t u r e , t h e r e have b e e n some i m p o r t a n t c h a n g e s i n t h e i n s t i t u t i o n a l s t r u c t u r e o f m a r k e t i n g w h i c h a f f e c t b o t h g r o u p s o f d e c i s i o n makers a n d t h e i r r e l a t i o n s h i p t o e a c h o t h e r . I n many l a r g e c i t i e s , t h e n o t i o n o f a s i n g l e downtown s h o p p i n g c o r e , a s c o n s i d e r e d by c e n t r a l p l a c e t h e o r y , h a s b e e n s u p e r s e d e d by a n o t i o n o f many a l t e r n a t e c e n t r e s o f r e t a i l i n g a c t i v i t y w i t h t h e r i s e o f t h e p l a n n e d s h o p p i n g c e n t r e a s a new f o r m o f r e t a i l i n s t i t u t i o n . Two o f t h e most i m p o r t a n t q u e s t i o n s w h i c h may be a s k e d by t h o s e i n v e s t i n g l a r g e sums o f money i n p r o m o t i n g and d e v e l o p i n g s u c h c e n t r e s w o u l d be i n r e g a r d t o where t h e c e n t r e s h o u l d be l o c a t e d , a n d how l a r g e t h e p h y s i c a l p l a n t s h o u l d be. I n a t t e m p t i n g t o answer s u c h q u e s t i o n s , R e i l l y ' s l a w h a s been a p p l i e d , w i t h m o d i f i c a t i o n s , t o e s t i m a t e t r a d i n g a r e a s o f p r o p o s e d s h o p p i n g c e n t r e s w i t h i n c i t i e s ( i n t r a - u r b a n ) . F o r 70 example, g i v e n a p r o p o s e d s i t e f o r a s h o p p i n g c e n t r e , t h e number, p r o x i m i t y , and s i z e o f c o m p e t i n g c e n t r e s c a n be d e t e r -m i n e d . U s i n g t h e m o d i f i e d g r a v i t y model, t h e b r e a k i n g p o i n t s f r o m t h e p r o p o s e d c e n t r e t o e a c h o f t h e e x i s t i n g c e n t r e s may be c a l c u l a t e d , a n d t h e r e s u l t s mapped t o d e l i n e a t e t h e p o t e n t i a l t r a d i n g a r e a o f t h e p r o p o s e d u n i t . S u c h a d e l i n e a t i o n c o u l d be i m p o s e d o v e r a c e n s u s t r a c t map t o d e t e r m i n e t h e number o f p e r s o n s w i t h i n t h e t r a d i n g a r e a . T h i s i n f o r m a t i o n may t h e n be u s e d a s a b a s i s f o r d e t e r m i n i n g t h e p u r c h a s i n g power p o t e n t i a l o f t h e t r a d i n g a r e a , a n d t h u s t h e p o t e n t i a l s a l e s a n d p r o f i t a -b i l i t y o f t h e p r o p o s e d s h o p p i n g c e n t r e . The same d e t e r m i n a t i o n may be made f o r o t h e r a l t e r n a t i v e l o c a t i o n s , a n d a c e r t a i n amount o f i n f o r m a t i o n may be d e v e l o p e d f o r c o m p a r i n g a l l t h e a l t e r n a t i v e s . The m o d i f i c a t i o n s made t o R e i l l y ' s law f o r i n t r a - u r b a n a p p l i c a t i o n s i n v o l v e c h a n g e s t o t h e f a c t o r s c h o s e n a s m e a s u r e s o f a t t r a c t i v e a n d d e t r a c t i v e f o r c e s . I n one m o d i f i c a t i o n , d e v e l o p e d by t h e C u r t i s P u b l i s h i n g Company, t h e f a c t o r s c h o s e n h a v e been t h e s q u a r e f o o t a g e o f e a c h r e t a i l c e n t r e i n p l a c e o f p o p u l a t i o n , a n d t r a v e l t i m e between r e t a i l c e n t r e s i n p l a c e o f 1 p h y s i c a l d i s t a n c e . D e d u c t i v e r e a s o n i n g may be u s e d t o s u p p o r t t h e c h o i c e o f t h e s e f a c t o r s i n t h e same manner a s R e i l l y a r g u e d f o r h i s o r i g i n a l s e l e c t i o n o f p o p u l a t i o n f o r h i s m o d e l . As i n R e i l l y ' s L e o n W. E l l w o o d , " E s t i m a t i n g P o t e n t i a l Volume o f P r o p o s e d S h o p p i n g C e n t r e s , " The A p p r a i s a l J o u r n a l ( O c t o b e r , 1954), 581-584. 71 m o d e l , t h e a r g u m e n t s , i n r e f e r e n c e t o s h o p p i n g goods o n l y , a r e d e v e l o p e d f r o m i n t r o s p e c t i o n a n d f r o m t h e r e s u l t s o f e m p i r i c a l r e s e a r c h . W i t h r e g a r d t o t h e s e l e c t i o n o f s h o p p i n g c e n t r e a r e a ( s q u a r e f o o t a g e ) a s a measure o f t h e a t t r a c t i v e power, i t may be a r g u e d t h a t i t r e a l l y i s t h e number o f i t e m s o f t h e g e n e r a l c l a s s w h i c h t h e consumer may d e s i r e t h a t a r e c a r r i e d by t h e v a r i o u s s h o p p i n g c e n t r e s t h a t i s a t t r a c t i v e . T h a t i s , i t i s t h e b r e a d t h a n d d e p t h o f t h e p r o d u c t a s s o r t m e n t o f a p a r t i c u l a r s h o p p i n g good a t t h e p a r t i c u l a r s h o p p i n g c e n t r e , m e asured i n a s u i t a b l e way, w h i c h i s t h e a t t r a c t i o n . The consumer d o e s n o t known i n a d v a n c e , g e n e r a l l y , w h e t h e r he ( o r she) may be a b l e t o o b t a i n a p a r t i c u l a r s h o p p i n g goods i t e m a t a p a r t i c u l a r c e n t r e , b u t may have f o r m e d some s o r t o f a p r i o r i o p i n i o n r e -g a r d i n g t h e r e l a t i v e p r o b a b i l i t y t h a t s u c c e s s i n o b t a i n i n g t h e 2 d e s i r e d i t e m may be a c h i e v e d i n one o r more p l a c e s . P r e s u m a b l y , t h e g r e a t e r t h e number o f i t e m s c a r r i e d by a p a r t i c u l a r s h o p p i n g c e n t r e , t h e g r e a t e r i s t h e consumer's e x p e c t a t i o n t h a t 3 a s h o p p i n g e x p e d i t i o n t o t h a t c e n t r e w i l l be s u c c e s s f u l . I t i s t h u s h y p o t h e s i z e d t h a t t h e d e g r e e o f e x p e c t a t i o n , o r r e l a -t i v e s u b j e c t i v e p r o b a b i l i t y i s d i r e c t l y p r o p o r t i o n a l t o t h e number o f i t e m s . However, i t i s n o t e a s y t o measure t h e number o f i t e m s , p a r t i c u l a r l y I n a way w h i c h r e p r e s e n t s t h e c onsumer's v i e w o f i t . T h u s , i t i s f u r t h e r h y p o t h e s i z e d t h a t t h i s i s The n o t i o n o f p r o b a b i l i t y i s h i g h l y s u b j e c t i v e i n t h i s c a s e . 3 W i l l i a m J . Baumol an d Edward A. I d e , " V a r i e t y i n R e t a i l i n g , " Management S c i e n c e , I I I ( O c t o b e r , 1956), 93-101. 72 m e a s u r e d ( o r a p p r o x i m a t e d a t l e a s t ) by u s i n g t h e s q u a r e f o o t a g e 4 o f s e l l i n g s p a c e , a g a i n i n d i r e c t p r o p o r t i o n . S i m i l a r l y , i t may be a r g u e d t h a t t h e consumer's p e r c e p t i o n o f t h e e f f o r t a n d e x p e n s e i n v o l v e d i n t r a v e l l i n g t o v a r i o u s s h o p p i n g c e n t r e s m o d i f i e s t h e s i t u a t i o n . The c o s t s o f t r a n s p o r t a t i o n , t h e e f f o r t i n making t h e t r i p ( a f o r m o f i n e r t i a ) , a n d o t h e r o p p o r t u n i t i e s f o r e g o n e , a l l t e n d t o d e t r a c t f r o m t h e consumer's p e r c e p t i o n o f a s h o p p i n g c e n t r e . I t i s n o t r e a l l y p o s s i b l e t o speak o f q u i t e t h e same p r o b a b i l i t y o r e x p e c t a t i o n a s ab o v e a f t e r d i s t a n c e a n d c o s t a r e i n t r o d u c e d . R a t h e r , t h e n o t i o n o f d i s t a n c e a n d c o s t r e q u i r e s t h e p r o b a b i l i t y t o be r e g a r d e d a s c o n d i t i o n a l , o r b e t t e r s t i l l t o be r e g a r d e d a s b e i n g m o d i f i e d by a measure o f l i k e l i h o o d r e l a t e d t o t h e d i s t a n c e i n v o l v e d ( a s i n B a y e s i a n s t a t i s t i c s ) . I t i s i m p l i c i t l y h y p o t h e s i z e d t h a t t h e l i k e l i h o o d measure s h o u l d be m u l t i p l i c a t i v e w i t h t h e o r i g i n a l p r o b a b i l i t y . The r e s u l t i s a measure o f t h e p r o b a b i l i t y t h a t t h e consumer w o u l d go f r o m h i s p a r t i c u l a r s t a r t i n g p o i n t t o t h e s t o r e i n q u e s t i o n . I t c a n be a r g u e d t h a t when a s t o r e i s c l o s e by, t h e e f f e c t o f d i s t a n c e on t h e consumer's e x p e c t a t i o n i s v e r y s m a l l . A s d i s t a n c e i n c r e a s e s , however, i t i s h y p o t h e s i z e d t h a t t h e n e t p r o b a b i l i t y w i l l d e c r e a s e r a p i d l y , a n d t e n d t o a p p r o a c h z e r o , b u t n e v e r q u i t e r e a c h i t . T h i s t y p e o f r e l a t i o n s h i p i s d e s -S q u a r e f o o t a g e may be some s o r t o f t o t a l , o r i t may be f u r t h e r r e f i n e d i n t o t h a t d e v o t e d t o t h e s a l e o f p a r t i c u l a r s h o p p i n g goods i t e m s . 73 c r i b e d by a d e c l i n i n g e x p o n e n t i a l f u n c t i o n , a n d t h e i n v e r s e d i s t a n c e - f a c t o r i n R e i l l y ' s l a w may be assumed t o be a s u i t a b l e f i r s t a p p r o x i m a t i o n . On t h e o t h e r hand, i t i s n o t t h e p h y s i c a l d i s t a n c e w h i c h i s r e l e v a n t i n t h e c o n t e m p o r a r y s c e n e , b u t r a t h e r d i s t a n c e a s measured i n u n i t s o f t r a v e l t i m e by a d i r e c t r o u t e . T h i s f o l l o w s c u r r e n t common u s a g e , and i s p r o b a b l y a g o o d measure o f t h e consumer's p e r c e p t i o n o f d i s t a n c e . ~* I t seems t o be r e a s o n a b l y l o g i c a l , i n a d e d u c t i v e s e n s e , t o s u b s t i t u t e s t o r e a r e a a n d t r a v e l t i m e i n t o t h e e q u a t i o n o f R e i l l y ' s l a w f o r u s e i n t h e i n t r a - u r b a n s i t u a t i o n . The b r e a k i n g p o i n t i s t h e n t h e p o i n t a t w h i c h t h e n e t p r o b a b i l i t i e s o f co n s u m e r s g o i n g t o one o r t h e o t h e r o f a p a i r o f c e n t r e s a r e e q u a l . Knowing t h e s t o r e a r e a s , a n d t h e t r a v e l t i m e f r o m one c e n t r e t o t h e o t h e r , t h e b r e a k i n g p o i n t c a n be d e t e r m i n e d a s b e f o r e , a n d t h e t r a d i n g a r e a p r o g r e s s i v e l y d e l i n e a t e d . One drawback o f t h i s a p p r o a c h seems t o be t h a t i t c o n s i d e r s o n l y t h e b e h a v i o u r o f p e r s o n s on o r n e a r t h e t r a f f i c a r t e r y j o i n i n g two c e n t r e s . I n t h e u s u a l c a s e , t h e b r e a k i n g p o i n t s a r e d e t e r m i n e d s e p a r a t e l y a n d t h e n j o i n e d on a map by s t r a i g h t l i n e s . I t may n o t be u n r e a s o n a b l e t o u s e main a r t e r i e s when t h e s e a r e i n t e r - u r b a n h i g h w a y s p a s s i n g t h r o u g h i n t e r m e d i a t e towns, b u t i t doe s n o t seems t o make good s e n s e t o i g n o r e i n e f f e c t , a v e r y l a r g e mass o f p e o p l e who happen t o T h i s d o e s n o t d i s t i n g u i s h between what t h e consumer b e l i e v e s t h e d i s t a n c e t o be, a n d what t h e d i s t a n c e a c t u a l l y i s when o b j e c t i v e l y m e a s u r e d . T h i s d i s t i n c t i o n i s q u i t e a n o t h e r i s s u e . 74 l i v e o r work between t h e s e p o i n t s . F u r t h e r , t h i s model seems t o assume that, consumers o n l y c o n s i d e r , a s s h o p p i n g a l t e r n a t i v e s , t h e two c e n t r e s i m m e d i a t e l y a d j a c e n t t o t h e i r s t a r t i n g p o i n t , a n d no o t h e r s . However, consumers do have a much w i d e r l a t i -t u d e o f c h o i c e , a n d t h e y f r e q u e n t l y e x e r c i s e i t . F i n a l l y , by t e r m i n a t i n g a s t o r e ' s r e t a i l t r a d i n g a r e a a t t h e b r e a k i n g p o i n t , t h i s a p p l i c a t i o n e x c l u d e s f r o m c o n s i d e r a t i o n t h o s e p e r s o n s whose p r o b a b i l i t y o f s e l e c t i n g t h e s t o r e i s l e s s t h a n 0.5* On t h e o t h e r hand, e v e n i f a l a r g e number o f consumers o n l y c h o s e a p a r t i c u l a r s h o p p i n g c e n t r e f o r one q u a r t e r o f t h e i r s h o p p i n g t r i p s , t h i s w o u l d s t i l l g e n e r a t e a n i m p o r t a n t volume o f t r a f f i c t o t h e c e n t r e . I t seems u n d u l y a r t i f i c i a l t o d e l i m i t t h e t r a d i n g a r e a a t t h e b r e a k i n g p o i n t . H u f f h a s commented on t h e s e , a n d o t h e r l i m i t a t i o n s o f g r a v i m e t r i c m o d e ls a n d t h e i r a p p l i c a t i o n t o r e t a i l t r a d e a r e a a n a l y s i s . ^ I n p a r t i c u l a r , he has q u e s t i o n e d t h e a s s u m p t i o n t h a t t h e e x p o n e n t o f t h e d i s t a n c e f a c t o r w h i c h R e i l l y h a d o r i g i n a l l y e s t i m a t e d a s 2 f o r i n t e r - u r b a n t r a d e movements w o u l d be t h e same f o r u r b a n a r e a s and u r b a n t r a d e movements. H u f f a r g u e s t h a t t h i s I s p a r t i c u l a r l y q u e s t i o n a b l e when o t h e r s t u d i e s ( d i s c u s s e d i n C h a p t e r I I ) have shown t h a t t h e e x p o n e n t ha s r a n g e d f r o m 1.5 to. o v e r 3> d e p e n d i n g on t h e t y p e o f t r i p , a s w e l l a s t h e g e o g r a p h i c a l s e t t i n g b e i n g a n a l y z e d . D a v i d L . H u f f , "A N o t e On t h e L i m i t a t i o n s i n I n t r a u r b a n G r a v i t y M o d e l s , " L a n d E c o n o m i c s , ( F e b r u a r y , 1961), 64-66. 75 Huff argues a l s o t h a t the g r a v i t y model does not have a sound t h e o r e t i c a l b a s i s ; that i t does not r e v e a l why the observed r e g u l a r i t i e s occur as they do. In view of the d i s c u s s i o n of R e i l l y ' s work i n Chapter I I I , t h i s argument may not be as v a l i d as a curs o r y glance might suggest. In terms of the accepted methodology of s o c i a l s c i e n c e , R e i l l y ' s law i s on f a i r l y f i r m ground. T h i s i s not to deny t h a t i t c o u l d be improved upon, however. Huff' s Approach to Consumer S p a t i a l Behaviour In view of the many l i m i t a t i o n s a t t a c h e d to the c u r r e n t fund of knowledge, Huff has developed and s t a r t e d to t e s t a 7 model which he b e l i e v e s overcomes some of these l i m i t a t i o n s . With regard to the a p p l i c a t i o n of h i s model, Huff w r i t e s : The model p r o v i d e s a t e n t a t i v e o p e r a t i o n a l b a s i s f o r understanding and determining the r e t a i l trade a r e a of a shopping c e n t e r . The r e t a i l trade area of an e x i s t i n g or proposed shopping c e n t e r can be a s c e r t a i n e d by: (1) d i v i d i n g the surrounding a r e a i n t o s m a l l s t a t i s t i c a l u n i t s ; (2) c a l c u l a t i n g the p r o b a b i l i t y of consumers from each of these u n i t s going to a p a r t i c u l a r shopping c e n t e r ; and (3) drawing l i n e s connecting a l l s t a t i s t i c a l u n i t s having l i k e p r o b a b i l i t i e s . A r e t a i l trade area i s thus not a f i x e d l i n e c i r c u m s c r i b i n g a shopping center,gbut r a t h e r a s e r i e s of z o n a l p r o b a b i l i t y contours. ' Huff, D e t e r m i n a t i o n of Intra-Urban R e t a i l Trade Areas (Los Angeles: U n i v e r s i t y of C a l i f o r n i a , Graduate School of Business A d m i n i s t r a t i o n , D i v i s i o n of Research, 1962); WA P r o b a b i l i s t i c A n a l y s i s of Consumer S p a t i a l Behavior," Emerging Concepts i n  Marketing, ed. W.S. Decker (Chicago: American Marketing Asso-c i a t i o n , Proceedings of the Winter Conference, December, 1962), pp. 443-461. Huff, Determination of Intra-Urban R e t a i l Trade Areas, p.5« 76 I n o r d e r t o d e t e r m i n e t h e s e p r o b a b i l i t i e s , H u f f s e t s o u t t o d e v e l o p a model w h i c h , w h i l e i t u s e s some o f t h e c o n c e p t u a l p r o p e r t i e s o f t h e g r a v i t y m o d e l , f o c u s e s on t h e consumer r a t h e r t h a n on t h e r e t a i l f i r m p e r s e . H u f f w r i t e s : S i n c e t h e consumer i s r e a l l y t h e p r i m a r y o b j e c t o f any t r a d e a r e a a n a l y s i s , an e x p l i c i t u n d e r s t a n d i n g I s n e e d e d n o t o n l y o f t h e f a c t o r s a f f e c t i n g h i s c h o i c e o f a s h o p p i n g c e n t e r , b u t a l s o o f t h e c h o i c e p r o c e s s i t s e l f w h i c h g i v e s r i s e t o o b s e r v a b l e s p a t i a l b e h a v i o r . Thus H u f f c o n c l u d e s t h a t t h e s p e c i f i c o b j e c t i v e s o f h i s s t u d y must be a s f o l l o w s : 1. A m a t h e m a t i c a l model w i l l be f o r m u l a t e d w h i c h r e p r e s e n t s a t h e o r e t i c a l a b s t r a c t i o n o f consumer s p a t i a l b e h a v i o r . M a t h e m a t i c a l c o n c l u s i o n s w i l l be d e d u c e d f r o m t h e model w h i c h , i n t u r n , w i l l be i n t e r p r e t e d i n t e r m s o f t h e i r b e h a v i o r a l i m p l i -c a t i o n s . 2. The b e h a v i o r a l i m p l i c a t i o n s stemming f r o m t h e model w i l l s e r v e a s a frame o f r e f e r e n c e f o r d e s i g n i n g a n e m p i r i c a l s t u d y t o t e s t t h e i r v a l i d i t y . A c t u a l consumer s p a t i a l b e h a v i o r w i l l be compared s t a t i s t i c a l l y t o t h e m a t h e m a t i c a l l y d e r i v e d b e h a v i o r , i . e . e x p e c t e d b e h a v i o r , i n o r d e r t o a p p r a i s e t h e s u i t a b i l i t y o f t h e model a s a p r e d i c t i v e and e x p l a n -a t o r y t h e o r y o f consumer s p a t i a l b e h a v i o r . 3. A method w i l l be d e v e l o p e d a n d e x e m p l i f i e d by w h i c h t h e b a s i c model o f consumer s p a t i a l b e h a v i o r c a n . be u s e d t o d e l i m i t i n t r a - u r b a n r e t a i l t r a d e a r e a s . I n s h o r t , H u f f , p r o p o s e s t o d e v e l o p h i s model, t e s t i t e m p i r i c a l l y , a n d t h e n a p p l y i t . T h i s t h e s i s i s c o n c e r n e d o n l y I b i d . , p.12. H u f f d e f i n e s " s p a t i a l b e h a v i o r " a s r e f e r r i n g t o t h e o b s e r v a b l e c o u r s e s o f a c t i o n t h a t consumers t a k e w i t h r e s p e c t t o t h e i r c h o i c e o f a s h o p p i n g c e n t r e . 