A...Ase Incidence Decisions

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Page 1

The “Shopping Basket”: A Model f or

Multicategory Purchase Incidence Decisions

Puneet Manchanda • Asim Ansari • Sunil Gupta

Graduate S chool of Business, Universit y o f C hicago, C hicago, Illinois 60637, [email protected]

Graduate School o f Business, C olumbia U niversity, N ew York, N ew Y ork 10027, [email protected]

Graduate School o f Business, C olumbia U niversity, N ew York, N ew Y ork 10027, [email protected]

AbstractConsumers make multicategory decisions in a variety of con-

texts such as choice of multiple categories during a shopping

trip or mail-order purchasing. T he choice of one category

may af f ect the selection of another category due to the com-

plement ary nat ure (e.g., cake mix and cake f rosting) of th e

two categories. Alternatively, two categories may co-occur in

a shopping basket not because they are complement ary but

because of similar purchase cycles (e.g., beer and diapers) or

because of a host of other unobserved f actors. While com-

plement arity gives managers some cont rol ov er consumers’

buying behavior ( e.g., a change in t he pr ice of cake mix could

change the purchase probability of cake f rosting), co-

occurrence or co-incidence is less controllable. Other f actors

that may af f ect multi-category choice may be (unobserved)

household pref erences or (observed) household demograph-

ics. We also argue th at no t accounting for t hese three f actors

simultaneously could lead to erroneous inf erences. We t hen

develop a conceptual framework t hat incorpo rates comple-

mentarity, co-incidence and heterogeneity (both observed

and unobserved) as the factors that could lead to multi-

category choice.

We th en translate this framework into a model of multi-

category choice. Our model is based on random utility theory

and allows f or simultaneous, interdependent choice of many

items. This model, the multivariate probit model, is imple-

mented in a Hierarchical Bayes framework. The hierarchy

consists of three levels. The first level captures the cho ice of

items f or the shopping basket during a shopping trip. T he

second level captures differences across households and the

third level specif ies the priors for t he unknown parameters.We generalize some recent advances in Markov chain Mon te

Carlo met hods in order to estimat e t he model. Specif ically,

we use a substitution sampler which incorporates techniques

such as the Metropolis Hit-and-Run algorithm and the Gibbs

Sampler.

Th e model is estimated on four categories (cake mix, cake

af f ect purchase incidence in related product categories. In

general, t he cross-price and cross-promotion ef f ects are

smaller th an th e own-price and own-promot ions ef f ects. T he

cross-ef f ects are also asymmetric across pairs of categories,i.e., related category pairs may be characterized as having a

“primary” and a “secondary” category. T hus these results

provide a more complet e description of the ef f ects of pro-

mot ional changes by examining them both within and across

categories. T he co-incidence results show the ext ent of th e

relationship between categories that arises f rom uncontrol-

lable and unobserved factors. These results are usef ul since

they provide insights int o a general structure of dependence

relationships across categories. T he het erogeneity results

show that observed demographic factors such as f amily size

inf luence the int rinsic category preference of households.

Larger f amily sizes also tend to make households more price

sensitive f or both the primary and secondary categories. We

f ind that price sensitivities across categories are not highlycorrelated at t he household level. We also f ind some evi-

dence that int rinsic preferences for cake mix and cake frost-

ing are more closely related than p ref erences f or f abric de-

tergent and f abric softener.

We compare our model with a series of null models using

both estimat ion and holdout samples. We show that both

complementarity and co-incidence play a signif icant role in

predicting multicategory choice. We also show how many

single-category models used in conjunction may not be good

predictors of joint choice.

Our results are likely to be of interest t o ret ailers and man-

uf acturers trying to opt imize pricing and promotion st rate-

gies across many cat egories as well as in designing micro-

marketing strat egies. We illustrat e some of these benef its by

carrying out an analysis which shows that t he “ true” impact

of complementarity and co-incidence on prof itability is sig-

nif icant in a retail sett ing. Our model can also be applied to

other domains. The combination of item interdependence

and individual household level estimates may be of part ic-

ular interest to database marketers in building customized



0732-2399/99/1802/0095$05.00C op yr ig ht 1 99 9, I nst i tu te f or Op er at i on s R esea rch

and the Management Sciences

MARKETING ScIENcE/Vol. 18, No. 2, 1999

pp. 95–114

f rosting, f abric detergent and fabric softener) using multica-

tegory panel data. T he results disentangle th e complemen-

tarity and co-incidence effects. The comp lementarity results

show that pricing and promotional changes in one category

“cross-selling” strategies in the direct mail and f inancial ser-

vice industries.

( M ulticategor y Models; Shopping Baskets; Retailin g ; M icromar-

keting ; M ultivariate Probit Model ; Hierarchical Ba yes M odels)

Page 2

THE “SHOPPING BA SKET”: A MODEL FOR MULTICATEGORY

PURCHASE INCIDENCE DECISIONS

1. IntroductionConsumers make multicategory decisions in a variety

of contexts such as grocery shopping trips , mail-order

purchasing, and financial portfolio choice. A common

thread that underlies these multicategory choice sce-

narios is that the chosen items may be related to each

other in some manner e.g., a pair of shoes may be or-

dered to go along with a particular dress. In the re-

tailing context, these multicategory decisions result in

the f ormation of consumers’ “shopping baskets”

which comprise the collection of categories that con-

sumers purchase on a specific shopping trip.1 Retailers

are interested in understanding the composition of the

shopping basket in terms of the cross-category depen-

dencies among the purchased categories. Industry ex-

perts have suggested that t he identification of “substi-

tute” and “complementary” categories is critical to

developing an understanding of how t hese shopping

baskets are put t ogether (Terbreek 1993). Besides the

cross-category f ocus, retail managers are also inter-

ested in implementing micromarketing programs

through the use of individual household level data

(Rossi et al. 1996; Food Marketing Institute 1994;

Mogelonsky 1994). A combination of cross-category in-

sights and individual household-level pref erences

could potentially lead to retailers’ partit ioning the

overall port f olio of retail categories into smaller inde-

pendent sub-p ortfolios, e.g., dairy products and clean-

ing products may be managed as independent sub-

port f olios. This would t hen allow retailers to make

pricing and promotion decisions to op timize prof its

across these sub-port folios for each individual

household.

The insights developed from cross-category analysis

coupled with micromarketing are also of interest tomanuf acturers and database marketers. M anuf actur-

ers, such as P & G, who market brands in related cate-

gories, could utilize these insights to rationalize mar-

keting expenditures across two or more categories,

e.g., the toothpaste and toothbrush categories. Data-

ty pically use knowledge about a sp ecif ic consumer’s

intrinsic pref erences or p urchasing patterns in one

category to induce purchase in the other, e.g., Ameri-

can Express cards may send a customer a coupon f or

golf clothing after observing a purchase of golf clubs.

In sp ite of significant industry interest in under-

standing cross-category purchase behavior of consum-

ers, most research in marketing has been on single

category choice decisions (see Meyer and Kahn 1991).

This research has focused on independent analyses of

consumer purchasing activities within categories with-

out explicitly capturing the interdependence across

categories. It is only recently t hat researchers have

started examining purchases across multiple catego-

ries (Russell et al. 1997).

A common industry practice to study cross-category

purchasing behavior is to cross-tabulate joint pur-

chases across multiple categories. However, this ap-

proach cannot distinguish among the possible mech-

anisms leading to joint purchasing activity. The choice

of one category may affect the selection of another cate-

gory due to the complementary nature (e.g., cake mix

and cake f rosting) of the two categories. Alternatively,

two categories may co-occur in a shopp ing basket not

because they are complementary but because of similar

purchase cycles (e.g., beer and diapers) or because of

a host of other unobserved factors (e.g., consumer

movement through a store). While complementarity

allows managers some control over consumers’ buying

behavior (e.g., a change in the price of cake mix could

change the p urchase probability of cake f rosting), co-

occurrence or co-incidence is less controllable.

In this research, we develop a framework f or inves-

tigating multicategory purchase decisions in the gro-

cery shop ping context. This framework incorporatesthree crucial elements that could inf luence these

choices—complementarity, co-incidence and hetero-

geneity. In more specific terms, this paper has three

objectives. The first objective is to develop a general

model of multicategory choice that explicitly allows for



96 MARKETING ScIENcE/Vol. 18, No. 2, 1999

base marketers are interested in discovering and con-

verting cross-category dependencies to targeted cross-

selling programs (Berry 1994). These p rograms

1We use the accepted notion o f “category” as def ined by managers

and data providers such as IRI and Nielsen.

dependence across the chosen categories, separates theef f ects of controllable drivers of multicategory choice

f rom the uncontrollable drivers, and accounts f or

household sp ecific preferences for each category. The

second objective is to test the model in the context of

consumers’ shopping baskets. The f inal objective is to

Page 3

MANCHANDA, ANSARI, AND GUPTA

The “Shopping Bask et”

provide retailers and manufacturers insights into t he

nature of the relationship between categories.

We translate our f ramework into a Hierarchical

Bayes model of multicategory purchases. Specif ically,

we use a multivariate p robit model and estimate it us -

ing Markov chain M onte Carlo techniques such as t he

Metropolis-Hast ings Algorithm and the Gibbs Sam-

pler. The pap er, therefore, makes both subs tantive and

methodological contributions.

