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On some analytical approaches to the study of consumer brand-switching behavior Windal, Pierre Marie 1977

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ON SOME ANALYTICAL APPROACHES TO THE STUDY OF CONSUMER BRAND-SWITCHING BEHAVIOR by PIERRE MARIE WINDAL D.E.S.C., L i l l e , France, 1972 M.B.A., The U n i v e r s i t y o f B r i t i s h Columbia, 1974. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the FACULTY OF GRADUATE STUDIES F a c u l t y of Commerce and A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming to the r e q u i r e d standard The U n i v e r s i t y of B r i t i s h Columbia December, 1977 P i e r r e Marie Windal 1978 In presenting th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f i n a n c i a l gain sha l l not be allowed without my wri t ten permission. Department pf C O H H E G C E The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT The purpose of t h i s r e s e a r c h i s to analyse, d i s c u s s and extend the a n a l y t i c a l methodology a s s o c i a t e d with the s t u -dy of consumer brand-switching behavior. As such, i t attempts to add to the e x i s t i n g understanding of the s t r u c t u r e o f the consumer brand c h o i c e p r o c e s s . R a t i o n a l human behavior may be viewed as a s u c c e s s i o n of c h o i c e s made among more or l e s s w e l l d e f i n e d a l t e r n a t i v e s . The problem we analyse i n t h i s study i s how to p r e d i c t these c h o i c e s when the a l t e r n a t i v e s are f i x e d i n advance. The a l t e r -n a t i v e s c o n s i d e r e d i n t h i s study are low-cost, f r e q u e n t l y pur-chased, brand i d e n t i f i e d consumer products. The u n i t of a n a l y s i s i s the i n d i v i d u a l consumer. S t o c h a s t i c models of brand c h o i c e are developed and used as c o n s t r u c t s f o r o r g a n i z i n g and i n t e r p r e t i n g brand c h o i c e data. These models are subsequently used to t e s t s p e c i f i c hypo-theses about brand l o y a l t y (the tendancy f o r consumers to h o l d a f a v o r a b l e a t t i t u d e toward - and concentrate t h e i r purchases on - a p a r t i c u l a r brand) and brand-switching (the tendancy f o r consumers to purchase more than one brand over a p e r i o d of t i m e ) . In t h i s r e s p e c t , t h i s d i s s e r t a t i o n f o l l o w s the framework of e a r l i e r brand c h o i c e s t u d i e s . In many dimensions, however, t h i s r e s e a r c h i s s i g n i -f i c a n t l y d i f f e r e n t from most s t o c h a s t i c models of brand-s w i t c h i n g behavior developed i n the p a s t . F i r s t , t h i s r e s e a r c h d e a l s e s s e n t i a l l y w i t h m u l t i - b r a n d s w i t c h i n g behavior as oppose to mere brand l o y a l t y . By c o l l a p s i n g the market i n t o an a r t i f i -c i a l two-brand market (to achieve mathematical t r a c t a b i l i t y ) , e a r l i e r r e s e a r c h e r s were f o r c e d to c o n c e n t r a t e on r e p e a t pur-chase behavior o n l y . A l l the i n f o r m a t i o n about brand s w i t c h i n g a c t i v i t y was l o s t i n the a g g r e g a t i o n p r o c e s s . In today's d i f f e -r e n t i a t e d markets, the c o m p e t i t i o n has to be monitored on a brand-by-brand b a s i s , and t h i s i s b e s t achieved through the use of models t h a t d e a l e x p l i c i t l y w i t h m u l t i b r a n d s w i t c h i n g , such as the one developed i n t h i s study. Second, t h i s r e s e a r c h views consumer brand c h o i c e behavior as both a c o g n i t i v e and a s t o c h a s t i c p r o c e s s . A m u l t i - d i m e n s i o n a l l y s c a l e d c o n f i g u r a t i o n i s used as a spe-c i f i c a t i o n of consumers' c o g n i t i v e s t r u c t u r e s . P e r c e p t u a l d i s t a n c e s d e r i v e d from t h i s c o n f i g u r a t i o n are then r e l a t e d to brand c h o i c e and b r a n d - s w i t c h i n g p r o b a b i l i t i e s through a model t h a t takes i n t o account the c o n s t r a i n t s imposed on the various p r o b a b i l i t i e s . The empirical results demonstrate that perceptions, preferences and cognitive structures are indeed s i g n i f i c a n t determinants of consumer brand-switching behavior. V TABLE OF CONTENTS Page LIST OF TABLES v i i LIST OF FIGURES i x INTRODUCTION 1 CHAPTER I THE EFFORTS OF THE PAST: BRAND CHOICE STUDIES 7 1.1 D e t e r m i n i s t i c Approaches To Brand Choice B e h a v i o r 8 1.1.1 Socio-Economic and Personality!..... 8 S t u d i e s .11 1.1.2 M u l t i - A t t r i b u t e A t t i t u d e Theory 11 1.2 S t o c h a s t i c Approaches t o Brand Choice B e h a v i o r 13 CHAPTER I I PRELIMINARY STATISTICAL ANALYSIS OF CONSUMER BRAND-SWITCHING DATA 22 2.1 D e s c r i p t i o n o f Consumer B r a n d - S w i t c h i n g Data 23 2.2 Choice o f a S t a t i s t i c a l Technique 27-2.3 The L o g - L i n e a r Model 29 2.4 Model F i t t i n g and H y p o t h e s i s T e s t i n g P r o c e d u r e s 35 2.5 D e s c r i p t i o n o f the E m p i r i c a l Data... 40 2.6 P r e s e n t a t i o n o f the E m p i r i c a l R e s u l t s 43 2.7 Comparison o f B r a n d - S w i t c h i n g B e h a v i o r A c r o s s Segments 45 2.8 Summary 60" CHAPTER III A NEW LEARNING MODEL OF BRAND CHOICE 3.1 L i n e a r L e a r n i n g and Markov Models o f Brand Choice iv 3.2 A New Purchase - t o - Purchase L e a r n i n g Model o f Brand C h o i c e : The P o l y a - L e a r n i n g Model 76 Page 3.2.1 The Polya Urn Model 77 3.2.2 Limitation of the Simple Polya Urn Model 79 3.2.3 The Polya-Learning Model 80 3.2.4 Model F i t t i n g Procedures and Data .. 90 3.2.5 Empirical Results 90 CHAPTER IV A MODEL OF CONSUMER BRAND-SWITCHING: MATHEMATICAL PRELIMINARIES 99 4.1 Problem D e f i n i t i o n 100 4.2 Mathematical Development 101 4 CHAPTER V THE DETERMINANTS OF CONSUMER BRAND-SWITCHING BEHAVIOR 119 5.1 Stochastic and Deterministic Theories 119 5.2 A Joint Space Theory of Brand Choice 121 5.3 Model Development 125 5.4 Estimation Procedure 141 5.5 The Data 142 5.6 Joint Space Construction 142 5.7 Analysis and Results 147 Conclusion 162 CHAPTER VI MULTI-DIMENSIONAL SCALING OF BRAND-SWITCHING DATA 162 6.1 Introduction 162 6.2 Methodological Implications 162 6.3 Multi-dimensional Scaling of Brand-Switching P r o b a b i l i t i e s 167 6.4 An Empirical I l l u s t r a t i o n 180 Summary 191 CHAPTER VII CONCLUSION 193 v i i BIBLIOGRAPHY 198 APPENDIX A: M o t i v a t i o n f o r the S i m p l i f y i n g A s s u m p t i o n Used i n C h a p t e r IV ( e q u a t i o n 4.5) 203 v i i i L I S T OF T A B L E S TABLE PAGE 1 :1 C l a s s i f i c a t i o n o f S t o c h a s t i c B r a n d C h o i c e M o d e l s 16' 1 1 . 1 E s t i m a t e s o f I n t e r a c t i o n E f f e c t s Among t h e P u r c h a s e O c c a s i o n s f o r t h e C o f f e e D a t a ( S a t u r a t e d M o d e l ) 6 i< 1 1 1 . 2 P a r a m e t e r E s t i m a t e s f o r t h e U n s a t u r a t e d M o d e l i n W h i c h t h e F i v e - W a y I n t e r a c t i o n i s A s s u m e d Z e r o 62 1 1 . 3 P a r a m e t e r E s t i m a t e s f o r t h e U n s a t u r a t e d M o d e l i n W h i c h A l l F o u r - W a y a n d t h e F i v e - W a y I n t e r -a c t i o n s A r e A s s u m e d Z e r o ( M 2 3 ) ^ 1 1 . 4 P a r a m e t e r E s t i m a t e s f o r t h e U n s a t u r a t e d M o d e l i n W h i c h A l l 3 , 4 a n d 5-Way I n t e r a c t i o n s A r e A s s u m e d Z e r o ( M 2 ) 64 1 1 . 5 P a r a m e t e r E s t i m a t e s o f T h r e e U n s a t u r a t e d M o d e l s 65 1 1 . 6 P a r a m e t e r E s t i m a t e s o f T h r e e U n s a t u r a t e d M o d e l s : M 2 3 4 * 5 , M ( 2 3 4 ) , a n d M ( 2 3 ) , 5 1 6 6 1 1 . 7 ' S u m m a r y o f t h e C h i - S q u a r e S t a t i s t i c s f o r V a r i o u s U n s a t u r a t e d M o d e l s 67 1 1 . 8 Summary o f t h e L i k e l i h o o d R a t i o C h i - S q u a r e S t a t i s t i c s f o r V a r i o u s U n s a t u r a t e d M o d e l s 68 1 1 . 9 E x a m p l e o f a F i v e - W a y C o n t i n g e n c y T a b l e 26 1 1 . 1 0 C o r r e l a t i o n M a t r i x o f t h e V a r i o u s S e g m e n t s B a s e d o n t h e M o d e l s ' R e s p e c t i v e P a r a m e t e r E s t i m a t e s f o r t h e S a t u r a t e d M o d e l a n d Two U n s a t u r a t e d M o d e l s 69 1 1 1 . 1 I n t e r m e d i a r y R e s u l t s t o C o m p u t e t h e L i k e l i h o o d s a n d t h e E x p e c t e d P r o b a b i l i t i e s o f P u r c h a s e i n The P o l y a - L e a r n i n g M o d e l ? 8 9 1 1 1 . 2 P a r a m e t e r E s t i m a t e s f o r t h e P o l y a - L e a r n i n g M o d e l ;91 , i x PAGE III. 3 Comparison of L i n e a r - L e a r n i n g , Brand L o y a l and P o l y a - L e a r n i n g G o o d n e s s - o f - f i t S t a t i s t i c s .. 96 V . l S o f t Drink Brands S e l e c t e d f o r the Study 143 V.2 P o s i t i o n on the D i s c r i m i n a n t Functions 146 V.3a Observed Average J o i n t P r o b a b i l i t y M a t r i x f o r the E i g h t S o f t Drink Brands 148 V.3b T h e o r e t i c a l Average J o i n t P r o b a b i l i t y M a t r i x . f o r Model 1.4 ~ 148 V.3c T h e o r e t i c a l Average J o i n t P r o b a b i l i t y M a t r i x f o r Model I I I . 3 148 V.4 Summary of Models 134 V. 5 Summary of Parameter Estimates f o r V a r i o u s Models . i 149 VI. 1 Parameter Estimates f o r Three Models, J o i n t Space C o n f i g u r a t i o n 183 VI.2 C o r r e l a t i o n M a t r i x of P e r c e p t u a l D i s t a n c e s Across the 2 8 P a i r s f o r V a r i o u s Models 18 6 VI.3 C o r r e l a t i o n M a t r i x of the Coordinate V e c t o r s f o r V a r i o u s Models 186 LIST OF FIGURES. FIGURE PAGE I I I . l D i a g r a m a t i c a l R e p r e s e n t a t i o n of the L i n e a r L e a r n i n g Model 73 VII , 2 P o s i t i o n of t h e E i g h t Brands on awTwo-Dimensional D i s c r i m i n a n t C o n f i g u r a t i o n 123 V.2 R e l a t i o n s h i p Between Brand Choice P r o b a b i l i t i e s and P e r c e p t u a l D i s t a n c e s 157 V.3 R e l a t i o n s h i p Between Observed Repeat Purchase P r o b a b i l i t i e s and P e r c e p t u a l D i s t a n c e s 173 VI.1 M u l t i - D i m e n s i o n a l S c a l i n g of the S o f t - D r i n k Experimental Brand-Switching M a t r i x : Model 1. 184 .ACKNOWLEDGMENTS -I t i s a pleasure to express my gratitude to a l l the people and organizations who d i r e c t l y or i n d i r e c t l y made i t possible for me to eventually write these words. My best thanks go to Professor Doyle L. Weiss. As my d i s s e r t a t i o n chairman, he i s to be credited for a l l those " r i g h t " things that happened at the " r i g h t " time. His u n f a i l i n g optimism, technical expertise and broad v i s i o n contributed much to this p a r t i c u l a r piece of research as well as to my understanding of research i n general. As a f r i e n d , he helped me step away from some of the many f a l l i n g rocks that have been aimed at generations of doctoral students' heads. My thesis' committee members made many valuable suggestions towards the ove r a l l improvement of this study ; Professor J. Claxton oversaw i t s marketing content while Professors M. Puterman, M. Schulzer and W. Ziemba made sure the mathematics were honest. I am also greatly indebted to Professor F.M. Bass, of Purdue University, who graciously provided me with the s o f t d r i n k e x p e r i m e n t a l d a t a upon which most of the e m p i r i c a l r e s u l t s p r e s e n t e d i n t h i s s t u d y are based. I c e r t a i n l y w i s h t o acknowledge here the gener-o s i t y o f the f o l l o w i n g o r g a n i z a t i o n s t h a t a l l o w e d me to s t a r t and complete my d o c t o r a l s t u d i e s . They i n c l u d e : The -Canada C o u n c i l The Killam.:. F o u n d a t i o n The U n i v e r s i t y o f B r i t i s h Columbia The Oppenheimer B r o s . & Company The P r o v i n c i a l Government o f B r i t i s h Columbia To a l l of them, I e x p r e s s my g r a t i t u d e . S e v e r a l i n d i v i d u a l s deserve a s p e c i a l mention f o r b r i n g i n g fun and e x c i t e m e n t i n t o my s t u d e n t l i f e . S i n c e they are too numerous t o be c i t e d h e r e , I w i l l j u s t s i n g l e out one o f them, M a r t i n Kusy, whose a b i l i t y t o e n j o y l i f e and w i l l i n g n e s s to share i t was d e f i n i t e l y an a s s e t f o r a l l h i s f r i e n d s . F i n a l l y , I owe a s p e c i a l debt o f thanks to my w i f e , Dominique, who s t u d i e d a l o n g w i t h me at U.B.C. and who p r o b a b l y knows more about t h i s d i s s e r t a t i o n than any-body e l s e . Her p r e s e n c e , and t h a t o f my two c h i l d r e n , Timothee and S t e p h a n i e (who by the way d e a r l y r e s e n t the a b r u p t c u t - o f f o f t h e i r d a i l y r a t i o n of j u n k computer p r i n t - o u t s which the completion of t h i s r e s e a r c h has caused) are g r a t e f u l l y acknowledged. xiv TO DOMINIQUE Foreword "I p e r c e i v e now t h a t the r e a l charm of the i n t e l l e c t u a l l i f e - the l i f e devoted to e r u d i t i o n , to s c i e n t i f i c r e s e a r c h , t o p h i -losophy, to a e s t h e t i c s , to c r i t i c i s m -i s i t s e a s i n e s s . I t ' s the s u b s t i t u t i o n of simple i n t e l l e c t u a l schemata f o r the com-p l e x i t i e s of r e a l i t y , of s t i l l and formal death f o r the b e w i l d e r i n g movements of l i f e . " Aldous Huxley, P o i n t Counterpoint. " J'en p a r l e r a i a mon c h e v a l " Anonyme. 1 INTRODUCTION The purpose of this research i s to analyse, discuss and extend the a n a l y t i c a l methodology associated with the study of brand-switching data. As such, i t attempts to add to the existing understanding of the structure of the consumer choice process. Rational human behavior may be viewed as a suc-cession of choices made among more or less .well defined alternatives. The problem we analyse in this paper i s how to predict these choices when the alternatives are fixed in advance. The alternatives Considered in this study are low-cost, frequently purchased, brand-identified consumer products. The unit of analysis i s the individual consumer. The data are made up of the purchase history of each individual over time. For any two successive purchase occasions, consumers are said to make a repeat-purchase i f the same brand was purchased on both occasions. Sim-i l a r l y , consumers are said to switch brands i f two di f f e r e n t brands were purchased on the two purchase occasions. Brand-switching data are the co l l e c t i o n s of such purchase h i s t o r i e s considering a l l possible brands and purchase occasions. The o b j e c t i v e o f t h i s r e s e a r c h i s t o develop s t o c h a s t i c b r a n d - s w i t c h i n g models t o be used as c o n s t r u c t s f o r o r g a n i z i n g and i n t e r p r e t i n g b r a n d - s w i t c h i n g d a t a . In t h i s r e s p e c t , t h i s d i s s e r t a t i o n f o l l o w s the framework o f brand c h o i c e a n a l y s e s o r i g i n a t e d by a s e r i e s o f a r t i c l e s w r i t t e n by Brown [1952] and Cunningham [1956] which s e t the pace o f the subsequent consumer c h o i c e s t o c h a s t i c m o d e l i n g a c t i v i t y o f the l a s t decade. T h i s a c t i v i t y was marked by the works o f Kuehn [1962 - 6 5 ] , Haines [ 1 9 6 4 ] , Massy [1965 - 66 - 6 7 ] , Montgomery [1966 - 7 - 9 ] , M o r r i s o n [1965 - 6 ] , Ehrenberg [1959 - 6 5 ] , Jones [1970 - 1 - 3 ] ) , H e r n i t e r [1973] and Bass [1974 - 6 ] . In many d i m e n s i o n s , however, t h i s r e s e a r c h i s s i g n i f i c a n t l y d i f f e r e n t from most s t o c h a s t i c models of b r a n d - s w i t c h i n g d e v e l o p e d i n the p a s t . F i r s t , i n c o n t r a s t to the approach o f t e s t i n g a - p r i o r i t h e o r i e s o f consumer c h o i c e b e h a v i o r , the method i n t h i s s t u d y i s t h a t o f d e s c r i p t i o n l e a d i n g to g e n e r a l i z a t i o n (see e.g., Simon [1968] and Ehrenberg [ 1 9 7 2 ] ) . To the e x t e n t t h a t the d e s c r i p t i o n s g e n e r a l i z e a c r o s s c i r c u m s t a n c e s and p r o d u c t c l a s s e s , knowledge i s g a i n e d about the e x i s t e n c e and s t r u c t u r e o f u n d e r l y i n g c h o i c e r e l a t i o n s h i p s . 3 Chapter II pro v i d e s an example of such an e x e r c i c e i n data a n a l y s i s . In that chapter, a set of consumer brand-s w i t c h i n g data are submitted to a p u r e l y s t a t i s t i c a l analy-s i s i n an attempt to uncover p o s s i b l e " r e g u l a r i t i e s " i n the data. These " r e g u l a r i t i e s " are the f t e x p l o i t e d i n chapter I I I to c o n s t r u c t a brand choice model based on the r e s u l t s of the p r e l i m i n a r y s t a t i s t i c a l a n a l y s i s . Second, t h i s r e s e a r c h deals e s s e n t i a l l y with m u l t i -brand s w i t c h i n g behavior as opposed to mere brand l o y a l t y . H e r n i t e r [1973] and Bass [1974] n o t w i t h s t a n d i n g , a l l of the authors mentioned above developed models of brand choice behavior r a t h e r than models of brand-switching behavior. By c o l l a p s i n g the market i n t o an a r t i f i c i a l two-brand mar-ket (to achieve mathematical t r a c t a b i l i t y ) , they were f o r c e d to concentrate on repeat purchase behavior only. A l l the in f o r m a t i o n about brand-switching a c t i v i t y was l o s t i n the aggregation p r o c e s s . Purchases of brands other than the brand under study were t r e a t e d as being purchases of "The Competitor's" brand without c o n s i d e r a t i o n f o r the v a r y i n g degree of co m p e t i t i o n between the v a r i o u s brands. In today's d i f f e r e n t i a t e d markets, the brand manager can no longer view h i s competitors' brands as being e q u a l l y t h r e a t e n i n g to h i s p a r t i c u l a r brand. The competition has to be monitored on a brand-by-brand b a s i s , and t h i s i s 4 best achieved through the use of models that deal e x p l i c i t l y with multi-brand switching such as the one developed in this study. Last, while most exis t i n g brand choice models treat consumer choice behavior as being completely stochastic or e n t i r e l y deterministic, this research views i t as both a stochastic and a cognitive process. Stochastic, since brand selection on a given t r i a l cannot be predicted pre-c i s e l y ; and cognitive because the steady state choice pro-b a b i l i t i e s observed over a sequence of choices reveal a choice pattern consistent with the consumer perceptions, preferences and b e l i e f s toward a p a r t i c u l a r set of brands. To acknowledge both the stochastic and deterministic'fea-tures of the brand-switching phenomenon, this study i n t r o - .. duces a class of models which implies aggregate brand-switching and repeat purchase p r o b a b i l i t i e s . In addition, i t also d i r e c t l y incorporates the impact of a - p r i o r i relevant exogenous variables into i t s structure. This approach, while preserving the important features associated with stochastic brand choice models from the past, allows re-searchers (at least those with more f a i t h in human r a t i o n -a l i t y ) to include in t h e i r models variables of behavioral and managerial significance. The research reported here i s presented in seven chapters. Chapter I relates this research to the relevant l i t e r a t u r e and i l l u s t r a t e s the multidiscxplinary nature of brand-switching modeling a c t i v i t y . The next two chapters are devoted to the analysis of consumer brand l o y a l t y (as opposed to consumer multi-brand switching behavior). In chapter II, we perform..a s t a t i s t i c a l analysis of consumer brand choice data in an attempt to discover possible "regu l a r i t i e s " in the data. In chapter III, we construct a sto chastic model of brand choice based on the " r e g u l a r i t i e s " uncovered by the s t a t i s t i c a l analysis of the preceding chapter. The major a n a l y t i c a l work on which this d i s s e r t a -tion rests i s contained in chapters IV to VI. Chapter IV lays the t h e o r e t i c a l base which guides the development and empirical investigation of the brand-switching models offered in the following two chapters. Chapter IV i s included to f a m i l i a r i z e . the reader with the p r o b a b i l i s t i c concepts and the model-building strategy that underlie the development of the operational formulations presented in the empirical chapters V and VI. In chapter V, a j^oint space theory of brand-choice i s offered to analyse consumer brand-switching behavior. The central concept underlying this theory i s 6 t h a t o f c o g n i t i v e c o n s i s t e n c y . The c h i e f h y p o t h e s i s h o l d s t h a t consumers s t r i v e t o m a i n t a i n an e q u i l i b r i u m between t h e i r p e r c e p t i o n s and p r e f e r e n c e s o f the b r a n d s , on the one hand, and t h e i r a c t u a l brand c h o i c e on the o t h e r hand. T h i s h y p o t h e s i s i s e m p i r i c a l l y t e s t e d w i t h the h e l p o f the ma t h e m a t i c a l t o o l s d e v e l o p e d i n c h a p t e r IV. Chapter VI d i s c u s s e s a pr o c e d u r e f o r b u i l d i n g a p e r c e p -t u a l map o f a market based on a c t u a l c h o i c e d a t a (brand-s w i t c h i n g p r o b a b i l i t i e s ) r a t h e r than the more w i d e l y used (see e.g., Green [1975] and Bouroche [1977]) p r e f e r e n c e and s i m i l a r i t y d a t a . F i n a l l y , c h a p t e r V I I c o n t a i n s a summary and an e v a l u a t i o n o f the d i s s e r t a t i o n f i n d i n g s . 7 CHAPTER I THE EFFORT OF THE PAST: BRAND CHOICE STUDIES An overview of analytic approaches employed in the study of consumer choice behavior necessarily r e f l e c t s the d i v e r s i t y of philosophies of the authors of the large and growing body of l i t e r a t u r e in this f i e l d . Since the focus here i s on a n a l y t i c a l approaches,the l i t e r a t u r e c i t e d w i l l be se l e c t i v e . While the number of published a r t i c l e s , combined with the d i v e r s i t y of approaches to the study of brand choice render any c l a s s i -f i c a t i o n attempt of the l i t e r a t u r e somewhat self-defeating, some kind of c l a s s i f i c a t i o n i s necessary. To this end, the following scheme has been retained for i t s s i m p l i c i t y . Past research w i l l be segmented according to i t s basic philosophy, i . e . whether i t views consumer brand choice as a deterministic or a stochastic process. Deterministic approaches w i l l be further d i f f e r e n t i a t e d on the basis of thei r position along the brand-specific versus person-s p e c i f i c continuum. Brand choice can be symbolically represented by the following r e l a t i o n : P i k * £ i k (Xk> Y 0 8 where P., = preference of individual k towards brand 1 i or p r o b a b i l i t y that individual k chooses brand i X, = a vector of household-related variables k Y. = a vector of brand-related variables 1 f.. = some mathematical function, ik The two most c r u c i a l problems facing the marketing research-ers are to specify: i ) the variables to be included in the X and Y vectors i i ) the form of The following discussion w i l l be wholly concerned with the f i r s t issue. As w i l l be seen l a t e r , most theories of consumer brand choice behavior can be c l a s s i f i e d according to t h e i r handling of this fundamental problem. 1.1 Deterministic Approaches to Brand Choice Behavior 1.1.1 Socio-economic and personality studies of brand  choice The most fundamental question that can be asked about consumer choice behavior i s whether that behavior is a l : least p a r t i a l l y stochastic or whether there exists causes and explanation for a l l such behavior. 9 Many b e h a v i o r a l s c i e n t i s t s s i n c e Freud have b e l i e v e d t h a t t h e r e e x i s t s an e x p l a n a t i o n f o r a l l human b e h a v i o r even i f the e x p l a n a t i o n must be sought i n the u n c o n s c i o u s . T h i s b e l i e f has shaped much o f the e a r l y r e s e a r c h concerned w i t h f i n d i n g the s o - c a l l e d brand c h o i c e c o r r e l a t e s and the demographic p r o f i l e i s p r o b a b l y the most • f a m i l i a r r e s u l t - o f t h i s e f f o r t . Examples o f attempts to l i n k the v a r i o u s components ( e . g . , age, income, e d u c a t i o n ) o f the demographic p r o f i l e t o consumer brand c h o i c e b e h a v i o r are g i v e n by the s t u d i e s o f Frank et a l . [1965,7,8,9]. In g e n e r a l , demographic p r o f i l e a n a l y s i s has had l i m i t e d s u c c e s s i n e x p l a i n i n g i n d i v i d u a l c h o i c e b e h a v i o r . As a r e s u l t , r e s e a r c h e r s have t u r n e d to o t h e r e x p l a n a t i o n s o f brand c h o i c e b e h a v i o r based on non-demographic consumer c h a r a c t e r i s t i c s such as s o c i o - e c o n o m i c and p e r s o n a l i t y f a c t o r s . As w i t h demographic p r o f i l e a n a l y s i s , the new stream o f r e s e a r c h met w i t h mixed s u c c e s s . Some s t u d i e s (see e.g., Day [1969] and Carman [1970]) d i d f i n d some r e l a t i o n s h i p s between brand l o y a l t y and c e r t a i n consumer c h a r a c t e r i s t i c s . For i n s t a n c e , Day [1969] found the brand l o y a l consumer to be v e r y c o n s c i o u s o f the need to econom-i z e when b u y i n g , c o n f i d e n t i n her judgment, o l d e r and l i v -i n g i n a s m a l l e r than average househ o l d . Other s t u d i e s 10 found t h a t h i g h brand l o y a l households a p p a r e n t l y have a p r o f i l e o f p e r s o n a l i t y and so c i o - e c o n o m i c c h a r a c t e r i s t i c s t h a t i s v i r t u a l l y i d e n t i c a l o f t h a t o f households e x h i b i t -i n g a lower degree o f brand l o y a l t y (see e.g., Frank e t • a l . [ 1 9 6 9 ] ) . Because o f t h e i r o f t e n c o n t r a d i c t o r y r e s u l t s , i t i s f a i r t o say t h a t t h e s e and o t h e r d e t e r m i n i s t i c s t u -d i e s o f i n d i v i d u a l consumer c h o i c e b e h a v i o r have c o n s i s -t e n t l y f a i l e d t o e x p l a i n a s u b s t a n t i a l p o r t i o n o f the v a r i a n c e i n the dependent v a r i a b l e . The i n c o n c l u s i v e o r even c o n t r a d i c t o r y r e s u l t s o f t h i s r e s e a r c h are due i n p a r t to the absence o f a w i d e l y a c c e p t e d r e s e a r c h t r a d i t i o n . T h i s makes comparisons between s t u d i e s d i f f i c u l t as d i f f e r e n t r e s e a r c h e r s use d i f f e r e n t d e f i n i t i o n s , c o n c e p t s and m e t h o d o l o g i e s . The f a i l u r e o f brand c h o i c e c o r r e l a t e s to e x p l a i n c h o i c e b e h a v i o r has l e d r e s e a r c h e r s t o i n v e s t i g a t e the r e l a t i o n -s h i p s between brand c h o i c e and c e r t a i n market c h a r a c -t e r i s t i c s , such as the a v a i l i b i l i t y o f ; b r a n d s , p r i c e f l u e - , t u a t i o n s and d e a l i n g a c t i v i t y (see e.g., Massy & Frank [ 1 9 6 5 ] , F a r l e y & R i n g [ 1 9 7 0 ] ) . In t h i s r e s e a r c h the f o c u s s h i f t s from the v a r i a b l e s (person s p e c i f i c ) t o the v a r i a b l e s (brand s p e c i f i c ) . T h i s approach has been w e l -comed by the b u s i n e s s community because i t r e l a t e s con-sumer d e c i s i o n making t o v a r i a b l e s t h a t are. c o n t r o l l a b l e by the f i r m . 11 The importance of these variable types was de-monstrated by Farley's study [1964] which found brand lo y a l t y to be influenced by pri c e , d i s t r i b u t i o n and pro-motional a c t i v i t i e s . Based on this finding, Farley con-cluded that much of the apparent differences in brand choice behavior across product classes could be explained on the basis of stru c t u r a l variables describing the market in which the product i s sold. Farley's results were sub-sequently strongly challenged by Massy and Frank [1965] and Anderson [1966] so that once again no invariant gene-r a l i z a t i o n s can be made. 1.1.2 Mult 1-Attribute' Attitude Theory A large number of recently published a r t i c l e s in the marketing l i t e r a t u r e have extended the attitude theory concepts developed in so c i a l psychology to the study of brand preference and brand choice (see e.g., Lehmann [1971], Pessemier et a l . [1971,2,2a], Bass et a l . [1972] and Wilkie & Pessemier [1973]). There i s no intent here to discuss the d e t a i l s of attitude theory studies in marketing"'". It w i l l be useful, however, to mention b r i e f -l y the underlying nature and basic structure of those studies. 1 For an excellent survey of psychological theories of consumer choice, see Hansen [1976]. 