1 0 I b i d . , p . 13. 77 w i t h the f i r s t two of h i s three steps. Huff's Concept of U t i l i t y and The Model Because he i s c r i t i c a l of the t h e o r e t i c a l b a s i s of R e i l l y ' s law, Huff seems to f e e l he would be on stronger ground to base h i s model on the theory of u t i l i t y a n a l y s i s as a p p l i e d to i n d i v i d u a l choice behaviour. In order to do t h i s , however, the d e f i n i t i o n of u t i l i t y i s modified from the o r d i n a l concept of economic theory to a view which regards u t i l i t y as being 11 measured, i n e f f e c t , on a r a t i o s c a l e . This i s perhaps best explained i n Huff's own words: ... I t i s assumed that the r a t i o between the proba-b i l i t i e s of a consumer's choosing any one of two p a r t i c u l a r shopping centers does not depend on the existence of other centers. This r a t i o i s c a l l e d the r a t i o of u t i l i t i e s of the two centers to a consumer. Such p o s t u l a t e d behavior d i f f e r s markedly from t r a d i t i o n a l economic theory. The l a t t e r maintains that a consumer e i t h e r always chooses one p a r t i c u l a r (the "most d e s i r a b l e " ) a l t e r n a t i v e w i t h p r o b a b i l i t y 1, or i s i n d i f f e r e n t between s e v e r a l "most d e s i r a b l e " a l t e r n a t i v e s (presumably chosen w i t h equal p r o b a b i l -i t i e s ) . As a consequence, a l l other a l t e r n a t i v e s possess zero p r o b a b i l i t i e s . This d i f f e r e n c e i n p o s t u l a t e s i s r e f l e c t e d i n the d e f i n i t i o n of the word " u t i l i t y " . I n the c l a s s i c a l theory, u t i l i t y i s i d e n t i c a l w i t h the rank of an a l t e r n a t i v e and t h a t a l t e r n a t i v e w i t h the highest rank i s always chosen. The d e f i n i t i o n being advanced here, however, a s s e r t s that u t i l i t y i s p r o p o r t i o n a l to the p r o b a b i l i t y of being chosen.12 This i s because, while c l a s s i c a l u t i l i t y theory deals i n ranked a l t e r n a t i v e s , Huff uses the r a t i o of two u t i l i t i e s . The d i s c u s s i o n of the v a r i o u s kinds of s c a l e s i n Chapter I I shows tha t t h i s i s only meaningful when usi n g r a t i o s c a l e s , since he i s making a proportionate t r a n s f o r m a t i o n . 2 I b i d * * P. 4. 78 This approach seems to introduce unnecessary confusion, c i r c u l a r i t y , and i n c o n s i s t e n c y i n t o the development of the model. A f t e r t r a c i n g Huff's reasoning i n a r r i v i n g at h i s model, i t w i l l a l s o be shown t h a t the same model can be devised without reference to the u t i l i t y n o t i o n , and the conceptual confusion which i t i n t r o d u c e s . Huff sets down some terms and d e f i n i t i o n s of elements on 1. A set of a l t e r n a t i v e shopping center choices which i s represented as set J ; 2. A subset of a l t e r n a t i v e shopping center choices which i s represented as J Q . The subset J 0 of a l t e r n a t i v e s represents a v a i l a b l e a l t e r n a t i v e s which are i n accord w i t h a consumer's t a s t e s and preferences. Any given a l t e r n a t i v e w i t h i n the subset J Q i s represented as j (where J = 1, ..., n ) ; and 3. A p o s i t i v e "pay o f f " f u n c t i o n u i s a s s o c i a t e d w i t h each a l t e r n a t i v e shopping center i n d i c a t i n g i t s " u t i l i t y " to a consumer. 13 From these b a s i c terms and d e f i n i t i o n s , Huff develops a set of b a s i c p r o p o s i t i o n s , which are summarized below: 1. The p r o b a b i l i t y Pr of a given a l t e r n a t i v e J being chosen from among a l l a l t e r n a t i v e s i n the subset J Q i s p r o p o r t i o n a l to UJ . That i s : which h i s model i s based, as f o l l o w s : (IV.1) 13 I b i d p. 14. • > 14 Adapted from I b i d . , pp. 14-16. 79 s u c h t h a t £ ft-, *i ; a n d °< fiy < l . i t seems h e r e t h a t what Hu f i s i m p l i c i t l y s a y i n g i s t h a t : Prj = k Uj a n d 2. The r a t i o between t h e p r o b a b i l i t i e s o f a c o n s u m e r ' s c h o o s i n g any one o f two p a r t i c u l a r s h o p p i n g c e n t r e s d o e s n o t d e p e n d on t h e e x i s t e n c e o f o t h e r c e n t r e s . T h i s r a t i o i s c a l l e d t h e r a t i o o f u t i l i t i e s o f t h e two c e n t r e s t o a consumer, an d t h u s : - ^ (IV.2) B u t , t h i s l o g i c i s u n n e c e s s a r y , b e c a u s e , by E q u a t i o n ( I V . 1 ) : P r t -- u> t h u s ; j = ££, a n d t h i s i s t r u e f o r j = 1, ...,n, r e g a r d l e s s o f t h e v a l u e o f n . 3. The p r o p e r t i e s o f t h e p a i r ( , Pr2 ) t h a t d e t e r -mine t h e u t i l i t y i n ( u, , ) a r e : (a) t h e s i z e Sy o f a g i v e n s h o p p i n g c e n t r e ; a n d (b) t h e d i s t a n c e 7// i n t i m e u n i t s , f r o m a consumer's t r a v e l b a s e / t o a s h o p p i n g c e n t r e j . T h e r e i s no argument w i t h h i s c h o i c e o f f a c t o r s , s p a c e a n d t i m e , s i n c e t h e d i s c u s s i o n r e g a r d i n g o t h e r i n t r a - u r b a n s t u d i e s h a s shown them t o be l o g i c a l l y a c c e p t a b l e . What i s d i s t u r b i n g a t t h i s 80 p o i n t i s the sudden switch from d e f i n i n g p r o b a b i l i t i e s i n terms of u t i l i t i e s and r a t i o s of u t i l i t i e s to d e f i n i n g u t i l i t i e s i n terms of p r o b a b i l i t i e s . What Huff means i s t h a t a r e l a t i v e u t i l i t y can be d e f i n e d i n terms of the r e l a t i v e frequency w i t h which c e r t a i n items (shopping c e n t r e s ) are chosen i n a s e r i e s 15 o f t r i a l s . The f a c t t h a t p r o b a b i l i t i e s are a l s o d e f i n e d as r e l a t i v e f r e q u e n c i e s i s immaterial to the d i s c u s s i o n . I t i s i n c o n s i s t e n t , c i r c u l a r , and unnecessary t o i n f e r t h a t p r o b a b i l -i t y u n d e r l i e s u t i l i t y when i t i s c l e a r l y the other way around. I t may be the consumer's s u b j e c t i v e p r o b a b i l i t y estimate of success which d e f i n e s or determines the u t i l i t y , but t h i s i s q u i t e d i f f e r e n t from the r e l a t i v e frequency n o t i o n which Huff seems to be d i s c u s s i n g . 4. The next p r o p o s i t i o n i s t h a t the u t i l i t y of a shopping centr e i s d i r e c t l y p r o p o r t i o n a l to the are a f a c t o r Sj , and i n v e r s e l y p r o p o r t i o n a l to the time f a c t o r 7~ij r a i s e d t o an exponent ^ , where X i s a parameter. Thus: "ij = £ Jiu (IV.3 ) where ^ij = the u t i l i t y of a shopping c e n t r e j to a consumer a t i . The t o t a l u t i l i t y of a l l shopping c e n t r e s r e l e v a n t t o the consumer a t would be: j=l 71 * T- * 7~~ * lT, SL_ (IV.k) V 15 There can be no doubt r e g a r d i n g what Huff i n t e n d s , as he d i s c u s s e s i t i n some d e t a i l , I b i d . , p. 1 6 . 81 T h u s , by E q u a t i o n s (IV.3) a n d ( I V . 4 ) , t h e r e l a t i v e f r e q u e n c y p r o b a b i l i t y t h a t a consumer a t i w i l l go t o a p a r t i c u l a r s t o r e j i s d e f i n e d a s : where t h e c o n s t a n t o f p r o p o r t i o n a l i t y £ i n b o t h E q u a t i o n s (IV.3) a n d (IV.k) i s assumed t o be t h e same, a n d t h u s v a n i s h e s f r o m t h e p r o b a b i l i t y d e f i n i t i o n . E q u a t i o n (IV.5) i s H u f f ' s m o d e l . The p a r a m e t e r A i s t o be e s t i m a t e d e m p i r i c a l l y t o r e f l e c t t h e e f f e c t o f t r a v e l t i m e on v a r i o u s k i n d s o f s h o p p i n g t r i p s , a s i t w i l l be r e c a l l e d t h a t H u f f d o e s n o t b e l i e v e t h a t t h i s s h o u l d be e q u a l t o 2 i n e v e r y c a s e . 5. One f u r t h e r p r o p o s i t i o n i s n e c e s s a r y i n o r d e r t o a p p l y t h e model, a n d a l s o f o r t h e c o m p u t a t i o n a l p r o c e d u r e u s e d t o e s t i m a t e ^ . T h i s p r o p o s i t i o n s t a t e s t h a t t h e e x p e c t e d number o f consumers a t a g i v e n p l a c e i , c h o o s i n g t o shop a t j , i s p r o p o r t i o n a l t o t h e number o f consumers a t i and t o t h e p r o b a b i l i t y t h a t a consumer a t 1 w i l l s e l e c t j f o r s h o p p i n g . T h u s : where £(d)j = t h e e x p e c t e d number o f consumers a t * c h o o s i n g s h o p p i n g c e n t r e j ; a n d Ci = t h e number o f consumers a t ^ . The d e f i n i t i o n o f O must change d e p e n d i n g on w h e t h e r t h e model i s b e i n g u s e d t o d e l i n e a t e t r a d e a r e a s , o r t o e s t i m a t e t h e p a r a m e t e r . I n t h e f i r s t c a s e , i t i s t h e t o t a l number o f (IV.5) (IV,6) 82 p o t e n t i a l consumers i n the area designated as i- . In the second case, i t i s the t o t a l number of consumers sampled from I . I t can now be demonstrated how t h i s same model, Equation (IV.5), can be developed without u s i n g the detour through a modified u t i l i t y theory. I t w i l l be r e c a l l e d that i n d i s c u s s i n g the b a s i s f o r s e l e c t i n g the f a c t o r s of store area and time f o r use i n the i n t r a - u r b a n a p p l i c a t i o n of R e i l l y ' s law, i t was argued that the consumer forms an o p i n i o n , a s u b j e c t i v e p r o b a b i l i t y , regarding the success of a shopping t r i p to a p a r t i c u l a r shopping c e n t r e . I t was explained that t h i s i s thought, or assumed, to be on the b a s i s of the breadth and depth of the s e l e c t i o n of items a v a i l a b l e , and t h i s i s assumed to be adequately represented by s t o r e area. Thus, the s u b j e c t i v e p r o b a b i l i t y i s taken to be p r o p o r t i o n a l to s t o r e area: SPSrj « ^ Sj (IV.7) where SPr-y = the s u b j e c t i v e p r o b a b i l i t y ; and 6j = a constant of p r o p o r t i o n a l i t y . I t i s assumed th a t ki i s the same f o r a l l j , otherwise there i s no hope f o r a model where i n d i v i d u a l s form opinions i n c o n s i s t e n t l y . I t was then assumed that a measure of l i k e l i h o o d i s attached to t h i s s u b j e c t i v e p r o b a b i l i t y to account f o r the f a c t that a l l shopping centres are not e q u a l l y d i s t a n t from consumers who do, under some circumstances, behave as economic men. This measure of l i k e l i h o o d i s assumed to be adequately 83 r e p r e s e n t e d by the i n v e r s e of 7 / / r a i s e d to an exponent. A c c e p t i n g the evidence presented i n Chapter I I tha't t h i s exponent d i f f e r s from 2 , depending on the t r i p type, i t can be r e p r e s e n -t e d as x . I t i s a l s o assumed t h a t the measure of l i k e l i h o o d m u l t i p l i e s the s u b j e c t i v e p r o b a b i l i t y . Thus: J 7—x (IV.8) where SLj = the s u b j e c t i v e l i k e l i h o o d t h a t a consumer would go to shopping centre j . There i s no necessary reason f o r a l l these s u b j e c t i v e measures to t o t a l 1 as i n the r e l a t i v e frequency case. T h e r e f o r e , on the assumption t h a t a l l o p i n i o n s are formed c o n s i s t e n t l y , each s u b j e c t i v e measure i s s c a l e d by a f a c t o r / 2 so t h a t they do t o t a l 1. Thus and, ^•2. 5 / -y ' = £j kz Sj T- x and thus, (IV.9) ^ z l ] Sly £j j-l J*X 7J.X I t i s then p o s s i b l e to d e f i n e a r e l a t i v e frequency measure of the p r o b a b i l i t y t h a t a consumer w i l l go from / to j , i n comparison t o a l l r e l e v a n t a l t e r n a t i v e s ( j = l , . . . , n ) : 84 PR-(IV.10) where PP>-ij = t h e r e l a t i v e f r e q u e n c y p r o b a b i l i t y measure. E q u a t i o n (IV.10) i s i d e n t i c a l t o E q u a t i o n ( I V . 5 ) , a n d was d e v e l o p e d w i t h o u t r e f e r e n c e t o t h e n o t i o n o f u t i l i t y . I t was m e n t i o n e d i n C h a p t e r I I t h a t m o d e l s may be d e v e l o p e d f o r p u r p o s e s o f e x p l a n a t i o n , p r e d i c t i o n , a n d / o r c o n t r o l . B o t h o f t h e p r e c e d i n g d e v e l o p m e n t s c o n t a i n t h e f i r s t two e l e m e n t s o f m o d e l s . However, e v e n i f t h e n e e d f o r e x p l a n a -t i o n i s d r o p p e d , t h e model c a n s t i l l be d e v e l o p e d . I t i s n o t e d on t h e b a s i s o f e m p i r i c a l e v i d e n c e t h a t t h e number o f p e o p l e a t c , g o i n g t o y , i s d i r e c t l y p r o p o r t i o n a l (more o r l e s s ) t o s t o r e a r e a , a n d i n v e r s e l y p r o p o r t i o n a l (more o r l e s s ) t o t h e d i s t a n c e , m e asured i n t i m e u n i t s , f r o m t o J . T h u s : where M y = t h e number a t c g o i n g t o a p a r t i c u l a r j ; a n d & = a c o n s t a n t o f p r o p o r t i o n a l i t y . The same i s t r u e f o r a l l j ( j = l , . . . , n ) , a n d t h e r e f o r e t h e t o t a l number o f p e o p l e (IV.11) 85 going to any j i s : E AJij * *Y, Sj_ (IV.12) J = J J?l 77 X V And the r e l a t i v e frequency of people a t c going to a p a r t i c u l a r j out of a l l p o s s i b l e J i s : Sj / x ^i/ = = l l i L . (iv.13) E /f E  s;/-r. x J j /'LJ where Cry = the r e l a t i v e frequency. E q u a t i o n (IV.13) i s i d e n t i c a l to Equations (IV.10") and ( I V . 5 ) . Whether i s i n t e r p r e t e d as a p r o b a b i l i t y measure i s i m m a t e r i a l . The model p r e d i c t s the r e l a t i v e frequency, and presumably c o u l d be t e s t e d as to whether or not i t agreed with a p a r t i c u l a r s e t of f a c t s . Whether or not any e x p l a n a t i o n i s i n v o l v e d i s a matter of o p i n i o n , and i n t e r p r e t a t i o n . H u f f ' s E m p i r i c a l T e s t and E v a l u a t i o n of the Parameter Huff chose to t e s t h i s model i n a suburban community w i t h i n the Los Angeles M e t r o p o l i t a n Area. A judgment sample, was used to determine which consumers would be asked to supply d a t a . As Huff e x p l a i n s : W i t h i n t h i s community three r a t h e r d i s t i n c t neighborhoods, i . e . , c l u s t e r s of households geo-g r a p h i c a l l y d e l i n e a t e d , were s e l e c t e d as consumer p o i n t s of o r i g i n — t h a t i s , the i ' s as d e p i c t e d i n the model ... The c r i t e r i a used to s e l e c t these neighborhoods were as f o l l o w s : 1. Each neighborhood should be approximately equal i n terms of p o p u l a t i o n d e n s i t y ; and, 86 2. Each of the three neighborhoods should be f a i r l y homogeneous i n terms of income. Because of the d i f f i c u l t y of o b t a i n i n g income data, the value of d w e l l i n g u n i t s was s u b s t i t u t e d as a c l o s e approximation.16 A f t e r s e l e c t i n g the three neighborhoods on t h i s b a s i s , a m a i l survey was conducted by q u e s t i o n n a i r e sent to a l l households known to be i n each a r e a . The q u e s t i o n n a i r e used i s reproduced as Appendix I . The number of q u e s t i o n n a i r e s sent and r e t u r n e d i s summarized i n Table IV. A d d i t i o n a l data were obtained f o r 14 planned shopping c e n t r e s which were to be the j 's i n the model. These shopping c e n t r e s were a l l w i t h i n an a r b i t r a r i l y d e f i n e d 20 mile r a d i u s from each of the sample neighbourhoods. For each shopping c e n t r e i n f o r m a t i o n was obtained r e g a r d i n g the s i z e of the c e n t r e , and i t s d i s t a n c e from each of the sample neighbour-hoods . Having meanwhile t a b u l a t e d the numbers of consumers i n each neighbourhood to determine, by neighbourhoods, how many had purchased what products a t which s t o r e s , i t was now p o s s i b l e f o r Huff to make e m p i r i c a l estimates of the exponen-t i a l parameter. T h i s c o u l d have been done by hand, but would have been very time-consuming. T h e r e f o r e , an i t e r a t i v e computer programme f o r s u c c e s s i v e approximation was d e v i s e d . A c c o r d i n g to Huff, the s e q u e n t i a l steps i n v o l v e d i n t h i s programme were: Huff, Determination of Intra-Urban R e t a i l Trade Areas, p.22. 87 TABLE I V H u f f ' s Sample D a t a P e r t a i n i n g t o t h e  N e i g h b o u r h o o d S t u d i e d N e i g h b o u r h o o d Item 1 2 3 I960 C e n s u s : p o p u l a t i o n 3812 4844 4559 i 9 6 0 C e n s u s : h o u s e h o l d 885 974 1199 M a i l i n g s i z e (no. o f 6 7 0 a 9 5 8 a 1 0 2 2 a h o u s e h o l d ) Number o f r e p l i e s 123 331 312 R e p l y p e r c e n t a g e 18 .3 34.5 30.5 a - The d i f f e r e n c e f r o m i 9 6 0 Census r e f l e c t s t h e f a c t t h a t some f a m i l i e s had moved s i n c e t h e t i m e o f t h e C e n s u s . S o u r c e : H u f f , D e t e r m i n a t i o n of I n t r a - U r b a n  T r a d e A r e a s , p. 23 . 88 1 . Assume a p a r t i c u l a r value of A which i s greater than unity.17 Correspondingly, s u b s t i t u t e the appropriate values f o r each of the appropriate a l p h a b e t i c characters noted i n the model and c a l c u l a t e the expected p r o b a b i l i t i e s . 1 fi 2 . Compare the expected p r o b a b i l i t i e s with the a c t u a l r e l a t i v e frequencies obtained from the survey data and c a l c u l a t e a c o r r e l a t i o n c o e f f i c i e n t . 1 9 3 . Continue to s u b s t i t u t e incremental values f o r A u n t i l the h i ghest c o r r e l a t i o n c o e f f i c i e n t i s obtained which w i l l represent the optimum value of the parameter. 2 0 Two a d d i t i o n a l r u l e s were a p p l i e d by Huff i n order to o b t a i n necessary data and to c a r r y out the computations: 1. The expected number of t r i p s (sample s i z e times the estimated p r o b a b i l i t y ) to any given shopping center had to be equal to or g r e a t e r than one-half a t r i p . F r a c t i o n a l t r i p s that were l e s s than t h i s minimum c r i t e r i o n were not i n c l u d e d i n c a l c u l a t i n g an estimate of the parameter. 