frecuency prices

The relationship between purchase regularity and propensity toaccelerate

Demetrios Vakratsasa,*, Frank M. BassbaMcGill University, Montreal, PQ H3A 1G5, CanadabUniversity of Texas at Dallas, Dallas, TX, USA

AbstractIdentifying random and regular-buying household segments and relating them to demographic and shopping characteristics has been the

focus of many marketing studies. Missing from the marketing literature, however, is a study that relates purchase regularity to marketingmix sensitivities. Such study could provide substantive implications since it would explore a practical dimension of a segmentation schemebased on purchase regularity. In this article, we investigate the relationship between purchase regularity and propensity to accelerate throughthe use of a mixture Weibull model of purchase timing. Applying this perspective to purchase timing data on four product categories(ketchup, sugar, bathroom tissue, margarine), we show that in the frequently purchased categories of bathroom tissue and margarine, randombuyers do not exhibit any propensity to accelerate while regular buyers do. In the occasionally purchased categories of ketchup and sugaron the other hand, random buyers exhibit at least as much propensity to accelerate their purchases as regular buyers do. Our rationale forthese results is based on information-theoretic arguments suggesting that propensity to accelerate depends on the frequency at which aproduct category is purchased. 2002 by New York University. All rights reserved.

Keywords: Purchase acceleration; Purchase regularity; Mixtures of hazard rates

Introduction

The regularity of household purchases is a topic that hasreceived a lot of attention from marketing researchers. Paststudies have mainly focused on identifying households asregular or random buyers (Schmittlein and Morrison,1983; Dunn, Reader and Wrigley, 1983; Gupta, 1988;Wheat and Morrison, 1990; 1994; Bawa and Ghosh, 1990).Households that make category purchases in relativelyfixed, clockwork-like, intervals are characterized as reg-ular, whereas households that make category purchases inrather irregular, seemingly random, intervals are character-ized as random (Bawa and Ghosh, 1990; Wheat and Mor-rison, 1990).What the marketing literature lacks, however, is a study

that relates purchase regularity to marketing mix sensitivi-ties. Such study could exploit practical aspects of a segmen-tation scheme based on purchase regularity, advancingtherefore research on the subject to issues beyond measure-ment. For example, the lack of regularity in the purchases of

a household identified as random may reflect a considerableflexibility in that households buying schedule. The routinepattern of the purchases of a household identified as regularon the other hand, may indicate that such household isbound by time constraints that forces it to buy in highlyregular time intervals (say, once every two weeks).The implied flexibility in the random households pur-

chasing behavior may allow such household to adjust thetiming of a purchase (buy earlier) in order to take advantageof a promotional event, a phenomenon called purchaseacceleration (Neslin, Henderson and Quelch, 1985). In suchcase, the retailer can identify random buyers as importantdrivers of store sales during promotional activities makingtherefore segmentation based on purchase regularity a use-ful tool. Identifying segments that accelerate their categorypurchases due to promotional activities is of considerableimportance especially after a recent multicategory study byBell, Chiang and Padmanabhan (1999), found that the effectof promotions on primary demand is substantial and greaterthan previously estimated.In this article we investigate the relationship between

purchase regularity and propensity to accelerate through theuse of a mixture Weibull model of purchase timing thatcharacterizes consumer segments both in terms of their

* Corresponding author. Tel.: !1-514-398-2052.E-mail address: [email protected] (D. Vakratsa).

Pergamon

Journal of Retailing 78 (2002) 119129

0022-4359/02/$ see front matter 2002 by New York University. All rights reserved.PII: S0022-4359(02)00068-4

purchase regularity and their response to price and promo-tional activities. An application to purchase timing data onfour product categories (ketchup, sugar, bathroom tissueand margarine) suggests that random buyers do not respondto prices and promotions uniformly across all product cat-egories.More specifically, in the frequently purchased categories

of bathroom tissue and margarine, random buyers do notexhibit any propensity to accelerate whereas in the occa-sionally purchased categories of ketchup and sugar, randombuyers exhibit at least as much propensity to acceleratepurchases as regular buyers. Our rationale for these resultsis based on information-theoretic arguments suggesting thatpropensity to accelerate depends on the frequency at whicha product category is purchased. Random buyers consis-tently purchased with a lesser frequency and exhibitedlower inventory sensitivity than regular buyers.The rest of the article is organized as follows: In Section

2 we provide a detailed definition of purchase regularity anda review of the relevant literature. Section 3 discusses themodeling approach and its advantages. Section 4 covers theempirical application by providing a short description of thedata and an extensive discussion of the results from theestimation of the mixture Weibull model. Section 5 carriesthe conclusions and directions for future research.