12 The central proposition in attitude theory i s that attitudes are composed of b e l i e f s about the a t t r i -butes of objects and the evaluative aspects of these b e l i e f s . Thus, for example, i f one were measuring a t t i -tudes towards d i f f e r e n t brands of toothpaste, one would determine the b e l i e f s which consumers have about the ex-tent to which each of the several brands possess proper-t i e s such as decay prevention, teeth whitening, taste and breath control. One would also determine the importance which consumers attach to each property for brand choice purposes. The rationale for using multi-attribute attitude (MAA) models to study consumer decision making can be t r a -ced to the work of Rosenberg [1960], Fishbe'in [1965] and Lancaster [1966]. The f i r s t two offered a psychological interpretation of the MAA approach while the l a t t e r pro-vided an economic foundation with his suggestion that "people buy not products but bundles of attributes that meet the i r needs". In general, i t i s established that the MAA model are good predictors of o v e r a l l evaluation or attitude, whereas t h e i r a b i l i t y to predict brand choice behavior i s more varied (see e.g., Wilkie & Pessemier [1973]). The data shows that even when stated preferences are unchangin 13 consumer brand choice does change (see e.g., Bass [1974]). Thus, stated preference i s not a good predictor of choice for a single choice occasion and most importantly a basic premise of much behavioral science theory i s brought into question. This conclusion has led several researchers to develop and test stochastic models of consumer brand choice to which we now turn. 1.2 Stochastic Models of Consumer Choice Behavior 1.2.1 General discussion When hope of understanding the motivations and subsequent actions of consumers i s gone, researchers turn to stochastic models of purchase behavior. The brand-switching l i t e r a t u r e of the past decade i s r i c h with such f r u i t s of the o r e t i c a l despair. T y p i c a l l y , stochastic models take the p r o b a b i l i s t i c nature of consumer brand choice as given and make l i t t l e or no attempt to model the underlying mechanisms of individual brand choice behavior. As a r e s u l t , they predict but do not explain the d i f f e r i n g brand purchase p r o b a b i l i t i e s which exist between i n d i v i -duals. This i s a ra d i c a l departure from t r a d i t i o n a l deterministic theories that purport to explain an i n d i v i -dual's brand preferences in terms of demographic or psy-chologic c h a r a c t e r i s t i c s of the in d i v i d u a l , or in terms 14 of his b e l i e f s and/or attitudes concerning the attributes of available brands and some sought-after " i d e a l " l e v e l of these a t t r i b u t e s . The underlying rationale behind the previously discussed " t r a d i t i o n a l " theories i s that behavior i s deterministic. Thus, i f object A i s preferred to object B, then, other things being equal, object A w i l l be chosen, and unless preferences change, choice w i l l not change. Any other pattern wouldcsuggest i r r a t i o n a l i t y . Proponents of the stochastic school have challenged this premise. They argue that there exists a substantial stochastic component in brand choice behavior, and as a result i t i s no more possible to provide an explanation for that com-ponent than i t i s to provide an explanation for the out-come of the toss of a coin. Massy et a l . [1970] have provided a detailed review of the issues and the structure of stochastic models developed before 1970. Their c l a s s i f i c a t i o n of the various models, reproduced in table 1.1, has been updated to incorporate the recent development i n the f i e l d . The table c l a s s i f i e s the models according to four c h a r a c t e r i s t i c s : (i) time e f f e c t s : whether the model deals with discrete time (purchase occasion) or continuous time; 15 (ii) purchase event feedback: whether the model assumes that the act of purchasing and using a product has a direct effect on the household's subse-quent purchase probabilities; (iii) population heterogeneity: whetiveir the model allows for the fact that consumers differ from one another in many ways (in terms of demographic and socio-economic factors, awareness, attitudes etc. . .) ; (iv) number of alternatives considered: whether the model can handle multi-brand markets or collapses the market into an artificial two-brand market. As can be seen from Table 1.1, the early Markov and Linear Learning models incorporate assumptions about the effects of purchase-event feedback on brand choice, but do not make provisions for time effects and popula-tion heterogeneity. The assumption of population homoge-neity has later been relaxed by Morrison [1 965a,b] and Massy [1965] who developed "compound" versions of the Bernouilli, Markov and Linear Learning models respectively. The term c^ompound" denotes the fact that an explicit pro-vision for a probability distribution of relevant parameter values is included in the model. TABLE 1.1 CLASSIFICATION OF STOCHASTIC BRAND CHOICE MODELS CHARACTERISTICS No Time E f f e c t s Time E f f e c t s T w o - a l t e r n a t i v e models Homogeneous p o p u l a t i o n No purchase event feedback B e r n o u i l l i Purchase event feedback Markov L i p s t e i n [1959] L i n e a r L e a r n i n g Kuehn [1958] No purchase event feedback none Purchase event feedback Semi-Markov Howard [1963] V a r i a b l e Markov T e l s e r [1963] Dynamic Markov L i p s t e i n [1965] Heterogeneous p o p u l a t i o n H o u s e h o l d - B e r n o u i l l i Frank [1962] Compound B e r n o u i l l i M o r r i s o n [1965a,b] Aaker's New T r i e r [1971] Household-Markov Massy [1966] Composite model Jones [1973] Response U n c e r t a i n t y Coleman [1964] Dynamic I n f e r e n c e Howard [1965] P r o b a b i l i t y D i f f . Montgomery [1966] Semi-Markov H e r n i t e r [1971] Household v a r i a b l e Markov Duhamel [1966] V a r i a b l e L e a r n i n g Kuehn e t a l . [1967] L e a r n i n g - D i f f . Jones [1969-71] M u l t i a l t e r n a t i v e models Heterogeneous p o p u l a t i o n Hendry model [1966] Entropy model H e r n i t e r [1973] Bass PNBD [1974-6] M a r k o v i a n P o l i c y model L e e f l a n g [1974] 17 Basic contributions to two-alternative brand choice models combining time effects and population het-erogeneity were made by Coleman [ 1 9 6 4 ] , Howard [ 1 9 6 5 ] and Montgomery [ 1 9 6 6 ] . In the work of Coleman, households' purchase p r o b a b i l i t i e s are i n i t i a l l y d i s t ributed over members of the population, and are then changed according to a time trend function. In contrast, Howard's work assumes a household to draw i t s purchase p r o b a b i l i t y from a d i s t r i b u t i o n not once, as in the Coleman model, but again and again at randomly dis t r i b u t e d points in time. Montgomery's Probabi l i t y Diffusion model [ 1 966]:" isran extension of Coleman's model. Herniter's semi-Markov model [ 1 9 7 1 ] extends Howard's and Mongomery's work and accounts for both purchase timing and brand selection. A f i r s t order Markov process i s used by Herniter to des-cribe brand selection and Erlang density functions are used to describe time between purchases. Models including time effects and purchase •• event feedback but not population heterogeneity have also been developed (see e.g., Howard [ 1 9 6 9 ] , Telser [ 1 9 6 3 ] and L i p s t e i n [ 1 9 6 5 ] . Relatively l i t t l e has been published on brand choice models including a l l four of the charac-t e r i s t i c s shown in Table 1.1. Exceptions are Duhamel [ 1 9 6 6 ] who estimated Telser's Variable Markov model using 18 household data, and Jones [1969-70] who extended Mont-gomery's P r o b i l i t y Diffusion model to include learning c h a r a c t e r i s t i c s . The recent trend i s toward building "composite" models which extend the notion of consumer heterogeneity to allow for the fact that d i f f e r e n t consumers may obey di f f e r e n t mechanisms of behavior. Previously proposed brand choice models have always assumed that each consumer in the population obeys the same underlying mechanism. In this context, Jones [1973] has discussed a model which does not assume a single behavioral mechanism for the entire population, but i s in fact a composite model which allows each consumer to obey one of three mechanisms: B e r n o u i l l i , Markov and Linear Learning. Jones provides:; for two types of heterogeneity in the composite model. The f i r s t type involves differences between consumers who obey the same behavioral mechanism (parameter heterogenei-ty) while the second type distinguishes the composite model from i t s components (model heterogeneity). This work, while s i g n i f i c a n t , has not been empirically tested. An approach similar to Jones has been suggested by Blattberg and Sen [1974-76] who described a Bayesian discrimination procedure which determines for each con-sumer the stochastic model of brand choice best supported 19 by his past purchasing behavior. Blattberg and Sen claim 0 that a market segmentation strategy.based on the consumer behavior mechanism provides better information to the mar-keting decision maker. Their work marks the re-emphasis of stochastic based research away from model-fitting and back to the more interesting diagnostic p o t e n t i a l , i . e . the extent to which the research can contribute to a bet-ter understanding of brand choice behavior. Meanwhile, zero order models of brand choice a< again become popular, es p e c i a l l y in the context of brand-switching analysis. Her.niter [1973] and Bass [1974] have employed zero-order models in a heterogeneous population for the study of brand-switching on adjacent purchase occasions. The distinguishing feature of these models rests on th e i r a b i l i t y to deal with multi-brand markets. Previously, a l l early brand choice models (such as the ones in the upper half of Table 1.1) had to - combine-the market into a two-brand market, i . e . the " f a v o r i t e " brand plus an " a l l - o t h e r " brand. Herniter's Entropy model [1973] deserves our attention i f only because i t i s the f i r s t multi-brand switching model to be published in the l i t e r a t u r e ^ . 2 Actually, the Hendry model antedates the Entropy model, but d e t a i l s of the model's derivation are not available. 20 Although the Entropy model i s based on sound mathematical and behavioral assumptions, i t s complexity becomes unwiel-dy as the number of brands goes past four. Moreover, the Entropy model i s a normative model and produces the brand-switching matrix as a function of market shares only. If there does not exist a one-to-one correspondence between a set of market shares and the associated brand-switching matrix, the model may not consistently y i e l d good predic-tions. Herniter, however, has supplied some empirical evidence that supports the model in one such case - that of an equilibrium market. The major weakness of the maximum entropy approach to the estimation of brand-switching p r o b a b i l i t i e s , as used by Herniter, i s i t s i n f l e x i b i l i t y . The entropy estimates of brand-switching depend only on the market shares and are therefore, independent of product category. This has led Bass [1974] to develop a model which can accomodate a much richer range of brand-switching situations. 1.3 In Summary After this b r i e f "tour d'horizon" of the va-rious approaches to stochastic and deterministic brand choice modeling, several comments are in order. F i r s t , most stochastic brand choice models are at the aggregate 2 1 l e v e l , even though they often employ a parameter to repre-sent heterogeneity across the population. Second, since stochastic models ra r e l y contain marketing variables (for some exceptions, see e.g.,: Haines [1 969], Kuehn and Rohloff [1967]; L i l i e n [1974] and Nakanishi [1973]) using them at the aggregatellevel has offered marketing managers l i t t l e insight into their problems. Third, both appro-aches to consumer brand choice modeling have weaknesses and advantages. The deterministic approaches (demographics, psychographics, and more recently, the multi-attribute attitude theory) are r i c h in their implications for mar-keting strategy but usually lack strong empirical support. With stochastic brand choice models, exactly the opposite circumstances occur. The models exhibit strong empirical support but offer l i t t l e in the way of marketing action and/or p o l i c y implications. In contrast, the model b u i l d -ing approach presented in chapter IV provides a mean of incorporating the influence of both behavioral and mar-keting variables, and represents a viable alternative to the existing brand-switching models. In preparation for t h i s a l t e r n a t i v e , the next chapter i s wholly concerned with a s t a t i s t i c a l analysis of consumer brand choice data. 2 2 CHAPTER I I PRELIMINARY STATISTICAL ANALYSIS OF CONSUMER BRAND-SWITCHING DATA In c o n t r a s t to the approach of t e s t i n g a p r i o r i t h e o r i e s of consumer c h o i c e behavior, the method i n t h i s chapter i s t h a t of d e s c r i p t i o n l e a d i n g to g e n e r a l i z a t i o n . To the extent t h a t the d e s c r i p t i o n s g e n e r a l i z e across circumstances and product;.; c l a s s e s , knowledge i s gained about the e x i s t e n c e of r e l a t i o n h i p s i f not about the reasons f o r the o b s e r v a t i o n . F o l l o w i n g Ehrenberg (1972), we s h a l l search f o r " r e g u l a r i t i e s " i n consumer brand-s w i t c h i n g data. These r e g u l a r i t i e s , to the extent t h a t they extend beyond the data from which they were generated can l e a d to the c o n s t r u c t i o n of explanatory t h e o r i e s from which the g e n e r a l i z a t i o n s can be d e r i v e d . Our methodology c o n s i s t s of two st e p s : 1. In t h i s chapter, we s h a l l perform a p u r e l y s t a t i s t i c a l a n a l y s i s of consumer brand-switching data i n an attempt to uncover p o s s i b l e " r e g u l a r i t i e s " i n the data. 2 3 2. I n t h e n e x t c h a p t e r , we s h a l l c o n s t r u c t s t o c h a s t i c m o d e l s o f b r a n d c h o i c e b a s e d o n t h e " r e g u l a r i t i e s " u n c o v e r e d b y t h e p r e c e d i n g s t a t i s t i c a l a n a l y s i s . T o t h i s e n d , we h a v e o r g a n i z e d t h e f o l l o w i n g d i s c u s s i o n a r o u n d - e i g h t m a j o r s e c t i o n s : - D e s c r i p t i o n o f c o n s u m e r b r a n d - s w i t c h i n g d a t a . - C h o i c e o f a s t a t i s t i c a l t e c h n i q u e . - T h e l o g - l i n e a r m o d e l . - M o d e l f i t t i n g a n d h y p o t h e s i s t e s t i n g p r o c e d u r e s - P r e s e n t a t i o n o f t h e e m p i r i c a l r e s u l t s - C o m p a r i s o n o f b r a n d - s w i t c h i n g b e h a v i o r a c r o s s s e g m e n t s . - Summary o f t h e r e s u l t s a n d c o n c l u s i o n . 2.1 D e s c r i p t i o n o f c o n s u m e r b r a n d - s w i t c h i n g d a t a . C o n s u m e r b r a n d - s w i t c h i n g d a t a d e n o t e t h e c o l l e c t i o n o f i n d i v i d u a l s ' p u r c h a s e h i s t o r i e s a g g r e g a t e d o v e r a l l p o s s i b l e b r a n d s a n d p u r c h a s e o c c a s i o n s . T y p i c a l -l y , t h e p u r c h a s e b e h a v i o r o f a s a m p l e o f N i n d i v i d u a l s i s o b s e r v e d o v e r a g i v e n t i m e p e r i o d T . F o r a g i v e n p r o d u c t c l a s s , e a c h i n d i v i d u a l ' s p u r c h a s e h i s t o r y c a n be r e p r e s e n t e d by a s t r i n g o f l ' s a n d G ' s . A "1" r e p r e s e n t s a p u r c h a s e o f t h e b r a n d u n d e r 24 c o n s i d e r a t i o n w h i l e a " 0 " r e p r e s e n t s a p u r c h a s e o f a n y o t h e r b r a n d 1 . A s s u m e t h a t d a t a a r e a v a i l a b l e f o r f i v e c o n s e c -u t i v e p u r c h a s e s f o r e a c h o f t h e N i n d i v i d u a l s i n t h e p a n e l , T h e N i n d i v i d u a l s c a n b e s e g m e n t e d b y t h e s e f i v e p u r c h a s e s 5 i n t o 2 o r 3 2 m u t u a l l y e x c l u s i v e a n d c o l l e c t i v e l y e x h a u s -t i v e c a t e g o r i e s . T h e p a n e l d a t a c a n now b e r e p r e s e n t e d i n t e r m s o f N , . , , , t h e n u m b e r o f i n d i v i d u a l s who p u r -i - j k l m e c h a s e d b r a n d i , j , . k , 1 a n d m r e s p e c t i v e l y o n f i v e s u c c e s -s i v e p u r c h a s e o c c a s i o n s , w h e r e i , j , k , 1, m = 1 o r'vO. T h e c o r r e s p o n d i n g p r o p o r t i o n p ^ j } , i m ± s j u s t t h e r a t i o o f N . . , , t o t h e n u m b e r o f i n d i v i d u a l s i n t h e p a n e l i . e . : l j k l m ( 2 . 1 ) P . . n = N.. , / N ' l j k l m i ] k l m We w i l l a l s o p a y p a r t i c u l a r a t t e n t i o n t o t h e p r o p o r t i o n o f i n d i v i d u a l s who p u r c h a s e d b r a n d m o n t h e f i f t h t r i a l g i v e n t h a t t h e y p u r c h a s e d b r a n d i , j , k a n d 1 o n t h e f i r s t f o u r p u r c h a s e o c c a s i o n s , w h e r e a g a i n i , j , k , 1, m = l o r 0. T h i s p r o p o r t i o n , w h i c h we s h a l l w r i t e P / . n i s b y d e f i n i t i o n e q u a l t o : m / i j k l . J I n t h i s c h a p t e r , we w i l l l i m i t o u r s e l v e s t o a t w o -b r a n d m a r k e t : b r a n d 1, t h e b r a n d u n d e r s t u d y a n d b r a n d 0, a n a l l o t h e r c a t e g o r y . T h e b r a n d u n d e r s t u d y i s o f t e n t h e i n d i v i d u a l ' s f a v o r i t e b r a n d . M o d e l s f o r m u l t i - b r a n d m a r k e t s w i l l b e i n t r o d u c e d i n c h a p t e r s I V . 25 P. i jklm (2.2) P m / i j k l + P. . i j k i r The P. . i j klm 1 s r e p r e s e n t a five-way contingency t a b l e . T h i s m u l t i d i m e n t i o n a l contingency t a b l e d e s c r i b e s the j o i n t d i s t r i b u t i o n of f i v e q u a l i t a t i v e dichotomous v a r i a b l e s . Q u a l i t a t i v e , s i n c e the v a r i a b l e s are nominal, and dichotomous because we are d e a l i n g with a two-brand market. These f i v e b i n a r y v a r i a b l e s r e p r e s e n t the purchase outcome observed f o r the corresponding f i v e purchase o c c a s i o n s . That i s , v a r i a b l e 1 stands f o r the purchase outcome a t purchase o c c a s i o n 1 . T h i s v a r i a b l e takes on a value of 1 or zero depending on which of the two brands was purchased. The remaining v a r i a b l e s are s i m i l a r l y d e f i n e d . a five-way contingency t a b l e r e l a t e d to c o f f e e consumption of a sample of 5 3 8 f a m i l i e s from the Chicago Tribune 2 Consumer Panel . As can be noted from the t a b l e , 3 7 . 0 6 % ( 1 9 6 8 / 5 , 3 1 0 ) o f the i n d i v i d u a l s purchased brand 1 on a l l f i v e purchase o c c a s i o n s . Of those who bought brand 1 on the f i r s t f o u r purchase o c c a s i o n s , 8 8 . 9 0 % ( 1 9 6 8 / ( 1 9 6 8 + 2 4 5 ) ) purchased i t a g a i n on the f i f t h o c c a s i o n . For i l l u s t r a t i o n purposes, Table .11.9 d i s p l a y s " T h e s e a n d o t h e r r e l a t e d d a t a w i l l b e d e s c r i b e d i n a l a t e r s e c t i o n . Example of -the five-way epn£ingen.ey table: Brand-switching data from the Chicago Tribune Consumer Panel data.// Purchase QU-feJcomeieatit occasion t-4 1 0 ft-3 1 0 1 0 ;t-2 1 0 1 0 1 0 1 0 t-1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 * 1 1968 203 196 90 222 67 75 64 276 79 84' 56 128 62 105 108 t 0 245 132 84 94 73 72 49 110 93 55 62 90' 59 68 85 156 Sample s i z e : 5310 * Read : 196 individuals purchased brands 1,1,0,1 and 1 at purchase occasions t - 4 , t - 3 , t-2,t-1 and t respectively. # These data are described at length i - n section 2.5 ho ON TABLE I I . 9 27 The corresponding p r o p o r t i o n f o r those who bought brand 0 on the f i r s t f o u r purchase o c c a s i o n s s h r i n k s to 40.9%. As a r u l e , we observe t h a t the p r o p o r t i o n of f a m i l i e s who bought brand 1 g i v e n t h e i r p a s t purchase h i s t o r y ) increase's with both the number of p a s t purchases of brand 1 and the recency of these purchases. For i n s t a n c e , we observe t h a t p i / ^ ^ > Pi / 0 0 0 0 a n < ^ that ]?1/0001 > p i / i o o o ' a n i n d i c a t i o n t h a t both the number and p o s i t i o n of the l ' s i n the purchase s t r i n g a f f e c t the observed p r o p o r t i o n s . I t i s the purpose of t h i s chapter to f o r m a l l y estimate the i n f l u e n c e , i f any, of past purchase outcomes on f u t u r e purchase d e c i s i o n s from brand-switching data such as the ones j u s t d e s c r i b e d . To t h i s end, we s h a l l f i r s t need to decide on an a p p r o p r i a t e s t a t i s t i c a l t e c h -nique. Hypothesis t e s t i n g , e s t i m a t i o n procedures and the e m p i r i c a l data w i l l then be presented and the r e s u l t s d i s c u s s e d . 2.2 Choice of a s t a t i s t i c a l technique. The r e s t r i c t e d range of the c r i t e r i o n (proportion) and the q u a l i t a t i v e nature of the p r e d i c t o r s v i o l a t e c r u c i a l assumptions of most t r a d i t i o n a l s t a t i s t i c a l t e c h -niques. Because the c r i t e r i o n v a r i a b l e i s a p r o p o r t i o n , 28 application of dummy-variable regression, Automatic Inter-action Detector or Anova would e n t a i l three l i m i t a t i o n s z i) the c r i t e r i o n variable (proportion buying brand 1 given past purchase history) i s not normally d i s t r i b u t e d , since i t i s sum-constrained; i i ) the c r i t e r i o n variable does not show constant variance (homoscedasticity) across variations i n the predictors (dummy-coded variables i n t h i s case); i i i ) the model's prediction could f a l l outside the range at 0 to 1. One way out of these d i f f i c u l t i e s i s to apply a transformation to the proportion data, whose value are constrained to range from'6 to 1, and then to use an estimation procedure that takes unequal variance error into account. The l o g i s t i c transformation i s one such trans-formation and the l o g i t model provides a useful and natural representation of the data (see e.g., Berkson [1944], Bishop [1969]). So i s the l o g - l i n e a r model of multidimensional contingency table which we s h a l l use i n t h i s study to detect meaningful patterns i n the brand-29 s w i t c h i n g data to be analysed . 2.3 The l o g - l i n e a r model . F o l l o w i n g B i r c h [ 1 9 6 9 ] and Goodman ( [ 1 9 6 8 ] , [ 1 9 6 9 ] , [ 1 9 7 0 ] , [ 1 9 7 1 ] , [ 1 9 7 2 ] ) , we can express the l o g a -r i t h m o f the observed p r o p o r t i o n s P ^ j ^ m i n t o main e f f e c t s ( f u n c t i o n s of a s i n g l e s u b s c r i p t ) and i n t e r a c t i o n e f f e c t s ( f u n c t i o n of two or more s u b s c r i p t s ) . The l a t t e r are, i n t u r n , d i s t i n g u i s h e d as b i v a r i a t e i n t e r a c t i o n s , t r i v a r i a t e i n t e r a c t i o n s and so f o r t h . The ( f u l l ) l o g - l i n e a r model by analogy w i t h Anova models i s d e f i n e d as: (213) l o g ( P i j k l m ) = 6 + X* + + A £ + A * + xl BC BD BE BF CD CE CF DE + A^ + X i k + X±1 + A i m + X j k + + A j m + X k l . DF EF , BCD , BCE , BCF , .BDE BDF km lm 13 k i ] l l.jm l k l lkm , BEF CDE , ,CDF , DEF , CEF , BCDE BCDF + X i l m + A j k l + Ajkm + Aklm + X i l m + X i j k l + A i j k m + CDEF + ^BDEF + ^BCEF + ^BCDEF jklm i k l m i j l m i j k l m The -logit. model i s a p a r t i c u l a r case of the more g e n e r a l l o g - l i n e a r model. In the marketing context, Kuehn [ 1 9 5 8 ] was the f i r s t to r e a l i z e the d i a g n o s t i c p o t e n t i a l of the f a c t o r i a l a n a l y s i s of v a r i a n c e technique to i s o l a t e the e f f e c t s of purchase d e c i s i o n s on each t r i a l i n the p a s t upon the p r o b a b i l i t y of p u rchasing the f a v o r i t e brand on the next t r i a l . Of course, the p r e s e n t s i t u a t i o n d i f f e r s from the a n a l y s i s of v a r i a n c e because we are d e a l i n g w i t h a sample from a c r o s s - c l a s s i f i c a t i o n r a t h e r than w i t h independent o b s e r v a t i o n s from p o p u l a t i o n s t h a t are normally d i s t r i b u t e d and homoscedastic. 4 where f o r uniqueness the A ' s s a t i s f y the u s u a l kinds of c o n d i t i o n s : (2.4) E X? = 0, Z : . ^ = 0, £. *BCD = E X BCD = E XBCD = E XBCDEE = . i j k k i j k U ' ' . i j k l m E XBCDEF = i j k l m m J The X's r e p r e s e n t the p o s s i b l e " e f f e c t s " o f the f i v e v a r i a b l e s (purchase occasions) on ^^im : t * i e B F main e f f e c t s are X ^ , X m ; the remaining X's r e p r e s e n t i n t e r a c t i o n e f f e c t s . The u p p e r s c r i p t on X stand f o r purchase o c c a s i o n s (B,;C, D, E, F i n order of i n c r e a s i n g recency) while the s u b s c r i p t s denote brand c h o i c e s . The number of s u p e r s c r i p t s (or s u b s c r i p t s ) on a p a r t i c u l a r X i n d i c a t e s the order of the i n t e r a c t i o n : one f o r main e f f e c t s , two f o r b i v a r i a t e i n t e r a c t i o n s , e t c . . . For BDE i n s t a n c e , the parameter ^-j^i q u a n t i f i e s the e f f e c t on P i j k l m °^ a P u r C j V i a s e °f brand i a t o c c a s i o n t - 4, k a t o c c a s i o n t - 2 , and 1 a t o c c a s i o n t - 1 . The e n t i r e s e t of X's must be estimated from the observed p r o p o r t i o n s . 4 See e.g., Goodman (1968). 31 In t h i s form, the log-linear model i s known as a saturated model because there i s a parameter for each data point. Making use of the set of constraints expressed i n (2.2), the l o g - l i n e a r model can be rewritten as: (2.5) Log ( P . . n , = eH A 6 . + \ „6 . + \ _<5 +' x „S -, + X c 6 ^ ljklm) l i . 2 ] 3 k 4°1 5m , + A 1 2 ( 5 i j + x 1 3 6 i k + x 1 4 6 i l + X 1 5 6 i m + X 236 jk t X 246 j l S+ A 256 jm + X 34 6kl + X 3 5 6 k m " X4.5 6lm +' X123 6 i j k + A235 6jlm + A 1 2 3 4 6 i j k l + + X 12456 i j l m + X 6 12345 ijklm where . _ 1 i = 1 8.. - , i f _ , l -1 0/w 6 _ 1 i+j . . - , i f . / i s even or zero 13 -1 0/w 6. .. = ^ i f ^ t " * + k i s odd i j k -1 0/w 1 .^ k+i+k+1 6. ., , = , i f n , i s even or zero i ] k l -1 0/w 6 . . . , _ 1 . c i+j+k+l+m . .,, l i k l m , i f „ / i s odd J -1 0/w If we further require equality between predicted and observed marginals, equation (2.5) reduces to: 2.6) P- M I = a. b.c,d,e M.. . M. . M, „ M. , , ' l j k l m i j k l m i t - 4 j t - 3 kt-2 i t - 1 M . A . .. . (0) mt i j k l m where a. = { E b.c.d-e M. . _ M. , 0 MT, , M , 1 . i n . j k l m i t - 3 kt-2 ..lt-1 mt j,k,l,m, J A . ., '(e) } ~ 1 i j k l m b. = { E a.c.d,e M., . M, 0 M, , M j - i n i k 1 m i t - 4 kt-2 l t - 1 mt J i , k , l , m A i j k l m ( * ) } _ 1 ( 2 - 7 ) c k = { . . \ - a i b j d l e m M i t - 4 M j t - 3 M l t - 1 Mmt i.,-j , 1 ,m. J A . . ( e ) } " 1 l j k l m d, ={ E a.b.c.e Ml4_ . M. _ M. M 1 . . . l j k m i t - 4 j t - 3 kt-2 mt l , j , K , m A . , (6) } _ 1 i j k l m e ={ E a.b.c.d, M.t , M.. , M. , , m . . , , I j k 1 i t - 4 j t - 3 kt-2 1 / J t K i -L A . •. -i ( e ) } ~ 1 i j k l m 3 3 where (2.8) M. . = E P.,,. i t - 4 . , , lnklm ;j,k,l,m J M . . _ = Z P. .. . ] t " 3 i , k , l , m i : ) k l m M.... „ = E P . , n kt-2 . . , l l k l m i , ] , l , r a l t _ 1 k i k m i : ] k l m M . = E : ; p , , n i , 3 , k , l J and Log (A..., (o)) = A , 0 6 . . ^ l j k l m 12 l j + X 1 3 6 i k + X 1 4 6 i l + A . J . 15 lm 23 j k + A 2 4 6 j l + 25 jm X 3 4 6 k l 3 5 km + X 4 5 6 l m + X 1 2 3 6 i j k + .... + X 2 3 5 6 j l m + X 1 2 3 4 6 i j k l + + X 1 2 4 5 6 i j l m + X 1 2 3 4 5 6 i j k l m The M's i n equation (2.6) are the s o - c a l l e d marginals of the j o i n t d i s t r i b u t i o n P. ., , . These marginals 3 l j k l m r e p r e s e n t the p r o p o r t i o n of consumers who purchased a g i v e n brand on a p a r t i c u l a r purchase o c c a s i o n . The a's, b's, c's, d's and e's are j u s t n o r m a l i z a t i o n constants t h a t f o r c e e q u a l i t y between p r e d i c t e d and observed 34 m a r g i n a l s . Note t h a t the parameter 6 and the parameters f o r the main e f f e c t s ( X^  through X^ ) are no longer p r e s e n t i n the f i n a l form of the l o g - l i n e a r model. T h e i r e f f e c t on P. , have been taken care of by the n o r m a l i z a t i o n l j k l m J c o n s t a n t s . I t may be a p p r o p r i a t e a t t h i s stage to remind the reader of the purpose of t h i s chapter, l e s t he be overwhelmed by the messy n o t a t i o n of the l o g - l i n e a r model. The purpose of t h i s chapter i s to search f o r " r e g u l a r i t i e s " i n a s e t of brand-switching data to be d e s c r i b e d l a t e r . In p a r t i c u l a r , we would l i k e to be able to q u a n t i f y the nature and e x t e n t of the i n f l u e n c e o f p a s t purchase out-comes on f u t u r e ones. Given the r e s t r i c t e d range of the c r i t e r i o n and q u a l i t a t i v e nature of the p r e d i c t o r s , we were l e d to propose the l o g - l i n e a r model of m u l t i - d i m e n s i o n a l c o n t i n -gency t a b l e as the most a p p r o p r i a t e model f o r the purpose at hand. The l o g - l i n e a r model f i r s t i n v o l v e s t a k i n g the n a t u r a l l o g a r i t h m of each P ^ j ^ m a n d e x p r e s s i n g i t as a f u n c t i o n of main and i n t e r a c t i o n e f f e c t s . In doing so, we hope to d e t e c t meaningful p a t t e r n s among the i n t e r -a c t i o n s t h a t would e i t h e r c o n f i r m or i n f i r m whatever hypotheses marketing r e s e a r c h e r s g e n e r a l l y h o l d with 35 r e s p e c t to consumer brand-switching behavior"". B u i l d i n g on the r e s u l t s of t h i s p u r e l y s t a t i s t i c a l a n a l y s i s , we w i l l then c o n s t r u c t s t o c h a s t i c models of brand-switching behavior and t e s t them wi t h consumer panel data. We now r e t u r n to the l o g - l i n e a r model to d i s c u s s a l t e r n a t i v e model f i t t i n g and hypothesis t e s t i n g procedures. 2.4 Model F i t t i n g and Hypothesis T e s t i n g Procedures. Formulae (2.6 - 2.9) d e s c r i b e the " s a t u r a t e d " model i n which a l l p o s s i b l e " e f f e c t s " are i n c l u d e d . In the s a t u r a t e d model, there are as many parameters as there are data p o i n t s , and the data w i l l be f i t t e d per-f e c t l y . Hence, f i t t i n g the s a t u r a t e d model i s o f l i t t l e v a l u e i n i t s e l f . Rather, i t s v a l u e l i e s i n p o i n t i n g out p o s s i b l e c l u e s t o unsaturated models, where some of the X's are s e t equal to zero, or equal to one another. S i n c e the f i v e v a r i a b l e s which c o n s t i t u t e the five-way contingency t a b l e r e p r e s e n t the purchase outcomes of i n d i v i d u a l s over f i v e s u c c e s s i v e purchase o c c a s i o n s , they are c l o s e l y r e l a t e d . T h i s dependence should t r a n s -l a t e i t s e l f i n t o a simp l e r s t r u c t u r e i n terms of i n t e r -5 Such as Kuehn's [1958] l e a r n i n g h y p othesis or Frank's [i960] zero-order h y p o t h e s i s . 36 a c t i o n e f f e c t s . That i s , some i n t e r a c t i o n e f f e c t s should e i t h e r v a n i s h or e x h i b i t some c l e a r p a t t e r n with one another. A l s o , brand-switching data have a l r e a d y been analysed w i t h the help of ( s t o c h a s t i c ) models t h a t were parsimonious i n terms of number of parameters. Kuehn's L i n e a r L e a r n i n g model [1958] and to a l e s s e r extent, Morrison's Brand L o y a l model [1970] appear to f i t a v a r i e t y of brand-switching data w i t h f a i r l y good accuracy, thus r e i n f o r c i n g our a - p r i o r i b e l i e f t h a t i t should be p o s s i b l e to " e x p l a i n " the data to a s a t i s f a c t o r y degree without i n c l u d i n g i n the l o g - l i n e a r model an overwhelming number of parameters. In the remainder of t h i s s e c t i o n , we s h a l l b r i e f l y o u t l i n e some- e s t i m a t i o n and t e s t i n g proce-dures to estimate and s e l e c t v a r i o u s "unsaturated" models. 2.4.1 E s t i m a t i o n procedure. Two d i f f e r e n t s e t s of parameters need to be estimated: i ) the i n t e r a c t i o n parameters: the A's; i i ) the n o r m a l i z a t i o n c o n s t a n t s : {a^}, , ^ e m ^ -S i n c e the n o r m a l i z a t i o n constants are f u n c t i o n s of the A's [see ( 2 \6) - ( 2 • .8 ) ] , a l l we need to do i s to estimate the l a t t e r from which the former w i l l f o l l o w . 