The structure of the p aper is as follows. We review

the previous work in this area in §2. Section 3 deals

with t he concept ual background which motivates our

modeling approach. Section 4 describes the model

structure. In §5 we use our approach to develop the

model of the shopping basket. Specif ically, we discuss

the data, model specification, variable operationaliza-

tion, estimation procedure, the results and the com-

parison with null models in this section. Section 6 dis-

cusses the managerial implications of our f indings. We

conclude the pap er with directions for f uture research

in §7.

2. Previous ResearchPrevious modeling research in the multicategory do-

main can be classif ied into three streams.2 The f irst

stream of research examines a phenomenon of interest

(say, promotion elasticity) across many categories and

“meta-analyzes” the results to obtain empirical gen-

eralizations. One of the earliest papers in this s tream

is by Fader and Lodish (1990) who showed that certain

consumer characteristics such as household penetra-

tion and f requency of purchase tend to explain the

pricing and promotional environment in retail stores

at the category level. An extension of this research by

Narasimhan et al. (1996) showed that promotionalelasticity of a category depends on category structure

and consumer characteristics. An earlier p aper by Ra ju

(1992) examined the variability in category sales and

2Mention must al so b e made of other related work i n an experimen-

linked it to both category characteristics (e.g., bulki-

ness) and marketing variables (e.g., promotion). Fi-

nally, Hoch et al. (1995) relate the store-level price elas-

ticities of various categories to the demographics of

consumers in the trading areas of the stores. Although

this stream of research generalizes across many cate-

gories, it ignores interdependence in consumers’ pur-

chases across multiple categories—which is the f ocus

of our research.

The second stream of research examines how a de-

pendent variable of interest (e.g., store choice) is af -

f ected by multicategory variables. For example, Belland Lattin (1998) examine how store choice is af f ected

by t he pricing strategy of a store across many catego-

ries. Though they do not specifically model depen-

dence across categories, they allow for a multicategory

f lavor through the use of a composite variable (total

amount spent on a shopping trip). Similarly, Bodapati

(1996) looks at the impact of feature advertising across

many categories on consumers’ store choice decision.

Again, this st ream of research does not explicitly cap-

ture int erdependence in consumer p urchase behavior

across categories.

The third s tream of research ex plicitlyallows f or de-

pendency across multicategory items. Walters (1991)

and Mulhern and Leone (1991) model store sales us ing

regression methods and show that sales of brands in

one category are affected by the p ricing and promotion

of brands in a predefined “complementary” category.

Chintagunta and Haldar (1998) examine purchase tim-

ing across predefined complement and subst itute cate-

gory pairs and show that the nature of the relationship

between category pairs influences interpurchase times.

Manchanda and Gup ta (1997) examine purchase inci-

dence and brand choice for pairs of categories by cre-

ating composite basket alternatives. A ma jor limitation

of these approaches is that, while they account f or com-

plementarity, they ignore either co-incidence or het-

erogeneity or both. As we will highlight in the next

section, this omission may lead to erroneous



MARKETING ScIENcE/Vol. 18, No. 2, 1999 97

tal setting that examines behavior across multiple categories. Menon

and Kahn (19 95) examine variety seeking behavior and sho w that

changes i n the cho ice context (a different category) can af f ect behav-

ior in t he target category. Ratneshwar et al. (1996) show that, given

goal directed behavior, consumers form consideration sets across

prod uct cat ego ries .

inf erences.

The f ocus of our paper is t o build a general f rame-

work that incorporates aspects of complementarity, co-

incidence and heterogeneity in modeling multicate-

gory consumer purchase decisions.

Page 4



Table 1 Co-incidence and Complementar ity Scenarios

Complementarity

3. Conceptual BackgroundA pair of categories can be bought together on a shop-

ping trip by a variety of reasons:

1. Purchase Complementarity or Cross-Ef f ects:

Two categories may be complementary. In such a sit -

uation, we can expect the marketing activity (price or

promotion) in one category to influence consumers’

purchase of the other category. For example, a p rice

discount on beer may cause the consumer to buy beer

as well as pot ato chips. We define two categories as

purchase complements if marketing actions (price and

promotion) in one category influence the purchase de-

cision in the other category.3 We do this to separate the

ef f ects of controllable and uncontrollable f actors. Price

and promotion are within managers’ control while

most of the other factors are either not controllable

(e.g., consumer heterogeneity ) or not observed by t he

researcher.

2. Co-incidence: We define co-incidence as the set of

all reasons except purchase complementarity and con-

sumer heterogeneity that could induce joint purchase

of items across categories. For example, consumers

may shop f or many items on the same shopping trip

because of economic reasons, (e.g., consumers canspread the cost of a trip over many items), or owing to

habit (Kahn and Schmittlein 1992). Similarly, other un-

observed f actors such as consumer knowledge of a

particular supermarket, t ime pressure on the consumer

during the t rip (Park et al. 1989) or consumer mood

(Donovan et al. 1994) may influence joint purchasing.

The physical environment of the store may also induce

joint purchases (Swinyard 1993). Co-incidence ref ers to

all the “residual” reasons that could cause joint inci-

dence af ter accounting for the impact of marketing

variables and heterogeneity.

3. Heterogeneity: Finally, consumer heterogeneity

may also result in aggregate data that shows high joint

purchasing between two categories. Accounting f or

heterogeneity may be crucial in understanding the true

nature of the association across categories. We p ostu-

late that households differ in their intrinsic utilities and

price responses for p urchase incidence in each cate-gory and that these utilities and responses may be re-

lated across categories. We also allow f or the f act that

household demographics may explain some p art of the

relationship among the intrinsic household ut ilities

and price responses.

It is crucial to account for the impact of both mar-

keting variables (i.e., cross-effect or complementarity)

and the unobserved factors (co-incidence) on cross-

category purchasing activity. Ignoring either of these

sets of f actors could lead to erroneous inf erences. To

illustrate this, we examine the relationship between a

pair of categories on both dimensions. In Table 1, on

the complementarity dimension, purchases between a

pair of categories may be negatively related (e.g., two

categories may be substitutes or “negative comple-

ments”), independent, or positively related (e.g., price

promotion in one category may increase the probabil-

ity of purchase incidence in the other category). Simi-

larly, a pair of categories may be related positively on

the co-incidence dimension, i.e., they may be bought

together due to similar purchase cycles, they may be

independent, or they may be negatively related. The

observed joint occurrences of p urchases in a pair of

categories is the net effect of these tw o f actors. For ex-

ample, when both co-incidence and complementarity

are positive, the observed data will show very st rong

evidence of joint purchases (cell 9). However, when

two categories have posit ive complementarity but neg-

ative co-incidence (cell 3) or vice versa (cell 7), it is not

clear whether the net effect will be positive, zero, or

negative.

In general, two ty pes of misleading inf erences may

arise f rom ignoring either co-incidence or complemen-

tarity . In the f irst type, we might inf er complementar-

ity or co-incidence when it was not actually p resent.




Co-incidence 0

1 2

/3

04

05

6

/7

8 9

3 No te th at th is d ef initio n applies to two categories being purchased

together as di stinct from two categories being consumed together.

We do not model consumption complementarity as we do n ot ob-

serve it.

Page 5



This could happen if cell 8 represents the “true” model

but we erroneously infer cell 6 to be the represented

behavior. In the second case, cell 3 may represent the

true model, but if the “” complementarity and “”

co-incidence ef f ects cancel out, we may erroneously

conclude that cell 5 is the correct sp ecification. In ot her

words, we may infer complementarity when it is not

there (f irst case) or we may infer no complementarity

when it truly exists (second case).

4. Modeling ApproachWe build our model in three stages. In the f irst stage

we model complementarity (or cross-eff ects) and co-

incidence f or a single household. In the second stage,

we account f or heterogeneity across households by

specif ying a population distribut ion over the house-

hold specif ic parameters. In the third stage, we specif y

priors over the parameters of the population

distribution.

4.1. Stage 1

We observe each household, h, making purchase inci-

dence decisions across a set of J categories on a store

trip t . These decisions can be represented by a vector

y ht { yh1t , yh2t ,..., yhJt }, of binary dependent variables.

In keeping with a random utility formulation, we

model this observed behavior in terms of latent ut ilities

f or the categories. Specifically, the underlying utilities

f or the J categories can be written as

u β β Own ef f ectsh1t h10 h11

β Cross ef f ectsh12 h1t

u β β Own ef f ectsh2t h20 h21

β Cross ef f ectsh22 h2t

M

u β β Own ef f ectshJ t hJ 0 hJ 1

β Cross ef f ects .hJ 2 hJ t

The link between the observed behavior and the latent

impact of purchase complementarity. The coef f icients

f or the cross-effects contained in βh j 2 represent t he

change in purchase utility of category j due to t he mar-

keting actions of other categories. The direct impact of

marketing actions of the same category is captured by

the coef f icients of the own effects βh j 1.

The utility equations for household h on store trip t

f or the J categories can be compactly represented as:

u X β , (1)ht ht h ht

where the jth row of the matrix X ht contains the causal

variables inf luencing the utility for the jth category and

βh

{βh0

, βh1

, βh2

} contains the household specif iccoef f icients.

The unobserved influences operating at the shop -

ping trip are represented by ht { h1t , h2t ,..., hJt }. As

these unobserved factors may be common across cate-

gories, we assume that

MVN [0, Σ], (2)ht

where Σ is a J J covariance matrix. This correlated

error structure of the purchase utilities captures co-

incidence. Thus if cov (hit , hjt ) 0, then an increase

in the purchase utility of category i will lead to an in-

crease in t he purchase utility of category j. In other

words, the error correlations capture t he linkages be-

tween the uncontrollable factors t hat drive joint

purchases.4

The above f ormulation results in a multivariate

probit model (Greene 1997, Chib and Greenberg 1998).

