Journal of Retailing 91 (1, 2015) 1933

An Analysis of Assortment Choice iKyuseop Kwak a,, Sri Devi Duvvuri b,

of Tecn, Bot

Iowa

Abstract

Consume odateparticular bu categresearch exa ss. Inless variety flexchoice. The itemselected, all ly theyogurt produ brandwe show tha d byfacts paint a ct qu 2014 New

Keywords: Assortment choice; Variety; Quality perceptions; Bundle models; Multivariate logistic model

Introduction

For freqconsumers

long-run prorder to balpreferenceseach shoppment, a buncategory (sbehavior pethat will beflexibility accommodhold (Cherunderstandments prov

CorresponE-mail ad

SDDuvvuri@(G.J. Russell)

that should be stocked in a given product category (Corstjens

http://dx.doi.o0022-4359/uently purchased products such as grocery items, can more easily anticipate short-run preferences thaneferences (Kahneman and Snell 1990; Luce 1992). Inance the trade-off between short-term and long-term, consumers assemble bundles of products duringing trip. Of particular interest is the product assort-dle consisting of items selected from a single productuch as breakfast cereals or yogurt). From a choicerspective, assortments allow consumers to buy goods

consumed over time. This provides the consumerin planning for future consumption events and inating the preferences of multiple users in the house-nev 2011). From a retail management perspective,ing the consumer decision process behind assort-ides guidelines for the number and variety of items

ding author.dresses: Kyuseop.kwak@uts.edu.au (K. Kwak),uwb.edu (S.D. Duvvuri), gary-j-russell@uiowa.edu.

and Corstjens 2002).

Variety of an Assortment

An assortment can be viewed as a choice strategy that allowsconsumers to pursue multiple decision objectives. Due to thetransaction costs of grocery shopping, it is more efficient forthe consumer to buy an assortment of products that will be con-sumed in the future. Assortments permit consumers to minimizedecision conflict on a given trip (Simonson 1990), account foruncertainty in future preference (Kahn and Lehmann 1991; Shinand Ariely 2004; Walsh 1995), and address attribute satiation inconsumption (Kim, Allenby, and Rossi 2002). All these studiesemphasize that assortments are constructed to maximize varietywith respect to item attributes.

Research in the store choice and consumer behavior litera-ture argues that assortment variety is a psychological perceptionthat plays a key role in the purchase decision (e.g., Broniarczyk2008; Hoch, Bradlow, and Wansink 1999). Variety perceptionhas been found to depend upon the size of the assortment(e.g., Broniarczyk, Hoyer, and McAlister 1998; Kahn 1995),

rg/10.1016/j.jretai.2014.10.004 2014 New York University. Published by Elsevier Inc. All rights reserved.a School of Marketing, Faculty of Business, Universityb School of Business, University of Washingto

c Tippie College of Business, University of

rs in grocery retailing commonly buy bundles of products to accommndle share common attributes (and are selected from the same product mines the impact of assortment variety on the assortment choice procefor higher quality items. To investigate this relationship, we employ a

model, based upon the assumption that the utility of purchase of oneows for a general utility structure across the assortment items. We appct category. Substantively, we show that yogurt choice is affected by

t reaction to reductions in variety (number of yogurt flavors) is mediate picture of a consumer who is willing to trade-off variety against produ

York University. Published by Elsevier Inc. All rights reserved.n Grocery Retailing, Gary J. Russell chnology, Sydney, Australiahell, United States, United States

current and future consumption. When all products in aory), the consumer is said to assemble an assortment. This

particular, we test the prediction that consumers demandible choice model, suitable for the analysis of assortment

in an assortment depends upon the set of items already model to household assortment choice histories from the

quality perceptions (quality-tier competition). Moreover, brand quality perceptions. Taken together, these empiricalality in assortment choice.

20 K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933

the distinctiveness of alternatives (Gourville and Soman 2005;Markman and Gentner 1993; Van Herpen and Pieters 2007),and the pBradlow, aically belieof alternati(Broniarczand Villas-effort tradechases (Hoitems are pprobabilityorganized)(Chernev 2(Kahn, Moand overallitem/brandceptions in

Variety and

Recent theory of chproduct quthe cogniti1980) and tthis theory.of items inquality of tto be a coneach new iThe consuincreasing variety.

Supposehigh qualitHamilton (marginal bis smaller tity assortma high quamarginal retion is quitassortmentresearch is variety and

Modeling ATo test

eral modelproduct bunstructed. Mthe researcthe bundleone anotheFor exampity model

code each attribute in terms of more is better (maximize thebundle mean) or heterogeneity is better (maximize the bun-

iancnce

temsbal tiall

osingd. By sca

promodureshougt sunmelly ht is ultaning cdel ac (Mtersing old

demtly, kage

iew o

rem

e a b, expl mostrury thtern

assu

oicedyin. Sub

yogerg ver,

yoguerallmiltrese

ulti

his sery

t upoent

chosroximity and display of alternatives (e.g., Hoch,nd Wansink 1999). While larger assortments are typ-ved to offer greater variety, decreasing the numberves may actually increase the purchase likelihoodyk, Hoyer, and McAlister 1998; Kahn 1995; KuksovBoas 2010). This may be attributed to the time and-off consumers resort to while making grocery pur-yer 1984). Assortments with greater similarity amongerceived to have lower variety, thus lowering the

of purchase. However, items that are displayed (or together are often perceived to offer greater variety011). Factors such as the marginal utility of itemsore, and Glazer 1987; Kahneman and Snell 1992)

assortment attractiveness (e.g., inclusion of favorite) (Botti and McGill 2006) supplement variety per-

motivating assortment preference and purchase.