37 However, g i v e n the n o n - l i n e a r i t y of the r e l a t i o n s h i p s between the n o r m a l i z a t i o n c o n s t a n t s , i t e r a t i v e procedures must be r e s o r t e d t o . The parameters were estimated by m i n i m i z i n g the usu a l c h i - s q u a r e goodness of f i t s t a t i s t i c , which measures the agreement between the observed p r o p o r t i o n s P ^ j ^ ^ ^ n the contingency t a b l e and the corresponding estimate P ^ j ^ m under a g i v e n h y p o t h e s i s : (2 10) X 2 = ' N ^ /p - p ) 2/P i , j , k , l , m i j k l m i j k l m ' i j k l m Where N i s the sample s i z e upon which the pro-p o r t i o n s P. ., , are based. For the s a t u r a t e d model (2.6), ^ i j k l m these minimum chi - s q u a r e estimates are a l s o maximum l i k e -l i h o o d e s t i m a t e s , s i n c e the f i t i s p e r f e c t . For an a l t e r -n a t i v e maximum l i k e l i h o o d procedure, see Goodman [1971]. For t e s t i n g purposes, Goodman [1970] advocates the use of y e t another c h i - s q u a r e s t a t i s t i c d e r i v e d from the l i k e l i h o o d r a t i o c r i t e r i o n , and denoted by X 2 r i n t h i s study. T h i s s t a t i s t i c i s d e f i n e d as: t 2 ' 1 1 * X L R = 2 . * N i j k l m l o 9 ^ i j k l m ^ i j k l m * i / 3 / K , i , m . where p ^ j ] c n m 1 S the model p r e d i c t e d p r o p o r t i o n . For convenience, the va l u e of both c h i - s q u a r e s t a t i s t i c s are r e p o r t e d i n the e m p i r i c a l s e c t i o n . 38 2.4.2 Unsaturated model se l e c t i o n . One (rather ad-hoc) way of selecting unsaturated model i s to compute a t - value for each c o e f f i c i e n t and to select as an i n i t i a l unsaturated model those interactions (A's) with s i g n i f i c i e n t t - values. However, we w i l l often be concerned with a set of A's rather than with a single A. The test which we s h a l l use to discriminate between two d i f f e r e n t models makes use of the fa c t that one i s a constrained version of the other (see Goodman [1970]). This w i l l be true of a l l the models entertained below. For example, the model i n which a l l the four-way and the five-way interactions are zero i s a constrained version of the model i n which only the four-way i n t e r -actions are assumed to vanish. Both models are i n turn "nested" i n the general saturated model, i . e . , they are constrained versions of i t . More generally, l e t Q denote the set of A's that are assumed zero under a given hypothesis H. E.g., i f we assume that a l l of the four-way interactions are zero, then a consists of the f i v e parameters appearing i n ( 2 . 5 ) - For a d i f f e r e n t model, say H', l e t 9,' denote the set of A's that are assumed zero under H'. Consider the case where 0, includes the set i n ft' (in addition- to 39 other sets), and l e t 0,* denote the set of A's that are included i n ft but not i n !!'. Let H* denote the hypothesis that the A's i n 9,* are zero, and l e t H*/H' denote the hypothesis that H* i s true assuming that H' i s true. (Note that H*/H' i s equivalent to the hypothesis that H i s true assuming that H' i s true). To test H, we calcu-late X 2 (H) by (2.11) . To test H*/H', we calculate: (2.12) X 2 (H*|H»")= X 2 (H) - X 2 (H' ) = 2 E N. .. , 1 J . . . , l i k l m i,],k,l,m J log (P?!. n IP?., , ), ^ l j k l m 1 ljklm ' which i s the chi-square s t a t i s t i c based upon the l i k e l i h o o d r a t i o c r i t e r i o n f or testing H assuming that H' i s true. The asymptotic d i s t r i b u t i o n of (2.12) i s the chi-square d i s t r i b u t i o n under H*|H', with degrees of freedom equal to., the number of A's i n ft*. For a more complete descrip-t i o n of these tests, see e.g., Goodman [ 1 9 7 0 ] . We are 1 now-> equipped to unravel some of the aspects of consumer brand-switching behavior that lay dormant i n the panel data that are described below. 40 2.5 Description of the empirical data. In the next sections, the models discussed in this chapter are applied to data obtained form the Chicago Tribune Consumer Panel. The product to be studied here i s regular coffee. Coffee was chosen for two reasons. F i r s t , the required data were readi l y available in Massy et a l . ([1970] pp. 126 - 128). And second, i t i s a product category that has been studied previously and thus tentative hypotheses on consumer behavior toward coffee have been formulated. These hypotheses w i l l constitute our p r i o r information regarding the coffee data and w i l l guide our selection of the various versions of the model to be f i t t e d to the data. The data and experimental design are best described in Massy et al.*s ([1970], pp. 128 - 130) own words: The purchase decisions cover the period January 1956 through February 1949. There were 531 fam-i l i e s who met the c r i t e r i o n of having at least 30 purchases of regular coffee during that three-year period. This c r i t e r i o n was set somewhat a r b i t r a r i l y . However, this was about the minimum number of pur-chases needed to run the tests, and a higher cutoff would have adversely affected the t o t a l sample size. Each family's purchase history was reduced to a 0-1 process, where a 1 indicated the purchase of that family's favorite brand of coffee; a 0 indicated the purchase of any other brand of coffee. The 0-1 pro-cess was defined with 1 representing the family's favorite brand and 0 representing a l l other brands instead of a p a r t i c u l a r brand being defined as a 1 for a l l families for two reasons: 1. It was f e l t that examining families' behavior toward t h e i r favorite brand was the better way to investigate the complex phenomenon of "brand l o y a l t y " . 41 2. For any p a r t i c u l a r brand there exist a few families who devote v i r t u a l l y a l l t h e i r purchases to t h i s brand, and a sizable portion of the sample that never buy the brand. These two groups of consumers make the beta d i s t r i b u t i o n (a unimodal di s t r i b u t i o n ) a very poor can-didate for the heterogeneity i n the population of consumers with respect to any p a r t i c u l a r brand. See Section 3.5.2. The f i r s t 30 purchases of each family were recorded, and t h i s sequence was then broken into two periods. Even i f a family had more than 30 purchases, only the f i r s t 30 were used. This was to give equal weightings to the purchase h i s t o r i e s of l i g h t as well as heavy buyers. Period 1 consisted of each family's f i r s t 15 purchases and period 2 the l a s t 15 purchases. The brand purchased most often i n period 1 was designated the f a v o r i t e brand for that period, and s i m i l a r l y for period 2. Thus, i t became possible to compare the behavior of "switchers" (those whose favorite brand i n period 1 was d i f f e r e n t from t h e i r favorite brand i n period 2) with the " l o y a l " consumers who kept the same favorite brand. Each family yielded two observations, each observation containing a past history of length four and the pur-chase decision that followed that past history. These ten observations per family were: Past History of Purchase Decision Length Four on the Next T r i a l Purchases 4 through 7 and 8 5 through 8 and 9 6 through 9 and 10 7 through 10 and 11 8 through 11 and 12 19 through 22 and 23 20 through 23 and 24 21 through 24 and 25 22 through 25 and 26 23 through 26 and 27 Purchases 14 through 18 were not used because the fa v o r i t e brand may have switched from period 1 to period 2, thus, the 0-1 process could not be defined. Also purchases 1 through 3 and 28 through 30 were not used i n order that the purchase decisions used would be roughly i n the middle of the sequence that determined the favorite brand. S t r i c t l y speaking, only two past h i s t o r i e s should have been taken (e.g., 6 through 9 and 10 and 21 through 24 and 25). The "overlap" of the past h i s t o r i e s makes the 42 past h i s t o r i e s within a family dependent. However, when a l l the 531 x 10 = 5,310 observations are aggre-gated, this s l i g h t dependence is not harmful to the models. Also, two observations per family would not have been s u f f i c i e n t to test the models adequately. The segments to be investigated and contrasted were defined as follows: ALL 100 PERCENTERS EXCEPT 100 PERCENTERS HEAVY LIGHT LOYAL NONLOYAL A l l 531 families who met the selection c r i t e r i a . The 142 families who bought only one brand of coffee during period 1 or period 2. (128 of them also had only one brand during both period 1 and period 2.) The remaining 389 families who had one or more zeros. Those families who purchased more than the median amount (for the sample) of coffee. Consump-tion was measured i n number of t r i p s . The other half of the sample which purchased less than the median amount. The 360 families whose favorite brand in period 1 was the same as the favorite in period 2. The remaining 171 families who switched favorite brands. This completes the description of the empirical data used in this chapter. The results of f i t t i n g various unsat-urated models to these data are presented in the next sections and t h e i r implications for consumer brand switching discussed. 43 2.6 P r e s e n t a t i o n of the /empirical r e s u l t s The s a t u r a t e d models d e s c r i b e d by equations (2.6 - 2.9) were f i t t e d t o the data f o r the s i x segments d e s c r i b e d by Massy e t a l . [1970]. The e s t i m a t i o n procedure 7 . o u t l i n e d above was f o l l o w e d . Table II.1 g i v e s the value ~^ ~ 2 of each A, i t s s t a n d a r d i z e d value (A'/S^ .) , v a r i a n c e (S^.) and the sample s i z e f o r each segment. The f i r s t column i d e n t i f i e s the 26 i n t e r a c t i o n terms. B i v a r i a t e i n t e r -a c t i o n s are denoted by two s u b s c r i p t s , t r i v a r i a t e i n t e r -a c t i o n s by three s u b s c r i p t s and so on. Equation (2.9) together w i t h the f i r s t column of Table IV.1 completely i d e n t i f y the parameters. For i n s t a n c e , the f i r s t row of the t a b l e e x h i b i t s f o r a l l segments the raw value of A^ a n d i t s s t a n d a r d i z e d v a l u e f o r the parameter A^ which denotes the i n f l u e n c e of the purchase outcomes of the f i r s t two purchase o c c a s i o n s on P^j^-n^ T n e estimate of A^ was .227 f o r the ALL segment and .234 f o r the LOYAL segment. We s h a l l now look f o r the presence of i n t e r e s t i n g p a t t e r n s among the parameters w i t h i n each segment and check whether they p e r s i s t f o r a l l segments. In so doing, we s h a l l o n l y c o n s i d e r those p a t t e r n s t h a t y i e l d The t a b l e s have been c o l l e c t e d a t the end of the chapter. 44 themselves to meaningful i n t e r p r e t a t i o n i n terms of b e h a v i o r a l s i g n i f i c a n c e . A p a t t e r n t h a t does not r e -p l i c a t e i t s e l f f o r d i f f e r e n t s e t s of data or cannot be r e a d i l y i n t e r p r e t e d i n terms of consumer h a b i t u a l brand s w i t c h i n g behavior i s of l i m i t e d i n t e r e s t . With t h i s o b j e c t i v e - i n mind, l e t us examine the r e s u l t s e x h i b i t e d i n Table I I . 1 . As i s o f t e n the case i n contingency t a b l e a n a l y s i s , the importance of the i n t e r a c t i o n e f f e c t s decreases as t h e i r o r d e r i n c r e a s e s . That i s , b i v a r i a t e i n t e r a c t i o n s tend to e x p l a i n a g r e a t e r p r o p o r t i o n of the v a r i a n c e observed i n the P. ., , than do the t r i - v a r i a t e i n t e r a c t i o n s 1 j klm which i n t u r n e x p l a i n more of the v a r i a n c e than the f o u r -way i n t e r a c t i o n s and so on. While t h i s p a t t e r n holds t r u e i n g e n e r a l f o r the c o f f e e data, there are noteworthy e x c e p t i o n s . For example, the five-way i n t e r a c t i o n proved to e x e r t more i n f l u e n c e on P. ., , than a l l of the f o u r -i ] klm way and h a l f of the three-way i n t e r a c t i o n s , as evidenced by the a b s o l u t e v a l u e s of the A's, f o r three segments: ALL, HEAVY and LIGHT. C o n s i d e r i n g the a b s o l u t e v a l u e s of the A's one a t a time, we see t h a t f o r a l l the segments, most of the b i v a r i a t e i n t e r a c t i o n s are s i g n i f i c a n t a t the .05 l e v e l or l e s s (A/ST^Z ftr. = 1.645) whereas none of the h i g h e r -A . U b order i n t e r a c t i o n terms i s s i g n i f i c a n t a t t h i s l e v e l except f o r the LOYAL segment where fo u r such i n t e r a c t i o n s proved to be of s i g n i f i c a n t magnitude. Some c a u t i o n i s r e q u i r e d i n the use of these x's as a simple guide i n s e l e c t i n g which hypotheses to f i t the data, s i n c e such guidance i s not always f o o l p r o o f . As w i l l be i l l u s t r a t e d l a t e r , i t would not be c o r r e c t to make i n f e r e n c e s about the s t a t i s t i c a l s i g n i f i c a n c e of a p a r t i c u l a r s e t of x's j u s t on the b a s i s of t h e i r i n d i v i d u a l s t a n d a r d i z e d v a l u e s For example, one cannot s t a t e t h a t the whole s e t of thr e e way i n t e r a c t i o n terms i s not s t a t i s t i c a l l y s i g n i f i c a n t even though each one, taken s e p a r a t e l y , does not reach s i g n i f i c a n c e . The s t a n d a r d i z e d v a l u e of a p a r t i c u l a r X gi v e s us a rough i n d i c a t i o n of whether t h i s parameter makes a s i g n i f i c a n t c o n t r i b u t i o n toward e x p l a i n i n g the v a r i a n c e observed i n the P .-• , , giv e n t h a t a l l of the x j k l m remaining parameters are i n c l u d e d i n the model. Thus, a parameter t h a t i s not s i g n i f i c a n t i n the s a t u r a t e d model may w e l l t u r n out to be so i n some unsaturated models, i . e . , models where some of the parameters are assumed to be zero. We s h a l l f i r s t assume t h a t a l l i n t e r a c t i o n s of a s p e c i f i e d order or high e r order (e.g., the 5-way, a l l 4-way and the 5-way, a l l 3, 4 and 5-way i n t e r a c t i o n s ) are 4 6 zero . Support f o r t h i s s e t of hypotheses can be found i n Kuehn [1958] whose f a c t o r i a l a n a l y s i s of v a r i a n c e d e s i g n l e d him to conclude t h a t f o r h i s data, a l l i n t e r -a c t i o n terms were n e g l i g i b l e . As mentioned above, the r e s u l t s e x h i b i t e d i n Table II.1 support the hypothesis t h a t a l l i n t e r a c t i o n s beyond the second-order are n e g l i -g i b l e when t e s t e d one at a time. Does t h i s c o n c l u s i o n s t i l l h o l d when they are a l l simultaneously assumed to be zero? To answer t h i s q u e s t i o n , the p r e v i o u s l y d e s c r i b e d c h i - s q u a r e t e s t s were a p p l i e d to the f o l l o w i n g models: Models I n t e r a c t i o n terms s e t equal to zero M 2 3 4 5-way M22 5-way and 4-way M 2 5-way, 4-way and 3-way For example, M ^ denotes the model i n which the 5-way and 4-way i n t e r a c t i o n s are s e t equal to zero. Thus, f o r t h i s model, the onl y i n t e r a c t i o n s t h a t must be Goodman [1971] has d e s c r i b e d stepwise procedures to s e l e c t unsaturated models t h a t f i t the data, u s i n g methods t h a t are, i n p a r t , somewhat analogous to the us u a l stepwise procedures i n r e g r e s s i o n a n a l y s i s . Such procedures w i l l not be f o l l o w e d here as they may not necessary l e a d to e a s i l y i n t e r p r e t a b l e p a t t e r n s . " 47 estimated from the data are the two and three-way i n t e r -a c t i o n s . Tables II.2 and II.3 e x h i b i t the parameter estimates f o r models ^234 a n d M 2 3 r e s P e c t i v e l y ' together with the s t a t i s t i c s r e q u i r e d to c a l c u l a t e the c h i - s q u a r e s t a t i s t i c d e s c r i b e d i n (2.11). Table II.7 summarizes the c h i - s q u a r e s t a t i s t i c s f o r a l l models and a l l segments, 2 1 while t a b l e II.8 c o n v e n i e n t l y e x h i b i t s the X (H*|H') value s needed to t e s t hypotheses about v a r i o u s nested models f o r a l l segments. A glance a t the f i r s t t h ree rows of Table II.8 r e v e a l s t h a t : i ) For a l l segments, the hypothesis of an i n s i g n i f i c a n t five-way i n t e r a c t i o n c o u l d not be r e j e c t e d . i i ) The hypothesis of i n s i g n i f i c a n t four-way i n t e r a c t i o n s c o u l d not be r e j e c t e d but f o r the LIGHT segment. i i i ) The hypothesis of i n s i g n i f i c a n t three-way i n t e r a c t i o n s (given t h a t a l l four-way and five-way i n t e r a c t i o n s are assumed to be n e g l i g i b l e ) was r e j e c t e d f o r three segments: ALL, HEAVY and LOYAL. U n f o r t u n a t e l y , these r e s u l t s are too n o n - s p e c i f i c to be of much i n t e r e s t . They do not shed much l i g h t onto consumer brand-switching behavior beyond suggesting t h a t 48 the behavior of LOYAL and HEAVY buyers tend to be more 9 complex than t h a t o f NON-LOYAL buyers: i t takes more i n t e r a c t i o n parameters (x's) i n t o the model f o r the former group than f o r the l a t t e r to o b t a i n a s i m i l a r f i t . In other words, past purchase outcomes e x e r t a s t r o n g e r i n f l u e n c e on f u t u r e purchase outcomes f o r LOYAL and HEAVY buyers than f o r the NON-LOYAL ones. There are, however, more i n t e r e s t i n g p a t t e r n s among the i n t e r a c t i o n s t h a t tend to support the l e a r n i n g h y p o t h e s i s 1 ^ o f consumer brand s w i t c h i n g advocated by Kuehn [1958]. A c l o s e look a t the numerical estimates of a l l two-way i n t e r a c t i o n s suggests the f o l l o w i n g p a t t e r n : the i n t e r a c t i o n parameters r e p r e s e n t e d by ( i , j ) , ( j , k ) , ( k , l ) and (l,m) are o.f s i m i l a r magnitude. T h i s p a t t e r n p e r s i s t s f o r a l l segments. S i m i l a r l y , the magnitude of the i n t e r -a c t i o n parameters denoted by the p a i r s ( i , k ) , ( j , l ) and (k,m) are a l s o q u i t e s i m i l a r as are t h a t o f the parameters r e p r e s e n t e d by ( i , l ) and (j,m). For convenience, Table II.4 d i s p l a y s the two-way i n t e r a c t i o n s i n a form t h a t i l l u m i n a t e s When complexity i s judged i n terms of the number of s t a t i s t i c a l l y s i g n i f i c a n t i n t e r a c t i o n e f f e c t s . The fundamental concept u n d e r l y i n g the l e a r n i n g h y p o t h e s i s i s t h a t of purchase event feedback. That i s , the a c t of purchasing and u s i n g a p a r t i c u l a r brand 1 the p a t t e r n s observed among the parameters. The matrix arrangement, reproduced below f o r the HEAVY segment, makes i t c l e a r what i s a c t u a l l y happening: HEAVY Segment j k 1 m i .25 .21 .08 .07 j .22 .18 .10 k .24 .18 1 .26 Each entry i n the above triangular matrix corresponds to a two-way i n t e r a c t i o n . E.g., the entry corresponding to the second row and t h i r d column g i v e s the parameter estimate of the ( j , l ) i n t e r a c t i o n ( i n terms of our e a r l i e r n o t a t i o n ) . As can be,.seen from the matrix, the d i a g o n a l elements are s t r i k i n g l y s i m i l a r . Taking advantage of t h i s f a c t , the number of two-way i n t e r a c t i o n terms to be estimated can be reduced from 10 to 4 by equating the f o l l o w i n g parameters: i s assumed to a f f e c t the p r o b a b i l i t y t h a t t h i s brand w i l l be s e l e c t e d again. 50 (2.12) ( i , j ) = (j,k) = (k,l) = (l,m) (i,k) = ( j , l ) = (k,m) (1,1) = (j,m) where i t i s understood t h a t the e q u a l i t y ( i , l ) = (j,m) means t h a t the i n t e r a c t i o n e f f e c t s r e p r e s e n t e d by the p a i r s ( i , l ) and (j,m) are assumed i d e n t i c a l . Before we t e s t the hypothesis expressed i n (2.12), l e t us pause b r i e f l y and t r y to r e l a t e the f i n d i n g to what we a l r e a d y know about brand-switching behavior. The q u e s t i o n i s : what do the i n t e r a c t i o n terms t h a t were observed to be of s i m i l a r magnitude share i n common? E.g., what i s the common "denominator" between the f o l l o w i n g f o u r i n t e r a c t i o n s : ( i , j ) , ( j , k ) , ( k,l) and (l,m)? Answer: They a l l r e p r e s e n t the e f f e c t of the outcome of two adjacent (successive) purchases I That i s , the i n t e r -a c t i o n ( i , j ) denotes the e f f e c t of having purchased brand i a t o c c a s i o n t-4 and brand j a t o c c a s i o n t-3. S i m i l a r l y , (j,k) denotes the e f f e c t of having purchased brand j a t o c c a s i o n t-3 and brand k a t o c c a s i o n t-2 and so on... Thus, the t en two-way i n t e r a c t i o n s can be p a r t i t i o n e d i n fo u r s e t s of i n t e r a c t i o n s on the b a s i s of t h e i r degree of "adjacentness". By "adjacentness" i s meant the extent to which the i n t e r a c t i o n r e p r e s e n t s the e f f e c t of s u c c e s s i v e purchase o c c a s i o n s . An i n t e r a c t i o n w i l l be s a i d to be 51 1-adjacent i f i t represents the inte r a c t i o n e f f e c t due to the p a r t i c u l a r outcomes of two successive purchase occasions. The interactions denoted by ( i , j ) , ( j , k), (k,l) and (l,m) are a l l 1-adjacent. S i m i l a r l y , the interactions denoted by ( i , k ) , ( j , l ) and (k,m) are 2-adjacent since they represent the in t e r a c t i o n effects due to the p a r t i -cular outcomes of purchases that are two occasions apart. We can now summarize our empirical observations by the following hypotheses: : Interaction e f f e c t s with the same degree of "adjacentness" are equal (see 2.12). The lower the degree of "adjacentness", the greater the in t e r a c t i o n e f f e c t . Hypothesis E^ states that the j o i n t influence of any two past purchases of brand 1 on the p r o b a b i l i t y of buying that brand again varies inversely with the number of purchase occasions that elapsed between those two purchases. This hypothesis i s consistent with Kuehn [1958]'s e a r l i e r finding that remote purchases exert less of an e f f e c t on the p r o b a b i l i t y of buying the brand again than do more recent ones. Hypothesis H^ states that the influence of successive purchases of brand 1 on the p r o b a b i l i t y of 52 buying brand 1 again i s independent of t h e i r p o s i t i o n i n the purchase s t r i n g . To t e s t and , we estimate a c o n s t r a i n e d v e r s i o n of M 2 where i n t e r a c t i o n s of same degree of "ad-j a c e n t n e s s " are f o r c e d to be n u m e r i c a l l y e q u i v a l e n t . Denoting t h i s model by JY^*, we see from Table I I . 7 and IT.8 t h a t i s s t r o n g l y supported by the c o f f e e data. Since model M 2* i s a c o n s t r a i n e d v e r s i o n of M 2, the t e s t i n g procedure d e s c r i b e d above a p p l i e s . The i n c r e a s e i n the c h i - s q u a r e s t a t i s t i c i n c u r r e d by c o n s t r a i n i n g i n t e r a c t i o n s of same degree of adjacentness to be equal i s marginal, g i v e n the d i f f e r e n c e i n the number of parameters of the two models (10 f o r M 2 versus 4 f o r M2*) . E.g., the c h i - s q u a r e s t a t i s t i c went up from 112.41 to 119.39 and 22.10 to 24.66 f o r the ALL and NON-LOYAL segments res p e c -t i v e l y . From the f o u r t h row of Table IV.8, we see t h a t f o r a l l segments, the i n c r e a s e observed i n the c h i - s q u a r e s t a t i s t i c i s not s t a t i s t i c a l l y s i g n i f i c a n t a t a l l l e v e l s of s i g n i f i c a n c e c o n s i d e r e d . Thus, on the b a s i s of the c o f f e e data, the "adjacentness" hypotheses and H 2 r e c e i v e d s t r o n g e m p i r i c a l support. Hypothesis H 1 allows us to substan-t i a l l y reduce the number of parameters to be estimated without harming the f i t of the model. 53 However, the model does not y e t p r o v i d e an adequate r e p r e s e n t a t i o n of the data, as evidenced by the magnitude of the c h i - s q u a r e goodness of f i t s t a t i s t i c s . To improve the f i t , we s h a l l have to i n c l u d e i n the model, i n t e r a c t i o n s of order h i g h e r than two. Consider the ten three-way i n t e r a c t i o n s ' . Once again, we can p a r t i t i o n them on the b a s i s of t h e i r degree of adjacentness as f o l l o w s : Degree of i n t e r a c t i o n s Examples of a d j a c e n t - parameter estimates ness EXCEPT 100 P. HEAVY 1 ( i j k ) , ( j k l ) , (klm) -.003 .009 .013 .004 .006 .004 2 ( i j l ) , (jkm), ( i k l ) .017 .026 .015 .025 .029 .016 (jlm) -.005 -.013 3 ( i j m ) , (ikm), (ilm) .036 .006 .032 .046 .049 .050 The f i r s t row of the t a b l e c o n s i s t s of the three 1-adjacent three-way i n t e r a c t i o n s , i . e . , i n t e r a c t i o n s which r e p r e s e n t the e f f e c t of the purchase outcomes a t three s u c c e s s i v e purchase o c c a s i o n s . The estimates d i s p l a y e d i n the l a s t two columns are reproduced from Table I I . 1 . For example, the three-way i n t e r a c t i o n e f f e c t denoted by (jkm) has been estimated as .026 and .029 f o r the EXCEPT 100 PERCENTERS and HEAVY segments r e s p e c t i v e l y . The p a t t e r n observed e a r l i e r for the two-way interactions tend to repeat i t s e l f for the three-way i n t e r a c t i o n s : Interactions of same degree of adjacentness tend to be of numerical equal importance. On the other hand, the hypothesis expressed by H 2 holds but i n a reverse fashion,i.e., the lower the degree of adjacentness, the lower the i n t e r a c t i o n e f f e c t . ^ As a formal test of H^, as applied to a l l i n t e r a c t i o n s t h i s time, the following models were estimated: NAME DESCRIPTION • ^2*5 This model i s similar to M2* but' includes the five-way i n t e r a c t i o n parameter as w e l l . M(23)*5 This model i s similar to M2*5 fc>ut includes three a d d i t i o n a l parameters f o r the three- .'. way i n t e r a c t i o n s . (The ten, three-way i n t e r -actions have been p a r t i t i o n e d i n three sets as explained above.) M(234)*5 Same as above but with an a d d i t i o n a l para- • meter for the f i v e four-way i n t e r a c t i o n s that are assumed to be equal M(234)* Same as above but without the five-way i n t e r -action . Table II.5 displays the parameters estimates for the three models M2, M.^ and-M^^. Recall that a * indicates that hypothesis has been acted upon. Thus, model M2 allows each two-way i n t e r a c t i o n .to have a d i f f e -rent numerical value whereas model M^ requires that a l l two-way interactions with the same degree of adjacentness (11) Given the magnitude of the estimates, such a reversal, may just be the spurious r e s u l t of random v a r i a t i o n s . 55 be equal. From Table I I . 8 , f i f t h row, we see t h a t the i n c l u s i o n of the five-way parameter (^2* v e r s u s M2*5^ b r i n g s about a s i g n i f i c a n t r e d u c t i o n of the ch i - s q u a r e s t a t i s t i c f o r three segments: ALL, HEAVY and LOYAL. The i n t e r a c t i o n e f f e c t s t h a t do make a d i f f e r e n c e , i n terms of goodness of f i t , are the three-way i n t e r a c t i o n s together w i t h e i t h e r one of the fou r or five-way i n t e r a c t i o n s . A glance a t Tables. II.6 and II.8 suggests t h a t e i t h e r one of M ( 2 3 ) * 5 o r M ( 2 3 4 ) * (koth 8-parameter models) p r o v i d e an adequate r e p r e s e n t a t i o n of the c o f f e e data. 2.7 Comparison of brand-switching behavior across segments. Although the o b j e c t i v e of t h i s chapter was not to d i s c u s s the segmentation s t r a t e g y f o l l o w e d by MASSY e t a l . [1970], the pre c e d i n g a n a l y s i s p r o v i d e s a f u r t h e r t e s t of t h e s e g m e n t s . d i s c r i m i n a n t v a l i d i t y , t h a t i s , the p o t e n t i a l o f the segments to i d e n t i f y any behavior pat-t e r n s t h a t s e t the buyers a p a r t from the market i n g e n e r a l . On the b a s i s of the r e s u l t s a f f o r d e d by t h e i r Brand L o y a l model, Massy e t a l . [1970] found no s i g n i f i c a n t d i f f e r e n c e s i n brand-switching behavior between the HEAVY and LIGHT segments. Our a n a l y s i s does not support t h i s c l a i m . The parameter val u e s of the segments f o r the three 56 models considered i n Table I I . 6 and reproduced (in part) below exhibit d i f f e r e n t patterns: Interactions ( i , j) , ( j , k ) , ( i , k ) , ( j , l ) , ( i , 1 ) , (j ,m) (i,m) ( k , l ) , (l,m) (k,m) Parameter values HEAVY .243 .192 .088 .075 Model M ( 2 3 4 )* LIGHT .277 .211 .159 .157 As the degree of adjacentness of the interactions increases, the importance of the inte r a c t i o n effects declines much more rapidly for the HEAVY than for the LIGHT segments. Thus, past purchases exert less influence on recent purchases for the HEAVY than for the LIGHT segments, a somewhat surprising r e s u l t given that the learning e f f e c t observed i n consumer brand choice i s generally assumed to be inversely related to interpurchase time: the greater the time lapse between successive purchases, the greater the possible erosion of the learn-ing e f f e c t s . The same difference i n pattern exists for the LOYAL and the NON-LOYAL segments, but t h i s time i n the expected d i r e c t i o n . Past purchases exert much more i n f l u e n c e on r e c e n t purchases f o r the LOYAL than f o r the NON-LOYAL buyers, an i n t u i t i v e c o n c l u s i o n g i v e n t h a t s u c c e s s i v e purchases of the former are probably h i g h l y c o r r e l a t e d . To f u r t h e r i n v e s t i g a t e the extent of the d i f f e r -ences between the d i f f e r e n t segments, the 26 parameter values of the s a t u r a t e d model (Table IV.1) were c o r r e l a t e d a cross segments. The r e s u l t s are summarized i n Table 11.10 On the b a s i s of the c o r r e l a t i o n matrix based on the whole s e t of i n t e r a c t i o n s , the v a r i o u s segments look very much a l i k e . The lowest c o r r e l a t i o n observed i s .800 (between LOYAL and NON-LOYAL) wh i l e the ALL and EXCEPT 100 PERCENTERS segments e x h i b i t near p e r f e c t c o r r e l a t i o n (.996). I t i s i n t e r e s t i n g to note t h a t the LOYAL and NON-LOYAL segments behave more l i k e the HEAVY segment than the LIGHT segment. The observed p a t t e r n s d i s c u s s e d above repeat themselves when the c o r r e l a t i o n s are taken w i t h r e s p e c t to the b i v a r i a t e i n t e r a c t i o n s o n l y . I t i s only a t the l e v e l of the t r i v a r i a t e i n t e r a c t i o n s t h a t the v a r i o u s segments do d i f f e r from one another. Many of the c o r r e l a t i o n s d i s p l a y e d i n the matrix are i n the (.10, .40) range, i n a b s o l u t e v a l u e . The c o r r e l a t i o n between the HEAVY and LIGHT segments drops from .96 58 for the b i v a r i a t e interactions to -.11 for the t r i v a r i a t e interactions. The corresponding jump for the LOYAL NON-LOYAL segments i s from .84 to -.42. The HEAVY segment that was found to behave very much l i k e the NON-LOYAL segment i s now hardly correlated with i t : .11. These results corroborate our previous finding that a l l i n t e r a c t i o n effects must be allowed for, i n order to achieve an adequate representation of brand switching data. Had we r e s t r i c t e d our attention to bi v a r i a t e interactions only, we would have reached the same conclusion as Massy et a l . [1970], namely that the various segments do not exhibit sharply d i f f e r e n t brand-switching behavior. 