Literature review

Before discussing conclusions from previous research onpurchase regularity, it is important to provide a comprehen-sive definition of purchase regularity especially since muchof past research has focused on its measurement. Wheat andMorrison (1990) note that regular (or routine) buyers makecategory purchases in clockwork-like time intervalswhereas random buyers make purchases in rather irregu-lar, seemingly random, time intervals. More specifically theinterpurchase times of a regular buyer exhibit a steadypattern consisting of intervals of similar lengths with verylittle noise.This similarity in the interpurchase times across purchase

occasions of a regular buyer further suggests that the timingof the next purchase of a regular buyer is strongly dependenton the time elapsed since the last purchase. The interpur-chase times of a random buyer on the other hand, producea pattern which exhibits significant noise consisting of bothshort and long interpurchase intervals, implying that thetiming of a random buyers purchase is weakly dependenton the time elapsed since the last purchase. The differencesin the purchasing patterns between a regular and a random-buying household are illustrated in Fig. 1, which comparesthe sequence of margarine interpurchase times for two suchhouseholds.The pattern of a households interpurchase times, char-

acterized by its noise and the dependence of the currentpurchases timing on the time elapsed since the last pur-

chase, is therefore what separates regular from randombuyers. Two distributions that produce different patterns forhousehold interpurchase times, and have been extensivelyused in the past to characterize random and regular purchas-ing behavior are the exponential and the Erlang-2 distribu-tions (Bawa and Ghosh, 1990; Dunn, Reader and Wrigley,1983; Gupta, 1988). Both these distributions are specialcases of the gamma distribution for different values of itsshape parameter. The exponential distribution exhibits thememoryless property suggesting that a consumers pur-chase timing does not depend on the time elapsed since thelast purchase. Furthermore, it has a coefficient of variation(ratio of the standard deviation to the mean) of one implyinghigh levels of variability in the data. Both these properties ofthe exponential distribution make it appropriate for describ-ing the pattern of a random buyers purchases and thuscharacterizing random purchasing behavior. The Erlang-2on the other hand, produces a low coefficient of variationand an increasing hazard rate implying that the propensity topurchase increases with the time elapsed since the lastpurchase. This makes this distribution appropriate for themodeling of regular purchasing behavior.Wheat and Morrison (1990) proposed an alternative

measure of purchase regularity called the M-statistic, de-fined as the ratio of a households randomly chosen inter-purchase time over the sum of all interpurchase times of thathousehold. Higher values of the M-statistic imply higherregularity. The major advantage of this statistic lies in thefact that it is robust with respect to the available number ofobservations (interpurchase times) per household and that itneeds at minimum only two interpurchase times to measurea households purchase regularity.Empirical studies have generally found that consumers

vary significantly in their purchase regularity for categorypurchases (Dunn, Reader and Wrigley, 1983; Gupta, 1988;Bawa and Ghosh, 1990; Wheat and Morrison, 1990). Dunn,Reader and Wrigley (1983), using data from England forpurchases of bathroom tissue and baked beans, reported thatthe hypothesis of exponentially distributed interpurchase

Fig. 1. Purchase sequences for a regular and a random household:margarine

120 D. Vakratsas, F. M. Bass / Journal of Retailing 78 (2002) 119129

times, and hence of random purchasing behavior, cannot berejected for most households. They found that the house-holds for which the random purchasing hypothesis is re-jected are more frequent (heavy) buyers, thus suggesting arelationship between purchase frequency and purchase reg-ularity. Wheat and Morrison (1990), using the M-statistic,and Gupta (1988), using the shape parameter of the Erlang-2distribution, on the same IRI coffee data set concluded thathouseholds appear to be more regular than random contrast-ing therefore the conclusion of Dunn, Reader and Wrigley(1983). Bawa and Ghosh (1990) took the issue of purchaseregularity beyond measurement and characterized randomand regular buyers in terms of their aggregate shoppingbehavior and demographics. Similarly, Dunn, Reader andWrigley (1983) found that random buyers purchase lessfrequently than regular buyers (i.e., they are lighter buyers),but also concluded that random buyers make proportion-ately more purchases on deal and are less brand loyal thanregular buyers.The results of Bawa and Ghosh (1990) regarding the