Product Quality

work by Chernev and Hamilton (2009) proposes aoice behavior that links assortment size to perceived

ality. This theory is based upon the trade-off betweenve effort in evaluating choice alternatives (Shuganhe desire for more variety. Two assumptions underlie

First, cognitive effort is proportional to the number the assortment, but has no relationship to perceivedhe items. Second, utility for item variety is assumedcave function. Due to decreasing marginal returns,tem in the assortment adds less incremental utility.mer will add new items to the assortment until themental costs overwhelm the added utility of more

that the consumer constructs assortments from eithery items or low quality items (not both). Chernev and2009) propose that for assortments of equal size, theenefit of adding an item to a high quality assortmenthan marginal benefit of adding an item to a low qual-ent. The argument, in essence, is that each item of

lity assortment is sufficiently valued that decreasingturns become binding more quickly. Thus, the predic-e simple: consumers will prefer a small high-quality

over a large low-quality assortment. The goal of ourto evaluate the evidence for this relationship between

product quality.

ssortment Choicethis prediction, it is necessary to construct a gen-

of assortment choice. The academic literature ondles provides insights on how a model might be con-ulti-attribute models of bundle choice assume that

her has a list of product attributes for each item in and understands how these attributes interact withr in determining consumer preferences (Rao 2004).le, Farquhar and Rao (1976) develop a bundle util-based upon the assumption that the researcher can

dle varprefereof all ithis glosequenof choselectegrocererence

utility proced

Altare no

envirotypicaover, ito simincludwe mo

logistiand Peof buyhousehfor thesequenthe lin

Overv

Theprovidmodelgenerawe con

categothe patwhichany chfor stuchoiceamong(BlattbMoreober of The ovand Hafuture

M

In tfor a vis buildependin the e). Preference for the bundle, then, reflects consumerfor global features that are derived from the attributes

in the bundle. Harlam and Lodish (1995) draw uponfeature logic, but assume that the bundle is assembledy (across grocery shopping trips), with the probability

the next product contingent on the products alreadyecause order of purchase is generally not reported innner datasets, the authors assume that higher pref-

ducts are purchased earlier. More generally, bundleels can be constructed using conjoint measurement

(Green, Jain, and Wind 1972).h these models have many attractive features, theyitable for empirical studies in a grocery shoppingnt. The key reason is that the researcher does notave an exhaustive list of product features. More-clear that the modeling framework must be ableeously accommodate various demand relationships,omplementarity and substitution. For these reasons,ssortment choice using a version of the multivariateVL) category incidence model developed by Russellen (2000). The model implies that the probabilityan assortment on a shopping trip depends upon thes valuation of each item in the assortment, adjustedand relationships among the items. As we show sub-the model provides a strong foundation for studying

between product quality and assortment variety.

f Research

ainder of this paper is organized as follows. We firstrief review of the multivariate logistic (MVL) choicelaining how the model can be extended to build adel of assortment choice. Drawing upon this theory,ct an assortment choice model for the yogurt productat links product characteristics (brand and flavor) toof household purchase decisions. This choice model,mes perfect substitution across brand names and pick-

among flavors within a brand, provides a structureg the role of brand quality perceptions in assortmentstantively, we show that the pattern of competitionurt items is affected by brand quality perceptionsand Wisniewski 1989; Sivakumar and Raj 1997).we show that reaction to reductions in variety (num-rt flavors) is mediated by brand quality perceptions.

pattern of our findings is consistent with the Chernevon (2009) theory. We conclude with suggestions forarch.

variate Logistic Model of Assortment Choice

ection, we describe an assortment model that allowsflexible pattern of product competition. The modeln the conditional utility of choice of a single item,

on its own marketing mix and the other item choicesen assortment. The development here, which builds

K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933 21

upon the discussion in Russell and Petersen (2000), yields achoice model specification that is a special case of the multi-variate logithe model The modeleconomicsaccommod

Model Des

We begithat an assmembers oh selects amultiple itters. We usattribute inthe laundryignations (la collection

Denote 1, 2, . . ., CDefine an acating the p

zbht =[z1h1t

where zchjtt, and equachoice, the(including out further probability

We nowgiven choictional utiliton item chitems in thetional on thconsiderati

U(jc|all els

where chjth, at time tbaseline utmarketing utility for t

These interm on thdenoted byFor purposfor [c, j] = later in thience of itehas positiv

c,chjk < 0 if item k negatively influences item js utility (sub-

stitution effect), and c,chjk = 0 if there is no influence between(independence). For reasons that will become clear sub-tly, we require that the c,chjk parameters be symmetric inc, j] and [c*, k].implify the presentation of the model, it is useful to definemetric matrix, h, containing all the elements of c,chjk ,

[1h

h

Ch

chj c of

can

ll els

umiditio

own bina

t|all

as n

hethe (5) i

reg Fro

be variaes o

asso

cond of t

f theal prockins prog in

ew otherehen stic distribution proposed by Cox (1972). However,is considerably richer than the derivation implies.

is consistent with theoretical frameworks in both and psychology, and has sufficient flexibility toate complex demand patterns among items.