2.8 Summary of the r e s u l t s . The re s u l t s of the preceding s t a t i s t i c a l analysis of brand-switching data are summarized below. i) Consumer choice behaviour i s a complex phenomenon. The degree of complexity embodied i n a contingency table can be measured by the extent to which higher order interactions add a s i g n i f i c a n t contribution toward explaining the observed variance i n the dependent variable. From the empirical results 59 presented here, i t appears that a l l in t e r a c t i o n effects must be allowed for, i n order to achieve an adequate representation of brand-switching behavior. i i ) The above analysis gives support to the learning hypothesis of brand-switching behavior i n that past purchases of a brand a f f e c t the p r o b a b i l i t y of that brand being bought again i n the future. i i i ) There i s also evidence of a "forgetting" e f f e c t i n that a purchase of a brand other than the i n d i v i d -ual's f a v o r i t e brand decreases the individual's p r o b a b i l i t y of buying his favori t e brand on the next purchase occasion. Support for t h i s "forgetting" e f f e c t comes from the non-rejection of hypothesis #2 which states that the influence of any two pur-chases of a brand varies inversely with the number of purchase occasions that elapsed between those two purchases. Conclusion The purpose of t h i s chapter was to perform a s t a t i s t i c a l analysis of a p a r t i c u l a r set of consumer brand-switching data. The model was formulated as a log-linear model of contingency table by decomposing the observed proportions into main and interaction e f f e c t s . Model f i t t i n g and hypothesis testing procedures were presented and the model applied to a product category (coffee) that has been previously studied by several researchers. The res u l t s indicate that (stochastic) models of consumer brand switching behavior should provide ways to accommodate the kind of adaptive behavior observed in the data. The purpose of the next chapter i s to present a new stochastic model of brand choice that builds on the " r e g u l a r i t i e s " uncovered i n th i s chapter. TABLE U - ; l ESTIMATES OF INTERACTION EFFECTS AMONG THE PURCHASE OCCASIONS FOR SIX SEGMENTS - COFFEE DATA. Segments ALL EXCEPT 100 Perc. HEAVY LIGHT LOYAL Interaction Raw Stand. R S IJ .227 3.857 .182 3.030 IK .181 3.070 .147 2.440 IL .078 1.321 .043 .717 IM .080 1.355 .039 .653 JK .200 3.390 .154 2.563 JL .154 2.604 .120 1.988 JM .100 1.688 .068 1.128 KL .224 3.796 .178 2.951 KM .170 2.875 .137 2.279 LM .244 4.146 .200 3.317 IJK .015 .258 -.003 -.054 U L .035 .589 .017 .280 IJM .050 .856 .036 .604 JKL .028 .468 .009 .155 JKM .042 .711 .026 .424 KLM .034 .570 .013 .219 IKL .033 .551 ' .015 .251 IKM .010 .177 .006 .100 ILM .050 .846 .032 .526 JLM .011 .193 -.005 -.077 IJKL .006 -.096 -.036 -.595 IJKM .008 -.143 -.033 -.548 IKLM .027 .452 .001 .023 JKLM .007 .124 -.025 -.419 IJLM .012 .202 -.006 -.108 I JKLM .027 .463 .009 .145 NON-LOYAL Variance (A) .00348 Sample siz e 5310 .00362 3890 R S R S R S R S .253 2.861 .277 3.329 .234 5.940 .178 2.097 .215 2.430 .221 2.656 .218 5.539 .106 1.248 .081 .917 .147 1.766 .061 1.564 .042 .499 .071 .798 .152 1.827 .115 2.936 -.000 -.004 .219 2.478 .253 3.043 .196 4.985 .166 1.953 .184 2.076 .198 2.380 .187 4.764 .080 .936 .099 1.122 .169 2.026 .081 2.062 .062 .733 .238 2.690 .282 3.385 .232 5.897 .189 2.215 .184 2.081 .234 2.692 .165 4.194 .126 1.476 .258 2.917 .302 3.635 .263 6.686 .200 2.355 .004 .047 -.041 -.499 -.020 -.498 .028 .323 .025 .284 -.024 -.293 .075 1.895 -.021 -.241 .046 .525 -.007 -.080 .035 .889 .036 .423 .006 .020 -.021 -.258 .007 .184 .032 .379 .029 .327 -.009 -.107 .091 2.301 -.016 -.191 .004 .046 -.003 -.036 .019 .483 .040 .472 .016 .179 -.019 -.227 .064 1.634 -.005 -.051 .049 .551 -.082 -.989 .009 .225 -.014 -.160 .050 .570 -.008 -.101 .033 .836 .059 .689 .013 -.151 -.030 -.361 .002 .051 -.010 -.115 .008 .086 .041 .489 -.040 -1.019 .011 .124 .023 -.260 .057 .689 .013 .333 -.046 -.539 .007 .078 .095 1.147 -.021 -.541 .058 .686 .022 .248 .052 .623 -.007 -.177 .006 .073 .024 .266 .056 .673 .039 .982 -.048 -.564 .031 .350 -.022 -.260 .052 1.321 .002 .028 .00782 2660 .00692 2650 .00155 3600 .00725 1710 T A B L E $1.2 PARAMETER ESTIMATES FOR MODEL M... IN WHICH THE FIVE-WAY INTERACTION IS ASSUMED ZERO - COFFEE DATA. Segments Interactions ALL EXCEPT 100 Percenters HEAVY LIGHT LOYAL N0N-L( IJ .227 .182 .252 .277 .232 .179 IK .181 .147 .214 .221 .216 .106 IL .079 .043 .082 .146 .068 .042 IM .080 .039 .070 .152 .113 .000 JK .201 .155 .221 .253 .199 .166 JL .153 .120 .185 .199 .186 ."080 JM .100 .068 .099 .168 .085 .062 KL .225 .178 .279 .281 .233 .189 KM .170 .137 .185 .224 .169 .176 LM .244 .199 .257 .303 .259 .201 IJK .019 -.002 .009 -.045 .012 .028 U L .036 .017 .027 -.026 .074 .020 IJM .054 .037 .050 -.010 .042 .036 JKL .027 .009 .005 -.022 .007 .032 JKM .041 .025 .027 -.009 .085 -.016 KLM .037 .014 .009 -.006 .024 .041 IKL .032 .015 .015 -.019 .059 -.005 IKM .012 .006 .051 -.084 .013 -.014 ILM .052 .032 .053 -.011 .095 .059 JLM .014 -.004 -.010 -.032 .009 . -.010 IJKL -.002 -.035 .012 .039 .029 .011 IJKM -.006 -.032 -.019 .056 .019 -.046 IKLM .030 .000 .011 .093 .011 .059 JKLM .010 -.024 .025 .050 .001 .006 IJLM .014 -.006 .027 .055 .046 -.048 X^ " x l r 2.145 .210 1.224 .684 3.636 .008 2.146 .210 1.225 .683 3.663 .008 N 5310 3890 2660 2650 3600 1710 ON TABLE II.3 PARAMETER ESTIMATES FOR THE UNSATURATED MODEL M. Segments ALL EXCEPT 100 HEAVY Interactions Percenters IJ .228 .181 .254 IK .182 .151 .216 IL .076 .046 .078 IM .080 .041 .071 JK .202 .163 .222 JL .153 .121 .180 JM .100 .069 .097 KL .220 .180 .234 KM .165 .139 .182 LM .248 .200 .260 IJK .017 -.018 .006 IJL .040 .009 .038 IJM .056 .029 .052 IKL .039 .008 .022 IKM .016 -.012 .048 ILM .066 .029 .063 JKL .016 -.002 .007 JKM .045 .012 .030 JLM .016 -.002 .007 KLM .050 .006 .018 X 2 2 § f 6.733 8.675 3.615 x l r 6.707 8.648 3.593 N 5310 3890 2660 WHICH ALL FOUR-WAY AND THE FIVE-WAY INTERACTIONS ARE 0. L I G H T L O Y A L NON-: .290 .233 .176 .222 .215 .110 .143 .062 .043 .158 .113 .003 .252 .198 .167 .195 .186 .081 .166 .083 .070 .275 .235 .185 .217 .166 .117 .310 .263 .195 •.026 -.015 .020 •.012 .082 -.028 .010 .060 -.023 .009 .050 .009 .057 .013 -.014 .013 .048 .055 .004 .000 .016 .009 .097 -.024 .007 -.000 .016 .023 .023 .049 37.421 8.046 8.406 38.302 8.048 8.472 2650 3600 1710 O N PARAMETER ESTIMATES OF THE UNSATURATED MODEL IN WHICH ALL 3,4 and 5-WAY INTERACTIONS ARE ASSUMED ZERO (M2> SEGMENTS ALL K L M I .227 .181 .078 .080 J .200 .154 .100 K .224 .170 L .244 EXCEPT 100 PERCENTERS J K L M .182 .147 .043 .039 .154 .120 .068 .178 .137 .200 HEAVY K L M .253 .215 .081 .071 .219 .184 .099 .238 .184 .258 SEGMENTS LIGHT LOYAL NON-LOYAL J K L M J K L M J K L M I .277 .221 .147 .152 .234 .218 .061 .115 .178 .106 .042 .000 J .253 .198 .169 .196 .187 .081 .166 .080 .062 K .282 .224 .232 .165 .189 .126 L .302 .263 .200 The parameter values displayed above are those of the saturated model. They are reproduced from Table IV.1 TABLE n . 4 PARAMETER ESTIMATES FOR THREE UNSATURATED MODELS: M„, M„. and M,^ , 2 2* 2*5 Segments Interactions J K L M I J K L ,254 ALL EXCEPT 100 PERC. Model M 2 Model_M2 .202 .096 .104 .184 .149 .052 .047 .220 .170 .117 .162 .123 .074 .240 .185 .180 .139 .272 .205 HEAVY Model M„ ,277 .233 .094 .099 .237 .193 .108 .247 .197 .279 ; 2gf v l r 108.92 112.41 16.77 16.90 40.78 41.22 Segments Interactions LIGHT J K L M I J K L Model M„ .286 ,214 ,248 .142 .155 .195 .170 ,276 .212 .314 LOYAL Model M 2 ,270 .236 .099 .136 .228 .205 .117 .252 .200 .287 NONLOYAL Model M 2 .177 .114 .042 .008 .167 .083 .070 .186 .118 .196 l r 45.02 45.91 73.15 74.27 21.90 22.09 SEGMENTS ALL EXCEPT P. HEAVY LIGHT LOYAL NON-LOYAL Models Interact. M2*5M2* M 2*5 M 2* M M M M 2*5 2* 2*5 2* M M M M 2*5 2* 2*5 2* j , j k , k l , l m .237 .245 .182 .182 i k , j l , k l .180 .184 .136 .136 i l , j m .100 .105 .061 .061 im .102 .110 .052 .052 ij k l m .096 - .008 .252 .259 .280 .280 ,200 .205 .205 .205 ,095 .099 .156 .156 ,100 .106 .160 .160 ,089 - -.003 .248 .258 .180 .181 .202 .211 .105 .104 .099 .116 .053 .053 .130 .140 .012 .013 .105 - .024 X 28f l r 77.5 116. 21.2 21.4 28.3 44.2 49.8 49.8 51.4 77.6 23.5 24.6 75.4 119. 21.3 21.5 27.2 45.0 50.7 50.8 49.6 78.5 23.6 24.7 Model M 2 B i v a r i a t e i n t e r a c t i o n s only. Model M2^ Bi v a r i a t e i n t e r a c t i o n s only with constraint (4.37) imposed, Model M 2 * 5 Same as above plus the five-way i n t e r a c t i o n term. T A B L E II. 5 PARAMETER ESTIMATES FOR THREE UNSATURATED MODELS M(234)*5' M ( 2 3 4 ) * a n d M(23)*5' SEGMENTS Models Interactions i j , j k , k l , l m ik,jl,km 11, jm im i j k , j k l , k l m i j l , j k m , i k l and jlm i j m , i k l , i l m i j k l . i j k m , iklm,j klm and i j l m i j k l m ALL EXCEPT 100 PERC. HEAVY LIGHT LOYAL 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 .224 .225 .223 .179 .179 .181 .243 .243 .242 .277 .277 .281 .231 .231 .231 .164 .164 .163 .131 .131 .134 .192 .192 .190 .211 .211 .207 .183 .184 .182 .089 .090 .088 .055 .055 .059 .088 .088 .086 .159 .158 .157 .075 .079 .075 .085 .085 .085 .044 .044 .045 .075 .074 .076 .157 .157 .163 .113 .111 .112 .023 .026 .027 .003 .003-.004 .003 .006 .008 -.024--.026- .004 -.005--.001-.004 .029 .029 .032 .012 .013 .005 .017 .017 .021 -.023--.024- .004 .056 .051 .057 .039 .041 .042. .024 .024 .017 .047 .049 .050 -.029--.031-.010 .036 .039 .037 .008 .011 -.018- .018 .010 .013 _ .062 .060 .002 .008 _ .026 - .031 .008 -.003 .025 - .032 -.020 - .011 .041 - .042 NON-LOYAL ,179 .180 .180 .102 .102 .103 .052 .052 .053 .010 .010 .010 .036 .037 .035 .014-.013-.015 .021 .022 .020 .006 .006 ,2gf K x ' l r „2 Morrison 11.6 13.7 12.8 11.4 11.6 17.1 4.42 5.27 5.23 11.7 12.2 48.5 12.2 14.6 12.2 15.9 16.0 16.0 11.6 13.7 12.7 11.4 11.6 17.0 4.44 5.35 5.19 11.6 12.2 49.3 12.4 15.1 12.4 15.8 15.8 15.8 8.0 7.8 2.61 10.1 8.5 13.5 Model 1 Model 2 Model 3 = !!(234)*5 = M( 2 3 4 ) * (23)*5 None of the i n t e r a c t i o n i s assumed zero, but the "adjacentness" constraint i s imposed Same as above but without the five-way i n t e r a c t i o n . Same as model 1 but without the four-way i n t e r a c t i o n parameter. TABLE I I - 6 ALL EXCEPT 100 P. HEAVY LIGHT LOYAL NON--LOYAL MODEL Parameters Degrees of set to 0 freedom X 2 X 2 gf X2 x l r X 2 X 2 x l r x 2 l r X 2 X gf X 2 *v 234 5-way 6 2.15 2. 14 .21 .21 1.23 1.22 .68 .68 3.67 3.64 .01 .01 23 4,5^ -way 11 6.71 6. 73 8.64 8.67 3.59 3.61 38.30 37.42 8.05 8.05 8.47 8.41 2 3,4,5-way 21 112.41 108. 92 16.90 16.77 41.22 40.78 45.91 45.02 74.27 73.15 22.10 21.91 2* 4,5-way 27 119.39 116. 18 21.53 21.20 44,96 44.24 50.83 49.82 78.52 77.62 24.66 24.56 2*5 3,4-way 26 75.37 77. 53 21.27 21.40 27.24 28.33 50.74 49.77 49.56 51.44 23.64 23.55 234*5 none 22 11.58 11. 64 11.41 11.43 4.44 4.42 11.61 11.67 12.38 12.25 15.78 15.91 234* 5-way 23 13.69 13. 71 11.58 11.59 5.35 5.27 12.16 12.24 15.10 14.62 15.83 15.97 23*5 4-way 23 12.69 12. 81 17.00 17.05 5.19 5.23 49.26 48.49 12.38 12.25 15,87 16.00 SAMPLE SIZE 5310 3890 2660 2650 3600 1710 SUMMARY OF CHI-SQUARE STATISTICS FOR VARIOUS UNSATURATED MODELS. TABLE I I . 7 SUMMARY OF THE LIKELIHOOD RATIO CHI-SQUARE STATISTICS FOR VARIOUS UNSATURATED MODELS. x 2 ( H | H ' ) = i Model H Model H Degrees ALL EXCEPT HEAVY of freed. 100 P. 234 2345 1 • 2.15 .21 1.23 23 234 5 4.56 8.43 2.36 2 23 10 105.70 8.26 37.63 2* 2 6 6.98 4.63 3.74 2*5 2* 1 44.02 .26 17.72 (234)* 234 17 11.54 11.37 4.12 (23)*5 (234)*5 1 1.11 5.59 .75 (234)* (234)*5 1 2.11 .17 .92 (234)*5 2345 17 11.58 11.41 4.44 2*5 (23)*5 3 62.68 4.27 22.05 This table exhibits the chi-square s t a t i s t i c s unsaturated models (see 4.36). T A B L : ( H ) - X 2 ( H " ) LIGHT LOYAL NON-1. CRITICAL LEVEL x .01 X 2 * .05 X 2 x .10 .68 3.67 .01 6.63 3.84 2.71 37.62 4.38 8.46 15.09 11.07 9.24 7.61 66.22 13.63 23.21 18.31 15.99 4.92 4.25 2.56 16.81 12.59 10.64 .09 28.96 1.02 6.63 3.84 2.71 11.48 11.43 15.82 32.90 27.75 24.79 37.65 .00 .09 6.63 3.84 2.71 .54 2.72 .05 6.63 3.84 2.71 11.61 12.38 15.78 32.90 27.75 24.79 1.48 37.18 7.77 11.34 7.81 6.25 X 2 ( H | H . ) used to discriminate between various II.8 ON CO CORRELATION MATRIX OF THE VARIOUS SEGMENTS BASED ON THE MODELS RESPECTIVE PARAMETER ESTIMATES FOR THE SATURATED MODEL AND TWO UNSATURATED MODELS. SATURATED MODEL. Except 100 Heavy Light percenters A l l .996- .985 .909 Except 100 p. .978 :874 Heavy .900 Light Loyal Loyal Non-Loyal .959 .960 .949 .853 .929 .922 .904 .834 .799 Model M 2 ( b i v a r i a t e i n t e r a c t i o n s only), Except 100 Heavy Light percenters A l l .997 .989 .988 Except 100 p. .993 .978 Heavy .956 Light Loyal Loyal Non-Loyal .953 .950 .957 .922 .965 .959 .938 .977 .845 MODEL M 3 ( t r i v a r i a t e i n t e r a c t i o n s only) Except 100 Heavy Light percenters Loyal Non-Loyal A l l Except 100 p. Heavy Light Loyal 966 .519 .784 .611 .423 .477 .790 .603 .346 -.111 .356 .107 .428 .435 -.425 TABLE H.IO CHAPTER III A NEW LEARNING MODEL OF BRAND CHOICE The purpose of this chapter i s to discuss a. new stochastic model <of brand choice that allows for the kind of adaptive behavior observed in consumer brand-switching behavior and confirmed by s t a t i s t i c a l analyses such as the one reported in the previous chapter. This adaptive behavior goes by the name of "learning" in the marketing l i t e r a t u r e . Learning models of consumer brand choice behavior have occupied an import tant position in the l i t e r a t u r e of brand choice ever since Kuehn [1958] adapted the work of Bush and Mosteller [1955] and applied the model to data on switching patterns for frozen orange j u i c e . The fundamental concept underlying a l l learning models of brand choice is that of purchase event feedback. That i s , the act of purchasing and using a p a r t i c u l a r brand is assumed to affect the pr o b a b i l i t y that this brand w i l l be selected again the next time the product class i s to be purchased. The model to be presented deals e x p l i c i t l y with the "learning" feature of consumer brand choice. I t attempts to p r e d i c t an i n d i v i d u a l ' s p r o b a b i l i t y o f s e l e c t i n g a p a r t i c u l a r brand on h i s next purchase o c c a s i o n given h i s past purchase h i s t o r y . The next s e c t i o n b r i e f l y d e s c r i b e s two s t o c h a s t i c models of brand c h o i c e which w i l l be used f o r comparison pur-poses to assess the performance o f the new purchase - to -purchase model of brand c h o i c e . 3.1 L i n e a r L e a r n i n g and Markov Models Of Brand Choice. S e v e r a l s t o c h a s t i c models p r o v i d e some ways to accommodate the k i n d of adaptive behavior observed i n brand c h o i c e data. Two of these w i l l be c o n s i d e r e d f o r comparison purposes. They are r e s p e c t i v e l y : - Morrison's Compound Markov Model (1965a) - Kuehn's L i n e a r L e a r n i n g Model (1962). The r a t i o n a l e f o r choosing the above two models i s based on p r a c t i c a l c o n s i d e r a t i o n s . They have been e x t e n s i v e l y t e s t e d i n t h e m a r k e t i n g l i t e r a t u r e a n d s t a n d o u t f o r t h e i r m a t h e m a t i c a l t r a c t a b i l i t y a n d " g o o d " e m p i r i c a l p e r f o r m a n c e i n t h e f a c e o f a c t u a l d a t a . M a s s y e t a l . [ 1 9 7 0 ] h a v e d i s c u s s e d t h e m a t l e n g t h - a n d r e p r o -d u c e d some p a n e l d a t a w h i c h w i l l f o r m t h e b a s i s o f t h e e m p i r i c a l c o m p a r i s o n p r e s e n t e d i n a l a t e r s e c t i o n . 3 . 1 . 1 The L i n e a r L e a r n i n g M o d e l K u e h n [ 1 9 6 2 ] p r o p o s e d a m o d e l o f c o n s u m e r b r a n d c h o i c e b e h a v i o r i n w h i c h P t , t h e p r o b a b i l i t y o f p u r c h a s e p u r c h a s e o c c a s i o n t , w a s a l i n e a r f u n c t i o n o f t h e p r o b a -b i l i t y a t o c c a s i o n t - 1 a n d t h e o u t c o m e o f t h e p u r c h a s e ( b r a n d s e l e c t e d ) a t t - 1 . I n s y m b o l s : a + 0 + yP* . 1 i f b r a n d 1 i s p u r c h a s e a t ( 3 . 1 ) / P t = W t - 1 , a + Y P t _ n i f a n y o t h e r b r a n d i s p u r c h a s e d a t t - 1 . T h e s e e q u a t i o n s c a n be r e p r e s e n t e d w i t h t h e f a m i l i a r d i a g r a m o f f i g u r e I I I . l w h i c h i l l u s t r a t e s many o f t h e p r o p e r t i e s o f t h e L i n e a r L e a r n i n g M o d e l . The two r e l a t i o n s e x p r e s s e d i n (3,.,1) a r e c a l l e d t h e p u r c h a s e o p e r a t o r a n d r e j e c t i o n o p e r a t o r , r e s p e c t i v e l y . A l l f a m i l i e s a r e a s s u m e d t o h a v e t h e same v a l u e s f o r t h e p a r a m e t e r s a , 6 a n d y , t h o u g h o f c o u r s e t h e p r o c e s s l e a d s t o d i f f e r e n t v a l u e s o f p ^ f o r d i f f e r e n t f a m i l i e s , due t o 7 3 D i a g r a m a t i c a l R e p r e s e n t a t i o n of the L i n e a r Learning Model FIGURE I I I . l 74 variations in realized purchased histories. "Learning" takes place because a purchase of brand 1 leads to a larger value of P t + 1 than a purchase of brand 0, for any-trial t. In addition to its intrinsic interest, the Linear Learning model can be viewed as a generalization of both the zero-order Bernouilli and the first-order Markov models. If y = 0, equation (3.1) becomes independent of P t .^ This is equivalent to a Markov model with purchase decisions as states having transitions matrix 1 0 1 a+R l-a-3 0 a 1 -a Similarly, if a = 3 = 0 and y = 1, then P t + 1 = Pt regardless of the outcome at time t, and so we have a zero-order Bernouilli model. The adaption of this model by Massy [1965] to include heterogeneity did not change the basic formulation, but involved only the method of estimating the model's parameters. Parameters a, B and y can be estimated without assuming that the entire popu-lation has the same initial probability of purchase. 75 3.1.2 A Compound Markov Model: Morrison [1965a]'s Brand  Loyal Model In the Compound Markov model, the l a s t purchase, and only the l a s t purchase, influences the current pur-chase decision. That i s , the Compound Markov model allows for f i r s t - o r d e r behavior. The Brand Loyal model developed by Morrison i s a special case of the general compound Markov model. For Morrison's model, the brand loyal population is defined as follows: 1. Each individual i s a f i r s t - o r d e r 0-1 process with t r a n s i t i o n matrix 1 0 1 p 1-p 0 kp 1-kp 2. Different individuals may have di f f e r e n t p. Thus, p i s a random variable distributed according to some density function f ( p ) . 3. k is a constant, the same for each i n d i v i d u a l . The Brand lo y a l model says that an individual with a high p r o b a b i l i t y of remaining with brand 1 w i l l also have a higher p r o b a b i l i t y of leaving brand 0 to buy brand 1 than an individual with a low pr o b a b i l i t y of remaining with brand 1. Thus, people with high p are also more apt to switch to brand 1 than are people with lower p. When k=l, the model becomes a compound B e r n o u i l l i model of brand choice. 76 The e x p e c t e d p r o b a b i l i t i e s o f p u r c h a s e , P ( l | x j are g i v e n by the f o l l o w i n g f o r m u l a (see Massy et a l . [1970] p. 69): / 0 p£(x|p).f(p)dp (3.2) P ( l | x ) = - f - — f0 ^ ( x | p ) . f ( p ) d p where Jt(x|p) i s the c o n d i t i o n a l l i k e l i h o o d o f the p a s t purchase h i s t o r y x g i v e n p, and f ( p ) i s the p r i o r d i s t r i b u t i o n o f p. T h i s completes the d e s c r i p t i o n o f M o r r i s o n ' s Brand L o y a l model. In the next s e c t i o n s , we develop a new l e a r n i n g model o f brand c h o i c e and compare r e s u l t s a c r o s s models. 3.2 A New P u r c h a s e - t o - P u r c h a s e L e a r n i n g Model o f Brand  C h o i c e : The P o l y a - L e a r n i n g Model The model, h e r e a f t e r r e f e r r e d to as the P o l y a - L e a r n i n g model i s based on the s o - c a l l e d P o l y a u r n model w i t h c o n t a g i o n (see F e l l e r [ 1 9 5 7 ] ) . The i d e a t o use u r n models t o d e s c r i b e a f t e r - e f f e c t s (such as purchase event feedback) seems t o be due t o P o l y a . H i s s i m p l e u r n scheme i s d e s c r i b e d below to m o t i v a t e the m o d i f i c a t i o n s t h a t were brought to. i t i n o r d e r to make i t more germane to the brand c h o i c e c o n t e x t . 77 3.2.1 The Polya Urn Model From an urn containing Np white and N(l-p) black b a l l s , a series of drawings i s made, but instead of replacing the b a l l drawn, 1 + Ng b a l l s of the colour l a s t drawn are l a i d into the urn before a new drawing i s made. Like p, the constant g is a multiple of 1/N, but while p is a value between zero and one, the constant 6 can take every pos i t i v e multiple of 1/N, zero included. However, in the l i m i t , i . e . , as N goes to °°, g and p can take any value in the real l i n e and unit i n t e r v a l , respectively. The unknown results of the individual drawings are random variables x n which a p r i o r i have the same pr o b a b i l i t y d i s t r i b u t i o n as in a B e r n o u i l l i model. However, unless g = 0, these variables are no longer independent, for, k white b a l l s having been obtained in the f i r s t n drawings, the urn w i l l , after these n drawings, contain N(p+kg) white b a l l s and N[(l-p)+(n-k)g] black b a l l s . Thus, the pr o b a b i l i t y of drawing white in the (n+l)st drawing w i l l under these conditions be equal to: (3.3) Pr, ^drawing white ^at t r i a l n+1 k white b a l l s have already been drawn ) P + kg 1 + ng It i s clear from (3.3) that in the Polya-urn model, results of the drawings made have a "contagious" influence on the results of the subsequent drawings. The strength 78 of this contagion i s measured by the parameter g. A value of zero for this parameter indicates that the process is B e r n o u i l l i , i . e . , the drawings are independent from one another. The application of this urn model to brand choice theory i s immediate. If drawing white and black b a l l s becomes purchasing brand 1 and brand 0, we have a "learning" model of brand choice. 3.2.2 Limitation of the Simple Polya Urn Model as Applied  to Brand Choice A special feature of the Polya urn model i s that the p r o b a b i l i t y of drawing white at the (n+l)st drawing only depends on the number of white b a l l s drawn at a l l previous t r i a l s , whatever the order in which they were drawn. In the brand choice context, this a n a l y t i c a l s i m p l i c i t y implies the absence of any "recency" e f f e c t , i . e . , a l l past purchases of brand 1 contribute to increase the purchase p r o b a b i l i t y on future purchase occasions, irrespective of th e i r occurrence time. The p r a c t i c a l implications are that the following groups of past purchase h i s t o r i e s y i e l d the same condi-t i o n a l p r o b a b i l i t y of purchase: P a s t Purchase h i s t o r y n Observed c o n d i t i o n a l " p r o b a b i l i t y o f pur-chase P ( l | x ) 0 1 1 1 . 748 1 0 1 1 .753 1 1 0 1 .700 1 1 1 0 .606 0 0 1 1 .684 0 1 0 1 .575 0 0 1 1 .684 1 0 0 1 .605 0 1 1 0 .590 1 1 0 0 .489 0 0 0 1 . 553 0 0 1 0 .477 0 1 0 0 .384 1 0 0 0 . 368 > > Model p r e d i c t e d c o n d i t i o n a l prob-a b i l i t y o f pur-chase P(1|x) P •+ 3 3 1 + 43 P + 23 1 + 43 P + 3 1 + 43 (1) These o b s e r v e d brand c h o i c e p r o b a b i l i t i e s were o b t a i n e d from p a n e l d a t a on c o f f e e consumption. See Massy e t a l . [1970, p. 126]. 80 The s t a t i s t i c a l analysis of the previous chapter made i t clear that both the number of past purchase of the brand and their position in the purchase st r i n g must be reckoned with in order to provide an adequate representation of brand choice data. The Polya urn model provides a mean to allow for learning behavior.in terms of purchase f r e -quency but f a i l s to recognize the impact of purchase recency. In the next section, we s h a l l extend the simple Polya urn model in several ways to accommodate the influence of purchase vrecency on subsequent purchase p r o b a b i l i t i e s . 3.2.3 Mathematical development of the Polya-Learning  Model Assumptions The assumptions underlying the model w i l l be divided into three basic classes: a) Model s p e c i f i c a t i o n assumptions. b) Response assumptions. c) Response p r o b a b i l i t y change assumptions. Each of these are discussed in turn. a) Model Spe c i f i c a t i o n Assumptions SI: Each respondent's response i s generated by the same stochastic process. S2: At any given response occasion n, there are two mutually exclusive and c o l l e c t i v e l y exhaustive responses: response 1 (purchase of the brand under study) and-response 2. S3: Each respondent i s associated with an urn containing Np black and N(l-p) red b a l l s . The urn composition in terms of black and red b a l l s may vary across individuals. That i s , we view p as a random variable d i s t r i b u t e d according to some density' f (p) . Assumptions SI and S2 deserve no comments. Assumption S3 provides a mechanism to allow for consumer heterogeneity through the consumer's i n i t i a l purchase pr o b a b i l i t y . This heterogeneity may result from d i f f e r -ences in perceptions, preferences and attitudes toward the brand. These differences are not taken into account ex-p l i c i t l y but are modeled vi a the parameter p that i s allowed to be a random variable. 82 b) Response Assumptions Rl: At any given purchase occasion, each individual's p r o b a b i l i t y of purchasing brand 1 i s equal to the proportion of black b a l l s in his urn. I n i t i a l l y , this p r o b a b i l i t y i s equal to p as stated in S3. R2: The actual response made by an individual at occasion n depends on his responses at a l l previous response occasions. The form of this dependence i s speci-f i e d in the response change p r o b a b i l i t y assumptions. R3: Individuals respond independently of one another. The l a s t assumption which i s required for testing purposes, assumes away the role of interpersonal influence in brand choice decisions. The nature of the data to be 2 used ensures that this assumption i s reasonable s a t i s f i e d in our p a r t i c u l a r case. 2) Consumer panel data. c) Response Probability Change Assumption CI: After each drawing (purchase occasion), the b a l l drawn i s replaced and, more-over, black b a l l s are added to the urn i f a black b a l l was drawn. If, instead, a red b a l l was drawn, \ n red b a l l s are added to the urn. Assumption CI constitutes'a marked departure from the simple Polya urn model, and provides us with a way to model the influence of purchase recency. The strength of the "learning" and "forgetting" effects depend on the absolute value of 3 R and ^ respectively. If 3 > A for a l l n, we assume that consumers "learn" faster n n ' than they "forget". In other words, i t w i l l take more purchases of brand 0 (drawing of a red b a l l ) to bring an individual's p r o b a b i l i t y of purchasing brand 1 down from p>l to p Q than i t took purchases of brand 1 to bring that same pr o b a b i l i t y from p Q up to p^. Also, brand lo y a l consumers w i l l tend to exhibit higher values for 3 n and lower values for X than less l o y a l consumers. w i l l exert more influence on subsequent purchase probabil-i t i e s than w i l l less recent ones. If 3 rt + 1 > 3^, recent purchases of brand 1 84 M a t h e m a t i c a l d e v e l o p m e n t To t u r n o u r p i c t u r e s q u e d e s c r i p t i o n i n t o mathe-m a t i c s , we s h a l l i n t r o d u c e some n o t a t i o n . L e t X ^ = j l i f b r a n d 1 was p u r c h a s e d a t o c c a s i o n n .0 o t h e r w i s e I (X ) be t h e i n d i c a t o r f u n c t i o n f o r t h e r a n d o m n v a r i a b l e X , i . e . n ' I , (X ) = 1 i f X = 1 1 v nJ n •:0 o t h e r w i s e a n d s i m i l a r l y , I (X ) = 1 i f X = 0 o ^ n^ n 0 o t h e r w i s e A c c o r d i n g t o t h e a b o v e a s s u m p t i o n s , t h e p r o b a -b i l i t y t h a t a n i n d i v i d u a l w i t h i n i t i a l p u r c h a s e p r o b a b i l i t y p p u r c h a s e s b r a n d 1 o n t h e (n + l ) s t t r i a l g i v e n h i s p a s t p u r c h a s e h i s t o r y f o r t h e f i r s t n p u r c h a s e s i s g i v e n b y (3.4) P r [ X n + 1 = l | X n = x n , . . . , X 1 = x 1 , p ] = 85 n p .+ >: ( 3 k l x (X k) } -H k 1 + I ( 3 k , I x (X k) + A k I 0 C X k ) } This probability; reduces to that of the Polya simple urn model when $ n and X^ equal some posit i v e constant, for a l l n. To reduce the number of parameters to be e s t i -mated from the data, we introduce a simplifying assumption for the non-stationarity of the learning parameters g n and X , namely: n' 3 (3.5) g n = ng X n = iU 3 Alternative formulations are of course possible . Support for the above formulation comes from i t s s i m p l i c i t y and the results from the previous chapter that demonstrated the importance of purchase re cency on subsequent purchase p r o b a b i l i t y . Substituting (3.5) i n (3.4) y i e l d s : Pr (l|x,p) d l f Pr(X n + 1=l|X n=x n,..., X ^ x ^ p ) n n = [ P + gz k i 1 c x k ) ] / [ i + E i a e i j C x ^ + x i 0 (x k )> ] (3) An alternative formulation 3 = a + nB i s the following: , n , , , 6 X = b + nX n It has the advantage over (3.5) to reduce to the Polya case for g = X = 0 and a = b. However, i t introduces another two parameters (a and b) and was thus discarded in favor of the more parsimonous formulation in (3.5). Updating the pri o r If we know an individual's i n i t i a l purchase pr o b a b i l i t y p, his pro b a b i l i t y of purchasing brand i on any t r i a l i s given by (3.6), provided we also know his past purchase history, say x. However, we never know a consumer's true value for p, so that i t i s for us a random variable with p r i o r d i s t r i b u t i o n f ( p ) . Once we observe a past history, x, our p r i o r d i s t r i b u t i o n gets updated to a posterior d i s t r i b u t i o n f(p|x) (See e.g., Massy et a l . [1970]). Hence, an individual with past history x picked at random w i l l have a . p r o b a b i l i t y of purchasing brand 1 equal to: (3.7) Pr(l|x) = / J Pr(l|x,p)f(p|x)dp. From Bayes' theorem f(p|x) = *(x|p).f(p)/*(x) = £(x|p).f(p)/ /J^(x|p).f(p)dp where £(x|p) i s the l i k e l i h o o d of the purchase sequence x given p. We f i r s t need to derive the l i k e l i h o o d &(x|p) for a l l x. They follow d i r e c t l y from (3.6). For example, consider the purchase sequence denoted by 0101. For each purchase occasion, the individual's p r o b a b i l i t y of purchasing brand 1 i s given by>.