aggregate shopping behavior of random buyers suggest thatthe lack of routine in the purchasing behavior of randombuyers may be associated with a response to prices andpromotions which is higher than that of regular buyers. Thislack of regularity therefore may indicate a greater flexibilityin the purchasing behavior of random buyers allowing themto exhibit opportunistic buying behavior where their cat-egory purchasing decisions are significantly affected bypricing and promotional activities (Bucklin and Lattin,1991; Bell and Lattin, 1998).In this study we attempt to take a closer look at a possible

relationship between random purchasing and response toprices and promotions on the purchase occasion rather thanthe aggregate level (as, for example, in Bawa and Ghosh). Itshould be noted that the intent of our study is to relatepurchase regularity to marketing mix sensitivities by mod-eling the regularity in timing of purchases of particularproduct categories for which we have full information onprice and promotional activities. Other studies (Kahn andSchmittlein, 1989; Kim and Park, 1997) have focused onstudying regularity in the timing of shopping trips (i.e.,visits to the grocery store regardless of the product catego-ries purchased). In the latter case, however, full informationon price and promotions on every product category pur-chased on every single shopping trip may not be available(e.g., Kim and Park, 1997), thereby prohibiting the study ofthe relationship between purchase regularity and marketingmix sensitivities.

Modeling approach

We assume that there are S household segments eachuniquely characterized by its purchase regularity and pro-pensity to accelerate. In order to effectively model eachsegments regularity and propensity to accelerate, we chose

the proportional hazard rate approach. This has been fre-quently used in studies of purchase timing behavior (Jainand Vilcassim, 1991; Helsen and Schmittlein, 1993; Gonuland Srinivasan, 1993). Under the proportional hazards for-mulation, each households hazard rate is the product ofthree factors:a) A baseline hazard rate, which captures the house-holds underlying purchasing pattern.

b) The covariate effects, which in our application referto the effect of marketing mix activities on the house-holds hazard rate.

c) An unobserved heterogeneity factor that captures thehouseholds intrinsic purchasing propensity.

We further assume that each segments baseline hazardrate follows the Weibull specification. The proportionalhazard model then takes the following form:

hi"s#tij$ ! exp#"0s# "1sln t$exp#Xij% $s$%s (1)Where:i & 1, . . . , I denotes the householdj & 1, . . . , Ji denotes the interval between purchases

for household is & 1, . . . , S denotes the segmenttij & duration of the jth interval (jth interpurchase time)

in days for the ith householdxij & vector of marketing mix variables in the end of jth

interval for household i%s & scale heterogeneity for segments.Under this formulation, acceleration effects are captured

in the ability of marketing mix activities (namely prices andpromotions) to increase the hazard rate, which translatesinto a decrease in the interpurchase time. This means that byincreasing a households hazard rate, a promotional eventdecreases the interpurchase time and causes an acceleratedpurchase.The density function fi"s and the cumulative distribution

function Fi"s corresponding to the Weibull hazard functionare:

fi"s#t$ ! exp#"0s# "1s ln t$ exp# x%ij $s$%s

& exp' ' exp# x%ij $s$%s!0

texp#"0s

# "1s ln u$du( (2)

FI"s#t$ ! 1 ' exp' ' exp# x%ij $s$%s!0

texp#"0s

# "1s ln u$du( (3)Depending on the value of its shape parameter "1s, the

Weibull hazard rate can produce from constant to sharplyincreasing hazard rates. If "1s & 0 for a particular segment