cription

n with a discussion of bundle choice, keeping in mindortment is a bundle of products, all of which aref the same product category. Suppose that householdn assortment b at time t. An assortment b containsems (zero, one or more items) from multiple clus-e the word cluster to mean a set of items with some

common. For example, in a study of assortments in detergent category, clusters might be submarket des-iquid detergents), family brand names (e.g., Tide), or

of product features (scented liquid detergents).jc as an item j (= 1, 2, . . ., Jc) in cluster c (=

). The total number of alternatives is M = Cc=1Jc.ssortment as a M 1 vector of binary variables indi-resence of the item in an assortment

, z1h2t , . . ., z1hJ1t , z

2h1t , z

2h2t , . . ., z

2hJ2t , . . ., z

ChJCt

](1)

= 1 if household h selects item j in cluster c at timels zero otherwise. Under the assumption of pick-anyre are 2M possible assortments that could be selectedthe null assortment containing no purchases). With-restriction, the model subsequently assigns a choice

to each of these 2M assortments. define the conditional choice probability for an item,es of all other items in the assortment. The condi-

y consists of two parts: (a) a baseline utility dependentaracteristics and (b) demand interactions with other

assortment. The utility of item j in cluster c, condi-e observed purchase outcomes of other items underon, is assumed to have the form

e) = chjt +c

k

c,chjk z

chkt + hjt (2)

is a baseline utility for item j in cluster c of household and hjt is a Gumbel distributed random error. Theility depends both upon household preferences andmix elements, and can be regarded as the inherenthe item, net of demand interactions with other items.teractions are represented by the double summatione right hand side of Eq. (2). Individual items are

the notation [c, j] for item j nested within cluster c.es of model identification, we assume that c,chjk = 0,[c*, k] (We discuss the necessity of this restrictions section). The parameter c,chjk captures the influ-m [c*, k] on [c, j]. Accordingly, c,chjk > 0 if item ke impact (complementarity effect) on item js utility,

items sequenitems [

To sthe symas

h =

=

where cluster(1), we

U(jc|a

Assthe conthe knof the

Pr(zchj

where,ting w

Eq.ogistic1993).seen tochase outcomin the a full and allcess o

a globinterlo

Thiworkinovervinotes, only w1, 1h2, . . .,

1hJ1 ,

2h1, . . .,

ChJC

]NJ1

0 1,1h12 1,Ch1JC1,1

21 0 1,Ch2JC.

.

.

.

.

.

.

.

.

.

.

.

,1JC1

C,1hJC2 0

NJNJ

(3)

is the column vector for the corresponding item j in the matrix h. Using matrix (3) and choice vector

now rewrite Eq. (2) as

e) = chjt + chjzbht + hjt (4)

ng that the error in (4) follows a Gumbel distribution,nal probability of purchasing item j in cluster c, givenpurchase outcomes for all other items, has the formry logit model

other zs) = {exp(chjt + chjzbht)}

zchjt

1 + exp(chjt + chjzbht)(5)

oted earlier, zchjt {0, 1} is an indicator variable repor-r or not item j is chosen.

s known in the spatial statistics literature as an autol-ression model (Anselin 2002; Besag 1974; Cressiem the standpoint of statistical theory, Eq. (5) can bethe full conditional distribution of one binary pur-ble (corresponding to item [c, j]), given the knownf the binary purchase variables for all other items

rtment. Note that each of the M possible items hasitional distribution in the general form of Eq. (5),hese M models collectively describe the choice pro-

household. The technical problem is to write downbability model that is consistent with this system of

g equations.blem has been analyzed and solved by researchers

spatial statistics (Besag 1974). We provide anf the key results in Appendix A. As Besag (1974)

exists a global probability model consistent with (5)the cross-effect matrix is symmetric (c,chjk = c,chkj ).

22 K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933

With this symmetry assumption, the global choice model takesthe form

Pr(zht = zbht) =exp(Vhbt)b exp(Vhbt)

exp( zb + 1 {zb zb })

where hM 1 colditional bahtz

bht is a

chosen asssystem of emodel givevalidity of consequencdistributionthe joint (mindicator vthe multivaas the mult

As discconstraintsond, the mensure thatEq. (6), it iseffect matrmain effecthe cross-eequations iway, if croconsistent

From a is a signedin the assorof an autolof the relatwith closer(Anselin 20the items inwhich closconsumers

and PeterseHowever, ubution (Coxoutcomes athe cross-efsary becausof the glob

1 Marketingand HildebranSeetharaman

The symmetry of the cross-effect matrix also has anotherinteresting interpretation. As explained in Appendix A, the MVLmodel deriassortmentment notwere actua

asso

etric ive ice is

Inte

MVctivempls in ution

Thi beha(6). Tcify ckinm thchoirst inse deAnse, calns a. Inual

en thwillrket

theoion

sec

mo

ture guntow

ey afor + (1ete e (ine actally cont

L citivertm

asso

Thiser b

leme= ht ht 2 hth ht

b exp(htz

bht + 12

{zb

ht hzb

ht

}) (6)

t =[1h1t ,

1h2t , . . .,

1hJ1t

, 2h1t , . . ., ChJCt

]is a

umn vector containing the individual (full) con-seline utilities for items in consideration. Here

summation of baseline utilities for items in theortment b. It is important to understand that thequations given by (5) implies the assortment choicen by (6). That is, once the researcher assumes thethe M equations in (5), Eq. (6) follows as a logicale. Conversely, (6) implies the set of full conditionals in (5). Using statistical terminology, Eq. (6) isultivariate) distribution of the M binary purchase

ariables zht. Because (6) is known to statisticians asriate logistic distribution (Cox 1972), we refer to (6)ivariate logistic (MVL) choice model.1ussed earlier, the cross-effect matrix h obeys two. First, the diagonal of the matrix is set to zero. Sec-atrix is symmetric. Both are technical restrictions to

the model is properly specified. Given the form of easily shown that zeros on the diagonal of the cross-

ix are necessary to allow the parameters in the htzbht

t terms to be identified. The symmetry restriction onffect matrix is necessary to conclude that the set ofn (5) implies the global model in (6). Put anotherss-effects were asymmetric, then no global model

with Eq. (5) exists (Besag 1974).more intuitive point of view, the cross effect matrix