The multivariate p robit model is particularly suited f or

our investigation as it allows for more than one cate-

gory to be purchased simultaneously. This is an ex-

tremely important feature since it allows us to model

both the size of the basket (how many items) and the

composition of the basket (which items). It is important

to not e that the multivariate p robit model is distinct

f rom the multinomial probit model (McCulloch and

Rossi 1994) which only allows the choice of one alter-

native f rom a set of mutually exclusive alternatives.

4 Not e that the co-in cidence term arises from a combin atio n of these




utility f or any category j can be represented as f ollows:

1, if u 0,hjt

yhjt

0, if u 0.hjt

In this utility specification, the cross-ef f ects capture the

residual reasons as well as biases du e to model misspecif ication

(such as omitted variables or not accounting f or consumer consid-

eration sets). This must be kept in mind whenever an attempt is

made to draw inf erences on the basis of the error correlation terms.

We thank the Area Editor for bringing this t o our attention.

Page 6



The binary nature of the observed dependent vari-

ables necessitates the introduction of approp riate scal-

ing constants to ensure identifiability of model param-

eters. In the multivariate probit model, the components

of the observed incidence profile, yht , remain unaltered

if each of the corresponding latent ut ilities in uht is

multiplied by a (dif ferent) positive constant. We can

theref ore f ix the scale of the utilities by dividing each

utility by its corresponding standard deviation,

thereby yielding an identified set of parameters. The

identif ication restrictions transform Σ to a correlation

matrix. The signs of the correlations in Σ p rovide in-

sights about t he qualitative nature of co-incidence. The

magnitudes provide a measure of the strength of the

impact of unobserved factors in inducing joint p ur-

chasing activity . In our subsequent discussion we will

retain the same set of symbols, ( β , Σ and u), to denote

identif ied parameters and ut ilities.5

4.2. Stage 2

In the s econd stage we model the observed and un-

observed sources of heterogeneity across households

by specif ying a p opulation (mixing) distribution over

the household specific coefficients. The household

level parameters can therefore be formulated as a lin-

ear f unction of demographic variables, i.e.,

β D µ λ , h 1 to H , (3)h h h

where βh {βh0, βh1,βh2}, λ h M V N [0,Λ], and Dh is a

matrix containing the demographic and other house-

hold specif ic variables.6 The vector, l, captures the im-

pact of demographic variables and thus provides in-

sights about the observed sources of heterogeneity

across households. The J dimensional vector, λ h { λh1,

λh2,..., λhJ }, represents the unobserved sources of het-

erogeneity across households. The variance-covariance

matrix, Λ, represents the residual covariation in the

household specif ic parameters after having ad justed

f or the variation due to observed sources of

heterogeneity.

The multivariate p robit model as sp ecif ied above is

empirically intractable. Greene (1997, p p. 911–912)

points out that models with more than three dimen-

sions are extremely difficult to estimate as it is hard

(computationally) t o evaluate the high order multivar-

iate normal integrals that are required in specif ying the

likelihood. Another source of difficulty arises in en-

suring a positive definite correlation matrix f or the er-

rors. We circumvent the need for high dimensional in-

tegration by using numerical Bayesian approaches that

build upon recent advances in Markov chain Monte

Carlo (MCM C) methods (Chib and Greenberg 1998,

Dey and Chen 1996). We use a hierarchical Bayesian

app roach, which is necessary for modeling observed

and unobserved sources of consumer heterogeneity .

Some examples of hierarchical models in the market-

ing literature include Allenby and Ginter (1995), Ross i

et al. (1996) and Ansari et al. (1996). Since the Bayesian

app roach requires priors over unknown quantities in

the model, the third stage of our model specif ies these

priors.

4.3. Stage 3

We provide a brief description of the priors in this sec-

tion (details are given in appendix A1). We specif y in-

dependent and diffuse (but proper) priors over the

pop ulation level parameters, Σ, µ, and K 1. Specif i-

cally, the prior f or µ is multivariate normal, MVN[h,

Θ], and the prior for t he precision matrix, Λ 1, is Wis-

hart, W [ ρ,( ρR) 1]. Finally, we assume that the nonre-

dundant correlation parameters, vec*(Σ) (σ 12,

σ 13,...σ ( J 1) J ), of the Σ matrix come from a truncated

multivariate normal population with a f ixed mean and

covariance matrix (Chib and Greenberg 1998).

4.4. Inf erence Procedure

The p robability of observing an incidence prof ile, yht

{ yh1t ,..., yhJt}, on a single observation is given by

Pr(Y y | β , Σ) ...ht ht h s1 sJ

1 1 1

exp R d , (4)ht ht ht 1/2

2(2π ) det(Σ)




5Further details on the identification for each set of parameters areavailable f rom the authors.

6This is the most general specification of heterogeneity. In our ap-

pl icati on, we impose a speci fic struct ure on the Λ matrix in order t o

reduce model complexity. This is discussed later.

where ht uht X ht βh, and(, 0), if y 0,hjt

S (5) j

(0, ), if y 1.hjt

The likelihood f or the household h is given by

Page 7



nh

Pr( y | β , Σ), (6)ht h

t 1

and the unconditional likelihood for an arbitrary

household is obtained as

nh

. . . Pr(y | β , Σ) f (β )d β . (7)ht h h h

t 1

This likelihood is clearly very complicated as it in-

volves computation of high order multidimensional

integrals making classical inference based on maxi-

mum likelihood methods difficult. In the Bayesian

f ramework, inf erence about t he unknown p arameters

is based on their joint post erior distribution. We use

simulation based methods to make many random

draws f rom this posterior distribution. Inf erence is

then based on the empirical distribution of this sample

of draws. MCMC methods are used to simulate the

draws. Specif ically, we use subst itution sampling (a

combination of the Gibbs sampler and the Metropolis

Hastings Hit-and-Run Algorithm) in tandem with data

augmentation (Albert and Chib 1993) to obtain a sam-

ple of parameter draws from the joint posterior

distribution.

Substitution sampling replaces one complicated

draw f rom the joint posterior with a sequence of rela-

tively simple draws from easy to sample f ull condi-

tional distributions of parameter blocks (Appendix A2

details t he f ull conditional distributions f or our

model). Since the f ull conditional dist ribution f or some

parameter blocks is not known in our app lication, we

cannot use a Gibbs sampling step for those p arameters.

We theref ore replace it by a Metropolis-Hastings step

(Hastings 1970). We generalize methods reported in

Chib and Greenberg (1998) and Dey and Chen (1996)

to p rovide an MCM C algorithm for a hierarchical

Bayes multivariate probit specification. In the context

of the model developed in the p revious section, the (m

1)th step of the substitut ion sampling algorithm in-

volves generating the following draws:

a. Generate βh draws from | Σ(m), µ(m),(m 1) (m)

p(β {{u }},h ht

Λ(m)) f or h 1 to H .(m 1)

d. Generate uht draws from p(uhjt | Σ(m), yht ) for (m 1)

βh

h 1 to H , t 1 to nh.

e. Generate a Σ matrix from p(Σ(m 1) | ,(m 1)

{{u }}ht

Σ0, G0) using a Metropolis-Hastings Hit-(m 1)

{β },h

and-Run step.

This sequence of draws generates a M arkov chain

whose stationary distribution is the joint posterior den-

sity of all unknowns. The initial output f rom the chain

ref lects a transient or “burn-in” period in which the

chain has not converged to the equilibrium distribu-

tion and is t herefore discarded. A sample of draws ob-

tained af ter convergence is used to make posterior in-

f erences about model parameters and other quantities

of interest.

5. The Shopping BasketIn this section we describe the modeling of the shop-

ping basket in the context of the prop osed approach.

We describe the data, the model specif ication and vari-

able operationalization, null models, and the est ima-

tion p rocedure. We then discuss the results of the pro-

posed model and compare these with the results f rom

the null models. We conclude the section with details

on model prediction.

5.1. Data

The data were drawn from a multicategory dataset

made available by A. C. Nielsen. The data span a pe-

riod of 120 weeks from January 1993 to March 1995

and are f rom a large metropolitan market area in the

western United States.

Four grocery categories were used f or this applica-

tion—Laundry Detergent (Det), Fabric Sof tener (Sof t),

Cake Mix (Cake) and Cake Frosting (Frost ). The cate-

gories were chosen to illustrate some of the f eatures of

our prop osed model and to check the f ace validity of

some of our results. For example, we expect t hat

detergent-sof tener and cake mix-frosting pairs to be

(purchase) complements while the remaining pairs are

likely t o be (purchase) independent.

Households that had a minimum of two purchases




b. Generate a µ draw from p(µ(m 1) | Λ(m), h,{β },h

Θ).

c. Generate a Λ 1 draw from p(Λ 1(m 1) |(m 1)

{β },h

{µ(m 1)}, ρ, R).

in each of the f our categories during the 120 week pe-riod were selected. This resulted in a sample of 205

households. We then randomly split this sample into

two p arts. The first p art consisted of 155 households

Page 8



Table 2 Frequencies of Joint (Pairwise) Incidence

Det Sof t Cake Frost

Det 1201 275 40 24

(estimation sample) and the second part consisted of

50 households (p rediction sample).