121D. Vakratsas, F. M. Bass / Journal of Retailing 78 (2002) 119129

s, the hazard rate in (1) becomes constant and the Weibullhazard rate reduces to the exponential which has been thebenchmark for characterizing random purchasing behavior.Large positive values of "1s on the other hand, producesharply increasing hazard rates implying that the instanta-neous probability of (re) purchase is lower immediatelyafter a purchase is made, allowing thus for significant in-ventory effects. Large positive values of the Weibull shapeparameter indicate more regular purchasing behaviorwhereas small (closer to zero) values are indicators of ran-dom purchasing behavior. The characterization of a randompurchasing segment is, therefore, relative (i.e., being the lessregular segment), consistent with previous empirical studies(Bawa and Ghosh, 1990; Gupta, 1988; Wheat and Morrison,1990).The choice of a Weibull mixture for modeling purchase

regularity raises a natural question: Why not use a mixtureof Erlang-2 and exponential instead, since they have beenused as benchmarks for identifying regular and randombuying segments respectively? There are several advantagesthat the Weibull mixture model offers in the context of ourstudy. First, the shape parameter of the Weibull is allowedto vary. As a result it can accommodate both increasing(Erlang-2) and constant hazard rates (exponential for "1s &0). In other words, Weibull is a more flexible distributionthan Erlang-2. Second, the Weibull hazard rate can berepresented in a proportional hazard formulation (Eq. (1))allowing therefore the modeling of marketing mix effects ona households hazard rate. Erlang-2 on the other hand, doesnot have a closed-form representation in the proportionalhazard framework (Jain and Vilcassim, 1991), thus prohib-iting the study of the relationship between purchase regu-larity and propensity to accelerate. The M-statistic (Wheatand Morrison, 1990) has a similar limitation since it cannotdirectly incorporate the effects of marketing mix activities.A third advantage gained from the use of a Weibull

mixture model is that it can produce a nonmonotonic ag-gregate hazard rate similar in shape to the ones previouslyobserved for purchase timing data in marketing (Jain andVilcassim, 1991; Gonul and Srinivasan, 1993). Nonmono-tonic hazard rates in the Weibull mixture specification maybe obtained as the result of aggregation of hazard rates withdifferent shape and scale parameters (Vaupel and Yashin,1985). Finally the Weibull mixture model can capture theeffects of marketing mix on time itself (in other words it canproduce time elasticities) rather than just on the hazardrate. This is extremely important since hazard rates areunobservable and in practice retailers would like to examinehow many days in advance a household is going to buy (i.e.,accelerate its purchase) due to a promotion. This ability ofthe Weibull mixture model to provide time elasticities sig-nificantly strengthens the practical benefits of our studysince random and regular purchasing behavior is explicitlyassociated with changes in the timing of a householdspurchase.

The time elasticity, the percentage change in time for aunit change in a marketing mix variable is given by:

"exp") $s"1s# 1# ' 1# & 100While for a promotional dummy variable such as Feature

or Display, a unit change can be meaningful (say from nodisplay to display) unit changes in price may not be realistic.In this case it is preferable to calculate the effect of apercentage cut in price on the timing of a purchase. In suchcase the effect of a 10% price cutoff a baseline price x0 onthe percentage change in the timing of a purchase is pro-vided by the following formula:

exp") $s0.9x0"1s# 1# ' exp") $sx0"1s# 1#exp") $sx0"1s# 1# & 100

A final aspect of the model that needs to be resolved isthe specification of the segment-specific scale heterogeneity%s. We employ the Gamma distribution to model the heter-ogeneity in the scale parameter of each segment. TheGamma distribution implies the following probability den-sity function gs for %s:

gs#%s$ !bsas

*#as$%sas)1 exp# ' bs%s$,%s( 0 (4)

Since we have already included an intercept parameterfor the hazard rate of each segment ("0s) we restrict thescale parameter of gamma heterogeneity a, to be equal to theshape parameter b to ensure model identification (Gonul andSrinivasan, 1993).An alternative way of modeling segment-specific scale

heterogeneity would be to use a nonparametric approachwith a finite number of support points. Proponents of suchan approach argue that it is a flexible method (Jain andVilcassim, 1991). The Gamma distribution however, isquite flexible since it does not impose any symmetry re-strictions and can take on a variety of shapes depending onits parameters. Such flexibility has been demonstrated innumerous household purchase timing studies incorporatinggamma heterogeneity (Jeuland, Bass, and Wright, 1980;Gupta, 1991; Gonul and Srinivasan, 1993; Kim and Park,1997). Furthermore Kim and Park (1997) found that a con-tinuous gamma heterogeneity approach is more parsimoni-ous than the finite mixture approach in the number ofparameters that uses.The likelihood function across spells for household i

conditional on the assumption that it belongs to segment s,is then given by:

Li$s! !+s

%j&1

Ji, fi$s#tij$xij,%s$du-)ij

& ,1 ' Fi$s#tij$xij,%s$-1))ijgs#%s$d%s (5)


Where:)ij & 1 if jth spell for the ith household is complete

& 0 if the jth spell for the ith household is right censored

and +s is the domain of %s.Alternatively, using the expressions (2), (3) and (4) one

could express the above likelihood function as follows:

Li$s! !0

. %j&1

JI,exp#"0s# "1slnt$exp# xij%$s$%s

& exp' ' exp# x%ij$s$%s !0

tijexp#"s0

# "s1ln u$du(-)ij (6)

& ,exp' ' exp# x%ij$s$%s!0

tijexp#"0s

# "1slnu$du(-1))ijbsas

*#as$%sexp# ' as%s$d%s

The likelihood in (6) has a closed-form expression,which makes its estimation relatively easy. The overalllikelihood (across households, segments and spells) is thengiven by:

L ! %i&1

I &s&1

SpisLi$s (7)

Where pis denotes the probability of household i belongingto segment s.We model pis by using the logit formulation. Since we

expect segments to vary both in terms of their regularity andtheir sensitivity to marketing mix variables, such differ-ences may be related to household demographics (Bawa andGhosh, 1990). We therefore express segment membershipprobabilities in terms of the demographic variables of thehousehold:

pis!exp#"! s# *! sdi$

&k&1

Sexp#"! k# *! k di$

s ! 1, . . . , S (8)

To ensure that the above probabilities add to 1, we stan-dardize with respect to the probability of belonging to seg-ment S:

pis!exp#"s# *sdi$

1 # &k&1

S)1exp#"k# *k dk$

s ! 1, . . . , S ' 1 (9)

Where di is a vector of demographic variables for house-hold i, with piS & 1 ) k & 1S ) 1 pis and "s & "! s ) "! S, *s &*! s ) *! S. Segment S therefore becomes the baselinesegment, with *s reflecting the difference in the effect of thedemographic variables on the probability of membership insegment s from the effect of the demographic variables onthe probability of membership in segment S (Gupta andChintagunta, 1994). The probabilities pis of assigning agiven household to one of the S segments can be updated bymeans of an empirical Bayes procedure based on the infor-mation on household purchase histories (Bucklin andGupta, 1992; Kamakura and Russell, 1989). The posteriorprobability of classifying household i to segment s given itsvector of purchase history xi and the estimated vector ofparameters $ is provided by the following formula:

pis$$, xI!pisLi$s

&m&1

SpimLi$m

(10)

These posterior probabilities can in turn be used to clas-sify households as regular or random buyers and calculaterelevant statistics for the random and regular purchasingsegments. We set the number of segments S to 2 to allow forthe presence of both a random and a regular purchasingsegment and maximize L (or equivalently logL) by imple-menting the BFGS (Broyden, Fletcher, Goldfarb and Sh-anno) iterative algorithm in GAUSS.

Data and results

Data

We used AC Nielsen scanner data from an ERIM market(Springfield, MO) on household purchases of four productcategories: ketchup, sugar, bathroom tissue and margarine.Consumers purchase two of the categories (tissue and mar-garine) frequently whereas the other two are purchasedoccasionally. We selected households that made at least sixpurchases (i.e., five completed interpurchase times) during a104-week period in all four product categories. While suchrule may result in a bias towards more frequent (heavy)purchasing households, it is necessary to ensure consistentestimation of the purchase timing distribution for eachhousehold on each category and allow for meaningful com-parisons across categories. The use of this rule resulted inthe identification of 272 households that made at least sixpurchases in all four categories. Price per ounce and ab-sence/presence of feature or display (a dummy variable wecall FoD) are the marketing-mix variables in our analysis.Relevant summary statistics for the four product categoriesare provided in Table 1.We included a volume variable in order to introduce