(plus or minus) measure of similarity across itemstment. In the spatial statistics literature, the elementsogistic cross-effect matrix are modeled as functionsive geographic distance between response locations,

locations having a larger (positive) cross-effect term02; Cressie 1993). Conceptually, it is useful to regard

the assortment as being located on a mental map iner items have stronger interactions with respect to the

buying behavior (Moon and Russell 2008; Russelln 2000). The elements of h are not correlations.sing the properties of the multivariate logistic distri-

1972), it can be shown that the correlations in choicemong the items in the assortment are determined byfect matrix. Thus, symmetry in cross-effects is neces-e of the link between h and the correlation structureal choice model.

science applications related to the MVL model include Boztugdt (2008), Moon and Russell (2008), Niraj, Padmanabhan, and

(2008), and Yang et al. (2010).

ues an

symmattractsequen

Model

Theperspean exa

procesdistrib1993).choicein Eq. to speinterlo

FroMVL The fipurchaitems. eraturefunctiolibriumindivid(5), thment) the manomica decis

Thechoicethe naChintaGentzktwo kutility htz

bht

a discrthe truthat thidenticin the the MV

Intuan asso

in the items. consum

If all evation assumes that the consumers valuation of an depends only on the final composition of the assort-

upon the order in which the items in the assortmentlly selected. Put another way, a consumer who val-rtment solely on the basis of its contents must have ah matrix. This order independence property is quiten grocery retailing applications in which purchase

not observed.

rpretation

L choice model can be viewed from a variety ofs. In the spatial statistics literature, the model is

e of a conditional autoregressive (CAR) stochasticwhich the researcher uses a set of full conditionals to derive the form of a joint distribution (Cressies interpretation emphasizes the fact that the localvior in Eq. (5) determines the global choice behaviorhe CAR approach provides a way for the researcher

a complex system by breaking the system into Mg parts, each of which can be developed separately.e standpoint of the marketing science literature, thece model can be given two different interpretations.terpretation treats Eq. (5) as a linkage between thecision for one item and the actual choices of all otherlin (2002), in a review of the spatial econometrics lit-

ls the system of equations represented by (5) reactionnd the implied global model in (6) the system equi-

other words, if the consumer simultaneously makesitem choices using a random utility process given bye observed global choice behavior (for the assort-

be observed to follow (6). In fact, recent work ining science literature by Yang et al. (2010) uses eco-ry to argue that MVL choice model in (6) representsprocess equilibrium.ond marketing science interpretation specifies thedel in (6) directly by making assumptions aboutof the utility function of an assortment. Song anda (2006), using a utility theory argument due to(2007), adopt this approach. These authors makessumptions. First, they assume that the expectedan assortment of goods Vhbt has the general form/2){zbht hzbht}. Here, Vhbt can be considered to be

variable second-order (quadratic) approximation todirect) utility of assortment b. Second, they assumeual utilities fluctuate around Vhbt due to independent,distributed, extreme value errors. These assumptions,ext of a random utility theory framework, also yieldhoice model in (6).ly, the MVL choice model states that the utility of

ent depends upon household valuations for each itemrtment, adjusted for interactions among the selected

theory exhibits clear relationships with work in theehavior literature dealing with product assortments.nts of h are set to zero, then Vhbt is just the sum of

K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933 23

the utilities of the items in the assortment. This type of modelwas proposed by Janiszewski and Cunha (2004) in a study ofconsumer r

that the vaeffects andthe marketRao 1976).

Restricting

The MVbe exploitetions (suchproduct cativariate lothe probabrandom vavariables (rwill be corof (6) that defined wimodel assudence of irrIt is importto assortmeIn general, (5), the relized by arb

The IIAform of a write the gin the choicbe observements). Thall remainithis pruninprocess. Foleast one itethat all assowhich no itis

Pr(zht = z

= b

subject to Implicitly, sible assorzero.