(3.6). Thus, we obtain: 87 s r m m f r n = H - T ^ P (i-p+*) lp+*ej ( 2 )since Pr(0|p) = 1-p P r ( l | X 1 = 0,p) = p / ( l - A ) Pr(0|X 2 = 1,X1 = 0,p) = '(l-p + A ) / ( l + A + 2g) P r ( l | X 3 = 0,X2 = 1, X 1 = 0,p) = (p + 2f3)/(l + 4A + 2g) The remaining likelihoods £(x|p) are s i m i l a r l y derived. Displayed in table I I I . l , they are a l l of the form: ' ( 3 - 8 ) ^ ( x | p ) = g - Z a, p x k=0 K x where the a v , k=0,..., 4 and d are functions of B , A kx> x and the p a r t i c u l a r purchase sequence x being dealt with, Substituting (3.8) in (3.7) y i e l d s : f i , ' 1 4 k (3.9) Pr(l|x) =^/0 Pr(l|x,p) { - j i a p K} x k = 0 K X f ( p ) d p . K / £ ( x ) From (3.6), we can express Pr(l|x,p) as: (3.10) Pr(l|x,p) = (p +b x)/c x where b and c are functions of 0, A and the p a r t i c u l a r purchase sequence being dealt with. Substituting (3.10) in (3.9) yields after some rearrangement and integrating p out: 88 (3.11) Pr(l|x) 4 U x C x k l Q akx H4-k ] where a k x = 0 for a l l k<0 1 k th yk = 0^ P .-f(P-)^P = k moment of p. 4 The values of {a v } , b , c and d for a l l purchase KX j^  = Q X X X sequencesx are displayed in table I I I . l . As in the Linear Learning model, the f i r s t f i v e moments of the random variable p are treated as parameters to be estimated from the data, together with the "learning" parameters g and A." The parameters must s a t i s f y the following constraints:. (3.12) 0 <=. 3, A (3.13) "0 < j i 5 <_. yi+ 4 y 3 < u 2 < vi <_ 1 (3.14) 0 < u 2 - u j . (3.15) 0 < y i l - y 2 ' Constraint (3.12) w i l l ensure that the predicted p r o b a b i l i t i e s l i e in the unit i n t e r v a l . The l a s t three 4 ineq u a l i t i e s must hold since the P^'s are the moments of the -random' variable p. 2 4) Note that - P-J. i s just the variance of p. These in e q u a l i t i e s can be,shown to hold my making use of the Jensen or the Cauchy i n e q u a l i t i e s . 89 INTERMEDIARY RESULTS TO COMPUTE THE LIKELIHOOD £(x p) AND P ( l x ) . Purchase seq. x a0x lx 2x a o 3x a^ 4x 1111 1 106 27B2 1863 0 0000 1 -(4+10X) [(2+10A+18A2)+ -[ (2+9A) (A+l) + (A+l) (1+9A+18A2) (2+A)(2+9A)] (2+A)(1+9A+18A2)] 0111 -1 (1-7B) 6(7-106) 1082 0 1011 -1 (1-5B) 6(5-46) 4B2 0 1101 -1 (1-4B) 6(4-36) 3B2 0 1110 -1 • (1-4B) 6(4-36) 36 2 0 0011 1 -(2+A-3B) [l+A-36(2+A)] 36(1+A) 0 0101 1 -(2+A-2B) [l+A-26(2+A)] 26(1+A) 0 1001 1 -(2+2A-B) [-6+(l+2A) (1-B)] 6(1+2A) 0 0110 1 -(2-2B+A) [1+A-2B(2+A)] 2B(1+A) 0 1010 1 -(2-B+2A) [l+2A-B(2+2A)j 6(1+2A) 0 . 1100 1 -(2+3A-B) [-6+(l+3A) (1-6)] 6(1+3A) 0 0001 -1 (3+4A) -(3+8A+3A2) (1+4A+3A2) 0 0010 -1 (3+4A) -(3+8A+3A2) (1+4A+3A2) 0 0100 -1 (3+5A) -(3+10A+4A2) (1+5A+4A2) 0 1000 -1 (3+7A) -(3+14A+10A2) (1+7A+10A2) 0 b c d X X X 1111 106 (1+106) (1+6)(1+36)(1+66) 0000 0 (1+10A) (1+A) (1+3A) (1+6A) 0111 96 (l+A+96) (1+A) (1+A+2B) (l+A+56) 1011 86 (1+2A+86) (1+6) (1+6+2A) (1+46+2A) 1101 76 (1+3A+7B) (1+6)(1+36)(1+3B+3A) 1110 66 (1+4A+6B) (1+B) (1+3B) (1+6B) 0011 76 (1+3A+76) (1+A)(1+3A)(1+3A+3B) 0101 66 (1+4A+6B) (1+A)(1+A+2B)(1+4A+26) 1001 5B (1+5A+56) (1+6)(1+6+2A)(1+B+5A) 0110 . 56 (1+5A+5B) (1+A)(1+A+2B)(1+A+5B) 1010 46 (1+6A+46) (1+B) (1+B+2A) (1+46+2A) 1100 36 (1+7A+3B) (1+B)(1+36)(1+36+3A) 0001 46 (1+6A+46) (1+A) (1+3A) (1+6A) 0010 36 (1+7A+36) (1+A) (1+3A) (1+3A+3B) 0100 26 (1+8A+26) (1+A)(1+A+2B)(1+4A+26) 1000 6 (1+9A+6) (1+6)(1+6+2A)(1+6+5A) TABLE 111 . 1 This completes the mathematical description of the Polya-Learning model. 3.2.4. Model f i t t i n g procedures and data A minimum chi-square procedure was u t i l i z e d to estimate the seven parameters of the model ( 3 , A and the., f i r s t f i v e moments of the random variable P ) . Given the non-linearity of the chi-square function to be minimized and the various constraints imposed on the parameters, an i t e r a t i v e method of constrained optimization had to be resorted to^. The coffee data that has already been described and analyzed in the previous chapter w i l l provide the empirical basis for testing and comparing the performance of the newly developed learning model with alternative models of brand choice. 3.2.5 Empirical results The parameter estimates for each of the six . . segments^ of coffee buyers are displayed in table 111-2 5) The algorithm that has been used in this study has been developed by M.J. Box [1965]. 6) These segments were described in chapter II page 40 . together with the corresponding chi-square goodness of f i t s t a t i s t i c s . TABLE 1 1 1 . 2 SEGMENTS ALL EXCEPT 100% HEAVY LIGHT LOYAL NON-LOYAL .er N. .mates 3 . .142 .120 .257 .051 .163 # A .082 .053 .047 .104 .050 vi .586 .487 .350 .776 .587 .357 .237 .122 .612 .373 ^3 .238 .121 .065 .491 .266 Vk .176 .064 .052 .403 .210 ^5 .142 .034 .046 .338 .278 . 2 X 8.053 7.340 3.962 8.310 12.300 p - level .43 .50 .86 .41 .15 degrees of freedom 8 8 8 8 8 Sample size 5,310 3,890 2,660 2.650 3,600 1,710 x2 • 8,:.. 0.5 = 15.507 # No estimates are available for this segment due to convergence problems. :92 The model f i t s the data quite well, as evidenced by the low values of the chi-square s t a t i s t i c s or. the 7 high p - values , except in the case of "Loyal" buyers. The magnitude of the two Learning parameters 3 A indicates the strength of the Learning of each segment. The greater the values of 3 and A, the greater the learning effect observed in the data. That i s , the lower the values of 3 and A, the less effect the la s t purchases have, or to put i t d i f f e r e n t l y : the lower.the values of 3 and A, the more " B e r n o u i l l i " the population of consumers becomes. Heavy versus Light consumers. Heavy and Light consumers exhibit d i f f e r e n t behavior patterns, as suggested by their parameter values: 7) The p - le v e l associated with a chi-square s t a t i s t i c i s defined as: p - l e v e l = f°° f(x) dx 2 X where f(x) i s the chi-square d i s t r i b u t i o n with the appropriate number of degrees of freedom. A low p - le v e l indicates that the model i s not a viable representation of the process. For models which have a d i f f e r e n t number of parameters, p - levels rather than chi-square values should be compared to correct for the di f f e r e n t degrees of freedom. 93 " L e a r n i n g " " F o r g e t t i n g " parameter 3 parameter A Heavy .257 .047 L i g h t .051 .104 The L i g h t , segment i s the o n l y group f o r which A > 3. For t h e s e consumers, the r e l a t i v e d e c r ease i n the p r o b a b i l i t y o f p u r c h a s i n g t h e i r f a v o r i t e brand f o l l o w i n g a purchase o f a competing brand i s g r e a t e r than the r e l a -t i v e i n c r e a s e f o l l o w i n g a purchase o f t h e i r f a v o r i t e b r a n d . The r e v e r s e h o l d s f o r the Heavy segment, where the l e a r n i n g e f f e c t p r o v e s t o be much s t r o n g e r than the f o r g e t t i n g e f f e c t . An example w i l l c l a r i f y those p o i n t s . Suppose we have two consumers, one " L i g h t " and one "Heavy", each w i t h an i n i t i a l p r o b a b i l i t y o f b u y i n g t h e i r f a v o r i t e brand e q u a l t o .5. A f t e r the f i r s t p u r c h a s e , t h e i r p r o b a b i l i t y o f p u r c h a s i n g t h e i r f a v o r i t e b r a nd a t the n e x t t r i a l i n c r e a s e s or d e c r e a s e s depending on the purchase outcome. The f o l l o w i n g t a b l e summarizes the p o s s i b l e outcomes. I n i t i a l p urchase Purchase p r o b a b i l i t y a f t e r Segments p r o b a b i l i t y the f i r s t p urchase Favorite brand Favorite brand was purchased was not purchased L i g h t .5 .524 ( 4 . 8 ) * .453 (10 .6) Heavy . 5 .602 (20.4) .478 (4. 4) F i g u r e s i n p a r e n t h e s i s c o r r e s p o n d t o r e l a t i v e i n c r e a s e or d e c r ease i n the purchase p r o b a b i l i t y f o l l o w i n g a p u r c h a s e . 94 F o l l o w i n g a purchase o f the f a v o r i t e b r a n d , the purchase p r o b a b i l i t y jumps form .5 t o .6 f o r the "Heavy" consumer as compared w i t h a mere .524 f o r the " L i g h t " consumer, or a 20.4% and 4.8% i n c r e a s e r e s p e c t i v e l y . I f the f a v o r i t e b r a nd was not s e l e c t e d , the c o r r e s p o n d i n g p r o b a b i l i t i e s d e c r ease t o .478 and .453 f o r the Heavy and L i g h t consumer r e s p e c t i v e l y . In o t h e r words, Heavy con-sumers e x h i b i t s t r o n g e r b r and l o y a l t y than L i g h t ones. Purchase o f competing brands w i l l n o t a f f e c t t h e i r p r oba-b i l i t y o f p u r c h a s i n g t h e i r f a v o r i t e b r a nd v e r y much, t h a t i s , they e x p e r i e n c e m i l d f o r g e t t i n g e f f e c t s and s t r o n g l e a r n -i n g e f f e c t s . The o p p o s i t e i s t r u e f o r L i g h t consumers., The d e f i n i t i o n of brand l o y a l t y i n terms of l e a r n i n g (g ) and f o r g e t t i n g ( x ) e f f e c t s i n t r o d u c e s a f i n e d i s t i n c t i o n between d i s l o y a l b e h a v i o r and mere brand s w i t c h i n g . T h i s d i s t i n c t i o n c a r r i e s much importance f o r the brand manager who would l i k e t o know the e x t e n t t o which b r a n d - s w i t c h i n g i s e v i d e n c e of d i s l o y a l b e h a v i o r or mere v a r i e t y s e e k i n g . For the c o f f e e d a t a , we c o u l d t e l l our manager t h a t a L i g h t consumer who s w i t c h e s brands i s l e s s " l o y a l " t h a n a Heavy consumer who does so. The l a t t e r i s more l i k e l y t o r e t u r n t o h i s f a v o r i t e brand as e v i d e n c e d by the low v a l u e o f the f o r g e t t i n g parameter. T h i s c o n c l u s i o n s stands at v a r i a n c e w i t h Massy e t a l . 95 [1970]'s who found the two segments to be virtually-i d e n t i c a l , but i s consistent with the findings of the previous chapter. Comparison of the Brand Loyal, Linear Learning and Polya Learning models. How does the Polya-Learning model perform as compared with other competing brand choice models? Table III.3 displays the values of three d i f f e r e n t models' chi-square s t a t i s t i c s and t h e i r corresponding p-levels. The models are: i) Kuehn's Linear Learning model. i i ) Morrison's Brand Loyal model. i i i ) The Polya - Learning Model. The two Learning models outperform the Brand Loyal (Markov) model. In most cases, the f i t i s far better as can be seen by comparing the p - values. There seems to be l i t t l e doubt that the Learning models provide a better representation of the coffee data than the Markov models, a conclusion already reached by Massy et a l . [1970]. 96 The two learning models (Linear Learning and Polya Learning) performed equally well. The p-values of the PL model were higher than those of the LL model for three segments (ALL, HEAVY and LOYAL) and lower for the remaining three. Thus, none ,of the two models i s dominated by the other. Comparison of Linear Learning (LI.) , Brand "r Loyal (BL) and Polya -Learning (PL) Goodness of F i t : Coffee data LL BL PL SEGMENTS \ X P X P X P ALL 8.9 .36 16 .7 .04 8.05 .43 EXCEPT 100% , 7.1 . 52 18 .1 .02 7 . 34 . 50 HEAVY 6.0 .65 15 .2 .06 3.96 .86 LIGHT 5.8 .67 9 .1 .37 8.31 .41 LOYAL 13.2 .11 14 .4 .08 12.3 .15 NON-LOYAL 3.8 .87 19 .0 .02 DEGREE OF FREEDOM 8 8 8 Table III.3 W h i l e the need to develop a l t e r n a t i v e models to accommodate the a d a p t i v e b e h a v i o r observed i n consumer brand c h o i c e i s not o b v i o u s , the P o l y a - L e a r n i n g model does e x h i b i t some i n t e r e s t i n g p r o p e r t i e s : 1) I t p r o v i d e s one p o s s i b l e way to accommodate a d a p t i v e b e h a v i o r i n t o s t o c h a s t i c models o f brand c h o i c e . A l t h o u g h b o t h the L i n e a r L e a r n i n g and the P o l y a - L e a r n i n g models d e r i v e from the same b a s i c assumption (namely, t h a t p a s t p u r c h a s e s i n f l u e n c e f u t u r e o n e s ) , the a c t u a l feedback mechanisms are d i f f e r e n t . As i n d i c a t e d by i t s name, the L i n e a r L e a r n i n g model assumes t h a t the a d a p t i v e p r o c e s s i s l i n e a r i n the p r o b a b i l i t i e s . W h i l e l i n e a r i t y i s not an u n r e a s o n a b l e assumption i n many s i t u a t i o n s , t h e r e are times where o t h e r s might be a p p r o p r i a t e . The P o l y a -L e a r n i n g model i s an example o f such a " n o n - l i n e a r " l e a r n i n g . 2) The l i m i t i n g form o f the P o l y a d i s t r i b u t i o n i s the s o - c a l l e d n e g a t i v e b i n o m i n a l d i s t r i b u t i o n (see F e l l e r [ 1 9 5 7 ] , p. 143), which i s the backbone o f bo t h Bass' [1974] brand c h o i c e and Ehrenberg's [1972] purchase i n c i d e n c e models. While b o t h a u t h o r s have j u s t i f i e d t h e i r use o f the n e g a t i v e b i n o m i a l d i s t r i b u t i o n on grounds o t h e r than a d a p t i v e b e h a v i o r , the good f i t they o b s e r v e d c o u l d be e x p l a i n e d i n terms o f l e a r n i n g b e h a v i o r . CONCLUSION. The purpose of t h i s chapter was to present and discuss a new model of brand choice based on learning assum-ptions. The model was a purchase-to-purchase pure brand choice model which allows an individual's purchase p r o b a b i l i t y to va-ry with his purchase history. Non-stationarity of purchase p r o b a b i l i t y was assumed to be caused by "learning" e f f e c t s . The d e f i n i t i o n of brand l o y a l t y i n terms of learning and forgetting e f f e c t introduced a fine d i s t i n c t i o n between d i s l o y a l behavior and mere brand-switching. For the coffee data, i t was shown that a LIGHT consumer who switched brands was less l o y a l than a HEAVY consumer who did so. The l a t t e r was more l i k e l y to return to his favori t e brand as evidenced by the low value of the forgetting parameter. The next three chapters are devoted to the study of consumer brand-switching behavior i n multi-brand markets. 99 CHAPTER IV A MODEL OF CONSUMER BRAND-SWITCHING":  MATHEMATICAL PRELIMINARIES The previous two chapters dealt with consumer brand choice as opposed to consumer brand-switching. The stochastic models developed in chapter III are limited to analysing consumer brand choice since by construction they collapse the market into two mutually exclusive categories: brand 1, or the brand under study, and brand 0, a catch a l l category for the remaining brands. For these models, the variable of interest i s the p r o b a b i l i t y that an individual chooses the brand under study at a pa r t i c u l a r purchase occasion. The pr o b a b i l i t y of that individual purchasing any other s p e c i f i c brand i s not known. What i s known i s just the individual's p r o b a b i l i t y of not purchasing the brand under study. As a r e s u l t , those models are inadequate for the study of consumer brand-switching behavior. In contrast, this chapter develops a general class of brand-switching models that acknowledge both the stochastic and deterministic features of the brand-switching phenomenon. We w i l l introduce a class of models which implies aggregate consumer brand-switching and repeat 100 purchase p r o b a b i l i t i e s , but also d i r e c t l y incorporates the impact of a p r i o r i relevant variables in i t s structure. This approach, while preserving some of the features associated with brand choice models, w i l l allow researchers (with more f a i t h in human r a t i o n a l i t y ) to include in the model variables of managerial and behavioral s i g n i f i c a n c e . 4.1 Problem D e f i n i t i o n Given a f i n i t e set of n brands, which includes a l l brands from which a given customer group makes i t s purchases, suppose that for each pair ( i , j ) of brands we are given a "datum" A ^ representing the s i m i l a r i t y , s u b s t i t u t a b i l i t y , association or in general proximity between them. This datum may be a function of the variables controlled by the s e l l e r s of the brands, e.g., the s e l l e r s ' advertising and promotional expenditures, the price of the brands, the reputation of the company etc... It may also be a function of the perceived s i m i l a r i t i e s between the brands as derived from standard composition (multi-attribute attitude approach) or decomposition procedures (multi-dimensional s c a l i n g ) . Indeed, A^j could be a function of both perceived and actual differences between the brands. 101 We postulate the existence of some functional r e l a t i o n between the observed aggregate brand-switching data, say (for the proportion of consumers purchasing brands i and j on two successive purchase occasions) and the s i m i l a r i t y measure so that a general class of brand-switching models can be written in the form: (4.1) T>±. = f±. (A i ; j), i , j =1, n for some function f ^^. In this section, we do not address ourselves to the problem of specifying meaningful general forms for the s i m i l a r i t y measure A... This i s dealt with in the next chapter. Our problem instead is that of deducing from some weak assumptions a functional form for f^. given A ^ j . Thus, we w i l l not derive any s p e c i f i c r esults about brand-switching behavior, but rather, some mathematical preliminaries. 4.2 Mathematical Development Let B = {1, 2 , . . . , n} be a set of n brands 102 P^j = P r e d i c t e d p r o p o r t i o n of consumers buying brand i and j on two s u c c e s s i v e purchase o c c a s i o n s . T h i s p r o p o r t i o n can be i n t e r -p r e t e d as the p r o b a b i l i t y of choosing at random from the p o p u l a t i o n of consumers, a consumer who purchased brand i and j on two adjacent purchase occasions.'*" p r o p o r t i o n of consumers buying and j on two s u c c e s s i v e purchase s . Observed p r o p o r t i o n of consumers who bought brand i on purchase o c c a s i o n 1. Observed p r o p o r t i o n of consumers who bought brand j on purchase o c c a s i o n 2. i , j' £ . B t . . = Observed brand i o c c a s i o n m. . - £ t . . i t . i i J J m. = E t . . 3 2 i i j (1) As no c o n f u s i o n can a r i s e , i t i s convenient to d e l e t e the time s u b s c r i p t form the P ^ ' s . A more r i g o r o u s n o t a t i o n would be P r i w ^ to r e f l e c t the time (J ,2) dependence (purchase occasion) of the p r o p o r t i o n s . 103 We now present two alternative methods that can be used to specify the function f^_. . The f i r s t one i s based on the maximum l i k e l i h o o d principle.The second one makes use of the concept of entropy developed by informa-t i o n t h e o r i s t s (Shannon [1949] and recently applied i n the so c i a l sciences (see e.g Thei l [1967] and Wilson [1970]). 4.2.1 The Maximum Likelihood Solution. Suppose we have available some panel data involving a t o t a l sample of N indi v i d u a l s . If n.. denotes the obser-13 ved number of consumers purchasing brands i and j on two 2 successive purchase occasions, then we must have: (4.2) Z n.. = N. • • 13 i,3 J We may wish to make inferences from the sample re-sults as to the d i s t r i b u t i o n of brand-switching for the whole population from which our sample was drawn. Since the random variables n^ .. (i,j=l,...,n) follow a multinomial 3 d i s t r i b u t i o n , the l i k e l i h o o d function i s proportional to : (2) For si m p l i c i t y ' s sake, the number of individuals i n the panel i s assumed to remain constant over time. (3) For the multinomial d i s t r i b u t i o n to hold exactly, i n d i v i -duals must make purchase decisions independently of one another. The n ^ , however, are dependent because of (4.2). 104 (4.3) L = n P "LJ . i» j I£ the s i m i l a r i t y measure A ^ i s a function of K parameters we can write: (4.4) P = f [A ( 0 , 0 )] i , j =1, n where the are parameters to be estimated from the data. 4 We assume : (4.5) f [A±. ( e 1 , e K ) ] = a / t K A ^ ( e 1 , G K ) = a. b. A.. ( 0 ) where a^ and bj are additional parameters to be estimated along with the 0 ^ . and 0 denotes the vector of parameters { 0 - p . .., ®K^' ^ e w a n t t o maximize equation (4.3) or equivalently, i t s logarithm, with the P-jj's expressed in terms of (4.5) subject to the r e s t r i c t i o n s : (4.6) I P . = 1 (4.7) P.. 5 0 . i n (4) Motivation for this assumption i s presented in Appendix ^ where equation (4.5) i s shown to resu l t from four weak assumptions. L e t t i n g t^.. = n^^ /N and A be a l a g r a n g e a n m u l t i -p l i e r , we f i n d v a l u e s o f ( a . } n , ( b . } n , { 0 V } ^ and A 1 i = l J j = l K k=l which maximize (4.8) H = E t . . l o g P . . - A [ E P.. - 1] . . i l 6 i i . . i i !,3 J J i , j J (9) E t [ l o g a . + lo g b + l o g A . , A [ E a . b . A i i ( e ) - l ] i , j J S e t t i n g the p a r t i a l d e r i v a t i v e s o f H, w i t h r e s p e c t to each model parameter, to z e r o , i . e . , (4.9) | I L = 0 V, (4.10) | £ - = 0 V, J (4.11) H- = 0 V k we o b t a i n r e s p e c t i v e l y : (4.9a) E t . . / a . - A E b.A..(9) = 0 i = 1, n J i i l - i i i v J ' ' (4.10a) E t i . / b , - A E a A (9) = 0 j = 1, n i J J i J 106 t 9A (0) 9A (0) L 4.iiaj i j _ ^ _ i j _ = a. b. _U_ k = l , . . . , K. ^ A i J O ) 3 9 k i , j 1 J " k Using (4.6), (4.9a) - (4.11a) become (4.12) AE P.. = E t. . • • • 13 • '"'ii V-j J I (4.13) A E P. . = I t. . V. . P. . 3A. .(5) t. . 9A. .(©) 1 > J 1 J K 1 > J 1 J K Since E P. . = E t.. = 1, we have . . i i • • i i (4.15) A = 1. Further manipulation of (4.9a)-(4.14) yields:^ (4.16) P.. = a.g. m., m.~ A.. (0) v J ij 1 3 i i j 2 13 v J where (4.17) a. = [E p. mj2 A.^. (0)]"1 (4.18) • = [E a^ m-1 A (0)]"1  J i J (5) The P^j's will all be non-negative as long as the similarity measure A^.. (0) is itself non-negative. 107 and as before (4.19) m n = l t.. (4.20) m., = E t . . . The normalizing constants and 3^ are such that the maximum l i k e l i h o o d constraints expressed in (4.12) and (4.13) are mechanically s a t i s f i e d as can be readi l y checked by summing (4.16) with respect to j and i respectively. The maximum l i k e l i h o o d method not only s p e c i f i e s the func-t i o n a l form of the a.'s and b.'s, but also indicates how to estimate the ©j/s: just solve equation (4.14) for e k, k = 1, ..., K. We w i l l now present an alternative derivation of equations (4.16)-(4.18) which makes uses of the concept of entropy. 4.2.2 Maximum Entropy Solution The concept of entropy was f i r s t applied in the study of thermodynamics (Jaynes [1957]) and has found more recent application in the f i e l d of information theory (Khinchin [1957]) where researchers are concerned with the measuring of the amount of information conveyed by a given message. In addition, i t has recently aroused additional 108 i n t e r e s t i n the s o c i a l s c i e n c e s , e i t h e r as a d e s c r i p t i v e measure o f u n c e r t a i n t y such as i n T h e i l [1972] and Carman [1 9 7 0 ] , or as a more s u b j e c t i v e concept used by the a n a l y s t as a m o d e l - b u i l d i n g t o o l to maximize the use o f i n f o r m a t i o n a v a i l a b l e t o him (see e.g., W i l s o n [1970] and H e r n i t e r [ 1 9 7 3 ] ) . 4.2.2.1 E n t r o p y and I n f o r m a t i o n To i l l u s t r a t e the n o t i o n o f e n t r o p y , c o n s i d e r an event E w i t h p r o b a b i l i t y o f o c c u r r e n c e p. At some p o i n t i n t i m e , we r e c e i v e a r e l i a b l e message s t a t i n g t h a t E i n f a c t o c c u r r e d . The q u e s t i o n i s : how s h o u l d one measure the amount o f i n f o r m a t i o n conveyed by t h i s message? Suppose t h a t p i s c l o s e t o one ( e . g . , p = -95). Then, one may argue, the message conveys l i t t l e i n f o r m a t i o n , because i t was v i r t u a l l y c e r t a i n t h a t E would t a k e p l a c e . But suppose t h a t p = .01, so t h a t i s i s almost c e r t a i n t h a t E w i l l not o c c u r . I f E n e v e r t h e l e s s does o c c u r , t h e message s t a t i n g t h i s w i l l be unexpected and hence c o n t a i n s a g r e a t d e a l o f i n f o r m a t i o n . These i n t u i t i v e i d e a s suggest t h a t t o measure the i n f o r m a t i o n r e c e i v e d from a message i n terms o f the p r o b a b i l i t y p t h a t p r e v a i l e d p r i o r to the a r r i v a l o f the message, we s h o u l d s e l e c t a d e c r e a s i n g f u n c t i o n o f p. The f u n c t i o n proposed by Shannon [1949] i s : 109 (4.21) h(p) = log . i = - log p which decreases from » ( i n f i n i t e surprise and hence i n f i n i t e information when the pr o b a b i l i t y p r i o r to the message is zero) to zero (zero information when the p r o b a b i l i t y i s one). In this instance the logarithmic d e f i n i t i o n for information in (4.21) can be shown to be the only possible d e f i n i t i o n where certa i n simple axioms are accepted (see e.g. Khinchin [1957]). 4.2.2.2 The Entropy of a Di s t r i b u t i o n The information received from the message which states that event E has occurred i s not the same as the information concerning the complementary event that E has f a i l e d to occur. If p i s the pr o b a b i l i t y of E, the information provided by the l a t t e r message i s : h(l-p) = - l o g ( l - p ) . Therefore, as far as event E i s concerned, the information to be received i s either h(p) or h(l-p) and we do not know which as long as the message of occurrence or non-occurrence has not been received. However, we can compute the expected information content of this message pr i o r to i t s a r r i v a l , i . e . , i i o (4.22) H = p h(p) + (1-p) h(l-p) = -P log(p)" (1-P) l o g ( l - p ) . The function H i s also known as the entropy of any d i s t r i b u t i o n that assigns p r o b a b i l i t i e s p and (1-p) to two di f f e r e n t events. It follows d i r e c t l y from (4.22) that the entropy function i s symmetric in p and 1-p. It is non-negative, takes the zero value at p=0 and p=l and reaches a maximum at p=l/2. F i n a l l y , the entropy function in (4.22) can be extended to the case of n events E^, E n with p r o b a b i l i t i e s p^, p n: n n (4.23) H = E p. h(p ) = " E p l o g(p ) i=l 1 1 i=l 1 1 4.2.2.3 Entropy as a Model-Building Tool This section presents the ideas of Jaynes [1957] and w i l l enable us to offer a second and more useful i n t e r -pretation of the concept of entropy. Let X be a random variable which can take on values x^, x 2 > • • • > x n.. with p r o b a b i l i t i e s p^,..., p n. The p r o b a b i l i t i e s are not known. A l l we know i s the expectation of some function f(X): (4.24) E[f(X)] = E p. f ( x ± ) i and I l l (4.25) a p. = 1, p. H -.Yi: Given this information only, what i s our best estimate of the p r o b a b i l i t y d i s t r i b u t i o n p.? Jaynes-writes: "Just as in applied s t a t i s t i c s , the crux of a problem i s often the devising of some method of sampling that avoids bias, our problem i s that of finding a p r o b a b i l i t y assignment which avoids bias while agreeing with whatever information is given. The great advance provided by information theory l i e s in the discovery that there i s a unique, unambiguous c r i t e r i o n for the 'amount' of uncertainty represented by a discrete p r o b a b i l i t y d i s t r i b u t i o n , which agrees with our i n t u i t i v e notions that a broad d i s t r i b u t i o n represents more uncertainty that does a sharply peaked one, and s a t i s f i e s a l l other conditions which make i t reasonable". This c r i t e r i o n is the one expressed in (4.23). Jaynes then writes: "It i s now evident how to solve our problem; in making inferences on the basis of p a r t i a l information, we must use that p r o b a b i l i t y d i s t r i b u t i o n which has maximum entropy subject to whatever i s known. This is the only unbiased assumption we can make; to use any other would amount to arb i t r a r y assumption of information which by hypothesis we do not have". Thus, to solve the problem posed above, we simply have to maximize entropy in (4.23) subject to equation (4.24) and (4.25), which represent what we know. This gives: 112 e x p [ - | if( x ) ] (4.26) p. = — . * i E e x p [ - y l ( x . ) ] J 3 where y i s a l a g r a n g e a n m u l t i p l i e r a s s o c i a t e d w i t h (4.24). In the m a r k e t i n g f i e l d , H e r n i t e r [1973] p r o v i d e s a good i l l u s t r a t i o n o f the use o f e n t r o p y as a model-b u i l d i n g t o o l t o maximize the use o f i n f o r m a t i o n a v a i l a b l e to the r e s e a r c h e r . B u i l d i n g on the i d e a s p r e s e n t e d above, he d e v e l o p e d a p r o b a b i l i s t i c model o f consumer purchase b e h a v i o r . The model i s c o m p l e t e l y d e t e r m i n e d by s p e c i f y i n g o n l y the market s h a r e s . A l l o t h e r brand s e l e c t i o n s t a t i s t i c s , such as r e p e a t purchase p r o b a b i l i t i e s (P^^) and brand-s w i t c h i n g p r o b a b i l i t i e s (P„ , are d e r i v e d from the model. The assumptions o f H e r n i t e r ' s model are b e s t ex-p r e s s e d i n h i s own words: " I t i s assumed t h a t each consumer has a s e t o f p r e f e r e n c e s f o r the brands i n the market, and t h e r e i s a d i s t r i b u t i o n o f p r e f e r e n c e s over the p o p u l a t i o n . The p r o b a b i l i t y o f a consumer p u r c h a s i n g a p a r t i c u l a r brand i s n u m e r i c a l l y e q u a l to her p r e f e r e n c e f o r the brand. R a t h e r than s p e c i f y i n g the j o i n t d i s t r i b u t i o n o f p r e f e r e n c e s and f i t t i n g the parameters to e m p i r i c a l d a t a , the concept o f e n t r o p y . i s employed and the d i s t r i b u t i o n i s s e l e c t e d t h a t maximizes the e n t r o p y o f the system s u b j e c t o n l y to the e m p i r i c a l market share v a l u e s . " By d o i n g so, H e r n i t e r d e v e l o p e d a genuine h e t e r -ogeneous consumer b e h a v i o r model based on m i c r o - t h e o r e t i c 113 assumptions. For a l l i t s elegance, his approach did not allow him to b u i l d into the model exogeneous variables of behavioral or managerial interest. The method to be presented below overcomes this deficiency but offers no postulates for the underlying individual behavior. 