The tot al number of p urchase occasions f or house-

holds in the estimation sample was 17,389. Out of these

at least one category was p urchased on 3,414 occasions

and no purchase was made on the remaining 13,975

occasions. The number of pairwise purchases (f or all

six category p airs) across the 3,414 occasions is detailed

in Table 2. An examination of this table shows high

dependence between detergent and sof tener and be-

tween cake mix and frosting and little dependence

across the other pairs. The purpose of our model is not

only to validate these obvious pairings but to also as-

sess t he relative effects of complementarity (cross-

ef f ects), co-incidence, and heterogeneity.

5.2. Model Specification and Variable Def inition

The deterministic component of the utility f or category

j f or household h at time t can be written as

u Constant Own Ef f ectshjt hjt hjt

Cross Ef f ects . (8)hjt

Direct Impact or Own Effects: These ef f ects f or cate-

gory j are specif ied as7

Own Ef f ects β * Price γ * Promo . (9)hjt h1 j hjt 1 j hjt

The Price and Promotion variables were constructed

as f ollows:

7There is a tradition of using household inventory as an explanatory

variable to predict purchase incidence in t he marketing lit erature

(e.g., Bucklin and Gupt a 1992). However, the use of this variable

necessitates some strong assumptions such as the use of a linear,

constant rate of consumption and raises concerns about endogeneity.

We theref ore do not in clude the inventory variable in the model

specif ication reported here though we estimated models with and

without i nventory variables.

Price: We used a “category price” variable (in $/Oz.)

f or each household on each shopping trip. T his was

done by constructing the category price f or each house-

hold as a weighted average price of brands where the

weights were the share of each brand bought by each

household.8

Promotion: A brand was considered to be on pro-

motion if it was either featured or displayed. The “cate-

gory p romotion” variable was const ructed in a similar

manner to that of the category p rice variable. This re-

sulted in a variable that took a value between 0 and 1.

Some descriptive statistics for the f our categories are

given in Tables 3a–3c.

Complementarity or Cross Ef f ects: We model thecross-ef f ects of all other categories on category j using

a similar set of variables.

Cross Ef f ects β * Pricehjkt h2 jk hkt

k

γ * Promo , (10)1 jk hkt

k

where k 1,..., J and k j.

Heterogeneity: The full model has 36 variables. A

completely unstructured representation of heteroge-

neity over the coefficients for these variables would

necessitate estimating 666 unique parameters of the 36

36 covariance matrix, Λ. In order to reduce model

complexity, we use a structured representation of het-

erogeneity . Specifically, we assume that the promotion

8Since our model is not based on an axiomatic f ramework, it does

not prescribe a “ correct” category price operationalization. In other

words, a weighted average household l evel price is jus t one of many

possi ble al ternat ives. We cho se th is o perati onalizat ion since, (a) it

allows f or hou sehold level differences by constructing a price based

on a household’ s past purchases and (b) it has been used in past

research (e.g., Krishnamurthi and Raj 1988 ). We also esti mated mod-

els with dif f erent operationalizations, e.g., price paid was used ascategory p rice when a brand was p urchased and a weighted average

otherwise. We f ound no directional differences among the results

and the estimated price elasticities were marginally dif f erent in mag-

nitude. Our modeling approach is in contrast to so me other model

f ormulations where a specific error structure results in a “ category”

level variable e.g., the Category Value term in a nested logit setting




Sof t 593 27 11

Cake 415 610

Frost 219

when the errors are distributed i.i.d Gumbel. In our f utu re researchwe plan to in vestigate model structures that do no t allow ad ho c

def initions of the price variable while allowing f or f lexible error

structures.

Page 9



Table 3a Summary Statistics

Category

Mean # o f

Purchases

# of

Brands

Mean Price

($/Oz.)

Mean

Promotion

Det 10 30 0.064 0.05

Sof t 6 7 0.054 0.06

Cake 8 4 0.063 0.19

Frost 6 4 0.103 0.10

Table 3b Price Correlation Across Cat egori es


Det 1.00 0.20 0.03 0.09

Sof t 1.00 0.09 0.11

Cake 1.00 0.15

Frost 1.00

Table 3c Promotion Correlat ion Across Categori es


Det 1.00 0.04 0.01 0.03

Sof t 1.00 0.04 0.05

Cake 1.00 0.22

Frost 1.00

variables have coefficients that are invariant across

households.9

We capture observed sources of heterogeneity by al-lowing household-specific base ut ilities, βh0, the own

price coef f icients, βh1, and the cross price coef f icients,

βh2, to be a f unction of demographic variables. We use

two demographic variables, Family Size (FS) and Total

Trips (TT). We hypothesize that larger households

(with more children) may consume cake mix and cake

f rosting more of ten and may consume larger quantities

of laundry p roducts—this would be reflected in more

f requent purchases of these categories. The mean f am-

ily size in t he data was 3.2 and t he range was f rom 1

to 8. Similarly, there is some evidence that households

who shop more often are different in their response

characteristics than those that shop less of ten (Ainslie

and Rossi 1998).10 We aggregated the total number of

shopping trips (across a large number of categories)

f or each household to construct t his variable. The av-

erage number of trips for these households in our data

was 139 with a range from 26 to 317. Theref ore, the

household specif ic parameters can be written as

f ollows:

β µ µ * FS µ * TT λ , (11)h0 j 01 j 02 j h 03 j h h0 j



where λh0 j, λh1 j and λh2 j are the error terms that repre-

sent the residual unobserved heterogeneity af ter ac-

counting f or observed heterogeneity .

The above specif ication of the ut ility equations f or

the J categories can be summarized as:

u β X β X β Z c , (14)ht h0 h1t h1 h2t h2 ht ht

β D l k , (15)h0 h 0 h0

β D l k , (16)h1 h 1 h1

9Our f inding across many model s pecifications was that ow n and

cross promotions ef f ects were less important than the ow n and cross

price ef f ects. Hence, in the interest of model parsimony, we chose to

keep these coef f icients, c, invariant across households . The prior f or

c is multi variate no rmal, MVN [g, C ]. c draws are included as a st ep

β D µ k , (17)h2 h 2 h2

where ht N (0, Σ), k h0 N (0, Λ0), k h1 N (0, Λ1), and

k h2 N (0, Λ2). The matrix X h1t is a J J matrix con-

taining the own price variables for the J categories, the

matrix X h2 t is a J J matrix containing the cross price

variables, and Z ht contains the own and cross promo-

tion variables.

5.3. Null Models

We compare our model with three null models (note

that all null models incorporate heterogeneity ):

Null Model 1 (Independent Probits): The simplestnull model is one that accounts for neither co-incidence

nor complementarity. This implies that the determin-

istic part of the model contains only own ef f ects. This

model also assumes that the errors are drawn f rom a

multivariate normal distribution with 0 mean and an




in th e substi tution sampler described earlier. More details are given

in Appendices 1 and 2.

10We thank one of the reviewers for bringing t his to our attention.

identity variance-covariance matrix. This is identical toestimating a series of J independent hierarchical probit

models, one f or each category.

Page 10



Null Model 2 (Independent Probits with Cross-

Ef f ects): In the next model, we add complementarity

to null model 1 but st ill do not account f or co-

incidence. This is achieved by adding the cross price

and promotion terms to the deterministic part of the

utility.

Null Model 3 (Multivariate Probit with Co-

Incidence): We then add co-incidence to null model 1

but do not add cross effects. This is achieved by allow-

ing the variance-covariance matrix of the error terms

to be an unrestricted correlation matrix.

This selection of null models would help in under-

standing the ef f ects of complementarity and co-

incidence alone as well as together.

5.4. Estimation

The models were estimated using C programs devel-

oped by the authors. As explained in § 4, repeated

draws were made from the series of full conditionals

to arrive at the joint posterior density of the unknown

quantities using the M CMC sampling scheme. The

substitution sampler was run for 50,000 iterations f or

the proposed model and null models. Convergence

was ensured by monitoring the t ime-series of the

draws f rom the full conditional distributions (cf .

Allenby and Ginter 1995). As indicated earlier, initial

iterations ref lect a “burn-in” period where the chain

may not have converged. We chose a burn-in length

of 45,000 iterations. Therefore, we retained only the last

5,000 draws f rom the posterior distributions f or inf er-

ence purposes. Since sequential draws f rom the joint

posterior may be highly correlated, we resort t o “thin-

ning the chain” (Geyer 1992, Raftery and Lewis 1995)

whereby we retain every fifth draw from the undis-

carded part of the chain. Therefore, 1,000 draws f rom

the posterior distribution of each parameter were re-

tained and used to make inferences.

5.5. Results

5.5.1. Proposed Model . Results from the pro-

posed model are given in Tables 4a–4f.11 As is usual

11As mentioned earlier, we estimated models w ith and without own

and cross inventory v ariables. The inclusion or exclusion of these

f or Bayesian inference, we summarize the posterior

distribution of the parameters by reporting the poste-

rior mean and 95% p robability interval.12

Own Ef f ects: The own price ef f ects (Table 4a) are in

the expected direction. The household specif ic price co-

ef f icients are negative and significant f or all f our cate-

gories, i.e., a reduction in category price increases the

probability of incidence for that particular category.13

There is a signif icant, p ositive effect of promot ions for

laundry detergent and fabric softener but not f or cake

mix and cake f rosting (Table 4b). This suggests that

perhaps cake mix and frosting purchases are more

planned than the purchases of detergent and sof tener

even though the promotion frequency f or these cate-

gories is higher than that of detergent and sof tener.Cross-Ef f ects: We first estimated a model where the

cross ef f ects of all three other categories were included

f or a p articular category. However, as expected, we

f ound that t he only significant set of cross-ef f ects were

f or detergent-softener and cake mix-cake f rosting

pairs. No cross-effects were found across any other

pair of categories. In the interest of reduced model

complexity and faster convergence we set the nonsig-

nif icant cross-effects to zero and reestimated the

model.14 The results from t his model are reported in

Tables 4a–4b.