inventory effects. This measures the amount (in ounces)


bought on the previous purchase occasion expressed as theproduct of number of units purchased multiplied by the sizeof the unit. Thus, the combination of the volume variableand the duration terms (scale and shape) of the Weibullhazard rate capture inventory effects in our model (Jain andVilcassim, 1991). In terms of demographic variables thataffect the segment membership probabilities, we chosehousehold size (number of household members), householdincome, female and male head of household working statusand female and male head of household college education.Small households, since they buy less frequently, are

expected to be more flexible in terms of their inventory andtheir purchasing schedule and should thus be more likely toexhibit random purchasing behavior. Although there hasbeen a lack of broad agreement on the effect of income onpromotional response (Blattberg and Neslin, 1990), somestudies (e.g., Blattberg et al., 1978) have suggested thatincome effects may be nonlinear with middle-incomehouseholds being in fact more deal prone. We thereforedivided the sample based on the income variable into bot-tom third (low income), middle third (middle income)and upper third (high income) and created a low and amiddle income dummy variable making therefore the highincome category the baseline. However, we did not for-mulate any hypotheses or conjectures regarding the effect ofincome on segment membership probability due to the lackof consistent empirical evidence.Both female and male head working status are variables

that can signal a households opportunity costs of the timerequired for purchasing activities. Households where thefemale and/or male head is not working may have moretime to evaluate promotions and therefore be more prone torespond to them. Such households may show higher flexi-bility in the timing of their purchases and thus be morelikely to exhibit random purchasing behavior. We usedummy variables for both female and male head workingstatus with 1 denoting full time employed and 0 otherwise.Finally, the male and female head education variables canalso affect purchase regularity. More educated householdsare more likely to be pressed for time due to more demand-ing work schedules, a factor that can limit their purchasingflexibility and force them into regular purchasing behavior.This variable too is expressed as a dummy with 1 noting thatthe head of the household had at least college education and0 otherwise.

Results

The estimation results for the Weibull mixture model forthe four product categories are reported in Table 2. In eachof the categories, the two segments differ considerably withrespect to their hazard shape parameters. Since we use theshape of the Weibull hazard rate to characterize random andregular buyers, those differences suggest that the two seg-ments in each product category vary considerably in termsof their purchase regularity. Thus, in each category thesegment with the smaller shape parameter is the randomsegment, and the segment with the larger shape parameter isthe regular segment.Regular buyers also appear to have higher hazard rate

intercepts (scales) than random buyers in all four categoriessuggesting that regular buyers have higher hazard rates.Since higher hazard rates are associated with shorter inter-purchase times this further implies that on average regularbuyers purchase more frequently than random buyers. Weaddress the issue of purchase frequency in greater detail inthe posterior analysis section where we calculate importantstatistics regarding the purchasing behavior of random andregular buyers. Fig. 2(a-d) compares the hazard rates ofrandom and regular buyers across all four categories.1 It isnot difficult to observe that hazard rates for the randomsegment for each of the four categories are flatter than thecorresponding ones for the regular segment suggesting amore exponential-like behavior for random buyers.The price and promotional parameter estimates of Table

2 suggest that the response of the regular and randomsegments to prices and promotions cannot be generalizeduniformly across the four categories. Random buyers appearto be responsive to price and promotional signals (FoD) forthe occasionally purchased categories of ketchup and sugarbut not for the more frequently purchased categories ofbathroom tissue and margarine. Regular buyers on the otherhand, appear to be (especially price) sensitive in their pur-chases of bathroom tissue and margarine but only for sugarfrom the occasionally purchased categories.As indicated by t tests, differences in marketing mix

sensitivities of random and regular segments are statisticallysignificant at least for one of the two variables (Price orFoD), with the exception of the sugar category (Table 3).We provide a detailed explanation of the empirical resultsconcerning the response of random and regular segments to

Table 1Summary of Relevant Statistics for all Four Categories

No. ofpurchases

Average Coefficient ofVariation (standarddeviation)

Average Interpurchasetime in days (standarddeviation)