4.2.2.4 Maximum Entropy Solution Let (X,Y) be a pair of random variables which can take on values ( x ^ y ^ ) , ( x 2 , y 1 ) , , ( x n'Y n) with p r o b a b i l i t i e s P^i> ^ 2 1 ' ' ^nn* "*"n t^ i e D r a n d . - s w i t c h i n g context, the random variable X= i w i l l represent a purchase of brand i at purchase occasion t-1, and s i m i l a r l y , Y=j w i l l represent a purchase of brand j at purchase occasion t, i , j=l,...,n. The p r o b a b i l i t i e s are not known. However, suppose we know the expected value of some function of the random variables X arid>Y,say A(X,Y): (4.27) E[A(X,Y)] = E P A[X=i, Y = j ] . i , j J To ease notation, l e t us write A ^ for A(X=i, Y=j). Suppose there i s available some panel data, involving a sample size of N individuals. If n.. denotes the observed number of consumers purchasing brands i and j on two adjacent purchase occasions, then we must have: 114 (4.28) E n i = N. i> j Let t .. = n../N. As before, we would l i k e to make inferences from the sample results as to the d i s t r i b u t i o n of brand-switching for the whole population from which our sample was drawn. Following Jaynes [1957], we s h a l l use that p r o b a b i l i t y d i s t r i b u t i o n P^ which has maximum entropy under the known constraints. That i s , we want to maximize the entropy of the j o i n t p r o b a b i l i t y d i s t r i b u t i o n P, 1 J (4.28) H = - E P., log (P ) i , j J J subject to (4.27.) E[A(X,Y) ] = E P. . A. . (4.29) E P... = E t i ; j = m.2 j = l , . . . , n (4.30) E P. - = E t.. i m., i = l , . . . , n j i j j i J i l ' where the symbol = means equality by d e f i n i t i o n . Constraints (4.29) and (4.30) require the predicted marginals (which can be interpreted as market shares) to be equal to the observed marginals. 115 The mathematical problem represented by (4.28) to (4.30) i s that of finding the maximum of a function subject to a set of equality constraints. This may be solved through the method of lagrange m u l t i p l i e r s as follows: Define the lagrangean (4.31) L = Z P i,log.(p .)- 1 . ^ ( 1 P.. - m n) i , j J J i j J - E y.(E P.. - m.9) - B[' E P.. A.. - E(A-i)] j 3 i x3 J 2 i } j i j 13 1 i 3 J J where (A^}, {y^} and 3 are lagrangean m u l t i p l i e r s , and E(A. . ) i s known. The P.- that maximize L are solution of: i j i j cA 7 T i 3L _ 3L _ 3L 3L _ n . . _-, C 4 ' 3 2 ) 9P7. - Jx7 - 31T7 3 3 " ° ' i»J-l»---» 13 i 3 From (4 . 32) : 3L 7 r B - = -log (P. .) - 1 - X. - y . - pA. . 8 P ± j 6 ^ 13 l 3 13 or (4.33) V±. = exp[-l -Ai" -v. -BA i j] Substituting in (4.30) for P.. yields 116 E exp[-l -A. -y. -3A..] - m.. j ^ 1 3 X 3 i i or exp (-A^) - m.. ^  {E exp[-l -y. ,-gA^.]} ^ i=l,...,n. 1 a . 3-> 13 Substituting in (4.29) for s i m i l a r l y y i e l d s : exp(-y.) = m--7{E exp[-l -A. -&A- 3=1,...,n, Upon defining a i = exp(-A i)/m i l bj = expf-A^/m^.^ one obtains (4.34) P.. = a.F.m.. on. 9exp (-gA. .) where (4.35) a. = [E m j 2 F j e x p ( B A i j ) ] " 1 (4.36) F, = [E m.1 a . e x p C - B A . , ) ] " 1 . i J Comparing equations (4 .16)-(4 .18) with equations (4 . 34)~(4 . 36) shows that the two approaches yield similar "looking" equations.6 This i s not surprising. It has always been recognized that (6) There are differences, however, due to the logarithmic d e f i n i t i o n of entropy. This explains the presence of the exponential function in (4.34). The two equations, however, are completely d i f f e r e n t i n terms of th e i r mathematical structure. 117 there i s a close connection between entropy maximizing methods and maximum l i k e l i h o o d methods (see e.g., Hyman [1969] and Wilson [1970]), since entropy i s the negative of the expected value of the l o g - l i k e l i h o o d function. The former method has the advantage of f l e x i b i l i t y . We have shown above that the entropy maximizing procedure i s a way of obtaining a p r o b a b i l i t y d i s t r i b u t i o n taking account of a l l the information available. If the available information i s modified or added to (e.g., i f some other constraints of the type expressed in (4.27) are shown to hold) then the estimate w i l l be changed to r e f l e c t the knowledge of the new information. F i n a l l y , the concept of entropy w i l l also provide us with a theoretical j u s t i f i c a t i o n for the problem of specifying an operational form for A ^ j , which i s the subject of the next chapter. 118 SUMMARY T h i s c h a p t e r has l a i d the t h e o r e t i c a l base f o r the development and e m p i r i c a l i n v e s t i g a t i o n o f the brand-s w i t c h i n g models o f f e r e d h e r e . A model b u i l d i n g strategy-has been o u t l i n e d t h a t e x p l i c i t l y i n c o r p o r a t e s i n t o the model v a r i a b l e s o f b e h a v i o r a l and m a n a g e r i a l s i g n i f i c a n c e to brand c h o i c e d e c i s i o n making. Three a l t e r n a t i v e d e r i v a t i o n s were p r e s e n t e d . Of p a r t i c u l a r importance f o r the remainder o f t h i s r e s e a r c h i s the e n t r o p y m a x i m i z i n g p r o c e d u r e t h a t p r o v i d e s a s e t o f r u l e s f o r model c o n s t r u c t i o n w hich w i l l guarantee c o m p a t i -b i l i t y w i t h known i n f o r m a t i o n and an i n t e r n a l c o n s i s t e n c y w hich i s not o t h e r w i s e e a s i l y a c h i e v a b l e . In the next c h a p t e r s , t h e concept o f e n t r o p y w i l l be used as a d e s c r i p t i v e s t a t i s t i c t o h e l p us i n t e r p r e t the e m p i r i c a l r e s u l t s . The maximum l i k e l i h o o d d e r i v a t i o n w i l l prove u s e f u l f o r e s t i m a -t i o n p u r p o s e s . The next c h a p t e r p r o v i d e s some o p e r a t i o n a l formu-l a t i o n s f o r the s i m i l a r i t y measure A^.(0) t h a t w i l l a l l o w a d e t a i l e d e m p i r i c a l comparison o f the model's performance i n terms o f goodness o f f i t and d i a g n o s t i c p o t e n t i a l w i t h t h a t o f a l t e r n a t i v e b r a n d - s w i t c h i n g models. 119 CHAPTER V THE DETERMINANTS OF BRAND SWITCHING BEHAVIOR Introduction In the l a s t chapter, a model testing strategy has been outlined that w i l l allow the researcher to express consumer brand-switching p r o b a b i l i t i e s i n terms of variables of managerial significance to brand choice decision making. The purpose of this chapter i s to develop a theory of consumer brand-switching behavior based on sound psycho-l o g i c a l premises and empirically test i t with the procedures developed in chapter IV. To th i s end, the central concepts underlying the proposed theory w i l l be presented i n the next section. The mathematical model developed in the preceding chapter w i l l then be operationalized, and the empirical data described. The chapter w i l l end with a discussion of the test results and some concluding comments about the general pertinence of the theory. 5.1 Stochastic and Deterministic Theories Best's [1976] study not withstanding, researchers have t r a d i t i o n a l l y treated brand choice behavior for an individual consumer as being completely stochastic or 120 e n t i r e l y deterministic. Quite recently, however, some studies^ have strongly suggested that brand choice is main-l y a stochastic process and that the outcome of any pa r t i c u -l a r choice decision cannot be predicted p r e c i s e l y . Their strong results have caused stochastic models to upstage the t r a d i t i o n a l brand choice correlates studies and have sanctioned the new s h i f t toward predicting rather than ex-plaining the actions of consumers. Indeed, the fundamental premise of s o c i a l psychology which postulates that a l l behavior i s ra t i o n a l and therefore can be explained has been put aside i f not put down. Clearly, consumer behavior i s both a stochastic and a cognitive (deterministic) process. Stochastic, since brand selection on a given t r i a l cannot be predicted pre-c i s e l y ; and cognitive because the steady state choice p r o b a b i l i t i e s observed over a sequence of choices reveal a choice pattern consistent with the consumers perceptions, preferences and b e l i e f s toward a p a r t i c u l a r set of brands. The next section w i l l develop and discuss a cognitive process model which incorporates the stochastic r e a l i t y of the consumer choice process. (1) See e.g. Herniter [1973] and Bass [1974]. 5.2 A J o i n t Space Theory o f Brand Choice A u s e f u l approach t o examining a brand's compet-i t i v e p o s i t i o n i s t o c o n s i d e r a s e t o f brands as p o i n t s l o c a t e d i n a space where the axes are d e f i n e d i n terms o f the " r e l e v a n t " c h a r a c t e r i s t i c s o f the brands. The concept o f a brand as a p o s i t i o n on a s e t o f a t t r i b u t e s has been suggested by economists ( L a n c a s t e r [ 1 9 6 6 ] ) , s o c i a l p s y c h o l -o g i s t s (Rosenberg [ 1 9 6 0 ] , F i s h b e i n [1963]) and m a t h e m a t i c a l p s y c h o l o g i s t s (Shepard [ 1 9 6 2 ] , K r u s k a l [ 1 9 6 4 ] ) . Numerous e x p l o r a t o r y and a few s i g n i f i c a n t a p p l i e d m a r k e t i n g s t u d i e s have been r e p o r t e d i n the m a r k e t i n g l i t e r a t u r e . They o f t e n appear under the t i t l e "Market S t r u c t u r e A n a l y s i s " (see e.g. S t e f f l r e [ 1 9 6 8 ] ) , " P e r c e p t u a l Mapping" o r " J o i n t -Space Theory" (see e.g. Best [ 1 9 7 6 ] ) . In the brand c h o i c e c o n t e x t , a j o i n t space c o n f i g -u r a t i o n c o n s i s t s o f a s e t o f brands and consumers p o s i t i o n e d i n the same " p e r c e p t u a l " space. The dimensions o f t h i s p e r c e p t u a l space r e f l e c t the consumers' p e r c e p t i o n o f those a t t r i b u t e s w h i c h they use i n making d i s c r i m i n a t e judgments among the brands. Consumers may be c h a r a c t e r i z e d as h a v i n g a unique s t i m u l u s o r i d e a l p o i n t i n the p e r c e p t u a l space. The i n t e r p r e t a t i o n h o l d s t h a t consumers p r e f e r some p a r t i c u l a r c o m b i n a t i o n o f v a l u e s on the p e r c e i v e d dimensions to a l l o t h e r c o m b i n a t i o n s . When b o t h the consumers' i d e a l p o i n t s 122 and the brands' c h a r a c t e r i s t i c s are mapped onto the same "perceptual" space, the l a t t e r i s often referred to as a "joint-space". The central concept underlying the j o i n t space theory of brand choice i s that of cognitive consistency. Consumers s t r i v e to maintain a cognitive equilibrium between the i r perceptions and preferences of the brands, on the one hand, and th e i r actual brand choice, on the other hand. While they may occasionally alternate between brands and exhibit varying degrees of brand l o y a l t y , they do so in a "long-term r a t i o n a l " fashion rather than in a purely random one. That i s , they tend to organize their choice behavior so as to achieve a cognitive equilibrium between thei r perceptions, preferences and brand choice. 2 Joint space configurations provide a geometric picture of consumers' perceptions and preferences for a p a r t i c u l a r set of brands. Inter-brands and ideal point -brand distances are meant to r e f l e c t consumers cognitive structure in terms of perceptions and preferences. A theory of j o i n t space brand choice postulates that: (2) A joint-space configuration for eight brands of soft brands of soft drink i s shown in figure V . l . II /N Diet Pepsi . Tab Coke Pepsi . 7-Up Like Fresca Sprite FIGURE V . l Po s i t i o n of the Eight Brands i n 2-Dimensional Discriminant Configuration i—1 OJ 124 i) Brand choice p r o b a b i l i t i e s are related function-a l l y to the distance between a consumer's ideal point and each brand in a p a r t i c u l a r choice set. The smaller the distance between a p a r t i c u l a r brand and the consumer's ideal point, the greater the l i k e l i h o o d of i t being chosen on a p a r t i c u l a r choice occasion. Moreover, the consumers may d i f f e r e n t i a l l y weight the dimensions of the perceptual space in terms of t h e i r r e l a t i v e impor-tance to them in an evaluative context. The distance of s p e c i f i c brands from his ideal point is assumed to r e f l e c t the d i f f e r e n t i a l weighting which he applies to the dimensions of interest. 3 i i ) Brand-switching p r o b a b i l i t i e s are related func-t i o n a l l y to the distances between the brands in the perceptual space. The premise i s that consumers w i l l tend to switch to similar rather than to d i s s i m i l a r brands. The smaller the perceived psychological distance between the brands, the greater t h e i r s i m i l a r i t y or substi-t u t a b i l i t y , hence the greater the switching between them. (3) Brand choice p r o b a b i l i t y i s defined as the p r o b a b i l i t y that a consumer selects a p a r t i c u l a r brand on a given choice occasion. Brand-switching p r o b a b i l i t y i s defined as the j o i n t p r o b a b i l i t y that a consumer selects any two d i s t i n c t brands on two adjacent purchase occasions. 125 To test these two hypotheses, the model developed in the preceding chapter w i l l be applied and elaborated upon in the next section. 5.3 Model Development In model form, brand-switching p r o b a b i l i t i e s can be written as: (5.1) P.. =f i, j(D i j, d i t ) , i , j =1, n, where P^j = j o i n t p r o b a b i l i t y that a consumer chooses brands i and j on two successive purchase occasions D^j = "perceptual" distance between brands i and j in the j o i n t space d^ = "perceptual" distance between the consumers' average ideal point and brand i in the j o i n t space, at purchase occasion t f^j 1 some mathematical function n = number of brands in the market Sim i l a r l y , brand-choice p r o b a b i l i t i e s can be expressed as : def 126 m. _ = IP. . = g(d..) i i j ij ^ i i def (5.2) n, j 2 - EP-. = g(d. 2) for some mathematical function g. Equations(5.1) and (5.2) merely restate in sym-bo l i c notation the two hypotheses described e a r l i e r , namely: i) brand-switching probabilities'^ ( P ^ , i t j) depend on inter-brand distances (D^.) in' the j o i n t space. i i ) brand choice p r o b a b i l i t i e s (E P--, EP-.) and repeat purchase p r o b a b i l i t i e s ( P ^) vary inversely with the distance between the brands and the consumers' average ideal point (d^ t) . An extensive rationale was provided in chapter IV to j u s t i f y the use of the following mathematical r e l a t i o n between the dependent variables [brand-switching (P-jj> i ^ j)> repeat purchase ( P^) and brand choice (m^t) p r o b a b i l i t i e s ] and the independent variables (joint space distance D^ ^ and d ^ t ) : (5.3) P... = a ib j g(d i ] L) g ( d j 2 ) h^±.t d^) i , j = l , . . . , n where (5.4) a± =[ E b. g(d, 2) h(D ., d ) ] _ 1 i = l , . . . , n j=l J J J n -1 (5.5) b. =[_E a. g ( d u ) h ( D i r d l t ) ] j = l , . . . , n g, h = some mathematical functions to be spec i f i e d l a t e r . 127 The n o r m a l i z i n g constants a. and b. are such that • ' i 3 i f equation (5.3) i s summed with r e s p e c t to e i t h e r i or j one o b t a i n s : (5.6) Z P £ m u = g(d ) and (5.7) Z P.. = m j 2 - g ( d . 2 ) . The above equations show t h a t brand choice prob-a b i l i t i e s are f u n c t i o n s of the d i s t a n c e s between the brands and the i d e a l p o i n t as p r e v i o u s l y hypothesized. Equation (5.3) c o n v e n i e n t l y expresses brand-switching and repeat purchase p r o b a b i l i t i e s as the product of brand choice prob-a b i l i t i e s and a f u n c t i o n that depends on j o i n t space d i s -tances. The l a s t step i n the m o d e l - b u i l d i n g procedure c o n s i s t s o f p r o v i d i n g o p e r a t i o n a l d e f i n i t i o n s f o r both the f u n c t i o n s g and h. T h i s task i s accomplished i n the next two s u b s e c t i o n s . When t h i s i s done, we s h a l l have an o p e r a t i o n a l model with which to t e s t our j o i n t space theory of brand choice and brand-switching behavior. 128 5.3.1 Sp e c i f i c a t i o n of 'the; function g While marketing theory suggests that brand choice p r o b a b i l i t y should vary inversely with the distance between the brands and the consumers' average ideal point, i t stops short of specifying the appropriate mathematical r e l a t i o n 4 between these two quantities . As there i s a p r i o r i no reason to suspect that one mathematical function i s better than the other, several functional forms w i l l be examined and f i t t e d to the empirical data. They include: (5.8) Hyperbolic model: t I n (5.9) Exponential model: exp[A d i t ] g ( ; d i t ) = E exp Ud.. J I t n (5.10) Polynomial model: k k l t n (4) Best [1976] provided some empirical evidence suggesting that d i f f e r e n t functions may be appropriate for dif f e r e n t i n dividuals. 129 The parameter(s) A measures the extent to which brand choice p r o b a b i l i t i e s vary with perceptual distances between the brands and the consumers' average ideal point. Since brand choice p r o b a b i l i t y i s assumed to be inversely related to perceptual distance, the parameter A should take on non p o s i t i v e values. When A vanishes, the hyper-b o l i c and exponential models y i e l d the same l i m i t i n g form: nu = 1/n, i . e . , a l l brands are equiprobable. As the absolute value of A becomes large, brand choice p r o b a b i l i t y decreases more sharply as perceptual distance increases. In the l i m i t , the brand which i s closest to the ideal point is chosen with p r o b a b i l i t y one and a l l the other brands are assigned zero p r o b a b i l i t y mass. In other words, the greater the absolute value of A , the greater the brand l o y a l t y to the most preferred brand. The polynomial model was included to accommodate possible non-monotonic relationships between brand choice p r o b a b i l i t y and perceptual distances. 5.3.2 S p e c i f i c a t i o n of the function h To specify the function h that controls the extent to which brand-switching p r o b a b i l i t i e s vary with j o i n t space distances, we w i l l use the concept of entropy as a model-building t o o l ^ . Our problem i s that of finding a p r o b a b i l i t y assignment which avoids bias while agreeing with whatever information i s given. In our p a r t i c u l a r case, we hypothesized that both brand choice and repeat purchase p r o b a b i l i t i e s were functions of the distance between the brands and the consumers' ideal point. We also held that brand-switching p r o b a b i l i t i e s vary inversely with i n t e r -brand j o i n t space distances. For s i m p l i c i t y ' s sake, l e t us assume that repeat purchase and brand-switching prob-a b i l i t i e s are proportional to (normalized) j o i n t space distances, that i s : (5.11) P i i = k1 d. Y 2, i = l , . . . , n and (5.12) P. . = k 0 D. - y 1 * 3 » . • ' * , j = 1, • • • ,n-where k-^  and k 2 are prop o r t i o n a l i t y constant, Y and y are distance parameters that control the extent to which brand-switching p r o b a b i l i t i e s vary with j o i n t space distances. As before, we require that: (5.13) m i l z E P... = g ( d n ) (5) The use of entropy as a model-building tool was reviewed in chapter IV. 131 (5.14) m j 2 = E P.. = g ( d j 2 ) Let us assume that equations (5.11) to (5.14) represent a l l the information we have about consumer brand-switching behavior. In making inferences about the P j j ' s on the basis of p a r t i a l information, we must, as for Jaynes [1957],use that p r o b a b i l i t y d i s t r i b u t i o n ( P j j ) which has maximum entropy subject to whatever information i s known. Thus, to solve the problem posed above, we simply have to: (5.15) Maximize E = - E p.. log p.. i , j '1;J 1 3 subject to the constraints expressed by (5.11) to (5.14). This maximization problem can be solved through the method of Lagrange m u l t i p l i e r s as'follows: Define the Lagrangean as: (5.16) L =- E P log P.. - E I' [ E P - g(d. )] i , j J J i J J - E u . r E D _ n f A . i _ E 3 H _ _ -Y • j [• P-- " g(d. 0)] " • ( P - d,-7 ' " k n) 3 J L i 13 Sl- i 2 ^ J i 1 ^11 i2 1J 2 Y• - (P-- D.."y - k~) • ; i ^ j 1.32 where cKj i s the Kroenecker delta, i.e. 1 for i = j 1 3 0 for i t j {A^}, { V j } , (6^1 and a r e the Lagrangean mu l t i p l i e r s associated with the four sets of constraints (5.11 to 5.14). Upon setting the p a r t i a l derivatives of L equal to zero and solving the r e s u l t i n g equations, one obtains: (5.17) P±. = a.bj g C d i l ) . g ( d i 2 ) exp [- p. 6±. d±2~y - a.. D . . " y ] 13 13 where the terms a^ and bj are defined as in (5-4) - (5-5) with the obvious modifications. When either one of the hyperbolic (5.8) exponential (5.9) or polynomial (5.10) models is substituted in equation (5.17) in l i e u of the brand choice functions g ( d ^ t ) , one obtains a f u l l y operational model which can be used to submit the j o i n t space brand-switching theory to an empirical test. For testing purposes, s i m p l i f i e d versions of the model expressed in (5.17) w i l l be submitted to empirical data. The need for simpler models arises from the desire to l i m i t the number of parameters to be estimated from the data. 133 As i t stands, the model (5.17) i s overparameterized: (n^ + 3) parameters (Avuy, y, {g.}^--, and{d..}. --?) i * j 2 for just (n - 1) degress of freedom. To reduce the number of parameters to be estimated without d r a s t i c a l l y a l t e r -ing the model's general character, some parameters had to be removed or set equal to one another. From this s i m p l i f i c a t i o n process, a number of models were retained for testing purposes. For convenience, they are summarized in table V.4 and b r i e f l y discussed in the next section. 5.3.3 Models for testing purposes The model are c l a s s i f i e d in three groups, in order of increasing complexity. a) Group 1 This i s the simplest class of models to be enter-tained in this chapter. Four d i f f e r e n t version w i l l be considered: SUMMARY OF MODELS 134 Basic Equation: P = a b m m h(D . •, d ) i , j = 1 n -LJ 1 J XX J Z X j I t h y = C i a i m i l h ( ' : ) ] _ 1 j = l , . . . , n MODEL BRAND CHOICE PROBABILITIES m , m ' Jh'(D , d ) ' i l ' j2 i j i t 1.1 Equality between predicted and observed exp (36 ) T o brand choice p r o b a b i l i t i e s i s forced by 1 . 2 exp ( 3 . 6 . . ) 1.3 se t t i n g m i t ( i = l , . . . , n , t = 1,2) equal to the observed brand choice p r o b a b i l i - : , e x p ^ 6 i c j D i j ^ I - 4 t i e s , exp [ 3 . ( 6 . . - D..) ] II m i t = d i i t 3 J t i = 1, ... ,n t = 1,2 exp [3. (6. . - D. .) ] I I I . l III.2 m. i t i t 3 J t i - l , . . t = 1,2 i ^ i j i2 exp [3 . (6 . .d.^ - D..)] l i j i2 " exp [3.(6..d + D..))] I I I . 3 exp [y6 12, J j2 B.D... TABLE V.4 135 Functional form for Model version 1.1 1.2 1.3 1.4 h (D. . , d . J ^ i j ' i t ' exp [3 6 ] exp [B. 6^.] exp [3(6.. - D..)] exp [ 3 . (6. . - D. .)] where <5„ i s the Kroenecker delta i . e . fi -. r l when i = j , i j 0 when i f j E( d i t ) Equality between predicted and observed brand choice probabil-i t i e s i s forced by setting g ( d i t ) in (5.17) equal to the observed brand choice p r o b a b i l i t i e s D. . "perceptual" distance between brands i and j in the jo i n t space. The set of a l l inter-brands distances i s an input to the model, and must be derived externally.^ The 3's are parameters to be estimated from the empirical data. The distinguishing features of those four simple models are two-fold: i) No attempt is made to "predict" or " f i t " the brand choice p r o b a b i l i t i e s . That i s , the observed brand choice p r o b a b i l i t i e s are substituted into equation (5.17) in l i e u of [ g ( d ^ ) , t = 1,2] so as to force equality between predicted and observed brand choice p r o b a b i l i t i e s . i i ) No attempt i s made to " f i t " the repeat purchase p r o b a b i l i t i e s . That i s no attempt i s made to express them as a function of the distance between the brands and the consumer's average ideal point in the jo i n t space. (6) Techniques to derive j o i n t spaces are mentioned i n the next section. 136 It should be noted at t h i s stage that part of the information contained in the j o i n t space i s deliber-ately ignored in the above formulations, in order to focus exclusively on consumer brand-switching. The perceptual distances between the brands and the consumers' average ideal point w i l l prove to be useful predictors for both brand choice and repeat purchase p r o b a b i l i t i e s in l a t e r applications. However, i t i s assumed that brand-switching p r o b a b i l i t i e s depend on the brands' respective position in the point space and not on th e i r p osition with respect 7 to the consumers' average ideal point . Models belonging to group I can be written as follows: (5.18) a. b . t. I 3 I I . t. where t. I . = E t. . 3 1 3 t = E t . . i 1 3 = observed brand-switching p r o b a b i l i t i e s = corresponding predicted brand-switching probab i l i t i e s = either one of the four functions expressed above (7) This i s an hypothesis which needs to be empirically v e r i f i e d 137 If one sums both side of (5.18) with respect to i or j , one obtains: Z P.. = t. Z P.. = t . In group I, predicted, and observed brand choice p r o b a b i l i t i e s are equal by construction. Thus, no attempt is made to express brand choice p r o b a b i l i t i e s in terms of j o i n t space distances, as w i l l be the case for the next group of models. b) Group II Models in group II d i f f e r from those in group I in that the brand choice p r o b a b i l i t i e s can now be expressed in terms of the j o i n t space distances between the various brands and the consumers' average ideal point. The three mathematical functions already discussed in (5.8) to (5.10) w i l l be f i t t e d to empirical data. While a l l of the four functional forms that were developed for the function h(D^j) could be used in conjunction with any of the three brand choice models described above (exponential, hyperbo-l i c and polynomial), i t was decided to confine the subse-quent analysis to the more general form afforded by model 1.4. That i s , models of group II can be written as: 138 (5.19) P.j = a.b. g ( d n ) g ( d j 2 ) exp [ ^ ( 6 ^ - D. .) ] where the expressions a^, b^ , <5^ j and D^j are defined above and the g(d^ t) ( i = 1, n, t = 1, 2) can take one any one of the three functional forms spe c i f i e d in (5.8) through (5.10). c) Group III This i s the most general group. In this group of models the brand-switching p r o b a b i l i t i e s are made a function of inter-brand perceptual distances, as in group I. Brand choice p r o b a b i l i t i e s are expressed in terms of the distances between the various brands and the customers' average ideal point, as in group II. The distinguishing feature of models in group III consists of allowing the repeat purchase p r o b a b i l i t i e s (the P-j^'s) t 0 depend on the perceptual distances between the brands and the ideal point by analogy with brand choice p r o b a b i l i t i e s . Four d i f f e r e n t versions were submitted to an 8 empirical test. They are respectively: (8) Actually, several other versions were empirically tested but due to t h e i r poor empirical r e s u l t s , they are not discussed in this study. 139 (5.20) P.. = a b 1 1 j 2 A h(D d ) where (5.21) h(.) = exp t B i (8±. d ^ - V^)] f o r model I I I . l (5.22) h(.) = exp [ 3 i ±. - l o g f S ^ + P . . ) ) ] . I I I . 2 d . | i 2 (5.23) h(.) = exp [.yfi.. ^ i ^ j - - 3 . D.-]J f o r model I I I . 3 13 L d A 1 i j / 3 J 2 The f o u r t h model, model 111.4, combines the f u n c t i o n a l form o f model I I I . l w i t h a s e t o f i n t e r - b r a n d i'. d i s t a n c e s based on the M i n k o v s k i m e t r i c w i t h p = 1 ( C i t y -B l o c k ) r a t h e r than the u s u a l E u c l i d e a n d i s t a n c e ( M i n k o v s k i 9 m e t r i c w i t h p = 2) . No new terms were i n t r o d u c e d i n the above f o r -m u l a t i o n s but an e x t r a parameter u t h a t c o n t r o l s the ex-t e n t to which the r e p e a t purchase p r o b a b i l i t i e s v a r y w i t h p e r c e p t u a l d i s t a n c e s i n the j o i n t space. When u = 0, model I I I ( v e r s i o n 1 and 2) reduces t o model I I . As the (9) In some i n s t a n c e s (see e.g. Bass et a l . [ 1 9 7 2 ] , the C i t y b l o c k m e t r i c proved to be s u p e r i o r to the u s u a l E u c l i d e a n m e t r i c i n terms o f e m p i r i c a l performance. I t i s i n c l u d e d here f o r completeness r a t h e r than f o r t h e o r e t i c a l r e a s o n s . d u y 140 absolute value of u gets large, the repeat purchase proba-b i l i t i e s decrease sharply as perceptual distances from the ideal point increase. Note that equations (5.21) through (5.23) can be rewritten as: (5.24) Model I I I . l h(.) = (5.25) Model III.2 h(.) ~ i i2 . _ . e 1 - 3 -3-D. . . , . e I 13 1 f 3 3-d,? 1 i2 1 = J -^ d.~/Ed^ 0 C y 1 2 . 3 2 . _ . (5.26) Model III.3 h(.) =)e 3 1 ~ J C e 1 13 1 f 3 This notation makes i t clear that while the repeat purchase p r o b a b i l i t i e s depend on the perceptual distances between the brands and the consumers' ideal point, the brand switching p r o b a b i l i t i e s are but functions of :' inter-brand perceptual distances. In models I I I . l and III. 2, the parameters 3^ ( i = l , . . . , n) act upon both the repeat purchase and the brand switching p r o b a b i l i t i e s . In model III. 3, the ;3 ' s are uniquely associated with brand switching p r o b a b i l i t i e s and the repeat purchase probabil-i t i e s are assumed to vary with r e l a t i v e rather than absolute weighted perceptual distances. 1.41 This completes the description of the models that w i l l be submitted to the empirical data to test the two hypotheses mentioned e a r l i e r " ^ . For convenience, the models are summarized in table V.4. 5.4 Estimation procedure As in chapter II, the model parameters were estimated by the minimum chi-square procedure. That i s , the parameters were chosen so as to minimize the usual goodness-of-fit chi-square s t a t i s t i c defined as: 9 (t - P - . ) 2 i y j i j where t ^ = observed brand switching p r o b a b i l i t i e s P^ .. = model predicted brand switching p r o b a b i l i t i e s N = sample size upon which the observed proba-b i l i t i e s were derived. (10) The hypotheses were: i) Brand choice and repeat purchase p r o b a b i l i t y are fu n c t i o n a l l y related to the perceptual distance between the brands and the consumers' ideal point. i i ) Brand-switching p r o b a b i l i t i e s are f u n c t i o n a l l y related to inter-brand perceptual distances in the j o i n t space. 