Constraini

A usefuconstraintswe are pres

F

ree its aren uns

gh t be ry, tholdtrict .g.,

to dee strmarkccom

tric (A, matr

,1

,

A

A

A,A21

, bos per

con

taskrandrix wutionssiblm thtain

B, oice ator of Eq. (6) becomes exp() = 0). With the struc-

Eq. (8), there are only 21 (obtained as 3 (23 1))ents for the consumer to consider. In model calibration,

ameters set to are not actually estimated. Rather, the specification in Eq. (6) is rewritten to exclude assortmentsero purchase probability.eactions to price anchors. More broadly, the notionluation of an assortment depends both upon main

interactions of item attributes has strong support ining literature (Chung and Rao 2003; Farquhar and

the Model

L choice model has an important property that cand to tailor the general model to particular applica-

as assortment choice with respect to a particulartegory). Given that Eq. (6) has the form of a mul-gistic distribution (Cox 1972), it may be viewed asility distribution of an M-dimensional multivariateriable. This implies, in particular, that these binaryepresenting the choice outcomes of different items)related. Interestingly, it is also clear from the formthe model can be viewed as a simple logit modelth respect to assortments. Accordingly, the MVLmes that choice is characterized by an IIA (indepen-elevant alternatives) property relative to assortments.ant to understand that this IIA property correspondsntsnot to the individual items within an assortment.given the interlocking full conditional expressions inationships between specific items will be character-itrary deviations from IIA.

assortment property makes it very easy to infer thechoice model subject to various restrictions. First,eneral form of the MVL choice model for all itemse set. Second, discard all assortments that will never

d (because they are not in the set of possible assort-ird, renormalize the model so that the probabilities ofng assortments sum to unity. Whatever remains afterg process must be the correct model for the choicer example, suppose that the choice task is such that atm must be chosen from a set of M items. This meansrtments are possible, except for the null assortment inems are chosen. Thus, the appropriate choice model

bht)

exp(htzbht + (1/2){zbht hzbht}) all nonempty bundles exp(htzb

ht + (1/2){zb

ht hzb

ht })(7)

the restriction that zbht cannot be the zero vector.the probability of any outcome not in the set of fea-tments in this case, the null assortment is set to

ng the Matrix

l way of understanding model restrictions is to place directly on the matrix. Suppose, for example, thatented with assortment choice task with three clusters

and thclusterhave aAlthouthelesscategoHousebut res(see, enames

then thof the

To asymmebrandsthe

2

31

0A

A

=

where 1 and 2impliesumers

choiceeach b matsubstitare po

Froing cer{Brandthe chnumer

ture inassortmthe parmodelwith zig. 1. Pick-any items nested within pick-one clusters.

ems each within each cluster. Suppose, as well, that considered substitutes, while items within a clusterpecified level of demand dependence on one another.his structure may not appear intuitive, it can never-useful in some settings. For example, in the yogurthere are multiple flavors within each brand name.s typically buy multiple flavors on a shopping trip,their purchases to items having the same brand nameKim, Allenby, and Rossi 2002). If we allow brandfine clusters and the flavors of a brand to define items,ucture shown in Fig. 1 is an appropriate descriptionet structure within the yogurt category.modate this structure, we impose restrictions on the

matrix in the following manner. Assuming threeB, C) with three flavors within each brand (1, 2, 3),ix takes the form

,32

,21

, ,31 32

,21

, ,31 32

0

0

0

0

0

0

0

0

A A

B B

B B B B

C C

C C C C

(8)

denotes the cross-effect parameter between flavorsth from brand A. The symbol (negative infinity)fect substitution. In words, Eq. (8) indicates that con-struct assortments by selecting one brand (pick-one) and then selecting an assortment of flavors within

(pick-any choice task). Because the parameters of theithin each brand are unrestricted, varying degrees of

and complementarity among flavors within brandse.e standpoint of the consumer, assortments contain-combinations of items (e.g., {Brand A, flavor 1} andflavor 2}) are excluded from consideration becauseprobability of the assortment equals zero (i.e., the

24 K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933

Summary

At this point, the important properties of the MVL choicemodel should be clear. From the standpoint of assortment choice,the model has two desirable features. First, it implies that themarginal ucharacterisMVL choicoutcome afied simultBecause thtive, variouimposed. Ican be excIn our empstudying th

As

We emppurchase hstructure ofEq. (8) aboto explore theory relation, we diswho are prcasts for thnext sectio

Data Desc

The datpurchase hhalf year passortmentand collapTaken togeeach brandanalysis areand Othersare able todecisions.

We splimonths (24specific SKsplit into twtrips over 5over 90 dabrands acrothere is noSKUs havwas no dis

2 Multiple p

Table 1Descriptive statistics.

Calibration data Holdout data

Number of observations 1443 185Number of households 113 81

Price 0.5464 (0.074) 0.5278 (0.070)Display 0.0569 (0.216) 0.0226 (0.158)Feature 0.1793 (0.366) 0.2542 (0.403)

rice ismean

ses. S Dann

same

nding

s. Ofolds

holdldoule 2 presents a contingency table showing items that aresed together. Diagonal elements are purchase frequenciesividual SKUs in the calibration data period. The table

no cross purchases across brands, but many instances ofurchasing of flavors within a brand. We assume the choices for Dannon and Nordica follows the pattern shown in

pick-one choice across brands, and pick-any choice for within brands. This type of demand pattern correspondsstructure of the matrix discussed earlier in relation to).alibrating the model (discussed below), we also make

her adjustments. Since the choice histories are restrictedy those instances in which either Dannon or Nordicachased, we never observe a null (empty) assortment.quently, we eliminate the null assortment from the denom-of Eq. (6), thereby fixing the probability of observing thesortment to zero. (As explained earlier, this procedure isd by the IIA property of the choice model with respect tot assortments.) In addition, some items are not available

eleven of 124 households made purchases across brands on a singleg occasion. This accounts for about 1% of total purchase events. Theselds were excluded from the dataset prior to model calibration. Kim,, and Rossi (2002) also report that almost all yogurt purchase bundlesf items having the same brand name.tility of each item in an assortment depends upon thetics of all other items in the assortment (Eq. (2)). Thee model (Eq. (6)) can be viewed as the equilibrium

t which all the marginal utility equations are satis-aneously. Second, the model is extremely flexible.e parameters of the matrix can be positive or nega-s patterns of substitution and complementarity can bef necessary, assortments that will never be observedluded, allowing a parsimonious model specification.irical work, we take advantage of this flexibility ine role of product quality in assortment choice.