2! 17 The s i g n i f i c a n c e of t h i s assumption, and i n t e r p r e t a t i o n of values l e s s than u n i t y , and even negative v a l u e s , w i l l be discussed i n Chapter V. 18 I t i s obvious here that i f these were c a l l e d expected r e l a t i v e f requencies, i n harmony wi t h the non-explanatory development of the model, i t would not a f f e c t the value of the e m p i r i c a l t e s t . 19 This i s not the same c o r r e l a t i o n c o e f f i c i e n t as i s u s u a l l y considered i n r e g r e s s i o n a n a l y s i s , and probably should not be so c a l l e d . The d e f i n i t i o n of t h i s c o e f f i c i e n t i s explained below. That i t i s not the same became very c l e a r during the study discussed i n Chapter V when values of /?* turned out to be negative. 2 0 I b i d . , p. 2 3 . 21 Why t h i s c r i t e r i o n was e s t a b l i s h e d i s not explained by Huff. Apparently, however, i t was thought t h a t i n samples as l a r g e as h i s , values l e s s than one-half were not s i g n i f i c a n t l y d i f f e r e n t from zero. I t w i l l be shown i n Chapter V that t h i s r u l e does change the value of the parameter. 89 2 . The p a r a m e t e r was e s t i m a t e d f o r e a c h n e i g h b o r -h o o d s e p a r a t e l y . T h e r e f o r e , t h e d i s t a n c e measure, i . e . t r a v e l t i m e , t o e a c h o f t h e f o u r t e e n s h o p p i n g c e n t e r s was u n i q u e f o r e a c h n e i g h b o r h o o d . The c e n t e r o f e a c h n e i g h b o r h o o d was u s e d a s t h e p o i n t o f o r i g i n a t i o n i n d e t e r m i n i n g t h e t r a v e l t i m e i n v o l v e d i n g e t t i n g t o e a c h o f t h e s e l e c t e d s h o p p i n g c e n t e r s . 2 The c o r r e l a t i o n c o e f f i c i e n t u s e d by H u f f t o d e t e r m i n e t h e optimum v a l u e o f h i s e x p o n e n t i a l p a r a m e t e r i s o f c o n s i d e r a b l e i n t e r e s t . T h i s c o e f f i c i e n t i s : C o n s i d e r i n g f i r s t t h e d e n o m i n a t o r o f t h e f r a c t i o n , w h i c h i s b e i n g u s e d a s a s t a n d a r d o f c o m p a r i s o n i n t h e c o e f f i c i e n t , i t seems t o be b a s e d on t h e L a P l a c e C r i t e r i o n ( s ometimes c a l l e d t h e P r i n c i p l e o f I n s u f f i c i e n t R e a s o n ) . What i t a r g u e s i s t h a t i f t h e r e were no f a c t o r s o p e r a t i n g t o make some consumers p r e f e r one s h o p p i n g c e n t r e t o a n o t h e r , t h e n c o n s u m e r s ' c h o i c e s w o u l d be random, a n d i t m i g h t be e x p e c t e d t h a t t h e y w o u l d be e v e n l y d i s t r i b u t e d among a l l a v a i l a b l e a l t e r n a t i v e s . T h i s a s s u m p t i o n i s compared t o t h e o b s e r v e d number o f p e r s o n s i n t h e s a m ple n e i g h b o u r h o o d by s u b t r a c t i o n , a n d t h e r e s u l t s a r e s q u a r e d a n d summed o v e r a l l s t o r e s . S i m i l a r l y , i n t h e n u m e r a t o r , t h e p r e d i c t i o n s o f t h e model a r e compared t o t h e o b s e r v a t i o n s by s u b t r a c t i n g , s q u a r i n g , a n d summing. I f t h e model p r o v i d e s no improvement o v e r t h e random b e h a v i o u r a s s u m p t i o n , t h e n t h e v a l u e s o f t h e f r a c t i o n w i l l be n e a r l y one, a n d t h e c o e f f i c i e n t o f d e t e r m i n a t i o n (>?*) , w h i c h m e a s u r e s t h e power o f a c c o u n t o f 1 -(IV.Ik) 22 I b i d P. 2 5 . 90 the model proposed, w i l l be h a r d l y d i f f e r e n t from z e r o . On the other hand, i f the model p r e d i c t s the e m p i r i c a l data e x a c t l y , 23 the numerator of the f r a c t i o n w i l l be zero, and Rz w i l l be 1. The c o r r e l a t i o n c o e f f i c i e n t R i s merely the square r o o t of the c o e f f i c i e n t of d e t e r m i n a t i o n and has no s i g n i f i c a n c e of i t s own. I t merely extends the range of numerical r e s u l t s , and tends t o spread the numbers out somewhat. As mentioned above, the programme c a l c u l a t e s a value f o r R i n each i t e r a t i o n , and the optimum va l u e of the e x p o n e n t i a l parameter i s the one which maximizes R. R e s u l t s of the E m p i r i c a l T e s t Huff makes some u s e f u l comments on the nature of h i s e m p i r i c a l t e s t : ... The e m p i r i c a l t e s t was designed merely as a s m a l l - s c a l e , experimental p i l o t study. An e f f o r t of t h i s type cannot v a l i d a t e or i n v a l i d a t e the t h e o r e t i c a l model. A l a r g e r number of more comprehensive i n v e s t i -g a t i o n s w i l l be r e q u i r e d to do so. I t was expected, however, that the s m a l l - s c a l e t e s t would p r o v i d e v a l u a b l e experience f o r the de s i g n of more compre-hensive e m p i r i c a l surveys and suggest d e s i r a b l e m o d i f i c a t i o n s of the model i t s e l f . The emphasis here i s on methodology r a t h e r than v e r i f i c a t i o n . 2 4 The r e s u l t s of the e m p i r i c a l t e s t conducted on t h i s p a r t i c u l a r sample are gi v e n i n Table V f o r c l o t h i n g , Table VI f o r f u r n i -t u r e , and Table VII which summarizes the oth e r two. 23 The e x p l a n a t i o n of how comes to be negative w i l l be g i v e n i n Chapter V when the de s i g n of the computer' programme i s d i s c u s s e d . 2 ^ I b i d . , p. 26. 91 TABLE V R e s u l t s o f H u f f ? s E m p i r i c a l T e s t  f o r Commodity " C l o t h i n g N e i g h b o u r h o o d 1 N e i g h b o u r h o o d 2 N e i g h b o u r h o o d 3 S h o p p i n g C e n t r e O b s e r v e d E x p e c t e d O b s e r v e d E x p e c t e d O b s e r v e d E x p e c t e d 1 71 70.76 148 144.28 143 141.49 2 0 1.27 19 25.99 6 9-78 3 0 1.04 4 3.10 2 .2.05 4 0 0.00 0 I . 3 6 2 4.02 5 5 2.60 38 13.73 21 2.07 6 l 0.77 0 2.36 7 1.41 7 0 0.00 2 2.03 6 3.22 8 0 0,00 0 I . 6 7 2 1.52 9 0 0.00 0 0.89 0 0.00 10 0 0.00 4 I . 8 7 3 0.00 n 1 0.99 2 3,44 3 1.52 12 0 0.00 0 1.09 2 0.00 13 1 0.78 0 10.58 6 35.92 14 0 0.79 1 5.61 0 0.00 R = 0.99 R = 0.98 R = 0.90 Lambda = 2.889 Lambda = 2.655 Lambda = 3.690 S o u r c e : H u f f , D e t e r m i n a t i o n o f I n t r a - U r b a n R e t a i l T r a d e  A r e a s , p. 26. 9.2 TABLE VI R e s u l t s o f H u f f ' s Emp i r i c a l T e s t f o r Commodity " F u r n i t u r e " S h o p p i n g C e n t r e N e i g h b o u r h o o d 1 O b s e r v e d E x p e c t e d N e i g h b o u r h o o d 2 O b s e r v e d E x p e c t e d N e i g h b o u r h o o d 3 O b s e r v e d Expecte 1 51 51.66 68 65.83 80 78.21 2 0 1 .50 4 16.72 1 7.43 3 0 1 .30 0 3.27 1 1.95 4 0 0.00 0 1.33 0 3 .05 5 3 3.43 24 14.06 11 2 .30 6 6 0.98 6 2.55 12 1.38 7 0 0.00 3 2.17 3 2.78 8 0 0.00 1 2.00 0 i . 5 0 b 9 0 0.00 0 1.10 0 0.00 10 2 1 . 3 1 a 16 2.02 4 0.00 11 0 0.00 0 - 3.88 1 1.58 12 0 o..ooa 0 1.50 1 0.00 13 0 0.91 0 7.28 8 21.82 14 0 0.91 6 4 .29 0 0.00 R = 0.99 R = 0.94 /? = 0.96 Lambda = 2.542 Lambda = 2.115 Lambda = 3.247 - These two v a l u e s seem t o be i n c o r r e c t l y i n t e r c h a n g e d i n t h e p u b l i s h e d r e s u l t s , a s was d i s c l o s e d by r e c a l c u l a t i o n ( s e e C h a p t e r V ) . b - T h i s v a l u e was i n c o r r e c t l y p u b l i s h e d as 15«0, b u t was c o r -r e c t e d i n H u f f . " P r o b a b l i s t i c A n a l y s i s o f Consumer S p a t i a l B e h a v i o r , p. 455* g o u r c e : H u f f , D e t e r m i n a t i o n o f I n t r a - U r b a n R e t a i l T r a d e A r e a s , p . 27. 9 3 TABLE V I I C o m p o s i t e R e s u l t s o f H u f f ' s  E m p i r i c a l T e s t E s t i m a t e d Commodity N e i g h b o u r h o o d Lambda C l o t h i n g 1 2 . 8 1 2 0 . 9 9 C l o t h i n g 2 2.604 0 . 9 8 C l o t h i n g 3 3 - 7 7 9 O .96 F u r n i t u r e 1 2.523 0 . 9 9 F u r n i t u r e 2 2.115 0.94 F u r n i t u r e 3 3*331 O .96 S o u r c e : H u f f , D e t e r m i n a t i o n o f I n t r a - U r b a n R e t a i l T r a d e A r e a s , p. 27. 9k H u f f makes some f u r t h e r comments r e g a r d i n g t h e r e s u l t s o f h i s t e s t : The e x p e c t e d b e h a v i o r d e r i v e d f r o m t h e model c o r r e s -ponds q u i t e c l o s e l y t o t h e a c t u a l b e h a v i o r o b s e r v e d f r o m t h e s u r v e y f i n d i n g s . However, c o n t r a r y t o what was e x p e c t e d , t h e e s t i m a t e s o f lambda v a r i e d f r o m n e i g h b o r h o o d t o n e i g h b o r h o o d ... ... t h e i n v e s t i g a t i o n was d e s i g n e d ... t o t e s t t h e h y p o t h e s i s t h a t lambda was p r i m a r i l y a f u n c t i o n o f t h e t y p e o f s h o p p i n g t r i p . E a c h o f t h e t h r e e sample n e i g h b o r h o o d s were s e l e c t e d on t h e b a s i s o f t h e i r h o m o g e n e i t y i n t e r m s o f income , a n d i t was t h e r e f o r e e x p e c t e d t h a t lambda w o u l d be a p p r o x i m a t e l y t h e same f o r e a c h n e i g h b o r h o o d w i t h r e s p e c t t o a g i v e n t y p e o f s h o p p i n g t r i p "... I t w o u l d be u n w a r r a n t e d t o d i s c a r d t h e e x t r e m e ... e s t i m a t e s ... u n l e s s i t c o u l d be shown t h a t t h e s e v a l u e s were so f a r removed f r o m t h e o t h e r e s t i m a t e s a s t o i n d i c a t e t h e p r e s e n c e o f f a c t o r s o t h e r t h a n sample v a r i a t i o n . T h e r e f o r e , a s t a t i s t i c a l c r i t e r i o n was c o n s t r u c t e d t o t e s t t h e h y p o t h e s i s t h a t t h e e x t r e m e v a l u e s ... were n o t due t o f a c t o r s o t h e r t h a n sample v a r i a t i o n . 25 The p a r t i c u l a r t e s t u s e d was a s i m p l e s t a t i s t i c b a s e d on c o m p u t i n g t h e r a t i o between t h e d i f f e r e n c e between t h e l a r g e s t 26 a n d s e c o n d l a r g e s t o b s e r v a t i o n s a n d t h e r a n g e . On t h e b a s i s o f t h i s t e s t , t h e h y p o t h e s i s was a c c e p t e d ( a t t h e one p e r c e n t l e v e l ) t h a t t h e e x t r e m e v a l u e s were due t o sample v a r i a t i o n . I t i s d o u b t f u l , however, w h e t h e r s u c h a s i m p l e t e s t i s m e a n i n g f u l . The numbers b e i n g c o n s i d e r e d h e r e a r e e x p o n e n t s , n o t a r i t h m e t i c m e a sures o f p r o p e r t i e s . S t a t i s t i c a l t e s t s o f I b i d . , pp. 26-28. 26 W i l f r e d J . D i x o n a n d F r a n k J . Massey, J r . , I n t r o d u c t i o n  t o S t a t i s t i c a l A n a l y s i s (New Y o r k : M c G r a w - H i l l Book Co., I n c . , 1957), PP. 272-278. 95 the type used were o r i g i n a l l y designed f o r use on t h i s l a t t e r type of measure. I t i s d i f f i c u l t to conceive how a v a r i a t i o n of one u n i t i n the exponent, which i s a t e n - f o l d change i n the un d e r l y i n g property (or p r o p e r t i e s ) , could be the r e s u l t of 27 sample v a r i a t i o n r a t h e r than other f a c t o r s . S a t i s f i e d that a l l values were accounted f o r by sample v a r i a t i o n , Huff c a l c u l a t e d a mean value of lambda f o r each of c l o t h i n g and f u r n i t u r e , 3-191 and 2.723 r e s p e c t i v e l y . I t was not p o s s i b l e to t e s t f o r s i g n i f i c a n c e of the d i f f e r e n c e of these two means since the numbers i n the sample are so s m a l l . Huff concludes: Despite the la c k of con c l u s i v e s t a t i s t i c a l evidence that the mean ... estimates are v a l i d r e p r e s e n t a t i o n s of the c l o t h i n g and f u r n i t u r e parameters, they do i n d i c a t e t h a t a consumer's s p a t i a l behavior i s a f u n c t i o n of the type of shopping t r i p . For example, the mean estimate f o r shopping t r i p s i n v o l v i n g c l o t h i n g purchases i s higher than the mean value estimated f o r f u r n i t u r e purchases, which confirms the observation that consumers are not w i l l i n g to t r a v e l as f a r f o r c l o t h i n g as they are f o r f u r n i t u r e purchases.28 Out of c u r i o s i t y , the exponents were converted to corresponding a r i t h m e t i c measure by viewing them as logarithms, and thus l o o k i n g up the corresponding a n t i - l o g a r i t h m s i n t a b l e s . The same t e s t s t a t i s t i c s were computed using these a r i t h m e t i c v a l u e s . Though the s t a t i s t i c s were higher i n value, the hypothesis that extreme values were caused by sample v a r i a t i o n s t i l l could not be r e j e c t e d a t the \% s i g n i f i c a n c e l e v e l . 28 I b i d . , p. 31. 96 Summary The d i s c u s s i o n of t h i s p a r t i c u l a r p r o b a b i l i s t i c model has been i n two p a r t s : the development of the model; and the e m p i r i c a l t e s t i n g of the model. C o n s i d e r i n g f i r s t the development of the model, i t was shown t h a t there seemed to be a c e r t a i n amount of i n c o n s i s t e n c y i n the l o g i c a p p l i e d by H u f f . The model was d e r i v e d a l s o from a s o r t of Bayesian approach u s i n g s u b j e c t i v e p r o b a b i l i t i e s and l i k e l i h o o d measures. Sin c e i t may be thought t h a t t h i s development i s s i m i l a r l y -not a c c e p t a b l e , i t was a l s o shown t h a t the same model c o u l d be d e v i s e d i n a manner which i s f r e e of any attempt a t e x p l a n a t i o n of behaviour, but i s aimed s o l e l y a t a b i l i t y t o p r e d i c t . Since i t was e s t a b l i s h e d t h a t the model was not necess-a r i l y based on any p a r t i c u l a r e x p l a n a t i o n , i t was p o s s i b l e to c o n s i d e r , s e p a r a t e l y and independently, the r e s u l t s of the e m p i r i c a l t e s t . The major c o n t r i b u t i o n of t h i s work l i e s i n the d e s i g n and use of a programme f o r making s u c c e s s i v e approximations to the value of a parameter, u s i n g minimum squared d e v i a t i o n from observed val u e s as a c r i t e r i o n , by means of a high-speed d i g i t a l computer. T h i s type of e s t i m a t -i n g procedure would not be p r a c t i c a l by any other, l e s s e r means of computation. While the a c t u a l r e s u l t s of the e m p i r i c a l work appear encouraging, they must n e v e r t h e l e s s be viewed with c o n s i d e r a b l e c a u t i o n . Huff emphasizes t h a t h i s work i s but a p i l o t study. 9 7 E v e n s o , he seems t o p l a c e a c o n s i d e r a b l e amount o f f a i t h i n 2 Q t h e r e s u l t s . The u s u a l t y p e o f s t a t i s t i c a l s i g n i f i c a n c e t e s t s do n o t seem a p p r o p r i a t e i n t h i s s e t t i n g b e c a u s e o f t h e f a c t t h a t t h e p a r a m e t e r s a r e e x p o n e n t s , r a t h e r t h a n a r i t h m e t i c m e a s u r e s . T h i s r a i s e s a n o t h e r q u e s t i o n r e g a r d i n g t h e d e s i g n o f a n a p p r o p r i a t e t e c h n i q u e f o r o b j e c t i v e l y e v a l u a t i n g t h e r e s u l t s o f t h i s t y p e o f e m p i r i c a l work. An a t t e m p t a t a n s w e r i n g t h i s q u e s t i o n i s a d v a n c e d i n C h a p t e r V o f t h i s t h e s i s . W h i l e H u f f m e n t i o n s t h a t a l a r g e - s c a l e s t u d y was b e i n g i n i t i a t e d , ( i b i d . , p.5)> no f u r t h e r e m p i r i c a l work ha s b e en p u b l i s h e d . On t h e o t h e r hand, H u f f h a s r e c e n t l y a d v o c a t e d t h a t t h i s same model be a p p l i e d i n a n o t h e r c o n t e x t : "The Use o f G r a v i t y M o d e l s i n S o c i a l R e s e a r c h , " M a t h e m a t i c a l  E x p l o r a t i o n s i n B e h a v i o r a l S c i e n c e , e d s . F r e d M a s s a r i k a n d P h i l b u r n R a t o o s h (Homewood, 1 1 1 . : R i c h a r d D. I r w i n , I n c . , a n d The D o r s e y P r e s s , I 9 6 5 ) , pp. 3 1 7 - 3 2 1 . T h i s r a i s e s a q u e s t i o n w i t h r e g a r d t o t h e t h e o r e t i c a l , o r e x p l a n a t o r y , c o n t e n t w h i c h H u f f i m p u t e s t o t h e model i n i t s o r i g i n a l s e t t i n g . CHAPTER V EMPIRICAL TEST OF THE PROBABILISTIC MODEL I N METROPOLITAN VANCOUVER The e m p i r i c a l t e s t o f t h e p r o b a b i l i s t i c model o f consumer s p a t i a l b e h a v i o u r , d i s c u s s e d i n C h a p t e r I V, was d e s i g n e d a n d I n t e n d e d t o be o n l y a p i l o t s t u d y . A l t h o u g h H u f f had s t a t e d t h a t he was i n i t i a t i n g a m a j o r e m p i r i c a l s t u d y , t h e r e a r e , a s y e t , no p u b l i s h e d r e s u l t s . F u r t h e r , i t was t h o u g h t t h a t t h i s m a j o r s t u d y w o u l d a g a i n be c o n d u c t e d i n t h e L o s A n g e l e s M e t r o p o l i t a n a r e a , a n d t h e r e f o r e , t h a t i t wo u l d be u s e f u l t o c o n d u c t a m a j o r s t u d y i n a d i f f e r e n t a r e a , namely M e t r o p o l i t a n V a n c o u v e r , b o t h f o r i t s own s a k e , and f o r p u r p o s e s o f c o m p a r i s o n s h o u l d H u f f p u b l i s h h i s r e s u l t s i n t h e i n t e r i m . I t w i l l be r e c a l l e d f r o m p r e v i o u s d i s c u s s i o n t h a t H u f f h a d d e s i g n e d h i s p i l o t s t u d y b o t h t o c h e c k on m e t h o d o l o g y , a n d t o t e s t t h e h y p o t h e s i s t h a t t h e e x p o n e n t i a l p a r a m e t e r was a f u n c t i o n o f t h e t y p e o f s h o p p i n g t r i p . The r e s u l t s o f t h e p i l o t s t u d y a p p e a r e d t o i n d i c a t e t h a t t h e h y p o t h e s i s s h o u l d be a c c e p t e d t e n t a t i v e l y . I t a p p e a r e d t o be d e s i r a b l e t o o b t a i n d a t a w h i c h w o u l d a l l o w e x a m i n a t i o n o f t h i s t e n t a t i v e h y p o t h e s i s on a much b r o a d e r b a s i s . S i n c e o t h e r s were d e s i r o u s o f c o n d u c t i n g a w i d e - r a n g i n g s u r v e y f r o m w h i c h t o examine o t h e r a s p e c t s o f s h o p p i n g t r i p s , i t was d e t e r m i n e d t h a t i t wo u l d n o t be d i f f i c u l t t o i n c l u d e i t e m s i n t h e s u r v e y q u e s t i o n n a i r e f r o m 99 w h i c h d a t a c o u l d be g e n e r a t e d i n t h e f o r m r e q u i r e d f o r u s e i n t h e m o d e l . I t w o u l d a l s o be n e c e s s a r y t o o b t a i n m e asures o f t h e s t o r e a r e a s i n v o l v e d , a s w e l l a s t r a v e l t i m e s f r o m t h e v a r i o u s n e i g h b o u r h o o d s d e f i n e d t o e a c h o f t h e r e s p e c t i v e s t o r e s . T h i s , a l s o , was n o t t h o u g h t t o be a d i f f i c u l t p r o b l e m . The Sample an d t h e S u r v e y To b e g i n t h e s t u d y , i t was n e c e s s a r y t o d e t e r m i n e t h e g e o g r a p h i c a l g r o u p i n g s w h i c h a r e r e f e r r e d t o a s n e i g h b o u r h o o d s . I t was d e c i d e d t o u s e C e n s u s T r a c t s f o r t h i s p u r p o s e f o r two r e a s o n s : (1) t h e y a r e a l r e a d y d e f i n e d ; and, (2) a number o f s u r v e y s h a d p r e v i o u s l y b e e n c o n d u c t e d on t h i s b a s i s , and i t was f e l t t h a t t h e y w o u l d p r o b a b l y y i e l d d a t a f o r f u t u r e e x t e n -s i o n o f t h i s s t u d y w h i c h w o u l d t h u s be on a c o m p a r a b l e b a s i s . The s u r v e y was c o n d u c t e d i n t h e M e t r o p o l i t a n A r e a o f G r e a t e r V a n c o u v e r , a n d a map o f t h i s a r e a s h o w i n g t h e C e n s u s T r a c t b o u n d a r i e s i s i n c l u d e d h e r e a s F i g u r e k. The p a r t i c u l a r C e n s u s T r a c t s i n c l u d e d i n t h e sample a r e shown s h a d e d i n t h i s f i g u r e . S i n c e n e a r l y f i f t y C e n s u s T r a c t s were c o v e r e d — t h o u g h n o t a l l y i e l d e d u s e f u l r e s u l t s — i t i s i m m e d i a t e l y o b v i o u s t h a t H u f f ' s c r i t e r i o n o f h o m o g e n e i t y o f income among n e i g h b o u r h o o d s i s n o t s a t i s f i e d . The t e c h n i c a l d e t a i l s o f how t h e sample was t a k e n a r e i n c l u d e d a s A p p e n d i x I I . A f t e r t h e method o f s a m p l i n g was d e t e r m i n e d , t h e d a t a were o b t a i n e d by p e r s o n a l i n t e r v i e w s u s i n g a q u e s t i o n n a i r e e s p e c i a l l y d e s i g n e d f o r t h i s s u r v e y . I n f a c t , two d i f f e r e n t 101 q u e s t i o n n a i r e s were u s e d , b u t t h e y d i f f e r e d o n l y i n t h e p a r t s b e i n g u s e d f o r o t h e r p u r p o s e s . T h u s , a l l t h e q u e s t i o n n a i r e s ( c o r r e c t l y ) c o m p l e t e d were p o t e n t i a l d a t a f o r t h i s s t u d y . The p a r t i c u l a r q u e s t i o n s o f i n t e r e s t a r e r e p r o d u c e d a s A p p e n d i x I I I . The r e s p o n s e s t o t h e s e , a n d o t h e r q u e s t i o n s , were s u i t a b l y c o d e d a n d t h e whole s u r v e y was t h u s r e d u c e d t o a d e c k o f p u n c h e d c o m p u t e r d a t a c a r d s , r e a d y f o r s u b s e q u e n t a n a l y s i s . I t may be n o t e d , on c o m p a r i n g t h e q u e s t i o n s a s k e d i n t h i s s u r v e y w i t h t h e q u e s t i o n s a s k e d by H u f f , t h a t t h e r e a r e marked d i f f e r e n c e s . H u f f b a s i c a l l y e l i c i t e d two s e p a r a t e r e s p o n s e s : (1) p l a c e o f l a s t p u r c h a s e o f i t e m s A, B, C, e t c . ; and, (2) p l a c e o f m a j o r i t y o f p u r c h a s e s o f i t e m s A, B, C, e t c . On t h e o t h e r hand, t h e p r e s e n t s u r v e y was o r i e n t e d t o w a r d e x a m i n i n g s e v e r a l a s p e c t s o f t h e s h o p p i n g t r i p a s a n e n t i t y . T h u s , t h e p e r t i n e n t q u e s t i o n s were f r a m e d i n s u c h a way t h a t t h e y d e t e r -m i n e d what i t e m s were p u r c h a s e d a t w h i c h s t o r e s on t h e l a s t  s h o p p i n g t r i p . S i n c e t h i s s t u d y i s p r i m a r i l y c o n c e r n e d w i t h s h o p p i n g g o o d s , w h i c h a r e by n a t u r e I n f r e q u e n t l y p u r c h a s e d , many i n t e r v i e w s d i d n o t g e n e r a t e u s e f u l d a t a i n r e s p o n s e t o t h e s e q u e s t i o n s . U n f o r t u n a t e l y , n o t a l l s h o p p i n g t r i p s a r e t a k e n i n q u e s t o f s h o p p i n g g o o d s , a n d t h o s e t h a t a r e , a r e n o t a l w a y s s u c c e s s f u l i n t e r m s o f a p u r c h a s e o f a n i t e m . T h i s h a d t h e n e t e f f e c t o f r e d u c i n g t h e s i z e o f t h e ,sample, w h i c h was a l r e a d y s m a l l r e l a t i v e t o t h e a r e a c o v e r e d . I t may a l s o be n o t e d t h a t whereas H u f f ' s s t u d y d e a l t o n l y w i t h p l a n n e d s h o p p i n g c e n t r e s , t h i s s t u d y i n c l u d e d t h e 102 major department s t o r e s c e n t r a l l y l o c a t e d i n downtown Vancouver. There are a t l e a s t two, r e l a t e d reasons f o r t h i s . F i r s t , Vancouver i s only a f r a c t i o n of the s i z e of an urban ar e a such as Los Angeles and Vancouver's c e n t r a l d i s t r i c t i s r e a d i l y a c c e s s i b l e to a l l suburban areas c o n s i d e r e d . I t was f e l t t h a t i t would be imp o s s i b l e t o f i n d a suburban ar e a i n Vancouver t h a t would not be s u b j e c t t o c o n s i d e r a b l e i n f l u e n c e by these major s t o r e s . Second, t h i s i n f l u e n c e i s r e f l e c t e d i n the survey r e s u l t s which showed t h a t many shopping goods purchases were made i n these s t o r e s . E x c l u s i o n of these s t o r e s would have reduced the p o r t i o n of the o r i g i n a l sample used i n the a n a l y s i s to a s m a l l amount, and, indeed, would have made the balance of the a n a l y s i s a q u e s t i o n a b l e e x e r c i s e . A l t o g e t h e r , ten s t o r e s and/or shopping c e n t r e s were used i n the study, and t h e i r l o c a t i o n s are shown on the map, F i g u r e 4. The next step r e q u i r e d i n p u t t i n g the raw data i n t o a form which c o u l d be used i n e v a l u a t i n g the model i n v o l v e d a g g r e g a t i o n , by neighbourhoods, of the p a r t i c u l a r shopping goods purchased a t p a r t i c u l a r s t o r e s . The q u e s t i o n n a i r e i n c l u -ded space f o r up to three s t o r e s on one t r i p , and the whole range of items f o r each s t o r e i n d i c a t e d . The data were a l s o coded i n t h i s manner such that s t o r e s were i d e n t i f i e d on three c a r d columns, and a s s o c i a t e d items i n three groups of columns. I t was thus a r e l a t i v e l y simple matter to s o r t the cards by neighbourhoods and apply the MVTAB programme from the Computing Centre L i b r a r y t o generate b i - v a r i a t e frequency t a b l e s of items 1 0 3 by s t o r e s f o r e a c h n e i g h b o u r h o o d . The c o n t e n t s o f t h e s e t a b l e s were m a n u a l l y a g g r e g a t e d t o p r o d u c e t h e C^y d a t a r e q u i r e d t o t e s t t h e m o d e l . T h i s was n o t q u i t e t h e end o f t h i s s t e p , however. The n u m e r i c a l v a l u e s o b t a i n e d were q u i t e s m a l l , and t h e r e was a s e r i o u s q u e s t i o n c o n c e r n i n g w h e t h e r t h e y w o u l d be s u i t a b l e f o r e v a l u a t i o n o f t h e m o d e l . T h u s , i t was d e c i d e d t h a t f u r t h e r a g g r e g a t i o n w o u l d be n e c e s s a r y , and so c e r t a i n r e l a t e d i t e m s were com b i n e d t o p r o d u c e t h r e e c o m p o s i t e c o m m o d i t i e s , A, B, a n d C. The i t e m s w h i c h d e f i n e t h e s e a r e i d e n t i f i e d i n T a b l e V I I I . T h i s p r o c e d u r e p r o d u c e d n u m e r i c a l r e s u l t s f o r £</ w h i c h , w h i l e s t i l l s m a l l r e l a t i v e t o H u f f ' s v i r t u a l l y c o m p l e t e sample o f a n e i g h b o u r h o o d , were c o n s i d e r e d a d e q u a t e t o p r o c e e d . The n e x t i t e m o f i n f o r m a t i o n r e q u i r e d was t h e a p p r o p r i a t e a r e a measurements f o r t h e s t o r e s . T h e s e were o b t a i n e d by p e r s o n a l c o n t a c t w i t h s u i t a b l e p e r s o n s employed i n t h e s t o r e s . The measurement u s e d was b a s e d on t h e a r e a d e v o t e d t o r e t a i l s e l l i n g , p l u s a d j a c e n t s t o r a g e a r e a . T h i s l a t t e r was i n c l u d e d b e c a u s e i t was t h o u g h t t o be m a t e r i a l i n d e t e r m i n i n g b r e a d t h a n d d e p t h o f s e l e c t i o n o f m e r c h a n d i s e . The measurements a r e shown i n T a b l e IX. The e x e r c i s e was s i m p l e b u t t i m e - c o n s u m i n g . MVTAB had t o be s e p a r a t e l y s e t up f o r e a c h n e i g h b o u r h o o d , a n d t h e r e were a p p r o x i m a t e l y f i f t y o f t h e s e . The s e t - u p t i m e g r e a t l y e x c e e d e d t h e c o m p u t i n g t i m e , w h i c h was g e n e r a l l y l e s s t h a n one m i n u t e f o r e a c h n e i g h b o u r h o o d on t h e C o m p u t i n g C e n t r e ' s IBM 7 0 4 0 m a c h i n e . TABLE V I I I  D e f i n i t i o n o f C o m p o s i t e Commodities Commodity Items A C l o t h i n g f o r s e l f C h i l d r e n ' s C l o t h i n g B F u r n i t u r e House f u r n i s h i n g s C h i n a , g l a s s w a r e A p p l i a n c e s C S p o r t i n g Goods Toys Hardware ( i n c l u d i n g p a i n t ) 105 TABLE IX Areas of Stores i n Metropolitan Vancouver Area (thousand- Store Number Store Location square f e e t ) a 1 Hudson's Bay- Downtown 617 2 Woodward' s Downtown 573 3 Woodward's Oakridge 225 4 Woodward's Park Royal 122 5 Eaton's Downtown 550 6 Eaton's Brentwood 210 7 Eaton* s Park Royal 125 8 Simpson-Sears Burnaby 240 9 Simpson-Sears Richmond 50 10 Army and Navy Downtown 42 a - These figures include only main store areas. They do not include food f l o o r s , or specialty shops i n the case of shopping centres. 106 The f i n a l measurement r e q u i r e d was t h a t o f t h e d i s t a n c e s , i n t i m e u n i t s , f r o m e a c h n e i g h b o u r h o o d t o e a c h s t o r e i n t h e s a m p l e . To a c c o m p l i s h t h i s measurement, t h e a r e a was l a i d o u t on a g r i d s y s t e m a l o n g m a j o r t r a f f i c a r t e r i e s . The l e n g t h o f e a c h p a r t o f t h e g r i d was d e t e r m i n e d by d r i v i n g a l o n g i t , a t n o r m a l s p e e d , and m e a s u r i n g t h e t i m e r e q u i r e d w i t h a s t o p w a t c h . A l l measurements were t a k e n i n t h e e a r l y e v e n i n g o r on weekends. W h i l e t h e t r a f f i c i s l i g h t e r a t t h e s e t i m e s t h a n d u r i n g n o r m a l b u s i n e s s h o u r s , i t i s c o n s i s t e n t l y so o v e r t h e whole s a m p l e . S i n c e i t i s t h e r e l a t i v e e f f e c t o f d i s t a n c e w h i c h i s s o u g h t , t h e f a c t t h a t t h e s e t i m e s a r e somewhat s h o r t e r t h a n t h o s e w h i c h c o n sumers a c t u a l l y e x p e r i e n c e was c o n s i d e r e d n o t t o o damaging t o t h e i n v e s t i g a t i o n . The d i s t a n c e measures t h u s o b t a i n e d were marked on t h e g r i d on a map. The most l i k e l y r o u t e f r o m e a c h n e i g h b o u r h o o d t o e a c h s t o r e was t r a c e d on t h e map, a n d t h e d i s t a n c e o b t a i n e d by a d d i n g up t h e a p p r o p r i a t e g r i d e l e m e n t s . A g a i n , a s i n . H u f f ' s s t u d y , t h e c e n t r e p o i n t o f e a c h n e i g h b o u r -h o o d was assumed t o r e p r e s e n t t h e p o i n t o f o r i g i n f o r a l l c o n s u m e r s i n t h a t n e i g h b o u r h o o d . The r e s u l t i n g measurements were t h e 77/ v a l u e s r e q u i r e d by t h e mod e l . Programme XLAMC f o r E v a l u a t i n g t h e E x p o n e n t i a l P a r a m e t e r I n o r d e r t o make u s e o f t h e h i g h - s p e e d c o m p u t i n g a b i l i t y o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a C o m p u t i n g C e n t r e ' s IBM 7040 m a c h i n e , i t was n e c e s s a r y t o p r e p a r e a programme i n a s u i t a b l e programming l a n g u a g e . The most u s e f u l l a n g u a g e f o r t h i s s t u d y was FORTRAN IV. A f l o w c h a r t was p r e p a r e d t o show 107 t h e s t e p s n e c e s s a r y i n e f f i c i e n t l y r e a c h i n g t h e d e s i r e d r e s u l t , 2 a n d i s p r e s e n t e d a s A p p e n d i x IV. The e q u i v a l e n t FORTRAN programme i s a l s o p r e s e n t e d a s A p p e n d i x V. T h e r e a r e two s i g n i f i c a n t p o i n t s t o be n o t e d a b o u t t h i s programme. F i r s t , i s t h e i n c l u s i o n o f t h e CALL PLOTS r o u t i n e . S e c o n d i s t h e i n c l u s i o n o f SUBROUTINE SRNEG. The CALL PLOTS r o u t i n e i s p a r t o f t h e C o m p u t i n g C e n t r e ' s IBM 7040 l i b r a r y , and i s u s e d when i t i s more c o n v e n i e n t t o d i s p l a y t h e r e s u l t s o f a c o m p u t a t i o n a l r o u t i n e i n a X-Y p l o t t h a n i n t a b u l a r f o r m . I n t h e p a r t i c u l a r c a l c u l a t i o n o f e m p i r i c a l r e s u l t s o f i n t e r e s t i n t h i s s t u d y , i t was t h o u g h t t h a t i t w o u l d be more i n f o r m a t i v e t o have t h e p a i r e d v a l u e s o f R a n d A d i s p l a y e d i n a n X-Y p l o t . T h i s i s p a r t i c u l a r l y so when t h e u s u a l s t a t i s -t i c a l t e s t s on t h e v a l u e o f A f o r e a c h maximum v a l u e o f /? a r e o f d o u b t f u l m e r i t . On t h e o t h e r hand, a g r a p h i c a l p r e s e n t a t i o n o f x a n d * shows t h e manner i n w h i c h R. v a r i e s w i t h A . By i n s p e c t i o n , i t c a n be s e e n how r a p i d l y ^ a p p r o a c h e s i t s maximum v a l u e w i t h e i t h e r i n c r e a s i n g o r d e c r e a s i n g v a l u e s o f A . On t h i s b a s i s , t h e r e s e a r c h e r c a n f o r m a n o p i n i o n a b o u t what may be t e r m e d t h e s e n s i t i v i t y o f t h e model, o r what amounts t o t h e same t h i n g , t h e u n i q u e n e s s o f t h e v a l u e o f A w h i c h d e t e r m i n e s t h e maximum v a l u e o f R . F o r example, i f t h e c u r v e o f A v e r s u s T h i s f l o w c h a r t was a d a p t e d f r o m D a v i d L . H u f f , D e t e r m i n a -t i o n o f I n t r a - U r b a n R e t a i l T r a d e A r e a s ( L o s A n g e l e s : U n i v e r s i t y o f C a l i f o r n i a , G r a d u a t e S c h o o l o f B u s i n e s s A d m i n i s t r a t i o n , D i v i s i o n o f R e s e a r c h , 1962), F i g u r e 5» P« 2 5 . 108 X has a d e f i n i t e and d e c i d e d p eak, t h e r e s e a r c h e r might be l e d t o c o n c l u d e t h a t t h e maximum R a n d A p a i r a r e q u i t e u n i q u e l y -d e t e r m i n e d , a n d t h a t t h e model i s q u i t e s e n s i t i v e t o t h e v a l u e o f t h e e x p o n e n t i a l p a r a m e t e r . . The c o n v e r s e w o u l d a l s o be t r u e . I f a l l , o r most, o f t h e A v a l u e s seemed t h u s u n i q u e l y d e t e r m i n e d , t h e r e s e a r c h e r m i g h t c o n s i d e r h i m s e l f on f i r m e r g r o u n d i n r e f e r r i n g t o a d i s t r i b u t i o n o f A v a l u e s . G i v e n s u c h a d i s t r i b u t i o n , i t m i g h t t h e n be a c c e p t a b l e t o p e r f o r m s t a t i s -t i c a l t e s t s o f s i g n i f i c a n c e on extreme v a l u e s . I f t h e v a l u e s a r e n o t so u n i q u e l y d e t e r m i n e d , t h e r e s e a r c h e r i s o n l y a b l e t o s a y t h a t t h e model i s i n s e n s i t i v e t o A ; t h a t i t i s m e a n i n g l e s s t o s p eak o f a p a r t i c u l a r v a l u e o f A a s b e i n g a p p r o p r i a t e ; and t h a t t h e model w i l l p r o b a b l y n o t p r o v i d e u s e f u l p r e d i c t i o n s . The n e e d f o r SUBROUTING SRNEG i s r e l a t e d t o t h e d i s c u s s i o f t h e s e n s i t i v i t y o f t h e model t o v a l u e s o f A , b u t h a s a d i f f e r e n t i n t e r p r e t a t i o n . T h o s e who a r e a c c u s t o m e d t o d e a l i n g w i t h c o e f f i c i e n t s o f d e t e r m i n a t i o n a s u s e d i n r e g r e s s i o n a n a l y s i s w i l l know t h a t - / ? * i s n e v e r n e g a t i v e . I t t h u s t o o k c o n s i d e r a b l e a n a l y s i s o f t h e programme r o u t i n e and t h e u n d e r -l y i n g m a t h e m a t i c s t o d e t e r m i n e why t h e computer r e p o r t e d n e g a t i v e v a l u e s f o r when p r o c e s s i n g e m p i r i c a l d a t a f o r V a n c o u v e r . R e f e r r i n g b a c k t o E q u a t i o n ( I V . 