Four interesting results emerge f rom these tables.

First, we f ind that all four cross p rice ef f ects, i.e., soft-

ener on detergent, detergent on sof tener, cake f rosting

on cake mix and cake mix on cake frost ing, are nega-

tive and significant. In other words, a decrease in cake

f rosting price increases the p robability of buying cake

ef f ects were negative and significant for all f our categories. The cross

inventory coef f icients were positi ve and signif icant f or the detergent-

so f tener and cake mix-frosting pairs. This indicates that consump-

tion complementarity may also play a role in indu cing joi nt pur-

chases.

12In our subsequent di scussion w e refer to a variable as “ not si gnif i-

cant” if 0 is included in the probability int erval of its coef f icient.

13We report the mean (across all hou seholds) and the range f or the

household specific price parameters (both own and cross) in Table

4a. More details on the βh parameters are provid ed when we dis cuss




variables did not af f ect the valu e of the ot her parameters in th e

model. For models t hat included the inv entory variables, we f ound

results that were consistent with our expectation. Own inventory

heterogeneity.14There was no sig nificant difference between the est imates f rom the

f ull and the reduced model.

Page 11



Table 4a Direct Impact and Complementarity (Pr ice)

Parameter Estimates

Incidence Category

Price of Det Soft Cake Frost

Det 25.75a

(30.98, 10.30)b

7.20a

(14.90, 0.21)b

Sof t 3.28a

(8.08, 0.36)b

10.81a

(20.42, 6.82)b

Cake 17.16a

(30.96, 10.30)b

13.42a

(29.59, 0.88)b

Frost 8.10a

(18.32, 1.86)b

11.70a

(20.89, 5.75)b

aMean across all hous eholds.

bRange across all households .

mix. The implications of this finding are signif icant f or

retailers and managers who manage brands across

categories. Retailers now have an additional marketing

variable that they can use to influence category inci-

dence and basket composition while managers can op-

timize promotion budgets across brands in dif f erent

categories. Second, the coefficients of the cross-price

ef f ect are smaller than the coefficients of the own-price

ef f ect. Third, we see a strong asymmetry in cross-

price ef f ects, e.g., the effect of a change in detergent

price on sof tener purchase is much larger than t he

other way around. Since the comparison of the “raw”

price coef f icients provides a rough estimate of the

cross-price ef f ects, we defer a detailed discussion of

this asymmetry to a later section when we discuss own

and cross-p rice elasticities. Fourth, although we ex-

pected the sign of the cross-promotion coef f icients to

be positive among complements, we find no signif icant

ef f ects except f or the effect of detergent promotion on

sof tener purchase. Once again, it shows the asymmetry

between the ef f ects of promotion across the two

categories.

Co-incidence: The co-incidence pattern as described

by t he correlation matrix of errors (Table 4c) shows

some interesting results. As expected, the correlation

coef f icients between cake mix and cake f rosting and

other p airs. The difference between the magnitudes of

the two correlations mentioned above could be be-

cause cake mix and cake frosting are purchased to-

gether more of ten in our data set than detergent and

sof tener. This might be due to t he f act that cake mix

and cake f rosting have higher consumption comple-

mentarity than detergent and soft ener. Also, as men-

tioned previously, the nonsignificant ef f ects of pro-

motions f or cake mix and cake frosting suggest that

perhaps the purchase of these two categories is

planned leading to high co-incidence. Of the other f our

category pairs, t wo of t he correlation coef f icients

(sof tener-cake mix and softener-frosting) are not sig-

nif icantly dif f erent from 0, and the remaining two are

about 0.05 ref lecting a very low level of co-incidence.

Category Specific Intercepts (βh0 j): We f irst discuss

the mean estimates of the category sp ecif ic intercept s

(Table 4d). We find that larger families tend to increase

the mean utility of purchase for the cake mix and f rost-

ing categories. This may result from the f act t hat the

primary consumers of prepared cakes in a household

are children. The total number of trips made by each

household do not seem to have a signif icant ef f ect on

the mean purchase utility for any category.

The covariance matrix, Λ0, of the residual mean cate-

gory specif ic utilities p rovides insights into the nature




detergent and sof tener are high (0.92 and 0.46 respec-

tively), when compared to the correlations between

of household heterogeneity across these f our catego-

ries. A transf ormation of the covariance matrix, Λ0,

Page 12



Table 4b Direct Impact and Complementarity (Promotion)

Parameter Estimates and 95% Probability Intervals1

Incidence Category

Promotion of Det Sof t Cake Frost

Det 0.75

(0.53, 0.98)

0.18

(0.02, 0.35)

Sof t 0.02

(0.20, 0.15)

0.35

(0.06, 0.61)

Cake 0.01

(0.13, 0.11)

0.05

(0.15, 0.05)

Frost 0.07

(0.26, 0.08)

0.14

(0.33, 0.10)

1Signif icant estimates are denoted in boldf ace type.

Table 4c Co-inc idence Result s (Correlation Matr ix R)

Parameter Estimates and 95% Probability Intervals

Det Soft Cake Frost

Det 1.00 0.46

(0.44, 0.47)

0.05

(0.01, 0.08)

0.04

(0.01, 0.07)

Sof t 1.00 0.03

(0.01, 0.07)

0.00

(0.02, 0.02)

Cake 1.00 0.92

(0.91, 0.93)

Frost 1.00

into a correlation matrix yields additional insights. The

mean utilities f or detergent and softener have a cor-

relation coef f icient of 0.24 while those of cake mix and

cake f rosting have a correlation coefficient of 0.58. This

suggests that, af ter accounting for f amily size and total

trips , there exist some more household sp ecif ic unob-

served variables that link the utilities of these two pairs

of categories at the individual household level. One pos-

sible hypothesis is that these correlations ref lect the ex-

tent of consumption complementarity amongst the

two p airs of categories. The fact that all the other f our

correlations are near 0 p rovides some additional sup -

port f or this hypothesis. Further, the correlation be-

tween cake mix and frosting is larger than the corre-

lation between detergent and softener, once again

suggesting stronger complementarity between cakemix and f rosting than between detergent and sof tener.

We now examine the effect of demographics on the

own and cross price coefficients (Tables 4e–4f ). We f ind

evidence that price sensitivity is affected by both

household demographics and household shopp ing be-

havior. For own p rice, the coefficients f or f amily size

and total trips are negative and signif icant across all

f our categories. This implies that large f amilies are

more price sensitive and that families that make more

f requent shopp ing trips are also more price sensitive.

Similarly, larger families tend to be more price sensi-

tive as f ar as cross p rice sensitivity is concerned,

though total trips do not seem to af f ect this. These f ind-

ings are consistent with those report ed in Ainslie and

Rossi (1998).

In summary, we find that bot h observed and unob-

served sources of heterogeneity (such as t he demo-

graphic and shopping variables used here) play a part

in explaining household response. Family size seems

to be a better predictor of price response than the total

number of trips (10 out of 12 coeff icients are af f ected

by f amily s ize while 4 out 12 are af f ected by total num-

ber of trips).

Overall, to put these results in the context of Table

1, we f ind that the detergent and softener and cake and

f rosting pairs occupy cell 9 (positive complementarity

and positive co-incidence) while all the other pairs

seem to be f rom cell 5 (independent in both comple-

mentarit and co-incidence).

expression f or determining the incidence elasticities

f or the multivariate probit model. We theref ore use

simulation to make 1,000 draws from a multivariate

normal distribution using the estimated mean β ’s and

the X ’s (e.g., price) and the mean Σ correlation matrix.

The resulting utility measures are used to estimate the

base incidence probabilities. We then make a 10%

change in the specific independent variable and recal-

culate the incidence probabilities in a similar manner.15

We calculate (purchase) incidence elasticities in the

usual manner. The estimated elasticities are reported

in Table 5. The own-price elasticities are dif f erent

15We also estimated the elasticities for a 5% and 15% change in the




5.5.2. Elasticity Analysis. There is no analyt icalindependent v ariable. There was n o appreciable dif f erence in the

estimated elasticities i n all three cases.

Page 13



Ta bl e 4 d Hetero genei ty R es ul ts (βh0 j )

Category Intercepts

βh0 j µ01 j µ02 j * FSh µ03 j * TTh λh0 j

Cat µ01 µ02 µ03

Det 0.69

(0.18, 1.18)

1.06

(0.51, 2.44)

0.04

(0.06, 0.02)

Sof t 1.06

(1.44, 0.57)

1.24

(0.28, 2.76)

0.00

(0.01, 0.02)

Cake 0.32

(0.65, 0.02)

4.40

(3.75, 5.05)

0.05

(0.12, 0.02)

Frost 0.94

(1.30, 0.58)

3.77

(2.66, 4.59)

0.01

(0.02, 0.03)

Ta bl e 4 e Hetero genei ty R es ul ts (βh1 j )

Own P rice Coef f icients


Cat µ 11 µ12 µ13

Det 16.88

(19.67, 10.92)

1.60

(2.97, 1.09)

0.03

(0.05, 0.01)

Sof t 4.62

(5.81, 3.13)

0.34

(0.58, 0.05)

0.04

(0.06, 0.02)

Cake 9.29

(10.92, 7.12)

2.33

(2.61, 2.05)

0.01

(0.03, 0.01)

Frost 2.53

(6.32, 0.32)

1.62

(2.11, 0.96)

0.03

(0.05, 0.02)

Ta bl e 4 f Hetero genei ty R es ul ts (βh2 j )

Cross Price Coef f icients


Cat µ21 µ22 µ23

Det/Sof t1 0.61

(1.33, 0.08)

0.87

(3.81, 0.13)

0.05

(0.23, 0.10)

Sof t/Det 6.04

(11.55, 1.15)

0.44

(0.98, 0.06)

0.01

(0.08, 0.10)

Cake/Frost 1.81

(3.22, 0.49)

2.04

(3.34, 0.61)

0.04

(0.36, 0.29)

Frost/Cake 4.69

(8.25, 0.88)

2.59

(3.52, 0.57)

0.14

(0.16, 0.41)

1An entry of Cat A/Cat B represents t he effect on Cat A p urchase due to a

change in price of Cat B.