sortment Choice in the Yogurt Category

loy the MVL choice model to analyze householdistories from the yogurt product category using the

the matrix corresponding to Fig. 1 (as depicted inve). Our main goal is substantive. We use the modelthe predictions of the Chernev and Hamilton (2009)ting demand for variety to product quality. In this sec-cuss technical aspects of model calibration. Readers

imarily interested in the implications of model fore-e ChernevHamilton theory may wish to skip to then.

ription

a are taken from an ERIM scanner panel of yogurtistories from the Sioux Falls area over a two and one-eriod. We focus on the low fat, fruit on the bottom. We select eight common flavors from both brandsse three flavors into one composite flavor (Others).ther, we have total of twelve SKUs (six SKUs for) in the analysis.2 The six common flavors used in the

Cherry, Mixed Berry, Peach, Raspberry, Strawberry,. By constructing the SKU set in this manner, we

study the role of brand and flavor in yogurt choice

t the data into three consecutive periods. The first 80 days) of the data were used to create household-U loyalty variables. The remainder of the data waso sets: a model calibration period (1443 shopping80 days) and a holdout period (185 shopping trips

ys). Table 1 shows descriptive statistics for the twoss these periods. Due to retailer marketing decisions,

variation in marketing mix activity across flavoring the same brand name at the same time. Thereplay activity at all for Dannon during the observed

urchases of a single SKU are considered to be one SKU choice.

Nordica

Dannon

Note: Pare the parenthebecausehave thecorrespo

periodhousehing thethe ho

Tabpurchafor indshowsjoint pprocesFig. 2:flavorsto the Eq. (8

In ctwo otto onlis purConseinator null asjustifieproduc

3 OnlyshoppinhousehoAllenbyconsist oPrice 0.6456 (0.049) 0.688 (0.008)Display NA NAFeature 0.0699 (0.247) 0.0222 (0.129)

the mean dollar per eight ounce container. Display and Feature of indicator (01) variables. Standard deviations are shown intatistics for the Dannon Display variable are not available (NA)on does not employ displays in this dataset. Individual flavors

means and standard deviations on marketing mix elements as the brand name.

Fig. 2. Assortment choice in the yogurt category.

the 113 households used for model calibration,3 32 did not make any purchases of these two brands dur-out period. For this reason, only 81 households enter

t data.

K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933 25

Table 2Joint purchase table.

on flavors

MX BY PCH RBY ST OT Total

Nordica avoCherry Mixed BerryPeach Raspberry Strawberry Others Dannon avoCherry Mixed Berry Peach Raspberry Strawberry Others

Total

Note: For cla odesPCH = peach,

in the marforth. We asuch occas

Model Spec

We first

phjt = phj

+ where p infor Nordicainfluence oture advertvariable thto buy the LOY

phj = l

ber of purcnh shoppinand NJ is name (NJ =are alwaysassume thahousehold.

Next, wh, followtwo brandsassortmentnever be obbetween an, thus to zero. Ho

r Dhjutionse refromonderal

[

Nh(NorNordica flavors Dann

CH MX BY PCH RBY ST OT CH

rs164 44 45 34 54 69 0

44 165 25 43 49 94 0 45 25 159 37 62 74 0 34 43 37 177 54 81 0 54 49 62 54 233 84 0 69 94 74 81 84 341 0

rs0 0 0 0 0 0 294 0 0 0 0 0 0 67 0 0 0 0 0 0 33 0 0 0 0 0 0 59 0 0 0 0 0 0 59 0 0 0 0 0 0 123

410 420 402 426 536 743 635

rity, row totals, columns totals and diagonal entries are shown in boldface. C RBY = raspberry, ST = strawberry, and OT = other.

ket at certain time points due to stock-outs, and solso adjusted the model specification to accommodateions.

ication

define individual SKUs baseline utility chjt as

+ p1,hPRICEpht + p2,hDISPpht + p3,hFEATphtp4,hjLOY

phj (9)

dicates a brand name, here taking on two values, 1 and 2 for Dannon. This specification allows for the

(Nhjk o

substitThe

ments correspthe gen

h =

where brand f price (PRICE), in-store display (DISP) and fea-ising (FEAT). The notation LOYphj denotes a loyaltyat adjusts for the households long-run propensityitem (flavor) j within a cluster (brand). We defineog((nphj + 0.5)/(nh. + 0.5NJ )) where nphj is the num-hases of the item j (in brand p) across the householdsg trips in the initial eight months of the dataset,the number of (flavor) SKUs within each brand

6). Because marketing mix variables (such as price) the same for all flavors of a particular brand, wet marketing mix parameters vary only by brand and

e specify the structure of the cross-effect matrixing the structural restrictions shown in Fig. 2. The

are treated as perfect substitutes, implying that any containing both Nordica and Dannon products willserved. That is, we restrict the cross-effects