1 4 ) i n C h a p t e r I V, i t i s r e p r o d u c e d h e r e f o r ^ r a t h e r t h a n R , a s f o l l o w s : (V.l) J -109 I t w i l l be r e c a l l e d t h a t the denominator of the f r a c t i o n i n t h i s d e f i n i t i o n was intended t o d e s c r i b e p u r e l y random behaviour based on the P r i n c i p l e of I n s u f f i c i e n t Reason. The numerator, on the other hand, measures the improvement of f i t of the valu e s estimated from the model. Thus, i t w i l l be s m a l l e r than the denominator and the f r a c t i o n i s l e s s than one. I f the model p r e d i c t s p e r f e c t l y , the numerator i s zero, and/? i s one. But what happens i f , f o r some reason, the model does not f i t the data as w e l l as p u r e l y random behaviour? E x a c t l y t h i s : the numerator i s g r e a t e r than the denominator s i n c e the sum of squared d e v i a t i o n s of estimated behaviour from the a c t u a l behaviour i s g r e a t e r than the sum of squared d e v i a t i o n s of random behaviour from a c t u a l behaviour. In t h i s event, the f r a c t i o n i s g r e a t e r than one, and / ? g i s n e g a t i v e . SUBROUTINE SRNEG merely t e l l s the computer to r e p o r t t h i s r e s u l t , take a l a r g e step forward, and t r y a g a i n . But how can the model p r e d i c t behaviour whose r e l a t i o n -s h i p to a c t u a l behaviour i s worse than a p u r e l y random p a t t e r n ? On l o g i c a l grounds, t h i s c o u l d only be the case when the model misrepresents the f a c t o r s which a c t u a l l y determine behaviour. In o t h e r words, the model must be i n c o r r e c t i n some very fundamental way. Some suggestions as to how t h i s might be w i l l be made i n Chapter VI, along w i t h some proposed m o d i f i c a t i o n s to the b a s i c model. At t h i s time, however, i t i s merely s t a t e d t h a t the occurrence of n e g a t i v e v a l u e s of R over the whole range of v a l u e s of ~\ being c o n s i d e r e d would appear t o be very 110 s t r o n g e v i d e n c e t h a t t h e model s h o u l d be r e j e c t e d , a t l e a s t i n i t s p r e s e n t f o r m . One f u r t h e r comment n e e d s t o be made r e g a r d i n g programme XLAMC. I n t h i s r o u t i n e , a l l v a l u e s o f e s t i m a t e s o f s h o p p i n g t r i p s a r e i n c l u d e d i n c a l c u l a t i n g t h e p a r a m e t e r r e g a r d l e s s o f t h e s i z e o f t h e e s t i m a t e . I t w i l l be r e c a l l e d f r o m t h e d i s -c u s s i o n o f H u f f ' s s t u d y i n C h a p t e r IV t h a t a l l e s t i m a t e s o f s h o p p i n g t r i p s l e s s t h a n one were e q u a t e d t o z e r o a n d d i s r e g a r d e d i n c a l c u l a t i n g t h e p a r a m e t e r . H u f f d o e s n o t s t a t e why t h i s was done, n o r d o e s he i n c l u d e any comment r e g a r d i n g t h e e f f e c t t h a t t h i s m i g h t have on t h e e s t i m a t e o f t h e p a r a m e t e r . T h i s s t e p m i g h t p o s s i b l y have a v e r y s m a l l e f f e c t i n h i s c a s e where t h e sample f o r e a c h n e i g h b o u r h o o d c o n t a i n s a h u n d r e d o r more c o n s u m e r s . F o r t h e V a n c o u v e r sample, however, i t was t h o u g h t t h a t t h i s w o u l d p r o d u c e v e r y m i s l e a d i n g r e s u l t s , a s t h e s a m p l e s a r e o f t h e o r d e r o f o n e - t e n t h o f t h e s i z e u s e d by H u f f f o r e a c h n e i g h b o u r h o o d . A p p l i c a t i o n o f Programme XLAMC t o H u f f ' s D a t a I t i s an a c c e p t e d p a r t o f s c i e n t i f i c p r o c e d u r e when a p i e c e o f r e s e a r c h i s i n t e n d e d t o e x t e n d t h e work o f o t h e r s , t h a t t h e new r e s e a r c h d e s i g n be a b l e t o r e p r o d u c e t h e p r e c e d i n g r e s u l t s . F o r t h i s r e a s o n , i t was d e c i d e d t h a t t h i s s h o u l d a p p l y a l s o t o programme XLAMC. The d a t a u s e d were t h o s e p u b l i s h e d by H u f f.3 The r e s u l t s o f t h e s e t r i a l s a r e p r e s e n t e d i n T a b l e X 3 H u f f , op. c i t . ; T a b l e I I I , p.24; T a b l e IV, p . 2 5 ; T a b l e V, p . 2 6 , a n d T a b l e V I , p . 2 7 . I l l f o r c l o t h i n g , T a b l e X I f o r f u r n i t u r e , a n d a r e summarized i n T a b l e X I I . The b e s t c h e c k on t h e f i t between t h e two s e t s o f e x p e c t e d v a l u e s i s g i v e n by t h e r e s u l t s f o r n e i g h b o u r h o o d 2 f o r e i t h e r commodity. I n t h i s c a s e , H u f f ' s programme was n o t r e q u i r e d t o e q u a t e any e s t i m a t e s t o z e r o . I t c a n be s e e n t h a t t h e e s t i m a t e s o f t h e number o f consumers g o i n g t o e a c h s t o r e d i f f e r g e n e r a l l y o n l y i n t h e s e c o n d d e c i m a l p l a c e . T h i s i s c o n s i d e r e d t o be a n i n s i g n i f i c a n t amount. More i m p o r t a n t a r e t h e s l i g h t d i f f e r e n c e s n o t e d i n t h e r e s u l t i n g v a l u e s o f /? a n d A . T h i s c a n o n l y be a t t r i b u t e d t o t h e d i f f e r e n c e i n t h e way s i g n i f i c a n t d i g i t s were s a v e d i n t h e two c o m p u t a t i o n a l r o u t i n e s . A l t h o u g h H u f f makes no e x p l i c i t s t a t e m e n t i n t h i s r e g a r d , i t seems f r o m t h e way i n w h i c h t h e r e s u l t s a r e r e p o r t e d t h a t he u s e d o n l y two s i g n i f i c a n t f i g u r e s f o r & a n d f o u r s i g n i f i c a n t f i g u r e s ( t h r e e p l a c e s o f d e c i m a l s ) f o r A . On t h e o t h e r hand, programme XLAMC u s e d f i v e s i g n i f i c a n t f i g u r e s f o r A a n d t h r e e f o r A . How t h i s makes a d i f f e r e n c e i s e x p l a i n e d b e l o w . C o m p a r i n g t h e r e s u l t s o b t a i n e d by e a c h programme f o r n e i g h b o u r h o o d s 1 a n d 3 shows t h a t H u f f ' s d e c i s i o n t o e q u a t e e s t i m a t e s l e s s t h a n o n e - h a l f t o z e r o d o e s make a d i f f e r e n c e t o t h e v a l u e o b t a i n e d f o r A . Some m i g h t a r g u e t h a t t h e d i f f e r e n c e i s s i g n i f i c a n t , t h u s H u f f s h o u l d j u s t i f y h i s d e c i s i o n . O t h e r s m i g h t a r g u e t h a t , i n v i e w o f t h e f o l l o w i n g d i s c u s s i o n , t h e r e a r e more f u n d a m e n t a l i s s u e s t o i n v e s t i g a t e . One o f t h e b e n e f i t s o b t a i n e d by p r o c e s s i n g H u f f ' s d a t a TABLE X Comparison of Huff's Evaluation of Parameter with Programme XLAMC Evaluation (Clothing) Shopping Neighbourhood 1 Neighbourhood 2 Neighbourhood 3 Centre ( j ) Qbs. Exp. Huff Exp. XLAMC Obs. Exp. Huff Exp. XLAMC Obs. Exp. Huff Exp. XLAMC 1 71 70.76 70.53 148 144.28 143.49 143 l 4 l . 1+9 143.73 2 0 1.27 1.15 19 25.99 26.10 6 9.78 9.05 3 0 1.01+ 0.93 4 3.10 3.15 2 2.05 1.77 4 0 0.00 0.19 0 1.36 1.38 2 4.02 3-72 5 5 2.60 2.29 38 13.73 13.96 21 2.07 1.71 6 1 0.77 0.69 0 2.36 2.40 7 l . i + l 1.21 7 0 0.00 0.27 2 2.03 2.07 6 3.22 2.87 8 0 0.00 0.22 0 1.67 1.71 2 1.52 1.30 9 0 0.00 0.11 0 0.89 0.91 0 0.00 0.10 10 0 0.00 0.19 4 1.87 1.90 3 0.00 0.10 11 1 0.99 0.87 2 3-44 3.51 3 1.52 1.29 12 0 0.00 0.14 0 1.09 1.12 2 0.00 0.24 13 1 0.78 0.71 0 10.58 10.65 6 35.92 35.49 14 0 0.79 0.72 1 5.61 5.66 0 0.00 0.42 R = O.99 O.99885 Lambda = 2.889 2.9&0 R = O.98 O.98OO6 Lambda = 2.655 2.660 R = O.96 O.96312 Lambda = 3.69O 3-840 TABLE XI Comparison of Huff's Evaluation of Parameter with Programme XLAMC Evaluation (Furniture) Shopping  Centre ( j ) Obs. Neighbourhood 1 Exp. Huff Exp. XLAMC Obs. Neighbourhood 2 Exp. Huff Exp. XLAMC Obs. Neighbourhood 3 Exp. Huff Exp. XLAMC 1 51 51.66 5 1 . 0 4 68 6 5 . 8 3 6 5 . 2 6 81 7 8 . 2 1 7 9 . 2 8 2 0 1 . 5 0 1 . 3 7 k 1 6 . 7 2 1 6 . 7 3 1 7.43 6 . 9 9 3 0 1 . 3 0 1 .17 0 3-27 3-31 1 1-95 1.74 4 0 0 . 0 0 0 . 2 5 0 1 . 3 3 1.34 0 3 . 0 5 2 . 8 8 5 3 3 . 4 3 3 . 0 7 2k 1 4 . 0 6 1 4 . 2 6 11 2 . 3 0 1 . 9 8 6 6 0 . 9 8 0.88 6 2 . 5 5 2 . 5 9 12 1 . 3 8 1 . 2 2 7 o 0 . 0 0 0 . 3 8 3 2.17 2 . 2 1 3 2 . 7 8 2.54 8 0 0 . 0 0 0 . 3 3 0.£8 1 2 . 0 0 2 . 0 3 0 1 . 5 0 1.33 9 0 0 . 0 0 0 1 . 1 0 1 . 1 2 0 0 . 0 0 0 . 1 3 10 2 0 . 0 0 0 . 2 7 16 2 . 0 2 2 . 0 5 4 0 . 0 0 0 . 1 4 11 0 1 . 1 3 1 . 1 7 0 3.88 3.94 1 1 . 5 8 1 . 3 8 12 0 0 . 0 0 0 . 2 3 0 1 . 5 0 1.53 1 0 . 0 0 0 . 3 1 13 0 0 . 9 1 0 . 8 3 0 7 . 2 8 7 . 3 0 8 2 1 . 8 2 21.64 Ik 0 0 . 9 1 0 . 8 3 6 4 . 2 9 4 .32 0 0 . 0 0 0.43 R = 0 . 9 9 Lambda = 2 . 5 4 2 0 . 9 9 2 4 9 2 . 6 2 0 R = 0 . 9 4 Lambda = 2.115 0.93387 2.120 R = O . 9 6 Lambda = 3-247 0 . 9 5 9 9 9 3 . 3 7 0 H 1 TABLE XII Comparison of Huff and XLAMC Parameter Estimates  Commodity Neighbourhood Huff XLAMC Lambda • R Lambda R Clothing 1 2.889 0.99 2.98O O.99885 V 2 2.655 0.99 2.660 O.98OO6 I I 3 3.690 O.96 3-840 0.96312 Furniture 1 2.542 0.99 2.620 0.99249 it 2 2.115 0.94 2.120 0.93387 I I 3 3-247 O.96 3.370 0.95999 115 w i t h programme XLAMC was t h a t t h e CALL PLOTS r o u t i n e p r o d u c e d g r a p h s s h o w i n g t h e b e h a v i o u r o f / ? w i t h c h a n g e s i n A o v e r t h e r a n g e f o r 3 o f 0 .5 t o 4 . 5 i n i n c r e m e n t s o f 0 . 0 1 . W h i l e t h e o r i g i n a l p l o t s more t h a n f i l l e d an 8|*» x 11" page, t h e i n t e r -e s t i n g p a r t s were t h o s e w h i c h show t h e c u r v e a t o r n e a r i t s p e a k . T h e s e c u r v e s a r e r e p r o d u c e d h e r e a s F i g u r e 5 f o r c l o t h i n g , a n d F i g u r e 6 f o r f u r n i t u r e , t r a c e d e x a c t l y f r o m t h e o r i g i n a l s . On e x a m i n a t i o n , t h e o n l y c u r v e t h a t l o o k s a t a l l l i k e one m i g h t e x p e c t i s t h a t f o r f u r n i t u r e f o r n e i g h b o u r h o o d 2, b u t t h e v a l u e a t t h e peak i n t h i s c a s e i s n o t a b l y l e s s t h a n i n e v e r y o t h e r c u r v e . I n e v e r y o t h e r c a s e , t h e model h a s p r o d u c e d c o n s i s t e n t l y h i g h v a l u e s o f R o v e r a v e r y wide r a n g e o f v a l u e s f o r X . T h i s i s i n d e e d a v e r y s u r p r i s i n g r e s u l t when f i r s t e n c o u n t e r e d . However, s e v e r a l c o n c l u s i o n s may be drawn f r o m i t . F i r s t , one c a n r a i s e t h e q u e s t i o n w h e t h e r H u f f ' s t e s t s f o r s i g n i f i c a n c e o f p a r t i c u l a r A v a l u e s a r e m e a n i n g f u l . I t would seem n o t . S e c o n d , t h e model i s i n s e n s i t i v e t o v a l u e s o f o v e r a v e r y wide r a n g e . How wide t h i s r a n g e m i g h t be was n o t d e t e r -m i ned, s i n c e t h e programme was i n s t r u c t e d t o s t o p a t 4 . 5 . R e g a r d l e s s o f t h i s , however, t h i s r e s u l t means t h a t i n t h i s m o d e l any v a l u e o f /\ o v e r t h i s wide r a n g e w i l l p r e d i c t a b o u t a s w e l l a s any o t h e r v a l u e . T h i s r a i s e s a v e r y s e r i o u s q u e s t i o n a b o u t t h e v a l u e o f t h e m o d e l . When t h i s r e s u l t i s c o n t e m p l a t e d f u r t h e r , i t becomes l e s s s u r p r i s i n g . R e c a l l f r o m C h a p t e r I I t h a t i n a p p l y i n g t h i s t y p e 116 117 118 of gravimetric model elsewhere, various researchers reported values of the exponential parameter varying from 1 . 5 to 3.0 and higher. This raises a serious question about gravimetric models i n s o c i a l science i n general. Is the exponential parameter r e a l l y a s t r u c t u r a l or behavioural parameter as Is postulated, or i s i t a function of research design and the method of processing data? Before dealing any further i n such general issues, however, i t may be well to review the r e s u l t s of using programme XLAMC to handle data obtained i n the survey of Vancouver. Application of Programme XLAMC to Vancouver Data As pointed out above, the data for Vancouver were grouped to produce three composite commodities: A, B, and C. On t h i s basis, s u f f i c i e n t data i n f i n a l form were obtained for further processing for 42 neighbourhoods for commodity A, 26 neighbour-hoods for commodity B, and 40 neighbourhoods for commodity C. Because of the large number of neighbourhoods involved, the r e s u l t s of processing a l l t h i s amount of data with programme XLAMC w i l l not be presented i n the same d e t a i l as above, but w i l l be given only i n summary form. Before t h i s i s done, one additional comment should be made. When the curves of/? versus A were examined for a l l these computations, i t was found that few of them resembled the ones previously obtained i n processing Huff's data. In f a c t , they could be c l a s s i f i e d into three d i s t i n c t types as 119 shown i n F i g u r e 7 . B o t h Type I a n d Type I I c u r v e s had d i s t i n c t maxima i n t h e r a n g e o f A c o n s i d e r e d . The Type I c u r v e was t h e one w h i c h w o u l d n o r m a l l y have been e x p e c t e d t o o c c u r . Some Type I c u r v e s were s h a r p l y p e a k e d ; o t h e r s were r e l a t i v e l y b r o a d . W i t h Type I I I c u r v e s , t h e r e were two v a r i a n t s . The f i r s t , d e s i g n a t e d I I I + , h a d a s t e a d i l y i n c r e a s i n g v a l u e o f / ? w i t h i n c r e a s i n g A , b u t d i d n o t r e a c h a maximum i n t h e r a n g e c o n s i d e r e d . C o n v e r s e l y , t h e s e c o n d , d e s i g n a t e d I I I - , h a d a s t e a d i l y i n c r e a s i n g v a l u e o f w i t h d e c r e a s i n g A . I n most c a s e s o f t h i s t y p e o f c u r v e , i t a p p e a r e d t h a t i f a maximum o c c u r s , i t w o u l d be w i t h n e g a t i v e v a l u e s o f A . T h i s r e q u i r e s some i n t e r p r e t a t i o n on l o g i c a l g r o u n d s , b u t t h i s w i l l be i n c l u d e d l a t e r i n d i s c u s s i o n o f t h e i m p l i c a t i o n o f a l l A v a l u e s l e s s t h a n 1 . 0 . I n a d d i t i o n t o t h e s e t h r e e t y p e s o f c u r v e s , t h e r e were t h o s e c a s e s where no c u r v e was o b t a i n e d f o r any v a l u e s o f A i n t h e r a n g e c o n s i d e r e d . T h i s i s d e s i g n a t e d a s u n d e f i n e d a s i t o c c u r r e d when a l l v a l u e s o f ^ z were n e g a t i v e ( a s d i s c u s s e d above) a n d t h u s / ? was u n d e f i n e d i n t h e m a t h e m a t i -c a l s e n s e . The summary r e s u l t s o f p r o c e s s i n g t h e V a n c o u v e r s u r v e y d a t a a r e p r e s e n t e d i n T a b l e X I I I f o r Commodity A, T a b l e XIV f o r Commodity B, and T a b l e XV f o r Commodity C. E a c h t a b l e g i v e s : ( 1 ) t h e C e n s u s T r a c t number; (2) t h e maximum v a l u e o f /? o b t a i n e d a n d i t s a s s o c i a t e d v a l u e o f 7\ ( e a c h t o t h r e e s i g n i f i c a n t f i g u r e s ) when t h e s e v a l u e s were o b t a i n e d ; (3) t h e t y p e o f c u r v e o b t a i n e d , w h i c h shows w h e t h e r /V w o u l d be 1 2 0 TABLE XIII Results of A p p l i c a t i o n of Programme XLAMC f o r Commodity A is Tract R max Lambda Curve Type C i j 00 0.735 k.32 I I 5 07 0.675 I.56 I I 10 08 0.910 I.96 I I 11 09 - - III+ 13 10 O.968 •1-51 I. 8 11 0.318 1.11 I 11+ 12 - - undefined 11 Ik - - I I I - 8 15 0.95k 2.1+5 I I 3 16 - - undefined 8 17 0.923 1.59 I I 12 19 0.795 2.93 I I 8 21 0.895 3.71 I I 19 23 - - undefined 3 2k - - I I I - 8 25 0.245 2.11+ I 5 26 - - I I I - 5 27 - - I I I - 13 28 - - I I I - 8 29 - - I I I - 11 31 0.841 2.09 I I 7 32 O.885 3.54 I I 6 33 O.582 0.57 I 6 3k O.962 1.1+0 I k 35 0.639 1.1+0 I 9 36 O.887 2.39 I I 11+ 39 - - III+ 5 ko O.895 1-75 I 11 TABLE XIII (cont'd) Census Tract 4l 43 44 45 46 47 48 49 51 52 53 54 55 56 R max Lambda 0.732 0.94 0.717 1.27 0-773 O.98 0.820 0.76 0.964 1-97 0.823 1.45 0.426 1.09 0.531 2.92 O.709 1-39 O.896 1.45 0.961 2.27 0.929 2.50 Curve Type C l j I 9 I 13 I 6 I 7 I 9 I 7 I 5 I I I - 12 I I 2 I I I - 8 I 8 I I 9 I. 15 I 8 TABLE XIV Results of A p p l i c a t i o n of Programme XLAMC f o r Commodity B Census Tract R max Lambda Curve Type Ci,1 00 _ _ I I I - 3 07 - - III+ 2 08 - - undefined 2 11 - - undefined 2 12 0.610 2.31 I 4 16 - - III+ 4 17 - - III+ 2 21 0.917 3-6o I I 8 23 - - I I I - 4 24 - - I I I - 3 25 0.337 1.80 .1. 4 27 0.416 2.00 I I 2 28 - - . I I I - 2 29 - - I I I - 3 31 - - I I I - 2 35 0.792 1.83 I. 2 36 0.848 1.30 I I 5 ko 0.475 1.71 I 3 kl O.776 1.54 I 12 43 - - I I I - 7 46 0.982 2.98 I 6 kl 0.758 1.04 I 3 48 0.501 1.79 I I 2;. 49 0.824 2.31 I 6 55 - - I I I - 5 56 b.701 3.19 I I 2 124 TABLE XV Results of A p p l i c a t i o n of Programme XLAMC f o r Commodity C  Census Tract R max Lambda Curve Type C i j 00 - - III+ 2 07 - - III+ 4 08 - - III+ 10 09 - - III+ 3 10 - - III+ 2 12 0.473 1.34 .1 . 5 14 0.646 1.22 I I 5 15 - - undefined 3 16 0.356 2.81 I I 5 17 - - I I I - 5 19 - - I I I - 4 21 - - I I I - 6 23 - - undefined 6 24 - - I l l - 5 26 - - undefined 5 27 0.337 0.88 I 7 28 - - I I I - 3 29 - - I I I - 6 31 - - III+ 3 32 - - III+ 11 33 0.959 2.17 I 4 34 - - • i n v a l i d 3 35 0.926 1.28 I 9 36 0.552 1.12 I I 7 39 - - III+ 8 40 - - III+ 6 1 2 5 TABLE XV (cont'd.) :us Tract R max Lambda Curve Type ci,i kl 0.944 2.89 I 12 43 0.944 2.70 I 13 44 - - i n v a l i d 5 45 - - i n v a l i d 7 46 - - i n v a l i d 3 47 O.852 1.60 I 6 48 - - undefined 3 49 - - I I I - 9 51 0.574 0.71 I I 5 52 - - undefined 5 53 0.654 3-17 I 2 54 0.752 3.31 I I 2 55 - - III+ 8 56 0.823 3-88 .1. 5 126 above o r b e l o w t h e r a n g e c o n s i d e r e d i n t h e c a s e o f Type I I I c u r v e s , o r may n o t e x i s t i n t h e c a s e o f u n d e f i n e d c u r v e s ; ^ a n d (4) t h e t o t a l number o f consumers whose r e s p o n s e s p r o v i d e d d a t a i n e a c h C e n s u s T r a c t . On e x a m i n i n g t h e r e s u l t s p r e s e n t e d i n t h e t a b l e s a s shown, i t seemed t h a t t h e r e m i g h t be s e v e r a l t e n d e n c i e s i n t h e r e s u l t s . F i r s t , t h e r e a p p e a r e d t o be some r e l a t i o n s h i p i n v o l v i n g i n c r e a s i n g v a l u e s o f t h e maximum A? w i t h i n c r e a s i n g v a l u e s o f 2 . S e c o n d , i t a p p e a r e d t h a t t h e r e m i g h t be some s i g n i f i c a n c e i n t h e d i f f e r e n c e o f t h e A v a l u e s o b t a i n e d f r o m T y p e I a n d Type I I c u r v e s , t h e l a t t e r h a v i n g h i g h e r v a l u e s . T h i r d , i n a n e g a t i v e way, t h e r e d i d n o t seem t o be any r e l a t i o n -s h i p w h a t s o e v e r between t h e number o f p e r s o n s i n t h e sample f r o m e a c h n e i g h b o u r h o o d a n d t h e t y p e o f c u r v e o b t a i n e d . T h i s i n c l u d e s a l s o w h e t h e r o r n o t a maximum/? was f o u n d i n t h e r a n g e o f ^ c o n s i d e r e d . To examine t h e f i r s t two t e n d e n c i e s more c a r e f u l l y , s c a t t e r d i a g r a m s were drawn o f t h e maximum^? and A p a i r s . T h e s e a r e p r e s e n t e d h e r e a s F i g u r e 8 f o r Commodity A, F i g u r e 9 f o r Commodity B, a n d F i g u r e 10 f o r Commodity C. I t One a d d i t i o n a l c l a s s i f i c a t i o n o c c u r s i n T a b l e XV w h i c h h a s n o t b e e n m e n t i o n e d . T h i s i s t h e c a t e g o r y c a l l e d I n v a l i d , w h i c h i s i n t e n d e d t o d e s c r i b e a p e c u l i a r r e s u l t . I n t h i s s i t u a t i o n , f o r s e v e r a l C e n sus T r a c t s l o c a t e d n e a r O a k r i d g e , ^ became 1.0 (meaning c e r t a i n t y ) f o r a l l v a l u e s o f /\ s l i g h t l y g r e a t e r t h a n 2.0. H e r e a g a i n , one c o u l d h a r d l y s p eak o f a m e a n i n g f u l R-7\ p a i r . 