Ta bl e 5 Mean Pri ce El as ti ci ti es

Probability of Incidence in Category

Price of Det Sof t Cake Frost

Det 0.40

(0.14)a

0.12

(0.05)

Sof t 0.06

(0.02)

0.70

(0.30)

Cake 0.17

(0.04)

0.15

(0.06)

Frost 0.11

(0.04)

0.21

(0.06)

a Numbers in p arenth eses rep resen t po st erior s tand ard dev iati ons .

across categories and range from 0.17 to 0.70.

Within the two related pairs, detergent and cake mix

have lower own p rice elasticities t han sof tener and

f rosting, respectively.16

As expected, the magnitude of each cross-price elas-

ticity is much lower than the magnitude of the asso-

ciated own-price elasticity . The cross-price elasticities

show an interesting asymmetric patt ern. Price changes

of detergent have a larger effect on sof tener purchase

(cross p rice elast icit y of 0.12) t han t he ot her w ay

around (cross p rice elast icit y of 0.06). Cake mix and

cake f rosting also show a similar pattern where price

changes of cake mix have a slightly large ef f ect on cake

f rosting p urchase (cross p rice elasticity of 0.15) than

vice versa (cross price elasticity of 0.11).

These results suggest t hat in each pair of categories,

there may be one “primary” category that drives p ur-

chase incidence f or the p air. These primary categories

(detergent and cake mix) have lower own-price elas-

ticities, but changes in their prices have a stronger ef -

f ect on t he purchase incidence of the associated “sec-

ondary” category. In other words, detergent and cake

mix prices are stronger drivers of sof tener and cake

f rosting purchases than vice versa. These results seem

to be consist ent with our a priori knowledge.

The promotion elasticities are much smaller in mag-

nitude than the p rice elasticities. The own-promotion

16 No te th at wh en we say on e elast ici ty i s lo wer th an ano ther , we are




ref erring to the absolute magnitudes of the estimates.

Page 14



Ta bl e 6 Pro po sed a nd Nul l Mo del s

Model

Accounts For

Nul l Mod el 1

(Independent

Probits)

Nul l Mod el 2

(Independent

Probits)

Nu ll Mo del 3

(Multivariate

Probit)

Proposed Model

(Multivariate

Probit)

Complementarity No Yes No Yes

Co-incidence No No Yes Yes

He e oge ei Yes Yes Yes Yes

elasticity f or detergent is 0.017 while the own-

promotion elasticity for softener is 0.005. The cross-

promotion elasticity of detergent p romotion on sof t-

ener incidence is 0.003. Again, we find that t he pat tern

of elasticities indicates detergent to be the primarycategory f or this pair.

A ma jor advantage of estimating household specif ic

parameters is that it enables us to examine response

across categories at t he household level. One possible

question could be whether each household is p rice sen-

sitive to a similar extent across all categories. This

question can be partly answered by converting the er-

ror covariance matrices of the own and cross price co-

ef f icients, Λ1 and Λ2, to correlation matrices, and ex-

amining the pairwise correlations. The error

correlations f or own p rice are: Det-Sof t (0.18), Det-

Cake (0.09), Det-Frost (0.08), Soft-Cake (0.09), Sof t-

Frost (0.07), Cake-Frost (0.25). The error correlations

f or cross p rice are: Det-Soft (0.11), Det-Cake (0.04),

Det-Frost (0.06), Soft-Cake (0.03), Sof t-Frost (0.06),

Cake-Frost (0.19).

There are two interesting findings here. First , it

seems that households do not have p rice sensitivities

that are similar across categories (the own and cross

price parameters show four p ositive correlations and

two negative correlations). Second, the magnitude of

the correlations is not very high (with higher correla-

tions f or the detergent-softener and cake mix-f rosting

pairs and lower correlations for the other f our pairs).

This suggests that p rice sensitivities may depend on a

combination of category or category groups and

household characteristics, e.g., a household may have

similar price sensitivity for the cleaning products,

which may be distinct from price sensitivity f or baking

products . Earlier studies have shown dif f ering results.

In the context of brand choice across unrelated cate-

gories, Kim and Srinivasan (1995) show that house-

holds do not seem to have similar p rice sensitivities.

In a s imilar context, however, Ainslie and Rossi (1998)

f ind evidence that household price sensitivities aresomewhat related. We speculate that this mixed bag of

results may arise from differences in the behavior be-

ing modeled (purchase incidence, conditional brand

choice) or the specific method that is used. Further re-

search is needed bef ore any definitive conclusions can

Another advantage of obtaining household level pa-

rameters is t hat these parameters may be translated

into own-p rice and cross-p rice elasticities f or all the

155 households in t he sample. These elasticities could

prove very usef ul in app lications such as databasemarketing.

5.6. Comparison with Null Models

As mentioned earlier, we estimated three null models

(Table 6). The prop osed model is compared with these

null models on price elasticities (Table 7), co-incidence

parameters and prediction using both the estimation

sample (Table 8) and a holdout sample (Table 9).

5.6.1. Price Elasticity and Co-incidence

Comparison.

Price Elasticities: We compare the complementarity

results across models by focusing on the own and

cross-price elasticities. Since simpler model specif ica-

tions (which omit complementarity or co-incidence)

are likely to overstate the effect of price, we expect

price elasticity t o be the highest for null model 1 and

the lowest f or null model 4. Similarly, we expect that

the cross-price elasticities would be higher f or null

model 2 than for the proposed model. Our hypothesis

is that the change in elasticities is likely to be higher

f or the cross price elasticities due to the direct t rade-

of f between complementarity and co-incidence.

The comparison of the elasticities is detailed in Table

7. For t he own p rice elasticities, we observe a drop

(ranging f rom 10% to 24%) in t he magnitudes between

null model 1 and the p roposed model. There app ears

to be no dif f erence between the est imates f rom null

model 1, 2, and 3. Consistent with our hypothesis, the

cross p rice elasticities show a larger drop in magnitude




be drawn.

Page 15



Table 7 Price Elastici ty Comparison Across Models

Price Elasticity Null Model 1 b Nu ll Model 2a Nul l Mod el 3 aProposed Model

Own

Det 0.44

(0.18)e

0.43

(0.17)

0.43

(0.17)

0.40

(0.14)

Sof t 0.78

(0.34)

0.76

(0.34)

0.75

(0.31)

0.70

(0.30)

Cake 0.21

(0.07)

0.21

(0.05)

0.20

(0.06)

0.17

(0.04)

Frost 0.24

(0.07)

0.23

(0.07)

0.23

(0.07)

0.21

(0.06)

Crossa

Det/Sof t 0.08

(0.03)

0.06

(0.02)

Sof t/Det 0.16

(0.07)

0.12

(0.05)

Cake/Frost 0.21

(0.08)

0.11

(0.04)

Frost/Cake 0.24

(0.11)

0.15

(0.06)

aAn entry of Cat A/Cat B represents t he effect on Cat A p urchase due to a

change in price of Cat B.

bIndependent probit models.

cIndependent probit models with complementarity.

d Multivariate probit model with co-incidence.

e Numbers i n paren theses rep resent pos terio r stan dard d eviat ions .

Table 8 Pred icted B askets on Es timation Samp le

Basket

Number

Basket

DescriptionaFrequency

Nul l

Model 1b

Nu ll

Model 2c

Nu ll

Model 3 d

Proposed

Model

1 N 13975 13387 13589 14239 13936

2 D 1201 1383 1319 1221 1250

3 S 593 724 697 612 570

4 C 415 851 643 417 399

5 F 219 596 519 210 207

6 DS 274 104 185 239 261

7 DC 40 93 53 37 38

8 DF 24 66 59 20 28

9 SC 20 62 55 29 35

10 SF 8 37 55 7 6

11 CF 502 61 155 268 535

12 DSC 7 9 15 10 13

13 DSF 3 5 14 3 4

14 DCF 60 7 6 44 64

15 SCF 30 5 24 23 2916 DSCF 18 1 2 10 15

“ Hit Rate” 0.85 0.90 0.96 0.99

a N impli es th at no catego ry is pu rchased . D, S, C, F imply t hat t here was

a purchase i n th e detergent, s oftener, cake mix or f rosting categories re-

spectively.

bIndependent probit models.

cIndependent probit models with complementarity.

d Multivariate probit model wi th co-incidence.

(f rom 33% to 75%) between null model 2 and the pro-

posed model. Interestingly, the pair with the higher co-

incidence (cake mix-frosting) shows a bigger ef f ect in

the drop in cross p rice elasticity magnitude.