(N,Dhjk

)y two items having different brand names to be

setting the choice probabilities of these assortmentswever, the cross-effects for flavors within a brand

must be syWe com

tions abouheterogeneeffect parahousehold ships

hN(, hN( , hN( ,

where hmarketing

In generfor the pcovariancefor simpliconal. Alth0 0 0 0 0 4100 0 0 0 0 4200 0 0 0 0 4020 0 0 0 0 4260 0 0 0 0 5360 0 0 0 0 743

67 33 59 59 123 635402 50 93 88 187 887

50 223 54 48 95 50393 54 521 90 184 100188 48 90 400 180 865

187 95 184 180 823 1592

887 503 1001 865 1592 8420

for flavors are as follows: CH = cherry, MX BY = mixed berry,

k

)are unrestricted (allowing for varying degrees of

, complementarity or independence).strictions reduce the total number of possible assort-

4095 (=212 1) to 126 (= 2 (26 1)). Theing cross-effect matrix for all brands and flavors has

form

Nh

Dh

]1212

(10)

and Dh are 6 6 matrices for the flavors within eachdica and Dannon, respectively). These sub-matrices

mmetric, but need not necessarily be equal.plete the model specification by developing assump-t the pattern of household response parameterity. Define the vector containing all unique cross-meters as h. Using this notation, we assume thatresponse heterogeneity is governed by the relation-

)))

(11)

and h are vectors of item specific intercepts andmix parameters.al, we can estimate any type of covariance structurearameters by properly specifying the variance-

matrices in (11). In this application, we assume,ity, that all variancecovariance matrices are diag-ough this assumption requires that parameters

26 K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933

Table 3Model comparison.

Model description k LL Cal 2 Cal AI 2

No Cross-Effects 58 4089.1 0.4109 82Same Cross-Effects 88 4053.1 0.4161 82Differential C 82

Note: Cal de defi2 = 1 (LL/ d of tto zero. The b

independenmarketing As we shothe fact thazero.

Model Cal

We introthe assortmhousehold standard fo

L =h

t

where, as nHowever, gto constructhe random

Lh =

t

where repgral can behousehold-

Lh 1R

r

which, in tan approximtime points

Maximito all modlihood (MSfashion arewe implemobtained u2003). Simcorrelated tribution. Bthe MSL p

tes ore ac

1,000 randexce

aramds. Ftion

Com

ng t prog the

com

t sub, ass

ting n thiclud

, call be sed sub-ird bes th, thale 3valut daian In, a

mo

l as hDiffeross-Effects 118 4019.1 0.4209 notes calibration data and Hold denotes holdout data. Fit measures are

LL null) where k is number of parameters, N is sample size, LL is log likelihoooldface entries denote the model with the best performance on each measure.

tly vary across households, it does not imply thatactions of one item have no impact on other items.w below, such cross-item dependencies exist due tot the cross-effect matrix in Eq. (10) is not equal to

ibration

duce the indicator variable Ybht , which equals one ifent b (zbht) is chosen and zero otherwise. Ignoring

parameter heterogeneity, the likelihood takes on thermb

{Pr(zht = zbht)}Ybht (12)

oted earlier, h denotes household and t denotes time.iven our heterogeneity assumptions, it is necessaryt the likelihood of each household by integrating out

components of the parameters as

b

{Pr(zht = zbht|)}Ybht f ()d (13)

resents all random elements in the model. This inte- approximated using R draws of simulated values ofvarying parameters. In this way, we obtain

L(r)h =

1R

r

t

b

{Pr(zht = zbht|(r))}Ybht (14)

urn, leads to LL(, , ) h log Lh(h, h, h),ation to the true log likelihood of all households and

.

estimabe mousing text ofreport level pmethoestima

Model

Usiof ouralterinmodelperfecEffectsrestriczero. Ieters inmodeleters toare baeffect The thassum

brandsTab

report holdou(BayescriterioEffectsas welto the zation of the approximate log likelihood with respectel parameters generates Maximum Simulated Like-L) estimates. Parameter estimates obtained in this

known to be consistent (Lee 1995). In this research,ented the MSL procedure with R = 250 simulates

sing Randomized Halton Sequence methods (Trainulates drawn from a Halton Sequence are negativelyand evenly cover the support of the parameter dis-oth features substantially improve the accuracy of

rocedure. Train (2003) reports that MSL parameter

Parameter

Table 4for the Difffor price, vor. Moreothe model in its promis, in geneC BIC LL Hold Hold

94.1 8600.1 670.4 0.243182.2 8746.4 678.3 0.234175.7 8898.9 665.0 0.2493ned as follows: AIC = 2LL + 2k, BIC = 2LL + log(N)k andhe model and LL null is the log likelihood with all parameters set

btained using 100 draws from a Halton Sequence cancurate than MSL parameter estimates obtained with

randomly drawn simulates. Moreover, in the con-om coefficient logit models, Huber and Train (2001)llent correspondence between estimates of householdeters obtained using MSL and Hierarchical Bayesor these reasons, MSL was deemed an appropriatemethodology for our model.

parison

he MSL methodology, we tested several variationsposed model on the yogurt purchase histories by

market structure implied by the h matrix. In allparisons, we retained the assumption that brands arestitutes. The first benchmark model, called No Cross-umes independence among flavors within each brand,all elements in the cross-effect matrix (h) to bes instance, we only estimate marketing mix param-ing item specific intercepts. The second benchmarked Same Cross-Effects, allows cross-effects param-estimated, imposing the restriction that cross-effectson flavor (not on brand). This makes the cross-

matrices identical across brands, that is, Nh = Dh .enchmark model, called Differential Cross-Effects,at cross-effect parameters vary across flavors andt is, Nh /= Dh .

displays the fit statistics for different models. Wees of the log-likelihood (for both calibration and

ta), 2, AIC (Akaike Information Criterion) and BICnformation Criterion). With the exception of the BICll fit statistics point toward the Differential Cross-del (shown in the last row of Table 3) in calibrationoldout data. Given these results, we restrict attentionrential Cross-Effects model.Estimates

presents parameter estimates for baseline utilitieserential Cross-Effects model. Recall that coefficientsfeature and display vary by brand, but not by fla-ver, the display variable for Dannon is omitted frombecause Dannon (in our dataset) never uses displayotions. The parameter values suggest that Dannonral, the preferred brand (larger brand intercepts).