142 5.5 The data The data used in this analysis were coll e c t e d as part of a laboratory experiment conducted in the summer of 1969 by Bass, Pessemier and Lehmann [1972]. The study focused on preferences for eight brands of soft drink which were selected to represent two major segments of the soft-drink product market: Lemon-Lime v e r s u s Cola, and Diet versus Non-Diet. The brands included in table V . l account form the bulk of the soft drink purchases in the area where the research was conducted. Two hundred and sixty four subjects chose one of the available soft drinks each Monday through Friday morning for three weeks, thus producing a t o t a l of 12 "purchase" occasions. The subjects also completed a t t i t u d i n a l and background questions at four points during the experiment. These questions included, among other items, ratings of the brands on eight attributes and paired s i m i l a r i t y judgments for the brands. 5.6 Joint space construction To test our hypotheses, we need a j o i n t space configuration from which the perceptual distances between each brand and the consumers' average ideal point together with a l l inter-brand perceptual distances can be obtained. 14 3 BRANDS OF SOFT-DRINK SELECTED FOR THE STUDY. NON-DIET DIET COLA COKE PEPSI TAB DIET-PEPSI LEMON-LIME 7-UP SPRITE LIKE FRESCA TABLE V . l 144 Several techniques"1""1' have been developed to i n f e r j o i n t space configurations from s i m i l a r i t y and preferences judg-ments, depending on the nature of the input data (metric versus non-metric). For convenience, the joint-space configuration that w i l l be used in this study is the one derived by Lehmann and Pessemier [1973]. The approach they followed was to develop a j o i n t space configuration based on determinant attr i b u t e s . They did this by submitting the attribute levels for each brand as judged by a l l the subjects in the study to a Cooley and Lohnes [1971] discriminant analysis program. The attributes used in the Lehmann and Pessemier study were: - carbonation - c a l o r i e s - sweetness - t h i r s t quenching - popularity with others - packaging - after-taste - flavor preference (11) For a review of multidimensional scaling techniques, see GREEN [1975]. 145 The res u l t i n g configuration provides a reduced space coordinate system in which the perceptual distances between the brands may be represented. Lehmann & Pessemier [1973] found two major dimensions which accounted for about half of the t o t a l variance and over 97% of the 12 explained variance . As can be seen in table V.2, the groups are c l e a r l y d i f f e r e n t . In order to better v i s u a l -ize the discriminant configuration, the position of the eight brands along the two dominant dimensions are plotted in figure V . l . Lehmann & Pessemier interpreted dimension 1 as overal l popularity and cal o r i e s content of the brands. This interpretation was supported by the discriminant structure ( i . e . , the " c a l o r i e s " and "popularity with others attributes were highly correlated with the f i r s t dimension .898 and .741 respectively). The second dimension is c l e a r l y flavor type: lemon-lime versus cola. Having derived a dimensional representation of the brands in the market, the l a s t step c a l l s for posi-tioning an average ideal point into the derived perceptual space. This task was accomplished with the aid of the Carroll-Chang PREFMAP algorithm [1970]. From the j o i n t (12) These figures represent the percentage of variance, explained accounted for by the f i r s t two discriminant functions. "14 6 POSITIONS ON THE DISCRIMINANT -FUNCTIONS BRANDS DIMENSIONS. 1 2 3 • 4 COKE 2. .74 .43 .17 . .15 7-UP. 1. .70 -.21 .15 -.29 TAB . -2. .33 .39 -.14 . .11 LIKE -1. .59 -.43 -.20 .11 PEP.SI . • 2. .42 .30 . -.17 .08 SPRITE .78 -.57 -.31 . - .07 D-PEP.SI -2. 49 .60 .05 .21 FRESCA -1 . 21 -.50 .46 .12 R e s p e c t i v e •%, of v a r i a n c e e x p l a i n e d by p a r t i c u l a r discriminant function 90.36 6. 92. 1.69 .54 1) From Lehmann and P e s s e m i e r [1973]. TABLE V.2 14 7 space so derived, a l l perceptual distances (inter-brand and between each brand and the average ideal point) were obtained. 5.7 Analysis and Results_ There were six time periods in which a l l of the brands were available and in which conditions were stable. From these six choice occasions i t was possible to compute fi v e j o i n t p r o b a b i l i t y matrices on the basis of the actual choice behavior of the 264 subjects who participated in each of the six choice occasions. The average of these j o i n t matrices i s presented in table V.3a. A summary of parameter estimates and goodness of f i t s t a t i s t i c s for various models is presented in table V.5, while tables V.3b and ¥.3c display the predicted brand-switching p r o b a b i l i t i e s for selected models. While the actual and predicted j o i n t p r o b a b i l i t i e s match remarkably well, the chi-square goodness-of-fit s t a t i s t i c i s s i g n i f i c a n t beyond the .01 le v e l for a l l models. However, caution should be exercized when interpreting the chi-square test r e s u l t s . Since the chi-square goodness-o f - f i t s t a t i s t i c i s proportional to the sample s i z e , small deviations from the observed j o i n t p r o b a b i l i t i e s get amplified as the sample size increases. (a) OBSERVED AVERAGE JOINT PROBABILITY MATRIX FOR SOFT DRINK BRANDS. Coke 7-Up Tab Like Pepsi Sprite D-Pepsi Fresca Marke' at t Coke .188 .033 .003 .010 .041 .017 .004 .011 .307 7-Up .032 .077 .001 .011 .024 .017 .002 .008 .172 Tab .002 .003 .004 .009 .002 .001 .002 .002 .025 Like .004 .007 .004 .007 .011 .002 .006 .005 .041 Pepsi .047 .035 .002 .008 . 137 .020 .007 .010 .266 Sprite .008 .013 .002 .005 .011 .023 .002 .006 .070 D-Pepsi .004 .002 .008 .004 .005 .004 .011 .005 .043 Fresca .017 .007 .004 .008 .011 .008 .005 .015 .075 Market share at .304 . 177 .028 .062 .242 .092 .038 .062 1.000 t share t+1 (b) THEORETICAL AVERAGE JOINT PROBABILITY MATRIX FOR SOFT DRINK BRANDS, MODEL 1.4 . ;.b4v);=exp{6i(6.. - D...)} Coke 7-Up Tab Like P o n e T_ Sprite -J - - R  F S C 3 Marke' at t Coke . 170 .034 .002 .008 .065 .016 .003 .009 .307 • 7-Up .028 .076 .002 .008 .028 .018 .003 .009 .172 Tab .002 .002 .004 .005 .002 .002 .004 .004 .025 Like .009 .006 .003 .006 .007 .004 .005 .006 .046 Pepsi .069 .036 .003 .008 .119 .017 .004 .009 .266 Sprite .008 .011 .002 .007 .008 .025 .003 .008 .070 D-Pepsi .004 .003 .005 .008 .003 .003 .009 .007 .043 Fresca .013 .009 .006 .012 .010 .007 .008 .010 .075 Market share .304 .177 .028 .062 .242 .092 .038 .062 1.000 at t+1 (c) THEORETICAL AVERAGE JOINT PROBABILITY'MATRIX FOR SOFT DRINK BRANDS, MODEL III.3 Coke 7-Up Tab Like Pepsi Sprite D-Pepsi Fresca Market share Coke .174 .030 .008 .010 .046 .020 .008 .011 at t .307 7-Up .022 .056 .004 .006 .026 .017 .004 .007 .172 Tab .002 .003 .013 .007 .003 .002 .010 .006 .025 Like .007 .007 .005 .007 .008 .006 .005 .005 .046 Pepsi .041 .030 .007 .009 . 140 .018 .007 .009 .266 Sprite .011 .017 .004 .006 .013 .035 .003 .007 .070 D-Pepsi .003 .004 .009 .007 .004 .004 .012 .006 .043 Fresca .007 .007 .005 .006 .008 .006 .005 .007 .075 Market share .304 . 177 .028 .062 .242 .092 .038 .062 1.000 , at t+1 TABLE V.3 Parameter S U M M A R Y O F P A R A M E T E R E S T I M A T E S F O R V A R I O U S M O D E L S - S O F T D R I N K D A T A . prmal Approxim. 7^  Degrees of Freedom 41 Sample Size: 1310 x 2.99,40 = 63.69 Estimates MODEL 1.2 MODEL 1.4 MODEL II MODEL I I I . l MODEL III.2 MODEL III.3 MODEL III.4 B l COKE 1.700 .564 .544 .318 .466 .248 .282 B2 7-UP 1.287 .609 .601 .285 .321 .430 .255 B 3 TAB .814 .201 .334. .273 .347 .404 .251 B4 LIKE .291 -.068 -.042 .040 .057 .042 .035 B 5 PEPSI 1.410 .559 .550 .315 .448 .298 .283 B6 SPRITE 1.359 .534 .539 .254 .283 .516 .219 B7 D-PEPSI 1.460 .214 .221 .216 .337 .313 .206 B8 FRESCA 1.070 -.020 -.012 .075 .120 .040 .064 . A - - -1.83 -1.84 -1.85 -1.82 -1.84 V - - .33 .35 4.17 .35 2 X 155.07 79.40 159.40 121.94 138.42 121.13 125.26 41 7.50 5-4 5.39 53 6.38 53 TABLE V.5 5.31 53 sionVS^ -j* For large values of df, the e x p r e s s n V ^ r _ y 2 ( d f ) - 1 may be used as a normal deviate with unit standard error. MODEL 1.1 1.378 176.54 5.58 53 MODEL 1.3 .376 155.58 150 An element by element comparison of the predicted with the actual j o i n t p r o b a b i l i t i e s indicates substantially accurate predictions for the , as i l l u s t r a t e d below for the repeat purchase p r o b a b i l i t i e s predicted by model 1.2: Repeat purchase Observed Predicted p r o b a b i l i t i e s (P i i D (MODEL 1.2) Coke .188 .190 7-up .770- .780 . TAB .004 .002 Like .007 .006 Pepsi .137 .138 Sprite .023 .023 D - Pepsi .011 .080 Fresca .015 .015 When the 64 observed and predicted j o i n t p r o b a b i l i t i e s were correlated, a l l models produced high c o r r e l a t i o n c o e f f i c i e n t s (as large as .98 for model 1.4). Therefore i t i s f a i r to say that the evidence i s in agreement with the theory and that consumers' cognitive structure (perceived s i m i l a r i t i e s among brands) i s a good predictor of actual brand switching. In other words, the findings of t h i s study are supportive of behavioral inferences commonly derived from interpretations of j o i n t space configurations. A more detailed analysis of the results afforded by the various models follows. 151 Models I The two one-parameter models (1.1 and 1.3) provided a reasonably good f i t to the data (chi-square values of 176.54 and 155.58 respectively). Model 1.3 tends to underestimate the repeat purchase p r o b a b i l i t i e s (P^) although i t does quite well in reproducing the brand switching p r o b a b i l i t i e s . In contrast, Model 1.1 which does not c a p i t a l i z e on the information afforded by the j o i n t space i s best at reproducing the repeat purchase probabil-i t i e s . Allowing the parameter 3 to vary across brands produced mixed r e s u l t s : i) Model I. l ' s performance i s not s i g n i f i c a n t l y enhanced by the introduction of the extra parameters. The decline in chi-square from 176.54 to 155.07 i s not s t a t i s t i c a l l y s i g n i f i c a n t at the usual l e v e l s . Note that model I.3's predicted power i s equivalent (in terms of chi-square) to that of model 1.3 based on the perceived distances between the brands, although the l a t t e r i s best at predicting brand-switching probabil-i t i e s whereas the former's streright rests on i t s a b i l i t y to reproduce repeat purchase 152 p r o b a b i l i t i e s , being in that respect similar to BASS' stochastic model [1974]. i i ) The dramatic improvement of model 1.4 over model 1.3".is worth noting. By allowing the parameter 3 to vary across brands, the c h i -square value shrank from 155.58 to 79.40, a decrease sharp enough to compensate for the loss of seven degrees of freedom. The estimates of the 3's confirm e a r l i e r conten-tions about t h e i r interpretation for models with or without perceptual distances as an input. One can readily check from the figures exhibited below that the 3's of model 1.2 are associated with the conditional repeat purchase 13 p r o b a b i l i t i e s (13) Conditional repeat purchase p r o b a b i l i t i e s are defined as: P^/^ = P i ; j / E P j i where P^. i s the j o i n t p r o b a b i l i t y . j They represent the p r o b a b i l i t y that a consumer w i l l purchase some brand, say i , given his la s t purchase was a purchase of brand i . 153 Conditional Corresponding value for g. Repeat purchase Model 1.2 Model I.4 probability h(.) = exp 6±.) M.)= exp [ B ^ f i ^ - D^-J COKE . .612 1,700 .564 PEPSI .515 1.410 .559 7-UP .448 1.287 .609 SPRITE .329 1.359 .534 D-PEPSI .256 1.460 .214 FRESCA .200 1.070 -.020 TAB .160 .814 .201 LIKE .152 .291 -.068 When the perceptual distances are used, as in model 1.4, the B's are s t i l l somewhat associated with the repeat purchase p r o b a b i l i t i e s . Non-diet brands (Coke, Pepsi, 7-Up and Sprite) which enjoy r e l a t i v e l y high condi-t i o n a l repeat purchase p r o b a b i l i t i e s exhibit higher values for B than do diet brands with low conditional repeat purchase p r o b a b i l i t i e s . But within each group (diet versus non-diet), the B's are no longer associated with repeat purchase p r o b a b i l i t i e s . Rather, they indicate the extent to which the observed brand-switching p r o b a b i l i t i e s are consistent with the derived j o i n t space. That this i s indeed the case i s best i l l u s t r a t e d by the following table: SWITCHING FROM / 'TO COKE PEPSI LIKE SPRITE 7-UP D-PEPSI TAB LIKE SPRITE TAB D-PEPSI 7-UP PEPSI COKE Switching inferred from Actual choice Joint space 7-UP PEPSI COKE FRESCA LIKE TAB 7-UP FRESCA PEPSI COKE LIKE TAB D-PEPSI D-PEPSI Actual choice Joint Space 155 If the derived perceptual map pictured in figure V . l i s a proper s p e c i f i c a t i o n of consumers', cognitive brand structure, and assuming that consumers would rather switch to similar than d i s s i m i l a r brands, one would expect regular consumers of Fresca to switch to brands such as Like and Sprite as indicated in the above table. It turns out, however, that Coke and Pepsi are the two brands which Fresca consumers switched Ito most of the time, i n spite of the fact they were perceived as being more unlike Fresca than any other brand (see figure V . l ) . In other words, consumers' switching behavior to and from Fresca (or Like) was not consistent with the joint-space theory of stochastic choice. Accordingly, the derived perceptual distances for those two brands were given n e g l i g i b l e (negative) weights: -.020 and -.068 respectively. In contrast, switching from and to Sprite agreed with what would be expected on the basis of consumers' perceptions. Brands that were perceived as being similar to Sprite, such as 7-Up, Pepsi and Coke were also those that were selected more often as second choice by regular Sprite consumers. As a r e s u l t , the 3 parameter associated with Sprite was large as were those of brands for which consumers' brand-switching behavior behaved according to expectations such as Coke, Pepsi and 7-Up. 156 Model II Of the three functional forms (hyperbolic, exponential and polynomial) that were experimented with, the hyperbolic function provided the best f i t to the observed brand choice p r o b a b i l i t i e s . Figure V.2 portrays the relationship between perceptual distances and brand choice p r o b a b i l i t i e s for one of the two choice occasions. As the perceptual distance between a brand and the consumers' average ideal point increases, the p r o b a b i l i t y of that brand being selected on any purchase occasion dwindles dramatically. The excellent match between observed and predicted brand choice p r o b a b i l i t i e s lends some additional support to the j o i n t space theory of brand choice. The g parameters that control the extent to which brand-switching p r o b a b i l i t i e s vary with interbrand percep-tual distances are very similar to those obtained under model 1.4. A l l comments about t h e i r interpretation extend to model II as well. 1 5 7 FIGURE V.2 Relationship Between Brand Choice P r o b a b i l i t i e s and Perceptual Distances Soft Drink Experiment. Hyperbolic Model Brand choice p r o b a b i l i t y distance Legend: « Predicted % Observed Observed brand choice p r o b a b i l i t i e s .307 .172 .025 .046 .266 .070 .043 .075 Functional form: m. = d. l Zd . A j 3 Predicted brand choice Perceptual p r o b a b i l i t i e s distance .309 34.04 .141 46.46 .046 82.21 .050 79.01 .262 35.58 .096 56.51 .047 82.49 .050 78.81 with X = -1.83 BRANDS COKE 7-UP TAB LIKE PEPSI SPRITE D-PEPSI FRESCA 158 Relationship Between Observed Repeat Purchase P r o b a b i l i t i e s and Perceptual Distances. Soft Drink Experiment. FIGURE V.3 159 Model III While a l l four versions of model III produced similar results in terms of goodness of f i t , model III.3 proved to be the only one for which no disturbing sign reversal occurred for the parameter y. By construction and from consumer choice theory, one would expect the parameter y in models I I I . l and III.2 to be negative. As the distance between the brands and the consumers' ideal point increases, the repeat purchase p r o b a b i l i t i e s are assumed to decrease, as i l l u s t r a t e d in figure V . 3 . The sign reversal for the parameter y i s a consequence of the dual role played by the g parameters as alluded to e a r l i e r . When these parameters are uniquely associated with the inter-brand distances as in model III.3 , no such sign reversal occurs. The parameter y, which in this model was expected to be p o s i t i v e , did turn out to be p o s i t i v e . Model III.3 i s therefore recommended as a most general and most appropriate model for the study of consumer brand switching behavior as applied to frequently purchased consumer goods. - 1 6 0 Conclusions This chapter investigated a stochastic theory of brand choice and brand switching derived from concepts of perception, preference, cognitive structure and consis-tency. A muti-dimensionally scaled configuration was used as a s p e c i f i c a t i o n of consumers' cognitive structure. Perceptual distances derived from this configuration were then related to brand-choice and brand switching probabil-i t i e s through a model that takes into account the constraints imposed on the various p r o b a b i l i t i e s (limited range and sum constraints). The following conclusions were reached: 1) There exists a relationship between brand choice p r o b a b i l i t y and joint-space distance. The hyperbolic function proved to be the best of the three mathematical functions that were a - p r i o r i postulated. 2) There also exists a relationship between repeat purchase and brand swtiching proba-b i l i t i e s on the one hand, and joint-space distance on the other. Several mathematical functions were experimented with and one of them was singled out as the most appropriate function for the data at hand. 161 The approach followed in this study represents a marked departure from e a r l i e r attemps at brand switching modeling such as the two stochastic models developed by Herniter [1973] and Bass [1974]. In these models, brand switching behavior i s assumed to be a pure random process. No attempt was made to relate brand switching p r o b a b i l i t i e s to variables of managerial significance such as perception, preference or other marketing variables of int e r e s t . It i s hoped that the methodology.developed in this chapter w i l l trigger future research toward greater internal and external validation.'*' 4 The next chapter i s devoted to the reverse problem that consists of i n f e r r i n g a j o i n t space configuration from a matrix of brand-switching p r o b a b i l i t i e s . (14) Such as the use of more r e a l i s t i c experimental set-tings, d i f f e r e n t scaling mechanisms, and more . disaggregate data. 162 CHAPTER VI MULTI-DIMENSIONAL SCALING OF BRAND SWITCHING DATA 6.1 Introduction This chapter discusses a procedure for deriving a joint-space configuration from brand-switching proba-b i l i t i e s rather than the more widely used judged s i m i l a r i t y or preference measures (Green and Carmone [1969]). Since a j o i n t space configuration implies a p a r t i c u l a r brand-switching structure, as demonstrated in the previous chapter, i t seems l i k e l y than a brand-switching structure w i l l in turn imply a joint-space configuration. In chapter V, we addressed ourselves to the problem of predicting actual brand switching p r o b a b i l i t i e s from an average j o i n t space configuration derived from the individuals' perceived s i m i l a r i t i e s between and preferences toward the various brands. Thus, we entertained the problem symbolically expressed by: £ < D i j ) brand-switching p r o b a b i l i t y between brand i and j (6.1) P.. where P-. 163 = perceptual distance between brands i and j in the j o i n t space In this chapter, we address ourselves to the reverse problem expressed by: (6.2) D.. = f " 1 (P.-) where f ^ denotes the inverse of the function f. Here, the concern i s with deriving a j o i n t space configuration that i s as consistent as possible with the observed brand switching p r o b a b i l i t i e s . The purpose of this chapter i s to present a methodology to perform such multidimensional scaling. To this end, the remaining discussion has been organized as follows: The next section demonstrates the i n a b i l i t y of t r a d i t i o n a l multidimensional scaling techniques to cope with the problem at hand. An alternative methodology i s then developed and i l l u s t r a t e d with the soft drink experi-mental data described in the preceding chapter. The discussion ends on some concluding remarks about the lim i t a t i o n s and future prospects for the recom-mended methodology. 164 6.2 Methodological Implications Since Shepard [1962 ] 's pioneering work, l i t e r a l l y scores of computer programs have been developed for metric or non-metric scaling of s i m i l a r i t i e s or preference data"1". But in spite of this intense algorithmic a c t i v i t y , none of the models produced so far can adjust to the r e s t r i c t e d nature of brand-switching data. The l i m i t a t i o n s of brand-switching p r o b a b i l i t i e s as an appropriate input to t r a d i t i o n a l multidimentional scaling (MDS) techniques are best appreciated when the properties of brand-switching p r o b a b i l i t i e s are compared with those of the usual distance and s i m i l a r i t y measures. If we l e t d.. stand for the distance between two objects i and j from a set of n such objects, say B. = {1,..., n }, the quantity d^j s a t i s f i e s the following axioms: (i) d i i = 0 V i e B ( i i ) ~d^. = dj ^  V i and j e B (symmetry) ( i i i ) d.^ £ d^^ + d^j V i j> j f k e B (triangle inequality) When axiom ( i i i ) i s not s a t i s f i e d , the quantity d.^ i s referred to as a measure of s i m i l a r i t y rather than (1) For an assessment of marketing applications of multi-dimensional scaling techniques, see GREEN £1975]. 165 as a measure of distance . Such s i m i l a r i t y measures are t r a d i t i o n a l l y used as input to metric multi-dimensional scaling algorithms. In contrast, brand-switching p r o b a b i l i t i e s gen-e r a l l y s a t i s f y none of the above axioms since: (i) the repeat purchase p r o b a b i l i t y P^^..need not and in general w i l l not be equal to zero, ( i i ) the brand switching matrix need not be symmetric, that i s P.. f P.. for ( i , j) e B vX B. ( i i i ) the.triangle inequality need not be s a t i s f i e d by the P j j ' s a n d i s void of any meaning in the brand-switching context. Moreover, (iv) brand-switching p r o b a b i l i t i e s are range constrained (P.• i 0 V. . e B), (v) brand-switching p r o b a b i l i t i e s are sum-constrained ( E P. . = 1) . Nevertheless, attempts have been made to submit 2 brand-switching p r o b a b i l i t i e s to ordinary MDS techniques . (2) See for instance Lehmann [1972]. 166 The p r o b a b i l i t i e s predicted by metric MDS models, however, may not necessarily l i e within the unit i n t e r v a l or add up to unity since there is no b u i l t - i n mechanism to ensure that they do. A l t e r n a t i v e l y , the brand-switching p r o b a b i l i t i e s could be transformed into a set of rank order input data. The set of ranked pairs would then be submitted to a non-metric scaling algorithm such as Kyst (Kruskal et a l [1973]) or Torsca 8 (Torgerson [1967]). While this approach would obviate the two l i m i t a t i o n s mentioned above, i t disregards the metric information provided by the brand-switching p r o b a b i l i t i e s and thus cannot be f u l l y e f f i c i e n t . Another d i f f i c u l t y arises from the possible asymmetry:of the brand-switching matrix. While s i m i l a r i t y measures can be constructed from brand-switching proba-b i l i t i e s , the question remains of which half of the matrix should be deleted or whether some kind of "data massaging" would be more appropriate. Last, none of the existing algorithms i s able to extract from the repeat purchase p r o b a b i l i t i e s the i n f o r -mation required to position the consumers' average ideal point into the joint-space. Such algorithms have to forgo the information contained in the repeat purchase proba-b i l i t i e s and l i m i t themselves to brand positioning only. 167 It i s clear at this point that existing metric or non metric multidimensional scaling techniques are not equipped to i n f e r j o i n t space configurations from brand-switching p r o b a b i l i t i e s . Failure to recognize the r e s t r i c t -ed nature of such p r o b a b i l i t i e s as an input to t r a d i t i o n a l 3 MDS techniques may consequently produce spurious results . What i s needed i s a methodology that makes ex-p l i c i t provision for the p a r t i c u l a r nature of brand-switching p r o b a b i l i t i e s . Such a methodology i s outlined in the following section. 6.3 Multidimensional Scaling of Brand-Switching Probabil-i t i e s The objective of multidimensional scaling of brand-switching p r o b a b i l i t i e s can be stated non rigourously as: Given a matrix of brand-switching p r o b a b i l i t i e s (similar to that shown in chapter V, table 3), fi n d a configuration whose distances - in a spec i f i e d dimensionality - best reproduce the o r i g i n a l brand-switching p r o b a b i l i t i e s . (3) Lehman's [1972] attempts to inf e r a perceptual map from brand-switching p r o b a b i l i t i e s provides such an example of spurious r e s u l t s . Using Bass et a l [1972] soft drink experimental data and a non-metric scaling algorithm (TORSCA), he found l i t t l e agreement between the perceptual maps derived from brand-switching p r o b a b i l i t i e s and s i m i l a r i t y measures. 168 The conceptual basis of using brand-switching matrices as a basis for perceptual mapping i s not d i f f i c u l t to grasp i n t u i t i v e l y . If one assumes that brand-switching p r o b a b i l i t i e s are a function of perceptual distances, i t is possible to work backward and inf e r a perceptual map from a brand-switching matrix. The chief d i f f i c u l t y rests upon the s p e c i f i c a t i o n of an appropriate model that w i l l cause the predicted p r o b a b i l i t i e s to s a t i s f y a l l requirements for l o g i c a l consistency. The model developed in the previous chapter provides a structural framework that brings about such l o g i c a l consistency. 6.3.1 Problem D e f i n i t i o n The problem i s to f i n d that j o i n t space config-uration which best reproduces the o r i g i n a l brand-switching p r o b a b i l i t i e s . That i s , one must select for each brand i and for each dimension k, the set of coordinates x ^ ( i = 1,..., n; k = 1, . . . , r) that provides the best f i t to the observed p r o b a b i l i t i e s . The coordinates of the consumers' average ideal point must also be derived simul-taneously with the brands' coordinates for each dimension. 169 To ease the subsequent discussion, l e t us intro-duce some notation. Let 3 . {1 , n) = a set of n brands t^.. = Observed proportion of consumers that "switched" from brand i to brand j on two successive purchase occasions, i , j e "B. P^ . = Corresponding model predicted proportion, i , j e B , X - [x-^, ... x^] x l l ' x 2 1 ' > x n l X-, , l r ' nr : i k = ^ ° i n t s P a c e coordinate of brand i dimension k, i e 'B,.? k = 1 , . . . on r = number of dimensions of the j o i n t space Y = [ y l f y 2 ] Xll' Xll * l r ' >^ 2r y t k - Joint space coordinate of the consumers' average ideal point on dimension k for purchase occasion t. k = 1 r t = 1, 2 170 = Joint space distance between brands i and I ' X i k " *jk 1/P = Minkovski metric i t Joint space weighted distance between brand i and the consumers' average ideal point at purchase occasion t, i e B;, t = 1, 2. m i t x i k >^ tk |P; 1/P\ i or j t^ j = Observed proportion of consum-ers who purchased brand i at purchase occasion t, t = 1, 2 As in chapter V, we express the brand-switching p r o b a b i l i t i e s (P-jj) i n terms of j o i n t space distances (D-. and d..) as follows: P i j = a i b j « C dil>. ^  « ( d v j 2 ^ 5 h (D-., d i t , e ) i , j eX Where b. z b . g ( D . 2 , x) h(d , d a ) j ^ a . g (d i ; L , : , x ) h (D i ; j , d i t , e ) -1 -1 i e B j e B; 171 and the functions g and h are as sp e c i f i e d in the previous chapter. The quantities A and 0 denote parameter vectors to be estimated along with the brands' coordinates by minimizing some function of the discrepancy between observed and predicted brand-switching p r o b a b i l i t i e s . Such a measure is the usual chi-square goodness-of-fit s t a t i s t i c defined as : (6.3) " x'2 = N E (t. . - P. .)2 / P. • where N i s the sample size from which the observed brand-switching p r o b a b i l i t i e s were obtained. The smaller the chi-square value, the better the f i t between observed and predicted p r o b a b i l i t i e s . Besides i t s potential usefulness as an indication of "correct" dimensionality, the chi-square s t a t i s t i c , un-l i k e the usual f i t measures of non-metric MDS l i k e Kruskal' "stress", proves very convenient for s t a t i s t i c a l inference. Since the sampling d i s t r i b u t i o n of the chi-square s t a t i s t i c i s known, tests can be carried out to determine whether the derived j o i n t space i s consistent with the observed brand-switching p r o b a b i l i t i e s . 172 6:3.2 Estimation Procedure For a spec i f i e d dimensionality r and a given Minkovski p - metric, the problem i s to: MINIMIZE x 2 = N E ( t . . - P..) 2/P.. i , 3 J J J with respect to X, Y, ^ ^ ^ = 1 , * a n d ®-This minimization process i s repeated in the next higher dimensionality u n t i l the decrease in the c h i -square value brought about by the additional dimension is not s t a t i s t i c a l l y s i g n i f i c a n t . The estimation of the model parameters was accomplished with the aid of a com-puter algorithm that i s described below. This i s followed by a discussion of the stopping rule that was adopted to select the "correct" dimensionality. 