Co-incidence: We expect the error t o decrease as the

models increase in s ophist ication and explanatory

power. In other words, we expect that the ef f ect of co-

incidence is reduced as we move from null model 3 to

the proposed model.17 For both t he correlations that

are of interest, i.e., detergent and softener and cake mix

and f rosting, we see some evidence of decline. The cor-

relation f or detergent and soft ener drops f rom 0.51 to

0.46, while f or cake mix and frosting it declines f rom

0.95 to 0.91. These results are consistent with our

expectations.

In conclusion, on the substantive f ront, the proposed

17 No te th at co-i ncidence is not model ed in nu ll model s 1 and 2.

model provides better diagnostics t han the null mod-

els. We f ind that the addition of controllable (price and

promotion) f actors can reduce the extent of co-

incidence f or a category pair (e.g., detergent and

sof tener).18

5.6.2. Model Comparison. We now compare

models by f ocusing on the predictive ability of the

models on both est imation sample of 155 households

as well as a holdout sample of 50 households. We gen-

erate the p robability of incidence f or each category by

using a f requency simulator based on 1,000 draws

f rom the estimated model. We use these category in-

cidence probabilities to generate the proport ions f or

each of the 16 baskets, e.g., detergent incidence only

18In other analyses not reported in this paper, we f ound some evi-

dence that accounting for uncontrollable factors such as unobserved

e e og e ei also e ds o ed ce e size o e e o co ela io .




Page 16



Table 9 Predicted Baskets on Ho ldout Sample

Basket

Number

Basket

Descriptiona Frequency

Nu ll

Model 1b

Nu ll

Model 2c

Nu ll

Model 3 d

Proposed

Model

1 N 3811 3707 3665 3686 3771

2 D 248 208 245 288 169

3 S 139 169 154 106 184

4 C 94 166 141 70 55

5 F 44 168 138 110 105

6 DS 37 12 55 53 47

7 DC 3 12 19 7 4

8 DF 0 12 18 9 7

9 SC 2 8 7 2 5

10 SF 0 8 7 3 4

11 CF 92 8 18 127 107

12 DSC 2 1 4 1 2

13 DSF 0 1 4 1 2

14 DCF 4 1 2 12 9

15 SCF 4 0 4 4 8

16 DSCF 1 0 0 2 3

“ Hit Rate” 0.88 0.90 0.92 0.93

aIndependent probit models.

bIndependent probit models with complementarity.

cMultivariate probit model with co-incidence.

corresponds to basket 2. We then sum up the propor-

tions across all the observations and translate the pro-

port ion into the number of baskets, e.g., an aggregate

prop ortion of 0.86 translates to 3854 baskets (0.88

4481 where 4481 is the total number of baskets). The

predicted baskets f rom each model are detailed in Ta-

ble 8 (est imation sample) and Table 9 (holdout sam-

ple). We then def ine a simple “Hit Rate” metric to com-

pare these aggregate level predictions across models

as f ollows:

Σ | B B |i i pred iact

1 (18) N

where Bi stands f or Basket i and N is the tot al number

of shopping trips. This metric lies between 0 (no pre-

dictive capability ) and 1 (perfect prediction).

As can be seen f rom Table 8, the proposed modelhas a hit rate of about 0.99 while the null models have

hit rates of 0.85, 0.90, and 0.96, respectively.

A more detailed examination of the predictions

shows that t he independent p robits model (null model

1) is poor at p redicting all baskets. To illustrate why

this may be so, let us examine the case of the null bas-

ket. It may be expected that p redicting the null basket

is st raightf orward since 80% of the trips result in the

null basket. However, recreating this null basket f rom

a series of independent models does not account f or

the joint probability of no purchase in each category.Thus, t hough null model 1 performs very well at p re-

dicting no purchase/purchase for each category taken

one at a time,19 it cannot do so when the joint purchase

is being predicted. This is because it ignores the com-

plementarity and co-incidence linkages that drive mul-

ticategory incidence. The addition of complementarity

(null model 2) and co-incidence (null model 3) both

lead to an improvement in the Hit Rate with better

predictions on all baskets. Of the two, complementar-

ity seems to have lower impact than co-incidence in

improving the Hit Rate. The combination of comple-

mentarity and co-incidence (proposed model) shows

considerable improvement in predicting both single

and joint category incidence (e.g., see baskets 4 and 11).

A similar pattern can be seen in the predictions made

by t he prop osed and null models using the holdout

sample (Table 9).

Some researchers have argued that if multi-item

choice models do not make better predictions t han

single-item choice models, the justif ication f or devel-

oping these models is not clear (Russell et al. 1997).

The results f rom the tests carried out above show that,

in the context of our research, there are signif icantgains to be made from such models.

6. Managerial ImplicationsOur model provides many p otential benef its to man-

agers, especially in the retailing context. First, the ex-

amination of the correlation structure can enable re-

tailers t o identif y traffic generating categories, i.e.,

categories which have a high amount of co-incidence

with many other categories but few complementary

ef f ects. Second, retailers can devise better p romotion

policies f or categories that are complementary. Third,

19There are very mino r differences across t he f our models in terms

of predi cti ng s ing le category p urchase. The propos ed model an d nu ll

model 1 make slightly better predictions than either null model 2 o r

null model 3.




Page 17



Table 10 Retail Prof its

Gross

Margin (%)

Nul l

Model 11

Prof it ($)

Nul l

Model 22

Profit ($)

% Increase

over Null

Model 1

Proposed

Model

Prof it ($)

% Increase

over Null

Model 1

20 235.33 243.74 4 259.92 10

30 460.68 480.98 4 507.85 10

40 695.05 723.33 4 766.76 10

50 921.43 966.58 5 1021.68 11

60 1119.81 1166.84 4 1232.89 10

1Independent probit models.

2Independent probit models with complementarity.

the pattern of correlations can help in reducing the en-

tire set of categories into more manageable “chunks”

f or f urther analysis of brand choice, purchase quanti-

ties, and other p henomena of interest. Finally, t he ca-

pability of generating household level estimates allows

retailers to design direct marketing programs such as

cross p romotions (e.g., cross couponing) or higher

across-the-board promotions to more price sensitive

households. Manuf acturers who have brands in com-

plementary categories can also get similar insights

f rom our model.

We demonstrate t he usefulness of our model by as-

sessing the ef f ect of retailer pricing policies under dif -

f erent model assumptions. We assume that the retailer

would like to quantify the effect of price promotions

on prof it. We f urther assume that t he retailer drops

price on each of the f our categories by 10% as his or

her promotional strategy. Since we have no knowledgeof the exact margins in these categories, we look at a

range of gross margins (20% to 60%). These margins

encompass numbers reported in industry publications

such as Progressive Grocer.20 We assume that the total

market consists of t he 155 households that we have

used in our calibration sample. We use the results f rom

null model 1, null model 2 and the p roposed model to

estimate the prof it impact (detailed in Table 10).

The results show that each model gives a dif f erent

estimate of the f inal profit. Looking at the numbers in

detail, we f ind that t he addition of cross-ef f ects (null

20Since the purpose of these analyses is to compare prof its across

models, the absolute numbers are not very important.

model 2) translates into an average increase in esti-

mated prof its of 4% relative to t he prof its estimated

f rom the independent probits model (null model 1).

Adding co-incidence to null model 2, i.e., using the

prop osed model, adds another 6% to the same base,

i.e., the proposed model shows that prof its may be in-

creased by about 10% over that of null model 1. We

should highlight t hat we are not opt imizing prof its,

but simply uncovering the multicategory prof itability

of promotions. If a retailer misjudges the prof it poten-

tial of a promotion, he/she may forego prof itable op-

port unities. For example, a manager may use null

model 1 to evaluate the profitability of a specif ic pro-

motion and may decide that it was not prof itable

enough. However, if the manager were to use the pro-

posed model, the incremental profit due to comple-

mentarity might t ip t he decision in f avor of running

the promotion.

7. Conclusions and Future ResearchIn this research we prop ose a general app roach to

modeling the shopp ing basket. The compos ition of the

basket is of signif icant interest t o retailers and manu-

f acturers. We also propose a general f ramework f or

classif ying multicategory choice situations in t erms of

complementarity and co-incidence (Table 1) and we

show how these problems may be modeled through

the use of an appropriate model form. The shopping basket is one of many multicategory choice phenom-

ena that are encountered by consumers. This modeling

approach is likely to prove very usef ul in other app li-

cations such as database marketing and micromarket-

ing. Finally, t he multivariate probit model may also be

app lied in any marketing situation in which there

could be more than one outcome simultaneously, e.g.,

it could be used to model the consideration set of con-

sumers in the brand choice process.