Average Interpurchasetime on a non-promotional occasion

Average interpurchasetime on a promotionaloccasion

Ketchup 3,383 0.69 (0.23) 70.7 (35.2) 72.4 (41.8) 66.9 (43.9)Sugar 5,414 0.82 (0.22) 46.6 (28.3) 47.7 (30.3) 39.9 (35.8)Bathroom Tissue 11,650 0.73 (0.23) 21.9 (12.1) 22.5 (13.0) 21.3 (15.8)Margarine 11,879 0.78 (0.24) 23.3 (17.9) 22.5 (16.7) 22.6 (26.6)


price and promotional activities after we characterize ran-dom and regular buyers in terms of some fundamentalstatistics of their buying behavior.The results also suggest that random buyers are less

inventory sensitive than regular buyers with the exceptionof sugar where both segments are equally insensitive toinventory. This low sensitivity to inventory exhibited by therandom buyers suggests that they should be more flexible interms of their purchasing schedule. Such flexibility mayhave been also inferred by the less regular pattern of pur-chases for the random buyers as we discussed in the intro-duction of the paper. However, as evidenced from the em-pirical results, flexibility does not always seem to translateinto a higher propensity to accelerate and can thus not fullyaccount for the response of that segment to promotionalevents.In the case of sugar, the hazard rate of both segments

appears to be positively related to the amount purchased inthe previous occasion. While this result is counterintuitive itmay be interpreted in the context of the particular category.Recall from the statistics reported in Table 1 that sugarpurchases exhibit the highest coefficient of variation amongthe four categories studied (0.82). This means that interpur-chase times for this category exhibit memory-less (expo-nential)- type patterns where sequences of short interpur-chase times are followed by sequences of longinterpurchase times and vice versa. The presence of a se-quence of short interpurchase times implies that households

keep repurchasing fast although volume is accumulated dueto recent purchases occurring in short intervals. Converselythe presence of a sequence of long interpurchase timesimplies that households delay their purchases although theymay run low on quantity due to few purchases occurring inlong intervals. Both cases suggest that the larger (smaller)the amount on hand, the higher (lower) the propensity tobuy the product, explaining the positive effect of inventoryon the hazard rate.Of the demographic variables potentially affecting mem-

bership in either segment, household size has consistently anegative effect on the probability of belonging to the ran-dom segment implying that random purchasing householdsare smaller than regular buying households. This is consis-tent with the finding that regular buyers have higher hazardrates (i.e., are more frequent buyers), since larger house-holds are more frequent buyers. Income does not appear tohave a significant effect on random segment membershipfor any of the categories and employment of either head ofthe household, whenever significant, is negatively related tomembership in the random segment with the exception ofsugar.This propensity of households where both heads are

employed to belong to the regular segment is consistentwith an opportunity cost of time explanation, according towhich such households are constrained in to a more regularpurchasing schedule due to their high costs of time. Thepositive effect of female head employment status on the

Table 2Parameter estimates for the two-segment model for all four categories (t-ratios)Variable Ketchup Sugar Bathroom Tissue Margarine

Random Regular Random Regular Random Regular Random RegularIntercept (scale) )2.15

()7.26))0.90()3.17)

)1.60()22.79)

)0.97()10.13)

)0.53()14.78)

0.57(6.46)

)0.83()9.20)

)0.41(4.68)

Ln(T) (shape) 0.27(7.39)

0.80(16.51)

0.14(5.38)

0.49(16.23)

0.37(25.25)

1.02(40.05)

0.24(13.20)

0.74(28.96)

Price )8.76()2.68)

)1.59()0.31)

)0.77()3.97)

)0.76()3.65)

0.04(0.38)

)0.62()2.16)

0.11(1.53)

)0.19()2.81)

Feature or Display 0.04(2.01)

0.07(0.66)

0.13(2.45)

0.14(2.50)

0.01(0.61)

0.08(2.82)

)0.03()0.74)

0.06(1.69)

Volume 0.37(0.34)

)3.43()6.93)

0.36(2.24)

0.36(2.14)

)4.22()5.35)

)10.65()11.16)

)1.65()1.54)

)29.80()12.02)

Heterogeneity Variance 0.20(9.04)

0.62(11.51)

0.21(11.33)

0.31(7.75)