128 129 i. ..: i . . f/GUfilE: Jo -t ' f ' I. I.. . + _ ) — i. . » : 4 • : ' i ; ; T ; r ; : l I I ; I . • I | 4 H f : P I; .1 . ...Li" I X i n i M r A H T T i T ~ •— o.Z 4 - -; r h :! -. if> • i ! 1.Tt - i - T — t -i- I T l LL. J.. i.JT . 1 U " ! H I T t r I 1 .l.L.-LJ. Tt: ,<3D 1- ! I 1 j(Ti .1 A J..J. — ! - -i 1 I T -pr— t™ 1 -. - -J .J. [!•!:;. TT t 1-HI I t ! • - i i I I P -r r r i -i ! i i i i: yr p i l l L i l .1 ..  j j 1 i ! .1.. + ,_ i ; '. t • 1 ;;t i I f. h i " . \.., ; u . i- i- i i U i " it;©. ! i - i -. j..; j ; . ;. j . -i . • . . t . l l r r : t i ! • i I . J j i . J i . i - i i ! ! |' • \\\'\ 1 * ' 1 Cornrno<// ttjr Q r I 1 i i ' I : • •• I *; • ' i i ! ' I i' ! J i~r'-T j ! j . : t i ! r i I n J : . t i i xr -t—i 4 t I ! H I j t i.L i ! I ' h ' : , ' I i ; ; ! i [••Ai .^4 -•• . ' i -I i 1.!":.' I l.. ; I I i i •f I-130 was o b s e r v e d t h a t t h e s c a t t e r was so g r e a t t h a t t h e t e n d e n c i e s n o t e d r e m a i n m e r e l y a s t e n d e n c i e s , a n d no f u r t h e r a c t i o n was t a k e n on t h e s e d i a g r a m s . W h i l e some m i g h t a r g u e t h a t i t w o u l d be p o s s i b l e t o f i t a s t r a i g h t l i n e o r a c u r v e t o t h e s e p o i n t s , o r p e r h a p s p e r f o r m a r a n k c o r r e l a t i o n , t h e i n t e r p r e t a -t i o n o f t h e r e l a t i o n s h i p d e s e r v e s some t h o u g h t . The i n t e r -p r e t a t i o n w o u l d be t h a t g r e a t e r v a l u e s o f 3 p r o d u c e g r e a t e r powers o f a c c o u n t ( e x p l a i n e d v a r i a t i o n ) i n t h e m o d e l . The e v i d e n c e d o e s n o t seem s t r o n g enough f o r s u c h a s t a t e m e n t t o be made w i t h t h e d e g r e e o f c o n v i c t i o n t h a t a r e g r e s s i o n c u r v e w o u l d i m p l y . I n a d d i t i o n t o t h e above t a b l e s a n d f i g u r e s , i t seemed u s e f u l t o b r i n g a l l t h e r e s u l t s t o g e t h e r i n one place.. T h u s , T a b l e XVI s h o w i n g t h e maximum a n d 7\ p a i r s f o r e a c h C e n s u s T r a c t , f o r e a c h commodity, was p r e p a r e d . A g a i n , on e x a m i n i n g t h e r e s u l t s p r e s e n t e d i n t h i s way, i t i s v i r t u a l l y i m p o s s i b l e t o d e t e c t any t r e n d s , n o t e v e n t h a t o f c e n t r a l t e n d e n c y . F o r t h i s r e a s o n , a s w e l l a s t h e r e a s o n s a d v a n c e d above i n d i s c u s s -i n g c u r v e t y p e s , t h e r e d o e s n o t seem t o be any j u s t i f i c a t i o n f o r c o n d u c t i n g any f u r t h e r t e s t s o f s t a t i s t i c a l s i g n i f i c a n c e b a s e d on t h e d i s t r i b u t i o n o f A v a l u e s . The r e s u l t s o f t h i s s u r v e y , and t h e d a t a p r o c e s s e d f r o m i t , must be examined w i t h r e s p e c t t o H u f f ' s h y p o t h e s i s t h a t t h e v a l u e o f t h e e x p o n e n t i a l p a r a m e t e r i s a f u n c t i o n o f t h e t y p e o f s h o p p i n g t r i p b e i n g c o n s i d e r e d . T h i s was a l s o t h e h y p o t h e s i s b e i n g t e s t e d by t h e V a n c o u v e r s u r v e y , s i n c e t h e TABLE XVI Table of Composite Results  Census Tract Commodity A Commodity B Commodity C R max Lambda R max Lambda R max Lambda 00 0.735 4-32 I I I - III+ 07 0.675 1.56 III+ III+ 08 0.910 I . 9 6 undefined III+ 09 III+ n i * III+ 10 O.968 1.51 n i . III+ 11 0.318 ; 1 .11 undefined n i 12 undefined 0.610 2.31 0.473 1-34 14 I I I - n i 0.646 1.22 15 O.954 2.45 n i undefined 16 undefined III+ O.356 2.81 17 O.923 1.59 III+ I I I -19 0-795 2-93 n i I I I -21 O.895 3-71 O.917 3.60 I I I -23 undefined I I I - undefined 2k I I I - I I I - I I I -25 0.245 2.14 0.337 .1.80 n i 26 I I I - n i undefined 27 I I I - 0.416 2.00 0.337 0.88 28 I I I - I I I - I I I -29 I I I - I I I - I I I -31 0.841 2.09 I I I - III+ 32 0.885 3-54 n i . III+ 33 O.582 O .57 n i 0.959 2.17 3k 0.902 1.40 n i i n v a l i d 35 O.639 1-40 O.792 I . 8 3 O.926 1.28 36 O.887 2-39 0.848 1.30 O.552 1.12 39 III+ n i III+ ko O.895 1-75 0.475 1.71 III+ kl O.732 O.94 0.776 1.54 0.944 2.89 k3 0.717 1.29 I I I - 0.944 2.70 kk O.773 O .98 n i i n v a l i d 132 TABLE XVI (cont'd) Census Tract Commodity A Commodity B Commodity C R max Lambda R max Lambda R max Lambda 45 0.820 O.76 n i i n v a l i d 46 0-964 1.97 O.982 2.98 inwalid 47 0.823 1.45 O.258 1.04 O.852 1.60 48 0.426 1.09 0.501 1.79 undefined 49 III- 0.824 2.31 I I I -51 0.531 2.92 n i 0.574 0.710 52 III- n i undefined 53 O.709 1.39 n i 0.654 3-17 54 0;896 1.45 n i 0.752 3-31 55 0.901 2.27 I I I - III+ 56 O.929 2.50 0.701. . 3.19 0.823 3-88 * The designation n i means t h i s Census Tract was not „ commodity. included f o r the p a r t i c u l a r 133 p u r p o s e was t o p e r f o r m a b r o a d e r t e s t on H u f f ' s m o d e l . I t c a n o n l y be c o n c l u d e d t h a t t h e r e s u l t s a r e so d i f f u s e a s t o n o t l e n d any s u p p o r t t o t h e h y p o t h e s i s . I t might be a r g u e d t h a t t h e s a m p l e s i n e a c h C e n s u s T r a c t were t o o s m a l l t o p r o d u c e any amount o f c e n t r a l t e n d e n c y . T y p i c a l l y i n s t a t i s t i c a l h y p o t h e s i s t e s t i n g , s a m p l e s o f l e s s t h a n 30 i t e m s a r e c l a s s e d a s s m a l l sample c a s e s , a n d a r e t r e a t e d w i t h a m o d i f i e d body o f t h e o r y a n d t e c h n i q u e . C e r t a i n l y , i t i s v e r y t e m p t i n g t o c o n c l u d e t h a t t h e s a m p l e s were t o o s m a l l compared t o H u f f ' s n e i g h b o u r h o o d s a t u r a t i o n t e c h n i q u e . B u t , i f t h e model r e q u i r e s t h i s t y p e o f h a n d l i n g i n e v e r y c a s e , i t i s much t o o e x p e n s i v e t o be g e n e r a l l y u s e f u l . F u r t h e r , b e f o r e r e a c h i n g any c o n c l u s i o n s r e g a r d i n g t h e v a l i d i t y o f t h e r e s u l t s o f t h e V a n c o u v e r s u r v e y , i t may be w e l l t o c o n s i d e r t h e r e s u l t s o f t r e a t i n g t h e d a t a somewhat d i f f e r e n t l y . A p p l i c a t i o n o f Programme XLAMS t o V a n c o u v e r D a t a One o f t h e o r i g i n a l h y p o t h e s e s u n d e r l y i n g t h i s t h e s i s — n o t s t a t e d p r e v i o u s l y — was t h a t t h e r e i s a s i g n i f i c a n t d i f f e r e n c e i n c o n s u m e r s ' a t t i t u d e s r e g a r d i n g s t o r e s l o c a t e d downtown v e r s u s s t o r e s l o c a t e d i n s h o p p i n g c e n t r e s . T h i s i s d e r i v e d f r o m t h e s t u d y by J o n a s s e n w h i c h s o u g h t t o compare a t t i t u d e s o f consumers t o w a r d s t h e two t y p e s o f l o c a t i o n s . - ' C T . J o n a s s e n , The S h o p p i n g C e n t e r v e r s u s Downtown (Columbus, O h i o : B u r e a u o f B u s i n e s s R e s e a r c h , C o l l e g e o f Commerce and A d m i n i s t r a t i o n , The O h i o S t a t e U n i v e r s i t y , 1955)» pp. 89-100. 134 I f t h i s i s s o , t h e r e s h o u l d be a s i g n i f i c a n t d i f f e r e n c e i n t h e e x p o n e n t i a l p a r a m e t e r o f a model f o r m a l l y t h e same a s H u f f ' s , b u t a p p l i e d t o s t o r e s r a t h e r t h a n n e i g h b o u r h o o d s . The h y p o t h e s i s was t e s t e d by, i n e f f e c t , t u r n i n g t h e c o m p u t a t i o n a l r o u t i n e u s e d p r e v i o u s l y on i t s s i d e t o p r o d u c e a programme named XLAMS. The programme i s t h e same a s XLAMC e x c e p t t h a t s t o r e s a n d n e i g h b o u r h o o d s a r e i n t e r c h a n g e d . Whereas XLAMC computed by n e i g h b o u r h o o d s a c r o s s a l l s t o r e s , XLAMS computed by s t o r e s a c r o s s a l l n e i g h b o u r h o o d s . The r e s u l t s o f a p p l y i n g programme XLAMS t o t h e same e m p i r i c a l d a t a f o r V a n c o u v e r a s b e f o r e a r e summarized i n T a b l e X V I I . Note t h a t t h e p a r a m e t e r i s now c a l l e d by a d i f f e r e n t name,^ , t o d i s t i n g u i s h i t f r o m t h e p r e v i o u s p a r a m e t e r . T h i s parameter<7 i s a c h a r a c t e r i s t i c o f s t o r e s , w hereas t h e p r e v i o u s p a r a m e t e r X was a c h a r a c t e r i s t i c o f n e i g h b o u r h o o d s . The r e s u l t s o f t h i s t r e a t m e n t a r e r e l a t i v e l y c o n c l u s i v e . W h i l e g r a p h s were n o t o b t a i n e d f o r e v e r y s t o r e and e v e r y commodity — s i n c e t h e sample was h e a v i e r i n some a r e a s t h a n o t h e r s due t o t h e a r r a n g e m e n t o f C e n s u s T r a c t s — t h e r e a r e a t l e a s t two t h i n g s t o be n o t e d . F i r s t , t h e v a l u e o f t h e p a r a m e t e r V. l i e s w i t h i n t h e r a n g e 0.51 t o 0.80 f o r m a j o r downtown s t o r e s . F o r s u b u r b a n s t o r e s , t h i s p a r a m e t e r l i e s i n t h e r a n g e 0.4-9 t o 2 . 4 9 . S e c o n d , t h e power o f a c c o u n t (measure o f e x p l a i n e d v a r i a t i o n ) i s a l s o s i g n i f i c a n t l y d i f f e r e n t b etween downtown a n d s u b u r b a n s t o r e s . F o r downtown, s t o r e s R*ls l e s s t h a n 10$ ( e v e n l e s s t h a n 5%), whereas f o r s u b u r b a n s t o r e s i t i s a s much 135 TABLE XVII Results of A p p l i c a t i o n of Programme XLAMS to Vancouver Data Store Location Commodity A Commodity B Commodity C R R R 1 c e n t r a l 0.232 O.51 0.137 0.79 0.209 0 . 8 0 2 c e n t r a l O.283 0 .61 undefined 0.21+8 0 . 6 8 3 suburb est. O.39 at 0.1+9 0.1+15 1.12 0.1+2 at O.51 k suburb undefined undefined 0.115 2.1+9 5 c e n t r a l undefined undefined 0.090 0.610 6 suburb 0.691+ 1.72 Note (c) 0.68327 2.21+ 7 suburb undefined undefined undefined 8 Note (a) 0.321+ O.99 undefined 0.112 0.64 9 suburb. undefined undefined undefined 10 Note (b) 0.227 1.01+ 0.61+3 2.83 undefined (a) This store i s suburban i n terms of Vancouver, but ce n t r a l to the outly i n g M u n i c i p a l i t y of Burnaby (north) (b) While t h i s store i s c e n t r a l l y located, i t i s l e s s than one-tenth the size of other c e n t r a l stores. Further, i t i s noted f o r i t s p r i c e appeals i n low-priced l i n e s of merchandise such as shoes, c l o t h i n g and other high volume items. (c) This commodity has a curve of Type III+, and maximum would be at a va&ue greater . „ than I+.5. 1 3 6 a s kO%. I n any c a s e , t h e r e s u l t s t e n d t o s u p p o r t J o n a s s e n ' s c o n c l u s i o n s , w h i c h were r e a c h e d by q u i t e a n o t h e r r o u t e . F u r t h e r , t h i s a g r e e m e n t t e n d s t o s u p p o r t t h e v a l i d i t y o f t h e e m p i r i c a l d a t a f o r V a n c o u v e r , i n s p i t e o f t h e r e l a t i v e l y s m a l l sample s i z e s . The p a r a m e t e r i n c l u d e s , i m p l i c i t l y , many o f t h e f a c t o r s w h i c h J o n a s s e n i d e n t i f i e s a n d examines e x p l i c i t l y . However, t h e r a t h e r low power o f a c c o u n t o f t h e H u f f - t y p e model i n t h i s c o n t e x t t e n d s t o i n d i c a t e t h a t i t o m i t s a g r e a t d e a l . E x a m i n a t i o n o f some i n d i v i d u a l e n t r i e s i n T a b l e X V I I shows some a n o m a l i e s , s p e c i f i c a l l y t h e v a l u e s of?7 d e t e r m i n e d f o r S t o r e s 3» 8, a n d 10. T h e s e w i l l e a c h be c o n s i d e r e d s e p a r a t e l y . F i r s t , l o o k i n g a t S t o r e 8 ( S i m p s o n - S e a r s i n B u r n a b y ) , i t was n o t e d i n t h e t a b l e t h a t t h i s s t o r e i s s u b u r b a n w i t h r e s p e c t t o V a n c o u v e r ' s c e n t r a l c o r e b u t i s c e n t r a l i n c e r t a i n r e s p e c t s w i t h i n t h e m u n i c i p a l i t y o f B u r n a b y . F u r t h e r , i t i s r e l a t i v e l y l a r g e compared t o o t h e r s t o r e s i n s h o p p i n g c e n t r e s , b e i n g t h e s o l e o c c u p a n t o f i t s l o c a t i o n and t h e l a r g e s t o u t l e t w h i c h t h i s c h a i n h a s i n B r i t i s h C o l u m b i a . T h u s , i t c a n n o t be c l a s s i f i e d w i t h e i t h e r o f t h e n e a t l y d i c h o t o m i z e d t y p e s o f c e n t r a l d e p a r t m e n t s t o r e s and s u b u r b a n s h o p p i n g c e n t r e s . The r e s u l t s o f programme XLAMS a r e s i m i l a r l y a m b i v a l e n t , t h o u g h t h e t e n d e n c i e s a r e i n a g r e e m e n t , a n d t h i s i s n o t u n e x p e c t e d . S e c o n d , on c o n s i d e r i n g S t o r e 10 (Army an d N a v y ) , l o c a t e d i n t h e c e n t r a l c o r e , t h e r e s u l t s a r e n o t c l e a r - c u t . T h i s s t o r e i s a b o u t o n e - t e n t h o f t h e s i z e o f o t h e r m a j o r d e p a r t m e n t s t o r e s 137 a n d h a s a m e r c h a n d i s i n g p o l i c y w h i c h i s q u i t e d i f f e r e n t f r o m t h e s e o t h e r r e t a i l o u t l e t s . T h i s p o l i c y may be d e s c r i b e d a s a i m i n g a t h i g h - v o l u m e u s i n g p r i c e a p p e a l on low p r i c e - r a n g e l i n e s o f m e r c h a n d i s e . T h i s p o l i c y i s i m p l e m e n t e d by b u y i n g t h e b r o k e n o r d i s c o n t i n u e d l i n e s o f w h o l e s a l e r s a n d o t h e r r e t a i l e r s a t o r a f t e r t h e peak o f t h e r e g u l a r s e a s o n , a t l i q u i -d a t i o n p r i c e s . T h u s , t h e p r o d u c t a s s o r t m e n t c a r r i e d i n t h e s t o r e i s f r e q u e n t l y o p p o r t u n i s t i c a l l y d e t e r m i n e d . Consumers may be somewhat more d u b i o u s o f t h e i r s u c c e s s i n s h o p p i n g a t t h i s s t o r e . As w i l l be d i s c u s s e d e x t e n s i v e l y below, t h i s may be r e f l e c t e d i n a somewhat h i g h e r v a l u e f o r Tf f o r Commodity A ( c l o t h i n g ) t h a n f o r o t h e r c e n t r a l s t o r e s . S i m i l a r l y , t h i s s t o r e was n o t n o t e d a s a f u r n i t u r e emporium, and t h i s seems t o be r e f l e c t e d i n t h e h i g h v a l u e o f TJ f o r Commodity B ( f u r n i t u r e ) . F i n a l l y , c o n s i d e r i n g S t o r e 3 (Woodward's a t O a k r i d g e ) , t h e r e s u l t s a r e somewhat c o n f l i c t i n g . The v a l u e s a r e as l o w a s f o r c e n t r a l s t o r e s b u t t h e A v a l u e s a r e m o d e r a t e l y h i g h e r . P e r h a p s t h i s anomaly c a n be a c c o u n t e d f o r by r e -i n t r o d u c i n g c e r t a i n i d e a s d e v e l o p e d i n C h a p t e r IV. I t may be r e c a l l e d t h a t t h e b a s i c model was d e v e l o p e d by p o s t u l a t i n g t h a t consumers f o r m a s u b j e c t i v e p r o b a b i l i t y e s t i m a t e o f s u c c e s s r e g a r d i n g a p a r t i c u l a r s t o r e f o r p a r t i c u l a r i t e m s , and t h a t t h i s c o u l d be r e p r e s e n t e d by s t o r e a r e a . However, t h i s w o u l d be m o d i f i e d by d i s t a n c e t o p r o d u c e a l i k e l i h o o d o f s h o p p i n g . B u t , i f t h e s u b j e c t i v e p r o b a b i l i t y e s t i m a t e s were t o be b a s e d on f i r m e r k n o w l e d g e , an d t h u s were more c e r t a i n , t h e d i s t a n c e f a c t o r 138 w o u l d have much l e s s i n f l u e n c e on l i k e l i h o o d . The model may f i t t h i s s t o r e b e t t e r t h a n downtown d e p a r t m e n t s t o r e s b u t t h e i n f l u e n c e o f d i s t a n c e i s l e s s t h a n f o r o t h e r s u b u r b a n s h o p p i n g c e n t r e s . T h i s e x p l a n a t i o n was s u g g e s t e d by e x a m i n i n g t h e sample o f consumers who h a d p a t r o n i z e d O a k r i d g e . I t was f o u n d t h a t t h e y t e n d e d t o r e s i d e i n f a i r l y w e l l - d e f i n e d n e i g h b o u r h o o d s known t o be o f m o d e r a t e l y h i g h a n d u n i f o r m s t a t u s i n a s o c i o -e c o n o m i c s e n s e . Many o f t h e s e p e o p l e have a p a t t e r n o f s h o p p i n g a t O a k r i d g e , a n d t h u s a r e p r o b a b l y more k n o w l e d g e a b l e r e g a r d i n g t h e a v a i l a b l e p r o d u c t mix. F u r t h e r , t h e O a k r i d g e c e n t r e , a s a c e n t r e , aims a c o n s i d e r a b l e amount o f a d v e r t i s i n g a t t h e p a r t i -c u l a r s o c i a l s t r a t a w h i c h a r e w e l l r e p r e s e n t e d among t h e n e i g h b o u r h o o d s h a v i n g a n o t a b l e t e n d e n c y t o w a r d p a t r o n i z i n g O a k r i d g e . T h i s , c ombined w i t h t h e e f f e c t s o f s t a t u s c o n g r u e n c y , may a c c o u n t f o r t h e c o n f l i c t i n g r e s u l t s o b t a i n e d f o r t h i s p a r t i c u l a r s h o p p i n g c e n t r e . Many o f t h e s e f a c t o r s , w h i c h a r e m e r e l y m e n t i o n e d h e r e , a r e examined i n more d e t a i l i n C h a p t e r V I . I t i s e m p h a s i z e d a g a i n t h a t w h i l e t h e m o d e l s a p p l i e d t o s t o r e s a n d a p p l i e d t o n e i g h b o u r h o o d s a r e t h e same i n a f o r m a l m a t h e m a t i c a l s e n s e , t h e y a r e q u i t e d i f f e r e n t i n t h e i r i n t e r -p r e t a t i o n s when a p p l i e d i n d i f f e r e n t c o n t e x t s . T h e s e d i f f e r e n c e s a r e , u n f o r t u n a t e l y , h i d d e n i n t h e many f a c t o r s w h i c h a r e i m p l i c i t i n t h e e x p o n e n t i a l p a r a m e t e r . However, t h e l o g i c a l argument b e i n g made i s t h a t t h e r e s u l t s o f u s i n g t h e e m p i r i c a l d a t a f o r V a n c o u v e r i n programme XLAMS a r e c o n s i s t e n t w i t h p r e v i o u s , 139 i n d e p e n d e n t l y c o n d u c t e d r e s e a r c h a s w e l l a s w i t h what i s b e l i e v e d t o be t r u e a b o u t V a n c o u v e r consumers an d r e t a i l i n s t i t u t i o n s . T h e r e f o r e , t h e r e s u l t s o f u s i n g t h e s e same d a t a i n programme XLAMC a r e h e l d t o be v a l i d a n d r e p r e s e n t a t i v e , d e s p i t e t h e i r i n c o n c l u s i v e n a t u r e . Thus, th e d i f f u s e n a t u r e o f t h e s e r e s u l t s c o n s t i t u t e s a t e s t o f H u f f ' s h y p o t h e s i s i n t h e a p p r o p r i a t e c o n t e x t b u t d o e s n o t s u p p o r t t h e h y p o t h e s i s , n o r t h e p r e d i c t i v e a b i l i t y o f t h e model when a p p l i e d t o n e i g h b o u r h o o d s i n V a n c o u v e r . C o n c l u d i n g Remarks I t was m e n t i o n e d above t h a t H u f f ' s c o m p u t a t i o n a l r o u t i n e i n c l u d e d o n l y e s t i m a t e s o f t h e e x p o n e n t i a l p a r a m e t e r g r e a t e r t h a n one. W h i l e H u f f d o e s n o t o f f e r any e x p l a n a t i o n f o r i n c l u d i n g o n l y v a l u e s g r e a t e r t h a n one, i t w i l l be s e e n b e l o w t h a t t h e a s s u m p t i o n t h a t a l l v a l u e s must be g r e a t e r t h a n one a v o i d s some d i f f i c u l t l o g i c a l e x p l a n a t i o n s . S i n c e i t h a d n o t b e e n d e m o n s t r a t e d t h a t t h e p a r a m e t e r c o u l d n o t have v a l u e s l e s s t h a n one, t h e programmes XLAMC a n d XLAMS began t h e i r i t e r a t i o n s by a s s u m i n g a v a l u e f o r t h e p a r a -m e t e r o f 0 . 