Our results show that it is possible to separate the

controllable drivers of multicategory choice f rom the

uncontrollable ones. We also show that, in some cases,

the addition of these controllable explanatory variables

reduces the ef f ect of the uncontrollable variables. In

addition to separating these effects, we are also able to

quantif y the magnitude of these effects. In general, as




Page 18



expected, complementarity effects are smaller in mag-

nitude than the own effects. However, since the gro-

cery retailing industry is highly competit ive with low

margins of around 5% ( Food Marketing I nstitute 1992),

even small demand (and profit) gains through cross

pricing may be of great significance. The nature of the

co-incidence patt ern can provide insights into how the

bigger problem of multi-category choice may be re-

duced to many smaller problems of interdependent

pairs of items. This is likely to be of much value to

retailers and manuf acturers. The use of hierarchical

models also allows us to capture heterogeneity at the

individual household level—this may enable manag-

ers to f ormulate and implement micromarketing strat-

egies. In terms of applicability of our results, we would

like to note that our sample consists of average to

heavier buying households across t he f our categories

that we have chosen. Thus, these insights are likely to

be limited to these kinds of households and may not

be applicable to all households that shop at a particular

store or stores.21

Our current and future work in this area is in f our

broad directions. In the first stream we plan to model

other multicategory decisions such as brand choice

and purchase quantity as well as build integrated mod-

els which link these decisions. In the second stream,

we p lan to develop a theoretical model of multicate-

gory consumer choice under an axiomatic, utility max-

imization f ramework. We would then t ransf orm the

theory’s predictions into an empirically est imablemodel which would use methods such as the ones p ro-

posed in this paper. For an excellent example of this

approach in the single category context, s ee Arora et

al. (1998). In the third stream, we investigate the nature

of household heterogeneity across categories in more

detail and try to build models for micromarketing ap-

plications. Finally, we are also investigating the exten-

sion of these models into other substantive areas such

21Though this (restricted) applicability may be seen as a li mitation,

retailers are keenly f ocussed on the buying behavior of these

“ heavier” buy ers. This is b ecause t hese cu stomers buy larger bask ets

on each visit and therefore have a higher impact on sales and prof it s

across categories, e.g., Ukrop Supermarkets labels customers who

spend about twice as much as the average customer during a store

trip as its “ Valued Customers” and provides them extra promotional

incentives ( F ood M ar k eting I nstitute1994).

as database marketing and the choice of households’

f inancial port f olios.22

Appendix 1. Prior Di stributions

1. The prior distributi on of c is a multivariate normal distribut ion

MV N[g , C ] where g 0 and C diag(103).

2. The prior distribution of l0 is a multiv ariate normal MV N[h 0,

Θ0] where h0 0 and Θ0 diag(103).





5. The prior distribution of is Wishart W[ ρ, ( ρR 0) 1] where ρ1

Λ0

NCAT 1 and R0 diag(10 3). NCAT is t he number of cate-

gories, i.e., in this application NCAT is 4.


Λ1

NCAT 1 and R1 diag(10 3).


Λ2

NCAT 1 and R2 diag(10 3).

8. The prior distribution for the free elements of the correlation

matrix (Σ), vec*(Σ) (σ 12 , σ 13 ,... σ ( J 1) J ), such that vec* (Σ) A, can be express ed as

π (vec*(Σ) | Σ , G ) exp0 0

1(vec*(Σ) vec*(Σ ))G (vec*(Σ) vec*(Σ )) ,0 0 0

2

where the region A is a subs et of the region [1, 1] J ( J 1)/2 that leads

to a prop er correlation matrix, Σ0 is a J J correlation matrix with

all diagonal elements equal to 0 and G0 is a ( J ( J 1)/2) ( J ( J

1)/2) precision matrix. Note that Σ0 and G0 are the hyperp arameters

f or the prior. We set Σ0 0 and G0 I where I is an id entity matrix

of dimensionality ( J ( J 1)/2) ( J ( J 1)/2).

Appendix 2. Description of Full Conditionals

1. The f ull conditional distribution f or c is multivariate normal,

E 1], where E 1 (C 1g R 1u˜ht ) and E H nh

N [ˆ c, ˆ c R R Z h 1 t 1t 1 ht

C 1 Σ 1 Z ht . The adjusted util ities, u˜ht , are defined H nh

Σ Σ Z h 1 t 1 ht

as uht bh0 X h1 t bh1 X h 2t bh2.

2. The f ull conditional distribution f or bh0 is multivariate normal,

], where ( Dhl0 Σ 1ûht ) and B0

1 1 1 nˆ ˆ h

N [b , B b B Λ Σ0 0 0 0 0 t 1

( Σ 1). The adjusted ut ilities, ûht , are def ined as uht

1 nh

Λ Σ0 t 1

Z ht c X h1 t bh1 X h 2t bh2 for all h and t .

22The authors w ould l ike to t hank Siddhartha Chib, Kamel Jedidi,

Don Lehmann, Sridhar Moorthy, Peter Rossi, and Gary Russell f or

f eedback, A. C. Nielsen for providing the data, David Bohl of A. C.

Niels en f or help on data iss ues and Monica Valluri f or help on com-

pu tati onal iss ues . The autho rs wo uld als o li ke to thank t he Ed itor,

Area Editor, and two anonymous reviewers f or their helpf ul sug-

gestions. The f irst author would like to acknowledge research sup-

po rt f rom the Graduate School of Business at the University of Chi-

cago.




Page 19



3. The f ull condition al distribution for bh1 is multivariate normal,

], where Σ 1ûht ) and1 1 1 nˆ ˆ h

N [b , B b B (Λ D l Σ X 1 1 1 1 1 h 1 t 1 h 1t

Σ 1 X h1 t ). The adjusted util ities, ûht , are de-1 1 nh

B (Λ Σ X 1 1 t 1 h 1t

f ined as uht bh0 X h 2t bh2 Z ht c for all h and t

4. The f ull condition al distribution for bh2 is multivariate normal,

], where ( Dhl2 Σ 1ûht ) and1 1 1 nˆ ˆ h

N [b , B b B Λ Σ X 2 2 2 2 2 t 1 h2 t

Σ 1 X h2 t ). The adjusted util ities, ûht , are de-1 1 nh

B (Λ Σ X 2 2 t 1 h 2t

f ined as uht bh0 X h 1t bh1 Z ht c for all h and t .

5. The f ull condition al distribution for l0 is multivariate normal,

], where ( bh0) and1 1 1 H 1 1

N [ˆ l , M ˆ l M Θ h Σ D Λ M 0 0 0 0 0 0 h 1 h 0 0

Dh.1 H 1

Θ Σ D Λ0 h 1 h 0

The f ull condition al distribution s for l1 and l2 are also multivariate

normal and may be obt ained in a similar manner.

6. The f ull conditional distribution for is Wishart and is given1

Λ0

by W {( ρR 0 (bh0 Dhl0)(bh0 Dhl0)) 1, ρ H } where H is H

Σh 1

the nu mber of households.

The f ull conditional distributions for and are also Wishart1 1

Λ Λ1 2

and may be obtain ed in a si milar manner.

7. The f ull condition al distribution for the identif ied set of utilities

on any observation is truncated multivariate normal, where the re-

gion of truncation is determined by the pattern of incidence across

the categories. The truncated multivariate normal variate can be gen-

erated by using a mini-Gibbs sampler (Geweke 1991). This involves

using J Gibbs sampling steps such that the jth component of uht , uhjt ,

comes f rom the univariate truncated condition al normal densi ty

. Here the univariate normal distri-(m) (m) (m) (m)

p(u | b , b , b , c, Σ ,y )hjt h0 h1 h2 ht

bu ti on i s tru ncated f rom below by 0 if a purchase was made in th e

category j, else it is truncated from above by 0.

8. The f inal step of the substit ution sampler involves sampling the

Σ correlation matrix. If the prior distributi on for the f reeelements of

the correlation matrix Σ, vec* (Σ) (σ 12 , σ 13 ,... σ ( J 1) J ), is π (vec*(Σ)

| Σ0, G0), then the f ull conditional distribution of Σ is proportional

to { L(Σ | , {{uht }}, y)}{τ (vec* (Σ) | Σ0, G0) where L(.) is(m) (m) (m)

b b , b ,h0 h1 h2

the conditional lik elihood and is gi ven as (ignoring the normalizing

constant).

n

1n /2

|Σ| exp (u β x β x β z γ )i 0 i 1 i1i 2 i2 i ii

2 i 1

1

Σ (u β x β x β z γ ) ,i 0 i 1 i1i 2 i2i ii

where vec* (Σ) A as in Appendix A1.

Since this f ull cond itional di stribution is complex, we generate Σ

using a Metropolis Hit-and-Run algorithm (Dey and Chen 19 96,

Chen and Schmeiser 199 3). If Σ(m) is the current value of the corre-

lation matrix, then in this step, a candidate matrix Σc is generated by

specif ying a random walk chain Σc Σ(m) H , where H (h ij) is

a matrix such t hat E (h i j) 0 and hii 0. Let ξ be th e smallest eigen -

value of Σ(m). Then th e elements of the increment matrix H can be

generated using the Hit-and-Run algorithm. This involv es the f ol -

lowing steps:

(a) generate a sequence of i.i.d standard normal deviates z 12 ,

z 13 ,..., z ( J

1), J , of length J ( J 1)/2,

(b) generate a deviate d from N (0, ) which is truncated to the2

σ d

interval ,( ξ / (2), ξ / (2))

(c) f ormulate the elements

dz i jh ij

(1/2)

J J 1 2

z l j 1 jl j 1

f or i j, hii 0, and hij h jifor i j.

Here is a tuning constant tha t needs to be chosen such that2

σ d

candidates are not rejected dis proportionately. If ξ c

is t he smallesteigenvalue of the candidate matrix, then once a candidate is gener-

ated, it is accepted or rejected based on the following Metropolis-

Hastings acceptance probability:

c c

ξ ξ c (m) (m) (m) c

L(Σ | b , b , b , {{u }}, y) p(vec*(Σ )) Φ Φh0 h1 h2 ht

2σ 2σ d d

min , 1ξ ξ

(m) (m) (m)

L(Σ | b , b , b , {{u }}, y) p(vec*(Σ)) Φ Φh0 h1 h2 ht

2σ 2σ d d

where Φ(.) is the standard no rmal cumulative dist ribution f unction.

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