6.3.2.1 Computer Algorithm The algorithm combines a gradient search with an i t e r a t i v e procedure similar to that used in Torgerson's [1967] non-metric MDS algorithm (TORSCA 8,) . Given the large number of parameters to be estimated, a gradient 173 s e a r c h a lone was c o n s i d e r e d unmanageable . F o r t u n a t e l y , t h e r e e x i s t s a s i m p l e h e u r i s t i c i t e r a t i v e p r o c e d u r e t o improve a s t a r t i n g c o n f i g u r a t i o n u n t i l i t converges to a s t a b l e v a l u e f o r a g i v e n d i m e n s i o n a l i t y . For e x p o s i t o r y p u r p o s e s , assume t h a t we have a se t o f s t a r t i n g v a l u e s ^ f o r Y, X and 9 , say Y°, to0, X° and 0 ° . The f i r s t s t e p c o n s i s t s o f f i n d i n g an i n i t i a l v a l u e f o r the m a t r i x o f brand c o o r d i n a t e s X, say X°. F i n d i n g an i n i t i a l c o n f i g u r a t i o n Assume t h e r e e x i s t s an i n t e g e r r < n and n v e c t o r s xn,... x eft such t h a t 1' n 1 P.. = a;D?. = (x. -x.)' (x. - *-) V i , j e -B i j 1 J i J i 3 (4) For the most g e n e r a l model (Model I I I ) , t h e r e are a l t o g e t h e r n + r (n + 3) + 2 parameters where n i s the number o f brands and r i s the number o f dimensions i n the j o i n t space. Even as few as e i g h t brands and two dimensions produce 32 p a r a m e t e r s . Ten brands and f i v e dimensions y i e l d a s t a g g e r i n g 77 p a r a m e t e r s . (5) For n o t a t i o n a l c o n v e n i e n c e , we s h a l l l e t <o s t a n d f o r { V k = l (6) And s e t t i n g p = 2 i n the d i s t a n c e f o r m u l a D... 174 That i s , we assume that the brand-switching p r o b a b i l i t i e s P^j are proportional to the square of inter-brand distances in the j o i n t space. This, of course, w i l l not hold exactly but w i l l allow us to derive an i n i t i a l set of values for X. The P^j are known. The coordinate matrix X = [x^, -x ] i s wanted up to a s i m i l a r i t y transform (rotation of the configuration about the o r i g i n , t r a n s l a t i o n of the o r i g i n , r e f l e c t i o n ) since such a transform preserves 2 proportionality between the . and D^j . Without loss of generality, we l e t a be equal to one and l e t the center of gravity of the 3C.. be the o r i g i n . It can be shown (Torgerson [1958]) that the scalar product between two vectors x- and x, , denoted by e^j, = Xj x^ ., depends only on the inter-point distances as follows: 1 (6.4) e j k = - 7 1 n 1 n 1 P., - - E P., - -. E P. + — 9 E P jk n e = 1 ck n e f t l je- n 2 ^ e Let E = [©j^] D e the matrix of scalar products between the unknown x.. Since the P.- are known, the matrix E can be computed from (6.4). The wanted i n i t i a l configuration can be obtained by factoring the known matrix E = X'X. Since E i s equal to the product X'X, E i s a posit i v e semi-d e f i n i t e matrix of rank r. As a r e s u l t , E can be written as 1:75 (6.5) E = Z' Z, 0 where the y's are the '...eigenvalues and Z i s the matrix of _ej.genve_cto.rs-associated with the matrix E. The d e s i r e d i n i t i a l c o n f i g u r a t i o n i s j u s t (6.6) X o _ /T 0 Searching the parameter space. From the s t a r t i n g values X°, Y°, A° and 0 ° , a chi-square value may be computed from ( 6 . 3 ) . Suppose, however, that t h i s chi-square value i s high ; that i s , the f i t between observed and p r e d i c t e d brand-switching proba-b i l i t i e s i s poor. The next step c o n s i s t s of f i n d i n g a new c o n f i g u r a t i o n X"*" and Y"*" that w i l l produce a b e t t e r match between observed and p r e d i c t e d p r o b a b i l i t i e s , keeping the other parameters (;to, A and 9) constant . From the new c o n f i g u r a t i o n X^ " and Y^, a g r a d i e n t search w i l l then be performed to o b t a i n improved values f o r !ur, A and 0. We f i r s t deal w i t h the e s t i m a t i o n of X"*" and Y"*". 176 Improving the configuration The objective i s to move the points (brands and ideal) around so that the new configuration yields a better match between observed and predicted p r o b a b i l i t i e s . In p a r t i c u l a r , consider a s p e c i f i c brand i and i t s r e l a t i o n -ship to each of the brands j in turn. We would l i k e to move brand i in the perceptual space so as to decrease the average discrepancy between the observed ( t ^ j ) and pre-dicted (Pjj) p r o b a b i l i t i e s . If P.. i s larger than t . . , we could move brand 13 6 13 i away from j by an amount which i s proportional to the size of the discrepancy. Conversely, i f t ^ j i s larger than P^, then brand 3 i s to be moved towards i by an amount proportional to the discrepancy. Let a represent the c o e f f i c i e n t of proportionality. To f i n d a new coordinate x^, for brand i on l k dimension k, as related to brand j , we can use the formula: 1 P- • <6-7) x i k ( j ) = x t k ( j ) + 01 ^ ~ try) C * j k - x i k ) -The above formula would move brand i in the appro-priate d i r e c t i o n with respect to brand j , but we must con-sider a l l (n - 1) brands insofar as th e i r effect on brand i i s concerned. When this i s done, equation (6.7) gener-a l i z e s into 177 J - 1 I J J That i s , we move brand i along dimension k in such a way as to take into account the discrepancies i n -volving a l l other brands. This i s , of course, done for a l l brands i n a l l dimensions and repeated u n t i l ' successive configurations converge. A similar procedure i s resorted to for the ideal point in order to select that set of coordinates that yields the best f i t between observed and predicted brand choice p r o b a b i l i t i e s . In p a r t i c u l a r , consider a s p e c i f i c brand, say brand i . We would l i k e to move the ideal point in the j o i n t space so as to decrease the discrepancy between the observed (m^t) and the predicted [g (cL t , X j ] brand choice p r o b a b i l i t i e s for brand i . 7 If m t^ i s larger than g(.) we could move the ideal point towards brand i by an amount that i s proportional to the size of the discrepancy. Conversely, i f g(.) i s larger than the ideal point i s to be moved away from brand i by an amount proportional to the discrepancy. (7) To ease, notation, we w i l l write g(.) instead of g C d , : ' , A 7 . 178 S i n c e a c h a n g e i n t h e i d e a l p o i n t l o c a t i o n a f f e c t s t h e e n t i r e s e t o f d i s t a n c e s b e t w e e n t h e b r a n d s a n d t h e i d e a l p o i n t , we w o u l d l i k e t o move t h e i d e a l p o i n t so as t o d e c r e a s e t h e a v e r a g e d i s c r e p a n c y b e t w e e n t h e o b s e r v e d a n d p r e d i c t e d b r a n d c h o i c e p r o b a b i l i t i e s . To do t h i s , we m e r e l y u s e t h e e x p r e s s i o n : y\t - ( i - f e e * . - *?k). w h e r e g r e p r e s e n t s t h e c o e f f i c i e n t o f p r o p o r -t i o n a l i t y ^ . T h i s i s , o f c o u r s e , d o n e f o r e a c h d i m e n s i o n a n d r e p e a t e d u n t i l t h e c o o r d i n a t e s o f t h e i d e a l p o i n t c o n v e r g e t o some s t a b l e v a l u e . S e a r c h i n g t h e p a r a m e t e r s p a c e f o r d i , A a n d e Once t h e b e s t c o n f i g u r a t i o n a f f o r d e d by t h e t e m p o r a r y p a r a m e t e r v e c t o r s <1) ° , A 0 a n d 0 ° i s o b t a i n e d , t h e n e x t s t e p c o n s i s t s o f s e a r c h i n g t h e p a r a m e t e r s p a c e f o r i m p r o v e d p a r a m e t e r s v e c t o r s X^ a n d 0 1 t h a t w i l l f u r t h e r r e d u c e t h e c h i - s q u a r e g o o d n e s s - o f - f i t s t a t i s t i c . g T h i s i s b e s t d o n e b y a g r a d i e n t p r o c e d u r e . (8 ) B o t h c o e f f i c i e n t s o f p r o p o r t i o n a l i t y (a a n d g) m u s t be s p e c i f i e d b y t h e u s e r . (9) The g r a d i e n t c o d e u s e d i n t h i s s t u d y was d e v e l o p e d b y BOX [ 1 9 6 5 ] . 179 The procedure then s t a r t s a f r e s h w i t h the l a s t c o n f i g u r a t i o n X"*", and the improved parameter v e c t o r s u>^~, X"*"and S"*" u n t i l the a b s o l u t e d i f f e r e n c e s between the valu e s of each parame-t e r f o r any two s u c c e s s i v e i t e r a t i o n s s a t i s f y some prespe-c i f i e d t o l e r a n c e l e v e l . S e l e c t i n g the Number of Dimensions i n the J o i n t Space. The goodness of f i t c h i - s q u a r e s t a t i s t i c p r o v i d e s v a l u a b l e i n f o r m a t i o n to decide whether or not to add an e x t r a dimension i n the j o i n t space. To s e l e c t the number of dimensions, one would keep up adding dimensions ( s t a r t i n g from a u n i d i m e n s i o n a l space) u n t i l the c h i -square s t a t i s t i c f a l l s below some p r e s p e c i f i e d t o l e r a n c e l e v e l . I f we l e t X2T denote the ch i - s q u a r e s t a t i s t i c and d f ( r ) denote the number of degrees of freedom asso-c i a t e d w i t h the model assuming a space of r dimensions, the s t o p p i n g r u l e can be formulated as f o l l o w s : (10) I t i s tempting to r e s o r t to an F - t e s t to decide whether the l o s s of degrees of freedom i n c u r r e d by adding an e x t r a dimension i s matched by a s i g n i f i c a n t r e d u c t i o n i n the c h i - s q u a r e s t a t i s t i c . T h i s temptation should,however, be r e s i s t e d s i n c e 180 S t a r t i n g from r = l , choose the f i r s t r f o r which ( 6 - 1 0 ) * 2 r <. x j f d f ( r ) , where a i s the p r e s p e c i f i e d t o l e r a n c e l e v e l . T h i s s t o p p i n g r u l e achieves a reasonable compromise between parsimony and the d e s i r e to achieve a good f i t between the d e r i v e d j o i n t space and the observed brand-switching p r o b a b i l i t i e s . 6.4 An E m p i r i c a l I l l u s t r a t i o n . To i l l u s t r a t e the use of brand-switching data to c o n s t r u c t j o i n t space c o n f i g u r a t i o n s , the s o f t d r i n k data d e s c r i b e d i n the p r e c e d i n g chapter have been submitted to the m u l t i d i m e n s i o n a l procedure j u s t developed. Three d i f -f e r e n t v e r s i o n s of the model were e n t e r t a i n e d . They a r e : (6.11) P.. = a.b. g(d..1^) g ( d . 2 , X ) h(D. ., d.^S ) the c h i - s q u a r e s o b t a i n e d a t s u c c e s s i v e i t e r a t i o n s are not independent (they are based on the same se t of d a t a ) . Lack of independence between the c h i - s q u a r e s combined wi t h the f a c t t h a t the F - t e s t i s very s e n s i t i v e to departure from independence p r e c l u d e s i t s use as a c r i t e r i o n t o decide on the space d i m e n s i o n a l i t y . 1-81 where MODEL I g ( d i t , M = m . t t = 1, 2 h C D . . , d . t , <S) - exp [9. ( 6 . . - D . . ) ] MODEL II d " t X S ( d i t ' A ) = T-571 t * 1 7 h ( D i j } d . t , 6) = exp [ 9 . (6.. - D..)] MODEL III g ( d u , i t t = 1, 2 1.2 h ( D i j 5 d . t , 9) exp [yS.. ^ - 9 . D. . ] j 3 ? The reader i s referred to the preceding chapter for a detailed description of and j u s t i f i c a t i o n for the functional forms spe c i f i e d for the functions g and h. The three models (I, II, and III) correspond to models 1.4, II and III.3 of chapter V. Model I i s appropriate for deriving a configuration without ideal points. The output of such a model consists of a set of coordinates for each brand. The other two models produce a configuration where both brands and ideal points (one for each time period) are simultaneously positioned. 182 Analysis and Discussion The results of the analysis are conveniently sum-marized in Tables VI. I to ". VI. 3 . Table VI. 1 exhibits for each model the coordinates of each brand for each dimension, the coordinates of the average consumers' ideal point for each time period, the estimates of the remaining parameters as well as some s t a t i s t i c s required for testing purposes. As in the preceding chapter, 0^ indicates the extent to which the observed brand-switching p r o b a b i l i t i e s are consistent with the derived j o i n t space. Two brands, LIKE and FRESCA were assigned n e g l i g i b l e and/or negative values for the parameter 0. In other words, consumers' switching behavior to and from LIKE and FRESCA was not consistent with the joint-space theory of stochastic choice. This lack of consistency, however, did not prevent a reasonably f a i r recovery of the consumers' perceptual map. As can be checked from figure VI.1 (map derived from brand-switching p r o b a b i l i t i e s ) and figure V . l of the previous chapter (map derived from judged s i m i l a r i t y mea-sures) , the two maps are in close agreement. In both con-figurations, the major dimension can be interpreted as calories content of the brands: diet versus non diet. The Parameter Estimates f o r Three Models - Joint Space Configuration Derived from Brand-Switching Data Soft Drink Experiment MODEL I BRANDS COKE 7-UP TAB LIKE PEPSI SPRITE D-PEPSI FRESCA 0 i .169 .228 .072 .024 .180 .214 .111 .008 Dimension I Dimension II 9.72 5.09 -10.42 -5.57 6.72 1.85 -6.73 -2.55 -.49 -.59 .36 -6.13 4.68 1.31 6.69 -5.89 0 i ,161 ,170 .144 .029 .161 .237 .145 ,038 MODEL II Dimension I Dimension II 9.59 5.35 -10.22 -5.85 6.77 1.76 -6.72 -2.57 -.46 -.61 .38 -6.14 4.73 1.29 6.70 -5.97 e. .145 .193 .055 .044 .151 .177 .113 .017 MODEL I I I Dimension I Dimension II 9.46 4.91 -11.47 -2.20 5.85 1.04 -7.38 -.08 2.24 .92 .07 -8.58 5.72 2.15 5.50 -9.47 Dimension Weights Ideal Point Coordinates Time 1 Time 2 2 32.23 -;732 2.84 8^54 8.67 43.22 .018 7.46 13.87 -.963 1.98 8.37 7.86 7.72 42.68 .101 8.07 14.29 Degrees of freedom 25 32 31 MULTI-DIMENSIONAL SCALING OF THE SOFT DRINK EXPERIMENT BRAND-SWITCHING MATRIX. MODEL I. D-PEPSI TAB II PEPSI SPRITE LIKE FRESCA 7-UP COKE FIGURE VI.1 oo 185 second dimension i s c l e a r l y flavor type X J" : lemon-lime versus cola. Table VI.2 exhibits the corr e l a t i o n matrix of inter-brand distances across the 28 pairs for various models. The magnitude of the corr e l a t i o n c o e f f i c i e n t s (.87 for MODEL I) indicates a f a i r l y good degree of agreement between the two configurations. There i s almost perfect c o r r e l a t i o n between the two maps (brand-switching versus judged s i m i l a r i t i e s ) for the f i r s t dimension (.98 for Model I) and moderate agreement for the second dimen-sion (~ .70) as evidenced in Table VI.3. There are differences, however. Based on the indivi d u a l s ' rating, the two diet brands Tab and Diet-Pep^si are perceived to be much more similar than they are on the basis of the brand-switching data. From the indivi d u a l s ' rating, we would think that consumers of Tab would switch to Diet-Pepsi, Like and Fresca respec-t i v e l y , should they be denied th e i r favorite brand. However, when faced with actual choice decisions, i t appears that they would rather switch to Like (36%), 7-Up (12%), or even to Coke, Pepsi and Fresca (8% each). Only 8% of them did switch to Diet-Pepsi. In the percep-tual map based on brand-switching data (see figure VI.2), (1 1) Notwithstanding COKE and SPRITE whose pos i t i o n i s somewhat in disagreement with the flavor interpre-tation . TABLE VI.2 CORRELATION MATRIX OF PERCEPTUAL DISTANCES (D..) ACROSS THE 2 8 PAIRS FOR VARIOUS MODELS MODEL II MODEL III Lehmann & * Pessemier MODEL I .999 .943 .870 MODEL II .940 .873 MODEL III .750 * Based on the perceptual distances obtained from Lehmann & Pessemier [1973]. TABLE VI.3 CORRELATION MATRIX OF THE COORDINATE VECTORS FOR VARIOUS MODELS (DIMENSION BY DIMENSION) MODEL II MODEL III Lehmann & Pessemier DIMENSION 1 2 1 2 1 2 1 . 999 .119 .972 . 352 .980 T.015 MODEL I 2 .126 . 999 -.0837 .947 .138 .687 1 .128 . 968 . 361 .982 -.008 MODEL II 2 -.082 .968 .1405 .690 1 .140 .932 -.160 MODEL III 2 .393 .671 18 7 7-Up i s closer to Coke than Pepsi i s , an unexpected result given that consumers of Coke switched to Pepsi more often than to any other brands and vice-versa. However, the predicted perceptual distance between Coke and Pepsi does not depend solely on the observed amount of switching between the two brands but also on that observed between a l l pairs of brands. The d i f f e r e n t i a l weights ( see the uj^ i n the expression cL^) assigned to each dimension reinforce the conclusion that the f i r s t dimension (diet versus non-diet) was the most determinant factor for brand choice purposes: dimension I was assigned a weight of 8.37 as against .101 for the second dimension. This result indicates that the cognitive process underlying the brand choice decisions of the subjects i s e s s e n t i a l l y unidimensional. This conclusion, however"; may not apply to the soft drink market as a whole, due to the r e s t r i c t i v e experimental setting used by Lehmann & Pessemier [1973] (laboratory experiment, student population, free consumption, e t c . . . ) . F i n a l l y , a most inter e s t i n g finding from this analysis of brand-switching p r o b a b i l i t i e s relates to the discrepancy observed for two brands, COKE and PEPSI between the perceptual distances as derived from judged s i m i l a r i t y measures on the one hand, and brand-switching p r o b a b i l i t i e s 188 on the other. As can be checked by a glance at figures V . l , VI.1 and table V.3a, the brand-switching a c t i v i t y between COKE and PEPSI observed from the subjects' brand choice over time was less dramatic than expected on the basis of thei r perceived s i m i l a r i t y . In other words, there should have been more "switching" between COKE and PEPSI for the j o i n t space theory of brand choice to hold exactly. While the subjects do perceive COKE and PEPSI as being two very similar soft drink brands, they do not consider them as being perfect substitutes for brand choice purposes. This finding bears far-reaching implications for brand positioning and advertising strategy. As a case in point, consider the advertising campaign launched by the Pepsi Cola company during the spring of 1977. The chief vehicle for this advertising campaign was a brochure urging the reader to perform a "blind t e s t " . The instructions read as follows: i) have somebody f i l l up two i d e n t i c a l glasses with Pepsi-Cola and Coca-Cola respectively while you are watching away; i i ) take a sip at the two glasses and decide which i s which. 189 In case the reader had trouble drawing the " r i g h t " conclusion, the brochure marveled at how d i f f i c u l t i t was to t e l l the difference and d i s c r e e t l y suggested that since this was so, the reader might as well drink Pepsi-Cola instead of Coca-Cola. The rationale underlying Pepsi-Cola Inc's advertising strategy rests on the i n t u i t i v e l y appealing theory that consumers tend to organize their choice behavior so as to achieve a cognitive equilibrium between their perceptions and brand choice. According to this theory a consumer who perceives two brands as being v i r t u a l -l y i d e n t i c a l would exhibit no p a r t i c u l a r preference for one or the other and thus would tend to act as though they were perfect substitutes. The analysis conducted i n this chapter, however, shows what appears to be diminishing returns on "psychic investment". In this context, psychic investment denotes a l l marketing expenses incurred in the process of a l t e r i n g the consumer perceptual space. As the perceptual distance between two brands gets small, one observes that further decreases in perceptual distances do not y i e l d propor-tionate increases in brand-switching p r o b a b i l i t i e s . At this stage, further psychic investment to a l t e r the consumers' perceptual space may y i e l d diminishing returns 190 in that decreases in perceptual distances are not matched by substantial increases in brand-switching p r o b a b i l i t i e s . As a result of these diminishing returns, the economic value of the Pepsi-Cola positioning campaign i s open to question. The results obtained in this study suggest that i t w i l l take more than a mere psychic investment to trans-from a Coke "addict" into a Pepsi zealot since perceptions and preferences are not the only determinants of brand choice behavior. 191 SUMMARY This chapter d e a l t with the problem of d e r i v i n g a j o i n t - s p a c e c o n f i g u r a t i o n form brand-switching proba-b i l i t i e s r a t h e r than the more widely used judged s i m i l a r i t y or p r e f e r e n c e measures. The shortcomings a s s o c i a t e d w i t h a p p l y i n g t r a d i t i o n a l MDS techniques to brand-switching p r o b a b i l i t i e s have been s t r e s s e d and a more a p p r o p r i a t e procedure d i s c u s s e d and i l l u s t r a t e d w i t h e m p i r i c a l data. This procedure s a t i s f i e s a l l c o n s t r a i n t s f o r l o g i c a l c o n s i s t e n c y a r i s i n g f o r the p a r t i c u l a r nature of the input data. Using brand-switching matrices as a b a s i s f o r p e r c e p t u a l mapping i s not without p e r i l . For example, brand-switching data r e q u i r e a sample of i n d i v i d u a l s or measures of an i n d i v i d u a l over time, and the i m p l i c i t assumption of homogeneity of p e r c e p t i o n over the group of i n d i v i d u a l s or time p e r i o d . Yet, these problems e x i s t i n any a p p l i c a t i o n o f p e r c e p t u a l mapping that i s not i n d i v i d -u a l l y based. L a s t , p e r c e p t u a l mapping from brand-choice data should not be co n s i d e r e d as a s u b s t i t u t e f o r p e r c e p t u a l mapping from judged s i m i l a r i t y measures, but r a t h e r as a u s e f u l complement. Comparison of the two maps d e r i v e d from p e r c e p t i o n s on the one hand and a c t u a l brand choice on 192 the other may give the brand manager valuable hints for brand positioning purposes. 193 'CHAPTER VII CONCLUSION The purpose of this research was to analyse, discuss and extend the analytic methodology associated with the stu-dy of brand-switching data by developing stochastic models to be used as constructs for organizing and interpreting brand-switching data. Models for both two-brand and multi-brand markets were constructed. The approach taken in this study was that of descrip-tion leading to generalization. In chapter II, a set of con-sumer brand switching data were submitted to a s t a t i s t i c a l analysis in an attempt to uncover possible " r e g u l a r i t i e s " in the data. These " r e g u l a r i t i e s " were then exploited in chapter III to construct brand choice models based on the results of the preliminary s t a t i s t i c a l analysis. The s t a t i s t i c a l analy-si s gave support to the learning hypothesis of brand-switching behavior in that past purchases of a brand affect the proba-b i l i t y of that brand being bought again in the future. While the need to develop alternative models to accommodate the adaptive behavior observed in consumer brand choice was not obvious, the model (Polya-Learning) developed in chapter II exhibits some interesting properties: 1 9 4 i); I t does not assume that the adaptive process i s l i n e a r i n the purchase p r o b a b i l i t i e s , i i ) The l i m i t i n g form of the Polya d i s t r i b u t i o n i s the s o - c a l l e d negative b i n o m i a l d i s t r i b u t i o n which i s the v backbone o f Bass' [19740 brand c h o i c e and Ehrenberg's [1972] purchase i n c i d e n c e model, i i i ) The model i n t r o d u c e s a f i n e d i s t i n c t i o n between d i s l o y a l b e havior and mere bra n d - s w i t c h i n g . T h i s d i s t i n c t i o n c a r r i e s much importance f o r the brand manager who would l i k e to know the extent to which brand-switching i s evidence o f d i s l o y a l behavior or mere^variety seeking. I f only f o r those c h a r a c t e r i s t i c s , the model c o u l d be c o n s i d e r e d as a v i a b l e a l t e r n a t i v e to e x i s t i n g two-brand s t o c h a s t i c choice models f o r the study of consumer brand c h o i -ce behavior. The d i s s e r t a t i o n ' s major c o n t r i b u t i o n i s contained i n chapters IV to VI which developed a general c l a s s of brand-s w i t c h i n g .models that acknowledge both the s t o c h a s t i c and d e t e r m i n i s t i c f e a t u r e s of the br a n d - s w i t c h i n g phenomenon. This approach stands at va r i a n c e with that of p r e v i o u s l y developed m u l t i - b r a n d s w i t c h i n g models (see e.g. H e r n i t e r [1973], Bass [1974] t h a t focused e x c l u s i v e l y on the p r o b a b i -l i s t i c components o f consumer bran d - s w i t c h i n g behavior. 195 A multi-dimensionally scaled configuration was used as a s p e c i f i c a t i o n of consumers' cognitive.structure. Percep-tual distances derived from this configuration were then related to brand choice and brand-switching p r o b a b i l i t i e s through a mo-del that took into account the constraints imposed on the various p r o b a b i l i t i e s . The empirical results demonstrated that perceptions, preferences and cognitive' structure were indeed s i g n i f i c a n t determinants of consumer habitual brand-switching behavior. F i n a l l y , chapter ¥ 1 1 dealt with the problem of deriving a joint-space configuration from brand-switching p r o b a b i l i t i e s rather than the more widely used judged s i m i l a -r i t y or preference measures. Comparison of the two maps derived from perceptions on the one hand and actual brand choice on the other was shown to provide the brand manager with valuable hints for brand positioning purposes. As with many other studies of consumer choice beha-vi o r , this d i s s e r t a t i o n raised more questions than i t solved, leaving open numerous possible avenues for future research. Among the topics i t i s important to explore are: i) Replication of the study with di f f e r e n t sets of data, i i ) Replication of the study at the individual l e v e l . 196 i i i ) Finding ways to discriminate between brand-switching as a result of mere variety seeking versus brand-switching as evidence of disloyal behavior. The conceptual framework of analysis developed in this study can handle aggregate as well as dis-aggregate input. The purpose of chapter V was to explain an aggregate brand-switching matrix with an aggregate perceptual map. Our under-standing of brand-switching behavior would be further enri-ched by dealing with each individual's brand-switching matrix and perceptual map in an attempt to identify different types of consumer behavior. While the constructing of perceptual maps at the individual level poses no conceptual nor opera-tional difficulties, data constraints severely limit the ga-thering of a brand-switching matrix at the individual level when the number of alternative brands is large, as in the soft" drink experiment. For an n-brand market, there are 2 n possible brand-switching probabilities to " f i l l " in with actual data. Given that one needs a minimum of 5 observa-tions per cell to have any confidence in the statistical tests 2 to be used, no fewer than 5n purchase occasions are required. In the soft drink experiment where a choice of eight brands was offered to the subjects, one would need a whopping 320 purchase occasions to carry out the analysis at the indivi-dual level, an unrealistic number for all practical purposes. 197 Traditional brand choice experiments, such as Bass et a l . [1972] and MacConnell [1968] beer experiments, w i l l y i e l d between 10 to 30 purchase occasions. The alternative i s to collapse the market into subsets of brands or collapse the individuals with the same pattern of purchase into homoge-neous groups. The l a t t e r procedure i s more appealing than the former that would involve comparing individuals with dif f e r e n t "evoked" sets of brands. Clustering techniques such as the ones suggested by Johnson [1967] or Lehmann and Pessemier [1973] could be employed to form homogeneous groups of con-sumers with respect to thei r purchase behavior. 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Weiss (1976), "Stochastic Response Models - Pers-pective and Comment", Proceedings of the Annual Conference of the Marketing Division of the Canadian Association of  Administrative Sciences, University Laval, Quebec. 207 APPENDIX A MOTIVATION FOR THE SIMPLIFYING ASSUMPTION USED IN CHAPTER IV (EQUATION 4.5) Let P^ = Proportion of consumers purchasing brands i and j on two successive occasions. A = [A_^j ] = n x n s i m i l a r i t y matrix whose ty p i c a l element i s A ^ j . B = {1,..., n} = a set of n brands. Assume A.. P. . = f. . (A) for some function f. . , i , i e B 1 13 13 v 1 13 ' ' J A- A.. § 0, E A.. ?0, 2 A.. ?0, V i , j e B 2 13 j 13 ' i 13 J A, f-. (A) = 0 i f f A.. = 0 V i , j A 4 f i ; j (A +• AE k l) - f±. (A) = g i j (A) V k, 1 / i , j e B k, ' l v* i , j •where E ^ i s a n x n matrix whose only non-zero element is the (kl)th. 208 Assumption Al postulates the existence of a func-t i o n a l relationship between the proportion P'^ . and a similar-i t y matrix A. By introducing this assumption, we need not worry about A provided i t s a t i s f i e s A2. In A2, we require that a l l entries be non-negative and for any fixed i , that A.j^0 for at least one j , j = l , . . . , n , and s i m i l a r l y for fixed j , that A^^O for at least one i , i=l,...,n. Thus, i f A. i s the i t h row of A, A2 states that A. i s not the zero vector. This rather inconsequential assumption i s used to simplify the analysis. Assumption A4 i s of c r i t i c a l impor-tance. It states that the change in the proportion of consumers switching from brand i to brand j due to a change A in the s i m i l a r i t y measure A-^ of some pair of brands k, l ^ i , j i s a function of A but not of k and 1. Proposition If P.. = f..(A) for some function f.. and some matrix A = [A^^], i , j = 1 .n, s a t i s f y i n g assumptions A 3 through A4, then ( A i j f {a i},{B j}j Where T'a ^ } = (a • _ > • • • » j e B fiy = (v..., v = E w. A . . i e B for somew. :> 0 and V;. 5 0 and not a l l zero. 1 ~l 209 Proof Consider the set S = {A: A^^ i s constant and E w^  A^j = , E v. A- . = a- , for some w- >, 0, v. >, 0 j J i j i ' 1 J and not a l l zero, i , j = 1,..., n} The non-negativity r e s t r i c t i o n on the weights w^  and V j together with assumption A2 ensure that the elements of the sets {a^}^_^ and ( B j } ^ = 1 are themselves non-negative and not a l l i d e n t i c a l l y zero. Let A, A e S and A t A component-wise. Then i t w i l l be shown that f ^ (A) = (A) from which i t may be concluded that f ^ (A) is a function only of A^j and the sets {a^}^, Let A 0 = min (A, A) taken component-wise and E^. be a matrix whose only non-zero element i s the ( i j ) t h one,""i.e. E k l ^0 i f k f i and/or 1_ f j 1 k = i and 1 = j Then i f b i s defined as the smallest non-zero o element of the two matrices : 2 1 0 (A - A°, A - A°), g some i , j , k, 1 such that we can define A 1 = A0•+ b°E.., A 1 § A I1 = A 0 + b° E k l , I1 i X where either A"!". = A. . or A?", - A, , I J i j k l k l By assumption A4 (A 1) = (A 1). Now define b"^  as the minimum non-zero element of (A - A 1, A - t1) and form A - A + b 1 E.. for some i , j , A s A —2 = 1 1 = 2 = A = A + b E k l for some k, 1 , A $ A Again, by A^ f n (A 2) = f n ( f 2 ) . — -k Since the number of zero elements of (A - A . = =k A - A ) increases by at least one at each i t e r a t i o n of this procedure, and f l l ( A k ) = £ 1 1 ( A K ) V K ' we have: A k = A and ? = I for k ^ n 2 - 1. 211-Thus (A) = f.^ (A) as required, establishing the claim that the function h.. (A) i s constant over the i j " set S and hence depends only upon the quantities A^^, {a^} and {Bj}. So we can write: 

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