5 . As i t t u r n e d o u t , t h e r e s u l t s o f t h e c o m p u t a t i o n s p r o d u c e d a s i g n i f i c a n t number o f e s t i m a t e s o f t h e p a r a m e t e r l e s s t h a n one f o r b o t h programmes. I n a d d i t i o n , programme XLAMC p r o d u c e d some Type I I I - c u r v e s w h i c h a p p e a r e d a s i f t h e maximum v a l u e o f R — i f i n d e e d t h e r e i s a maximum — w o u l d o c c u r w i t h n e g a t i v e v a l u e s o f t h e p a r a m e t e r . E a c h o f t h e s e 140 s i t u a t i o n s m e r i t s i n d i v i d u a l i n t e r p r e t a t i o n . C o n s i d e r i n g f i r s t t h e c a s e o f programme XLAMC where e s t i m a t e s o f t h e p a r a m e t e r were l e s s t h a n one, b u t g r e a t e r t h a n z e r o , i t w o u l d seem t h a t t h e model was b e h a v i n g a s i f d i s t a n c e ( t i m e ) were a r e l a t i v e l y m i n o r f a c t o r . T h i s f o l l o w s f r o m t h e f a c t t h a t any p o s i t i v e e x p o n e n t l e s s t h a n one r e d u c e s t h e v a l u e o f t h e number w i t h w h i c h i t i s a s s o c i a t e d . S i n c e t h i s d i s t a n c e measure a p p e a r s i n t h e d e n o m i n a t o r o f a f r a c t i o n , s u c h a n e x p o n e n t would t e n d t o i n c r e a s e t h e e f f e c t o f f a c t o r s w h i c h a r e i n c l u d e d i n t h e n u m e r a t o r — i n t h i s c a s e , s t o r e a r e a . I t w i l l be r e c a l l e d a l s o t h a t t h e power o f a c c o u n t , & , showed a t e n d e n c y t o be l e s s w i t h l o w e r v a l u e s o f t h e p a r a m e t e r . T h i s i s c o n s i s t e n t w i t h o b s e r v a t i o n t h a t t h e model i s d e -e m p h a s i z i n g d i s t a n c e a s a f a c t o r , a n d w e i g h i n g more h e a v i l y o t h e r f a c t o r s w h i c h a r e o n l y p a r t i a l l y i n c l u d e d i n t h e n u m e r a t o r o f t h e model a s f o r m u l a t e d . The s e c o n d s i t u a t i o n i n v o l v e s programme XLAMC i n t h o s e c a s e s where t h e p a r a m e t e r w o u l d a p p e a r t o be n e g a t i v e . M a t h e m a t i c a l l y , a n e g a t i v e e x p o n e n t o c c u r r i n g i n t h e d e n o m i n a t o r becomes p o s i t i v e when t h e whole e x p r e s s i o n i s moved t o t h e n u m e r a t o r . T h i s means t h a t t h e d i s t a n c e f a c t o r t h e n m u l t i p l i e s t h e s t o r e a r e a f a c t o r . Two i n t e r p r e t a t i o n s may be p u t on t h i s . F i r s t , i n t e r m s o f b e h a v i o u r , t h i s w o u l d s u g g e s t t h a t t h e f u r t h e r away a s t o r e was f r o m t h e consumer, t h e more a t t r a c t i v e i t w o u l d become. W h i l e some i n d i v i d u a l s u n d o u b t e d l y e n j o y j o u r n e y i n g t o shop f o r i t s own s a k e , t h e s i t u a t i o n a p p e a r e d a s 141 i f i t w o u l d o c c u r t o o f r e q u e n t l y f o r t h i s e x p l a n a t i o n t o be l o g i c a l l y a c c e p t a b l e . A more t e n a b l e e x p l a n a t i o n , t h o u g h a r a t h e r m e c h a n i c a l one, i s a s f o l l o w s . The programme XLAMC was w r i t t e n i n s u c h a way a s t o d e t e r m i n e o n l y a g r o u p o f numbers, t h e / ^ ' s , w h i c h a r e i n t e n d e d t o measure t h e power o f a c c o u n t o f t h e model w i t h r e s p e c t t o c e r t a i n e m p i r i c a l o b s e r v a t i o n s s u p p l i e d t o i t . I t may be t h a t i f t h e programme had been a l l o w e d t o r a n g e downward o v e r v a l u e s o f t h e p a r a -m e t e r , t h r o u g h z e r o , a n d i n t o n e g a t i v e v a l u e s o f i n c r e a s i n g a b s o l u t e m a g n i t u d e , i t wo u l d have p r o d u c e d two i d e n t i c a l p e a k s . The f i r s t peak m i g h t have o c c u r r e d w i t h a s m a l l p o s i t i v e v a l u e o f t h e p a r a m e t e r a n d t h e s e c o n d w i t h a n e g a t i v e v a l u e o f more s u b s t a n t i a l a b s o l u t e m a g n i t u d e . I n e i t h e r c a s e , t h e p a r a m e t e r s w o u l d be t h e same i n t h e i r e f f e c t on t h e n u m e r a t o r o f t h e f r a c t i o n ; one by d i v i s i o n by a s m a l l number, t h e o t h e r by m u l t i p l i c a t i o n by a l a r g e number. The i n t e r p r e t a t i o n o f t h i s m e c h a n i c a l e x p l a n a t i o n w o u l d be t h e same a s f o r o t h e r v a l u e s o f t h e p a r a m e t e r l e s s t h a n one; namely, more w e i g h t w o u l d be s h i f t e d t o o t h e r f a c t o r s b e s i d e s d i s t a n c e . F i n a l l y , c o n s i d e r i n g t h e r e s u l t s o f programme XLAMS, i t seems t o be l o g i c a l t h a t d i s t a n c e i s much l e s s i m p o r t a n t i n d e t e r m i n i n g t h e a t t i t u d e s t h a t c o n sumers h o l d t o w a r d m a j o r downtown s t o r e s t h a n i n t h e c a s e o f s h o p p i n g c e n t r e s . I t i s s u s p e c t e d t h a t V a n c o u v e r r e s i d e n t s a r e n o t v e r y d i f f e r e n t i n t h i s r e s p e c t t h a n r e s i d e n t s o f o t h e r c i t i e s w h i c h have a w e l l -d e f i n e d c o r e a r e a f o r s h o p p i n g . J o n a s s e n ' s s t u d y t e n d s t o 142 support t h i s view, as he s t a t e s that "the e f f e c t of distance was minimized or overcome under c e r t a i n s i t u a t i o n s and c o n d i t i o n s by the presence of other v a r i a b l e s " . Why then might Huff have obtained such encouraging r e s u l t s with h i s model i n Los Angeles, d e a l i n g only i n suburban shopping centres? Perhaps i t i s because the area i s so d i f f e r e n t i n i t s s t r u c t u r e that the model works reasonably w e l l i n i t . Los Angeles i s known to be a c i t y made up of a conglomeration of suburbias. I t does not have a w e l l - d e f i n e d and e s t a b l i s h e d c e n t r a l core, l i k e Vancouver and many other c i t i e s . I n t h i s way, i t i s much more analogous as an urban s e t t i n g to the in t e r - u r b a n s i t u a t i o n w i t h f a i r l y uniform t r a d i n g centres as o r i g i n a l l y contemplated by R e i l l y . On the other hand, Converse's unbalanced model, having one or s e v e r a l r e l a t i v e l y l a r g e t r a d i n g c e n t r e s , i s more c l o s e l y analogous to c i t i e s having well-developed cores. I t seems that a w e l l -developed urban shopping core d i s t o r t s the s i t u a t i o n such that the model r e q u i r e s other f a c t o r s beyond space and distance to account f o r observed behaviour. Some of these f a c t o r s were mentioned b r i e f l y above and w i l l be discussed more f u l l y i n Chapter V I . Jonassen, op. c i t . , p. 91-143 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS Restatement of the Problem As s t a t e d i n Chapter I, the problem to which t h i s t h e s i s was addressed was th a t of making an e m p i r i c a l t e s t of a p r o b a b i l i s t i c model of i n t r a - u r b a n r e t a i l trade i n t e r a c t i o n s . S p e c i f i c a l l y , the model which was examined was t h a t put forward by Huff and termed, by him, a p r o b a b i l i s t i c model of consumer s p a t i a l behaviour. R e c a p i t u l a t i o n of Purposes i n View of R e s u l t s Obtained The purpose of t h i s study was to d i s c o v e r whether e m p i r i c a l data gathered i n Vancouver would tend to support a model formulated and t e s t e d i n a p i l o t study elsewhere. I f the data supported the model, or more s p e c i f i c a l l y , an hypo-t h e s i s about the model, then the purpose would have been accomplished. On'the other hand, i f the data d i d not support the model, the purpose would encompass the d e t e r m i n a t i o n of whether the data or the model appeared t o be d e f i c i e n t . I f the data were d e f i c i e n t , the manner of i t s c o l l e c t i o n would have t o be reviewed, and a new r e s e a r c h d e s i g n developed. I f the model appeared to be d e f i c i e n t , then a t t e n t i o n would have t o be turned to the model I t s e l f . 144 The r e s u l t s o f t h e e m p i r i c a l r e s e a r c h , r e p o r t e d i n C h a p t e r V, showed t h a t t h e d a t a d i d n o t s u p p o r t t h e model, b u t t h a t t h e d a t a seemed t o be r e p r e s e n t a t i v e s i n c e t h e y t e n d e d t o s u p p o r t a n o t h e r i n d e p e n d e n t h y p o t h e s i s — t h a t c o n sumers have d i f f e r e n t a t t i t u d e s t o w a r d downtown v e r s u s s u b u r b a n s t o r e s . A t e s t o f t h e s e n s i t i v i t y o f t h e model was a p p l i e d t o t h e d a t a o f t h e o r i g i n a l p i l o t s t u d y . The model was shown t o be q u i t e i n s e n s i t i v e . T h e r e f o r e , i n l i n e w i t h t h e p u r p o s e s s t a t e d , a t t e n t i o n must be t u r n e d t o t h e model, and c o n s i d e r a t i o n g i v e n t o b o t h f a c t o r s a n d r e l a t i o n s h i p s . T h i s c h a p t e r i s d e v o t e d t o t h i s f i n a l e l e m e n t i n t h e s t a t e d p u r p o s e . The Need f o r a New M odel W h i l e some s u c c e s s e s have been a c h i e v e d i n t h e p a s t w i t h t h e g r a v i m e t r i c model i n s o c i a l s c i e n c e i n g e n e r a l , a n d m a r k e t i n g i n p a r t i c u l a r , t h e r e s u l t s o f t h i s s t u d y t e n d t o show t h a t i t may n o t be e n t i r e l y s u i t e d t o t h e i n t e n d e d p u r -p o s e s i n t h e new c o n t e x t s t o w h i c h i t h a s b e e n a p p l i e d . W h i l e t h e g r a v i m e t r i c model ha s a p r i o r i a p p e a l , i t w i l l n o t n e c e s s -a r i l y l e g i t i m a t e t h e r e s u l t s o f s o c i a l s c i e n c e r e s e a r c h by f i n d i n g t h a t t h e same models a p p l y a s i n p h y s i c a l s c i e n c e . S o c i a l s c i e n c e must make i t s own a b s t r a c t i o n s , i n t e r m s o f f a c t o r s a n d r e l a t i o n s h i p s , b a s e d on e v i d e n c e a v a i l a b l e t o i t j u s t a s p h y s i c a l s c i e n c e d i d i n t h e f i r s t i n s t a n c e . The e v i d e n c e a v a i l a b l e t o s o c i a l s c i e n c e i n g e n e r a l , o r any p a r t i c u l a r s o c i a l s c i e n c e , may a d m i t o f a n i n f i n i t u d e 14-5 of p o s s i b l e models. Some method of determining c r i t e r i a f o r choosing among models must be e s t a b l i s h e d . These must be based on the c a p a b i l i t i e s of the models themselves, as w e l l as the purposes f o r which the research i s undertaken. One b a s i s f o r d i s t i n g u i s h i n g among models i s to determine whether the model i s intended f o r purposes of explanation (or d e s c r i p t i o n ) , p r e d i c t i o n , or c o n t r o l , as was s t a t e d e a r l i e r . Not a l l models are s u i t e d to a l l these purposes. Other bases may be the d i s t i n c t i o n between microscopic and macroscopic models, or between s t a t i c and dynamic models. I d e a l l y , the model p e r t a i n i n g to a p a r t i c u l a r phenomenon should be macroscopic i n the sense of ap p l y i n g to the whole of the r e l e v a n t world, and dynamic i n the sense of e x h i b i t i n g the c o r r e c t behaviour with the passage of time. I t should s a t i s f y a l l the purposes of e x p l a n a t i o n , p r e d i c t i o n and c o n t r o l . Because such a model would i n c l u d e a l l r e l e v a n t v a r i a b l e s , a l l t h e i r necessary i n t e r - r e l a t i o n s h i p s , and a l l appropriate time-dependencies, i t would l i k e l y have to be viewed as a set of i n t e r - r e l a t e d microscopic models c o n t a i n i n g the necessary aspects. An economic model of t h i s k i n d , u s i n g a " b u i l d i n g -block" approach, has r e c e n t l y been developed, and i s c u r r e n t l y 1 being t e s t e d and extended. Such a model i s an u l t i m a t e g o a l , however, and considerable research must be performed to s p e c i f y the b a s i c b u i l d i n g b l o c k s . Guy H. Orcutt, M a r t i n Greenberger, John K o r b e l , and A l i c e M. R i v l i n , M i c r o a n a l y s i s of Socio-economic Systems; A S i m u l a t i o n  Study (New York: Harper and Brothers, 1961). 146 The g r a v i m e t r i c model, such as used i n t h i s study, can be viewed as an e f f o r t of b a s i c r e s e a r c h intended to s p e c i f y f a c t o r s and r e l a t i o n s h i p s which w i l l d e s c r i b e or e x p l a i n the s p a t i a l a spects of consumer behaviour. In the p i l o t study conducted by Huff, the model had the appearance of being q u i t e d e s c r i p t i v e i n the sense of the amount of v a r i a t i o n i n d ata " e x p l a i n e d " by the model was h i g h . The e x p l a n a t o r y nature of the model may be q u i t e i l l u s o r y , s i n c e the model was shown to be i n s e n s i t i v e to the parameter which was c o n s i d e r e d to be so i n d i c a t i v e of behaviour, and upon which the major h y p o t h e s i s was based. The b a s i c model of the g r a v i m e t r i c type c o n t a i n s only two v a r i a b l e s — s t o r e a r e a as a proxy measure of product assortment, and d i s t a n c e measured i n time u n i t s . The e m p i r i c a l d a t a presented i n Chapter V showed t h a t the model d i d not work w e l l i n Vancouver. The very low measures of e x p l a i n e d v a r i -a t i o n g e n e r a l l y obtained suggest t h a t a t l e a s t one other f a c t o r needs to be e x p l i c i t l y i n c l u d e d . However, the fundamental problem may be such t h a t i t cannot be remedied by merely i n c l u d i n g other f a c t o r s . I t may be r e c a l l e d t h a t i n e a r l i e r d i s c u s s i o n s the consumer was viewed as being i n a s i t u a t i o n of decision-making under c o n d i t i o n s of u n c e r t a i n t y . In t h i s s i t u a t i o n , the consumer was seen as making a s u b j e c t i v e estimate of the p r o b a b i l i t y of h i s success i n shopping a t v a r i o u s s t o r e s , but h i s i n f o r m a t i o n was l i m i t e d . I t seems a c c e p t a b l e to suggest t h a t any source which p r o v i d e s 147 t h e consumer w i t h a d d i t i o n a l i n f o r m a t i o n w i l l i n f l u e n c e t h e s u b j e c t i v e p r o b a b i l i t y e s t i m a t e s f o r m e d . W h i l e t h e r e a r e many s u c h p o s s i b l e s o u r c e s , t h e consumer's own e x p e r i e n c e o r l e a r n i n g , i n t e r - p e r s o n a l c o m m u n i c a t i o n s w i t h f a m i l y o r s o c i a l c o n t a c t s , a n d t h e p r o m o t i o n a l a c t i v i t i e s o f t h e v e n d o r o r s t o r e a r e p e r h a p s among t h e more i m p o r t a n t . I t may be t h a t t h e l a t t e r two s o u r c e s have t h e i r m a j o r i m p a c t on sonsumer b e h a v i o u r by a c t i n g on t h e f i r s t s o u r c e . E v e n i f t h i s were n o t s o , e a c h o f t h e s e s o u r c e s h a s t h e common p r o p e r t y o f b e i n g a n o n - g o i n g p r o c e s s . The g r a v i m e t r i c model i s , however, a s t a t i c framework o f a n a l y s i s . T h e r e i s a s i g n i f i c a n t d i f f e r e n c e i n k i n d between t h e dynamic e q u i l i b r i u m , o r s t e a d y s t a t e , o f a n o n - g o i n g p r o c e s s , a n d t h e unmoving, c r o s s - s e c t i o n a l a s p e c t o f a s t a t i c a n a l y s i s . I t was s u g g e s t e d above t h a t a l l t h e v a r i o u s s o u r c e s o f i n f o r m a t i o n may e x e r t t h e i r i m p a c t on t h e consumer t h r o u g h a l e a r n i n g p r o c e s s , and t h u s t h e s u b j e c t i v e p r o b a b i l i t y e s t i m a t e s w h i c h t h e consumer f o r m s a t any p a r t i c u l a r t i m e , a r e t h e r e s u l t o f some c o m b i n a t i o n o f a l l t h e f a c t o r s r e l e v a n t t o t h e p a r t i c u -l a r consumer. What t h e s e f a c t o r s m i g h t be, and how t h e y a r e i n t e r - r e l a t e d , becomes t h e p r o p e r s u b j e c t f o r s t u d y i n e x a m i n i n g consumer b e h a v i o u r , s p a t i a l o r o t h e r w i s e . The