K. Kwak et al. / Journal of Retailing 91 (1, 2015) 1933 27Ta

ble

4M

odel

coef

ficie

nts.

Inte

rcep

t

Loya

lty

Pric

e

Disp

lay

Feat

ure

Mea

n

SD

Mea

n

SD

Mea

n

SD

Mea

n

SD

Mea

n

SD

Nord

ica

Cher

ry

3.21

4**

(0.62

8)

0.81

0**

(0.14

9)

2.45

5**

(0.30

6)

0.28

9**

(0.08

6)

3.8

52**

(0.55

2)1.

707*

*(0.

144)

0.22

3(0.

137)

0.18

0(0.

134)

0.73

7**

(0.10

4)0.

409*

*(0.

119)

Mix

ed

Ber

ry

2.60

8**

(0.53

9)

0.08

9

(0.15

7)

2.09

9**

(0.25

4)

0.01

2

(0.08

3)Pe

ach

2.44

7**

(0.58

6)

0.63

0**

(0.16

3)

1.98

1**

(0.28

7)

0.33

5**

(0.07

4)R

aspb

erry

2.21

8**

(0.46

5)

0.22

3

(0.16

8)

1.60

8**

(0.24

9)

0.16

1*

(0.07

9)St

raw

berry

1.73

9**

(0.46

5)

0.51

8**

(0.15

1)

1.18

4**

(0.20

4)

0.31

0**

(0.08

6)O

ther

s

2.55

8**

(0.46

3)

0.00

1

(0.11

6)

1.39

1**

(0.21

3)

0.11

4**

(0.08

9)Dan

non

Cher

ry

4.72

9**

(0.77

4)

0.39

0

(0.19

1)

1.66

8**

(0.19

0)

0.36

1**

(0.07

9)

5.4

30**

(1.01

1)1.

866*

*(0.

147)

NA

NA

1.01

8**

(0.19

5)0.

678*

*(0.

167)

Mix

ed

Ber

ry5.

229*

*

(0.79

5)

0.54

0**

(0.11

5)

1.99

3**

(0.23

7)

0.49

6**

(0.08

9)Pe

ach

4.62

6**

(0.82

3)

0.00

4

(0.17

0)

2.02

1**

(0.27

3)

0.48

7**

(0.09

9)R

aspb

erry

5.60

0**

(0.74

6)

0.79

1**

(0.18

3)

2.16

8**

(0.19

5)

0.27

6**

(0.08

2)St

raw

berry

4.00

6**

(0.77

0)

0.18

5

(0.16

5)

1.31

2**

(0.22

2)

0.56

7**

(0.07

6)O

ther

s

4.83

8**

(0.71

5)

1.08

9**

(0.13

7)

1.65

8**

(0.18

4)

0.35

8**

(0.05

3)No

te:

The

no

tatio

n

SD

indi

cate

s sta

ndar

d

dev

iatio

n

of t

he

hous

ehol

d

hete

roge

neity

distr

ibu

tion.

Sign

ifica

nce

level

s are

deno

ted

as

p

0 for complements,

for substitutes, and S(j, k) = 0 for independence.pute aggregate elasticities, we first define the over-hares as MSj =

h

tPr(j)ht/NT where NT (the

useholds (N) times average number of shopping tripssehold) is the number of total observations in a par-

set. Following procedures outlined in Russell and(1994), we differentiate this expression and use thelevel derivatives in (B5) to infer market share elastic-rocess yields the expressions

MS(jp)RICE

pj

=[{

hphPr(jp)h[1 Pr(jp)h]

}/N

MS(jp)

]PRICE

pj

MS(jp)PRICE

qk

= [{

hqhPr(jp)hPr(kq)h

}/N

MS(jp)

]PRICE

pk

MS(jp)RICE

pk

=[{

hph

[Pr(jp, kp)h Pr(jp)hPr(kp)h

]}MS(jp)

]PRIC

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An Analysis of Assortment Choice in Grocery RetailingIntroductionVariety of an AssortmentVariety and Product QualityModeling Assortment Choice

Overview of Research

Multivariate Logistic Model of Assortment ChoiceModel DescriptionModel InterpretationRestricting the ModelConstraining the MatrixSummary

Assortment Choice in the Yogurt CategoryData DescriptionModel SpecificationModel CalibrationModel ComparisonParameter Estimates

Variety and Quality in the Yogurt CategoryQuality Tiers in the Yogurt CategoryProduct Quality and Assortment VarietySummary

ConclusionsSubstantive ContributionsModeling Assortment ChoiceLimitations and Extensions

Appendix A Derivation of Assortment ModelAppendix B Derivation of Price ElasticitiesReferences