main p o i n t seems t o be t h a t i t i s u n w a r r a n t e d t o t r e a t t h e c h o i c e o f a s t o r e f o r s h o p p i n g a s s e p a r a t e a n d d i s t i n c t f r o m any o t h e r a s p e c t o f consumer d e c i s i o n - m a k i n g b e h a v i o u r . L e a r n i n g i s an i n d i v i d u a l p r o c e s s . I t i s n e v e r t h e same f o r any two p e r s o n s . T h i s f a c t r a i s e s a s e r i o u s q u e s t i o n 148 r e g a r d i n g t h e v a l i d i t y o f a n a l y s e s o f b e h a v i o u r u s i n g a g g r e g a t i v e p r o c e d u r e s , s u c h a s h a s b e en done i n a l l s t u d i e s o f g r a v i m e t r i c m o d e l s , i n c l u d i n g t h e p r e s e n t s t u d y and i t s p r e c u r s o r s . I t does n o t seem l o g i c a l t o i g n o r e t h e i n d i v i d u a l -i s t i c n a t u r e o f t h e u n d e r l y i n g p r o c e s s e s i n o r d e r t o d e v i s e a more manageable r e s e a r c h m e t h o d o l o g y . A g g r e g a t i v e m o d e ls i m p l y t h a t t h e f u n d a m e n t a l m a t h e m a t i c a l a x i o m o f a d d i t i o n i s a p p l i c a b l e , b u t i t i s n o t d e m o n s t r a t e d , a n d h i g h l y d o u b t f u l , t h a t t h e f u n d a m e n t a l a s s u m p t i o n o f h o m o g e n e i t y among e l e m e n t s i s s a t i s f i e d . L e a r n i n g i s a l s o a t i m e - d e p e n d e n t p r o c e s s , and t h i s f a c t a l s o h a s a s i g n i f i c a n t i m p a c t on t h e m e t h o d o l o g y o f r e s e a r c h on s u c h p r o c e s s e s . C r o s s - s e c t i o n a l a n a l y s e s , w h e r e i n a l l d a t a a r e g a t h e r e d a t one p o i n t i n t i m e , do n o t seem a p p r o p r i a t e e i t h e r , b u t a r e t h e n a t u r a l c o n s e q u e n c e o f u s i n g s t a t i c , a g g r e g a t i v e m o d e l s . The l i t e r a t u r e on consumer b e h a v i o u r i n g e n e r a l i s e x t e n s i v e . So, a l s o , i s t h e l i t e r a t u r e on b o t h l e a r n i n g a n d c h o i c e p r o c e s s e s i n p a r t i c u l a r . T h i s l i t e r a t u r e c a n n o t be r e v i e w e d h e r e , a s t h i s i s more p r o p e r l y t h e s u b j e c t o f a n o t h e r s t u d y . What m i g h t be done c a n be s u g g e s t e d i n g e n e r a l t e r m s , however. C o n s i d e r i n g an i n d i v i d u a l , i t may be p o s s i b l e , , t o e numerate t h e v a r i o u s f a c t o r s w h i c h m i g h t i n f l u e n c e h i s c h o i c e o f s t o r e s f o r s h o p p i n g . T h e s e v a r i o u s f a c t o r s may be combined i n t o some s o r t o f w e i g h t e d f u n c t i o n w h i c h c a n p r e d i c t t h e p r o b a b i l i t y o f s h o p p i n g a t a p a r t i c u l a r s t o r e . Many o t h e r 149 i n d i v i d u a l s may "be s i m i l a r l y c o n s i d e r e d , a n d t h e i n d i v i d u a l f u n c t i o n s examined t o d e t e r m i n e i f a p a t t e r n , w h i c h m i g h t be t e r m e d c o l l e c t i v i t i e s , emerge. Many p o s s i b l e s t a t i s t i c a l t o o l s m i g h t be u s e d i n s u c h a n a l y s e s , b u t t h e m u l t i p l e d i s c r i m i n a n t a n a l y s i s p r o c e d u r e seems t o h o l d c o n s i d e r a b l e 2 p r o m i s e . W h i l e i t i s n o t i n t e n d e d t o d i s c u s s e i t h e r t h e t h e o r y o r t e c h n i q u e o f m u l t i p l e d i s c r i m i n a n t a n a l y s i s i n t h i s s t u d y , i t may be p o i n t e d o u t t h a t t h e w e i g h t i n g s o f v a r i o u s f a c t o r s i n t h e d i s c r i m i n a n t f u n c t i o n w i l l t e n d t o show w h i c h f a c t o r s have t h e g r e a t e s t e f f e c t on c h a n g i n g t h e c l a s s i f i c a t i o n o f i n d i v i d u a l s f r o m one c o l l e c t i v i t y t o a n o t h e r . The d i s c r i m i n a n t f u n c t i o n i s d e s c r i p t i v e i n t h e s e n s e t h a t i t s p e c i f i e s i m p o r t a n t f a c t o r s a n d t h e m a g n i t u d e o f t h e i r e f f e c t s on t h e phenomenon o f c o n c e r n . I t i s b u t a f i r s t s t e p -- a m a c r o s c o p i c g e n e r a l i z a t i o n — t o w a r d s p e c i f y i n g t h e n e c e s s a r y m i c r o s c o p i c mechanisms w h i c h a c t u a l l y d e t e r m i n e b e h a v i o u r . I t does t r e a t t h e i n d i v i d u a l , however, an d i t does n o t assume h o m o g e n e i t y among members o f a p a r t i c u l a r c o l l e c t i -v i t y , b u t r a t h e r s e e k s t o measure i t . B e c a u s e o f i t s g e n e r a l f o r m , i t may be p o s s i b l e t o i n c l u d e l e a r n i n g p r o c e s s e s by i n c l u d i n g f a c t o r s w h i c h w o u l d measure t h e t r a n s i e n t s t a t e o f t h e consumer, s u c h a s l a g g e d v a r i a b l e s , f o r e xample. A l t o g e t h e r , T h e r e a r e s e v e r a l s o u r c e s w h i c h t o u c h on t h i s a n d r e l a t e d a r e a s , however, f o r p r a c t i c a l p u r p o s e s , t h e b e s t seems t o be W i l l i a m W. C o o l e y a n d P a u l R. L o h n e s , M u l t i v a r i a t e P r o c e d u r e f o r t h e B e h a v i o r a l S c i e n c e s (New Y o r k : J o h n W i l e y and S o n s , I n c . , 1962). 150 I t seems t o o f f e r c o n s i d e r a b l e hope f o r a v o i d i n g t h e many-s h o r t c o m i n g s , i n t e r m s o f v a l i d i t y , i m p o s e d by t h e g r a v i m e t r i c model on t h e d a t a g a t h e r e d f o r i t and t h e r e s u l t s o b t a i n e d f r o m i t . T h i s s t u d y h a s shown t h a t g r a v i m e t r i c m o d els have s e v e r e s h o r t c o m i n g s i n t e r m s o f g e n e r a l v a l i d i t y , a n d has s o u g h t t o s u g g e s t t h e r e a s o n s f o r t h e s e . S u g g e s t i o n s were a l s o made f o r new a p p r o a c h e s t o t h e a n a l y s i s o f t h e phenomenon o f consumer s p a t i a l b e h a v i o u r , . p r i m a r i l y by v i e w i n g i t a s b e i n g c l o s e l y r e l a t e d t o o t h e r k i n d s o f d e c i s i o n - m a k i n g a n d c h o i c e b e h a v i o u r . S t a t i s t i c a l t o o l s , s u c h a s m u l t i p l e d i s c r i m -i n a n t a n a l y s i s , o f f e r c o n s i d e r a b l e hope f o r u s e f u l q u a n t i t a t i v e m o d e l s . R e s e a r c h i n m a r k e t i n g d i r e c t e d a l o n g t h e s e l i n e s w o u l d seem t o o f f e r much g r e a t e r o p p o r t u n i t y f o r s u c c e s s t h a n h a s b een a c h i e v e d w i t h g r a v i m e t r i c m o d e l s . 151 BIBLIOGRAPHY Books A c k o f f , R u s s e l l .L. 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"The Hypothesis of the 'Minimum Equation' as a Unifying Social. .Principle.: with Attempted Synthesis," American S o c i o l o g i c a l Review, XII (December, 1947), 627-650. 157 APPENDIX I HUFF'S CONSUMER BEHAVIOR QUESTIONNAIRE A t what SHOPPING CENTER d i d y o u LAST make a p u r c h a s e o f the f o l l o w i n g i t e m s ? H o u s e h o l d C l o t h i n g F u r n i s h i n g F o o d ... . Hardware C o s m e t i c s o r Drugs 2. How many members o f your, h o u s e h o l d n o r m a l l y work i n e x c e s s o f f o u r h o u r s p e r day? 3 . What t y p e o f WORK does t h e m a j o r income e a r n e r o f y o u r h o u s e h o l d do? • S e l f Employed? YES ( ) NO ( ) 4 . What a r e t h e APPROXIMATE AGES o f t h o s e p e r s o n s l i v i n g w i t h i n y o u r h o u s e h o l d ? MALE (...) (...) (...) (...) FEMALE (...) (...) ( . . . ) ( . . . ) 5 . How many CARS i n t h e f a m i l y ? 6. A t what s h o p p i n g c e n t e r do you.NORMALLY MAKE THE MAJORITY o f t h e f o l l o w i n g f a m i l y p u r c h a s e s ? H o u s e h o l d C l o t h i n g F u r n i s h i n g s F o o d Hardware C o s m e t i c s o r Drugs 7. What b r a c k e t a s i n d i c a t e d b e l o w most c l o s e l y f i t s y o u r t o t a l FAMILY INCOME? \, 0-1999 f5000-5999< 2000-3999 6000-7999. 4000-4999 8000-9999. $10,000 & o v e r S o u r c e : H u f f , D e t e r m i n a t i o n o f I n t r a - u r b a n R e t a i l T r a d e A r e a s , p. 39. 158 APPENDIX I I TECHNICAL DETAIL OF VANCOUVER SAMPLE S a m p l i n g The g r o u p w h i c h c o n d u c t e d t h e s u r v e y was s y n d i c a t e d , a n d e a c h s y n d i c a t e was s u p p l i e d w i t h a l i s t i n g o f sample " b l o c k s . The i n t e r v i e w i n g p l a n i n v o l v e d a s y s t e m a t i c sample w i t h a n i n t e r v a l o f s e v e n , t h e r e f o r e i t was n e c e s s a r y t o a p p r o a c h e v e r y s e v e n t h d w e l l i n g i n t h e b l o c k s a s s i g n e d . I n t e r v i e w i n g was begun w i t h a random s t a r t , c o u n t i n g i n a c l o c k w i s e d i r e c t i o n commencing a t t h e N o r t h w e s t c o r n e r o f a b l o c k . On c o m p l e t i o n o f a b l o c k , c o u n t i n g was c o n t i n u e d w i t h o u t i n t e r r u p t i o n a t t h e n e x t b l o c k a s s i g n e d , e a c h t i m e b e g i n n i n g a t t h e N o r t h w e s t c o r n e r a n d c o u n t i n g i n a c l o c k w i s e d i r e c t i o n . T h r o u g h o u t t h e l i s t o f b l o c k s , e v e r y s i x t h d w e l l i n g was a p p r o a c h e d . No one c o n t a c t e d i n a n i n s t i t u t i o n o f any k i n d was i n t e r v i e w e d . I f a n a p a r t m e n t b l o c k was e n c o u n t e r e d , e a c h s u i t e was t r e a t e d a s a d w e l l i n g , a n d c o u n t i n g was c o n t i n u e d s y s t e m a t i c a l l y . I n t e r v i e w e r s were i n s t r u c t e d t o c o n t a c t a n a d u l t f e m a l e , p r e f e r a b l y t h e f e m a l e h e a d o f t h e h o u s e h o l d . I f t h e r e was no woman i n t h e h o u s e h o l d , a n a d j a c e n t d w e l l i n g was s u b s t i t u t e d . One c a l l - b a c k was r e q u i r e d . I f t h e s e c o n d c a l l was u n s u c c e s s f u l i n m aking c o n t a c t , a n a d j a c e n t d w e l l i n g was s u b s t i t u t e d . I n v i e w o f t h e b l o c k s a t u r a t i o n s a m p l i n g t e c h n i q u e u s e d by t h e D o m i n i o n B u r e a u o f S t a t i s t i c s , i t was d e s i r e d n o t t o u s e 159 a b l o c k a l r e a d y c o v e r e d by them. An a d j a c e n t b l o c k was s u b s t i t u t e d i n t h i s e v e n t . D e f i n i t i o n s The f o l l o w i n g d e f i n i t i o n s o f t e r m s were p r o v i d e d f o r g u i d a n c e o f t h e i n t e r v i e w e r s : D w e l l i n g : a d w e l l i n g i s a s t r u c t u r a l l y s e p a r a t e s e t o f l i v i n g p r e m i s e s , w i t h a p r i v a t e e n t r a n c e f r o m o u t s i d e t h e b u i l d i n g , o r f r o m a common h a l l w a y o r s t a i r w a y i n s i d e . E n t r y i s n o t t o be t h r o u g h anyone e l s e ' s l i v i n g q u a r t e r s . H o u s e h o l d : a p e r s o n o r g r o u p o f p e r s o n s o c c u p y i n g one d i a l l i n g i s d e f i n e d a s a h o u s e h o l d . A p e r s o n i s a member o f t h e h o u s e h o l d i n whose d w e l l i n g he n o r m a l l y a n d r e g u l a r l y s l e e p s . Q u e s t i o n n a i r e O n l y one m u l t i - p a r t q u e s t i o n i n t h e s u r v e y was r e l e v a n t t o t h i s s t u d y ( s e e A p p e n d i x I I I ) . T h i s q u e s t i o n was a s c r e e n -i n g q u e s t i o n , t h u s some answer w o u l d u s u a l l y be o b t a i n e d . The q u e s t i o n was i n i t i a l l y c o n c e r n e d w i t h a v i s i t t o a l o c a l d e p a r t m e n t s t o r e , w h e t h e r o r n o t a p u r c h a s e was made. I f s u c h a v i s i t o c c u r r e d w i t h i n t h e p r e v i o u s t h r e e months, i n t e r v i e w e r s were r e q u i r e d t o d e t e r m i n e : (a) w h i c h s t o r e s were v i s i t e d , w h e t h e r o r n o t a p u r c h a s e was made; 160 (b) f o r p u r c h a s e s made, t h e names o f t h e d e p a r t m e n t s t o r e s v i s i t e d were t o be r e c o r d e d , t o g e t h e r w i t h t h e t y p e s o f p u r c h a s e s i n t e r m s o f p r e d e f i n e d c a t e g o r i e s ( n o r m a l l y d e p a r t m e n t s ) . The e m p i r i c a l d a t a were d e v e l o p e d f r o m r e s p o n s e s t o t h i s q u e s t i o n . 161 APPENDIX I I I QUESTIONS FROM VANCOUVER SURVEY The f o l l o w i n g q u e s t i o n s were u s e d t o d e v e l o p t h e d a t a f o r t e s t i n g t h e model c o n s i d e r e d i n t h i s t h e s i s : When was t h e l a s t t i m e y o u went t o any one o f t h e l o c a l d e p a r t m e n t s t o r e s ? a . What s t o r e s d i d y o u v i s i t on t h a t t r i p ? Hudson*s Bay .... Woodwards - O a k r i d g e .... Downtown .... P a r k R o y a l .... E a t o n s - Br e n t w o o d .... Downtown .... P a r k R o y a l .... S i m p s o n - S e a r s - B u r n a b y .... Richmond .... Army & Navy .... O t h e r ( s p e c i f y ) .... b. What d i d y o u buy a t e a c h d e p a r t m e n t s t o r e y o u v i s i t e d ? ( W r i t e i n s t o r e a n d c h e c k m e r c h a n d i s e p u r c h a s e d ) C l o t h i n g ( i n c l u d i n g h a t s a n d s h o e s ) f o r y o u r s e l f C h i l d r e n ' s c l o t h i n g F u r n i t u r e House f u r n i s h i n g s , p i e c e g o o d s , f l o o r c o v e r i n g s o r h o u s e -wares J e w e l r y C o s m e t i c s o r d r u g p r o d u c t s C h i n a o r g l a s s w a r e S p o r t i n g goods A p p l i a n c e s * ( i n c l u d i n g T.V. a n d r a d i o ) G r o c e r i e s T o y s Hardware ( i n c l u d i n g p a i n t ) O t h e r N o t h i n g 163 . - . . . i a.... •-- r r f - j - f •I -r - I APPENDIX IV PROGRAMME XLAMC FLOW CHART FOR SUCCESS IVE : APPROXIMATION OF LAMBDA .1 'Si i . ' Read r S j J C i j , T i r j , K, ;N • • - - i n i t i a l value of X-"]" " increment .'of X . ' . _, , ^  upper l i j r . i t of A i r J .. . 1 . Set = 1 ' ; • •-where i i = I,K. : i:. m t j . ::n: J u u C f T T T i ; "r 1 . Set ! jX =' i n i t i a l ; •j value of A ; . - i • -; i . f t ! ! 1 I 4.1. Set Bl = 0 P r i n t ' i . L iU.;.! r : • • i • —,—-- r + _J..J_ Set £ S i _ ! - = 0 -•1J i ; ; -h"J-1 - H -- r - 1 " ' r.. -TT -I— i l t t •i rr 1 -i ! i T t »r Li J _t-..u.. i i i L' "vr i • ':"Tt .":-Set"!7TT!' | where; j •= 1 ,N ;-4 i-rH-i •! i-i 1 f ru: :i.t: x't :i_L_L'.f: ri : r TT" 7 4-; .!. : ' X i : . t _ t . _ (_ 164 ! 1 XL • i - i • t : -...Li i j . X i ii., "T7 i ; H — r --rx~ . i - j - U j . j - . r j - p " j r-t-+ T|t;-T.ttx -fx' - X j x i ; : A ? P E N D I X IV (cont'd) i ' i x I • 4 • PROGRAMME XLAMC : i •  • • ' I : • * : FLOW CHART FOR SUCCESSIVE 'APPROXIMATION OF LAMBDA TTT i X . .1- 4-" T i .xxxx i i 0 - i - f -i i T - r - , , 4 X i L j X X X . . . Set ~ O .. ,_i„ j i 1 T X X H 4 4 i ™ )• • - • 4 - t — Sat •J-J ! A / J . ' j i T ; x l 1 i ; .|.T |.| L i ' L iTrF : J T t - T - f -1 r - r ~ -rr . ! IT \ ' ! ' i i • i i i -t-r -t f """*" 7 ' a 1 <: I ! 4 T f-r-r T T T f X ' I i I i i . ; i i j ' j X r l - j X f -r 4 - i . - ^ - t - r -.. . t _ . „ " 1 " ' T • 7 • f f ! I i r ' T i T"T T - I -. - ( - . . . : , • ' i : i i r [ " ! ~ i •r T2. T X Q X X r ! - U . ; i IX: 4 • 4 1 1 i :..H.. r r i i i i r : v x - ^ h 42 i r . j I r IX r: •HI ! H"" -Q.LL T' i T-ri-I I T : 4 t T i i : TTF EiiXI !" r T T 1 -• ; -r -lH-i; +4-f ui i t : n i i ; 1 1 -t- 4- -— -rr ~rr -r-.; -X. r. | X T" i . . U ; . . "4'4" XI •n 4 fFt 4 - 4 J.. X X i j. -I-K- .-f -1 -i - 4 TIXL 4-1 i i - l - 4 i l l . . 7 _ r _ J 1 • r r -L-I....U.! 444 i • i . X T i 4-1 4 - f -.4-4 ; r t 165 ~ T " ~ I T ' T T ! ..•).• i APPENDIX IV (cont'd) • i • I ' < - • ] • PROGRAMME XLAMC ': ' i i - j FLOW CHART FOR SUCCESSIVE;APPROXIMATION OF LAMBDAi • ;'I - I T ±fcfc trtT .it!. - r » - •• i -1 T -I i J. -r i r J. A J . . U . I ! • n • Y , L ! . .. : 2 <—>. • j=i - j • • • • r t 1 ,.i4,,:-t-H J i TC: :. r: r x x x u . n x _ . , J _ : , 3 ; J : ; : L !;^ |hiii::T:.ri#.;.r. r r _,„X ... ;-| -* ; r 1 ! I Jn£ A X 2\ : T"T i-i -t TT T I I -J- I-' ;. i i. . •T • 4-i 4--Tf'T i 7 i I T--1-166 167 APPENDIX V Programme XLAMC : Fortran 0 * SIBFTC XLAMC \ ; 1 OIMENSLON S(25),C( 50,25,) ,T ( 50, 25 ) , EC ( 50 ,25) ,XECI(50) ! 2 * CALL PLOTS ! 3 1 FORMAT { 12F6.2 ) ! .4 2 FORMAT (216) i i " "5" 3 FORMAT (8F10.4) i 6 * 4 FORMAT (3F6.3) j ! 7 * READ (5,2) K,N i 12 R F AD (5,3) (S(J),J=1,N) ! 1 7 * READ (5,1) ( ( C (I,J),J=1,N) , 1 = 1,K) ! 30 READ (5,1) ( (T{ I, J ) , J = L N > , 1 = 1,K) j' 41" READ (5,4) XL AMO,DL AM »XLA Ml : 42 * 6 FORMAT (6X,20H NEIGHBORHOOD NUMBER 13) 43 9 FORMAT (/6X,F10.5,\4X,F6.3//(4X,F6.2 ,4X, F6.2 , 4X ,F6.21/) 44 00 70 1=1,K . ! 45 WRITE (6,6) I 1 i 46 CALL AXIS (.0,.0,1HR,+1,10.,90.,.0,.1) 1 4 7 CALL AXIS (.0,.0,6HLAMBDA,-6,9.,0.,.5,. 5) i ! 50 SUMC=0. i 5 1 * DO 10 J=1,N 5 2 * SUMC=SUMC+C(I,J) t-i i 53 * 10 CONTINUE I ! 5 5 XLAM=X LAMO j 5 6 Rl = 0. i -) 1 57 * IPEN=3 i 1 60 * 15 SUMST=0. i 6 1 DO 20 J = 1,N ; 6 2 SUMST=SUMST+S(J)/T(I,J)**XLAM i 63 * 20 CONTINUE 1 * "65 XN=N 66 * SUMCEC=0. 67 SUMCCM=0. 70 DO 25 J=1,N i 71 EC(I,J)=(S(J)/T(I,J)**XLAM)*SUMC/SUMST i i i 7 2 CEC=(C( I»J:)-EC (I ,J) )**2 j ' 7 3! * SUM.CEOSUMCEC + CEC 74 * CCN=(C( I,J)-SUMC/XN)**2 1 ^ 75 * SUMCCN=SUMCCN+CCN 1 • 76 * 25 CONTINUE | . 168 APPENDIX V (cont'd.) Programme XLAMC : Fortran i O O * SR = l . - S U M C E C / S U M C C i ' 10:1 * IF ( SR')33, 30,30 102""*' 3 3 C A L L S R N E G ( X L A M ) 103 * GO TO 36 104 * 30 R = S Q R T ( S R ) 105 .106 110 I I I 112 * 50 36 CALL PLOT ( 2 . * X L A M - l . , I 0 . * k , I P E N ) I F ( R - R 1 ) 3 5 , 5 0 , 5 0 XLA M=XLAM+DLAM "I.PEN=2 I F (XLAM1-XLAM)60»15,15 Kl=R 113 114 115 116 117 * * 51 Rr.AX = R l XLAMM=XLAM DO 5 1 J=1,N X E C I { J } = E C ( I , J ) CONTINUE 121 GO TG 35 : 122 * 6 0 WR I TE(6,9) RMAX.XLAMM, ( X E C t ( J ) , C ( I , J ) « T ( I , J ) t J = U N ) • 127 * CALL SYMBOL ( 2.*XLAMM-1. ,10.*RMAX,0.21,4,0 . , -1) : .13 0 * CALL PLOT (1 1 . , 0 . , - 3 ) 131 TO CONTINUE 133 * 72 CALL PLGTNO 134 80 STOP 135 * END 0 *• SIPFTC AUX 1 * SUBROUTINE SRNEG(XLAM) 5 * 6 * 2 * WRITE ( 6 , 1 1 ) XLAM 3 * 11 FORMAT I/6X,32H SR IS UNDEFINED AT LAMBDA EQUAL F6./1) 4 * XLAM=XLAM+Q. 5 / RETURN END 

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