0.29(17.54)

0.64(12.02)

0.37(14.76)

0.35(13.67)

Effects on probability of belonging to the random segmentKetchup Sugar Bathroom Tissue Margarine

Intercept 3.09 (2.87) 1.99 (2.86) 2.43 (4.12) 2.25 (4.39)Household Size )0.42 ()2.37) )0.46 ()2.47) )0.20 ()3.55) )0.33 ()2.38)Low Income )0.49 ()0.92) )0.02 ()0.57) 0.03 (0.26) 0.19 (0.51)Middle Income 0.33 (0.67) 0.57 (1.09) 0.16 (0.15) 0.25 (0.72)Female Head Working Status )0.82 ()1.81) 1.26 (3.09) )0.40 ()1.17) )0.29 ()0.83)Male Head Working Status )1.07 ()1.69) )0.86 ()1.23) )0.84 ()1.92) )1.11 ()2.56)Female College Education )0.09 ()0.34) 0.11 (0.76) )0.59 ()1.91) )0.41 ()1.26)Male College Education 0.22 (0.68) 0.34 (0.55) 0.24 (1.64) 0.37 (1.10)Log-Likelihood )8,504.7 )11,571.7 )15,491.6 )15,814.2


random segment membership for sugar may be once againexplained on the basis of category usage. Since randombuyers are less frequent (light) buyers of the category andsugar can be used in time consuming baking activities towhich working heads of the household cannot commit, suchhouseholds tend to belong to the less frequent buying ran-dom segment. College education is generally not signifi-cantly related to segment membership.

Characteristics of random and regular buyers

In order to relate segment membership to purchase be-havior characteristics we first calculated the posterior prob-ability of segment membership for each household accord-ing to Eq. (10). While a standard practice for assigninghouseholds to either segment is to use the highest probabil-ity rule, where each household is assigned to the segmentfor which it produces the highest posterior probability(Gupta and Chintagunta, 1994), such rule can be fuzzy.Consider for example the case where a large number ofhouseholds have posterior probability of belonging to therandom segment of 60%. The highest probability rule willassign all such households to the random segment althoughthere is a considerable likelihood that they belong to theregular segment (40%). Such practice can potentially lead tobiases when calculating important purchase behavior statis-tics such as purchase frequency and time elasticities.To overcome the fuzziness of the highest probability

rule, we include all households when calculating the statis-tics for a particular segment (regular or random) and use theposterior probabilities of belonging to that segment asweights.2 Thus, in order to calculate the average interpur-chase time of the random (regular) segment we calculate theaverage of the interpurchase times of all householdsweighted by their corresponding posterior probability of

Table 3Statistical significance tests (t-ratios) for differences of price andpromotional coefficients between regular and random segmentsCoefficient Ketchup Sugar Bathroom Tissue MargarinePrice )2.28* 0.46 2.08 2.62FoD )0.28 )0.1 )1.82 )3.04* The entry should read: t-ratio for the difference between the estimated

price coefficients for the random and regular segments in the ketchupcategory.1 Since the coefficients for both the regular and random segments are

based on the same sample of households, we used a paired t-test based onthe following formula: t ! $1 ' $2'var#$1$# var#$2$' 2 cov#$1, $2$

where $1and $2 are the estimated parameters corresponding to the random andregular segments. The variances and covariance of the parameters arebased on the estimated variance-covariance matrix.

Fig. 2. (a) hazard rates for regular and random segments: ketchup; (b) hazard rates for regular and random segments: sugar; (c) hazard rates of regular andrandom segments: bathroom tissue; (d) hazard rates for regular and random segments: margarine


belonging to that segment (Table 4). Similarly, the size ofeach segment was calculated as the average of the corre-sponding posterior probabilities across all households. Fi-nally price cut elasticities for each segment were calculatedas the weighted average of each households elasticity for a10% price cutoff the average price paid by that household.There are some clear patterns emerging from the exam-

ination of Table 4, some of them having already beenpointed out in the description of the empirical results above.First, across all four categories buyers appear to be morerandom than regular since the size of the random segment ishigher than that of the regular buyers. This finding tends tobe more in agreement with Dunn, Reader and Wrigley(1983) and less with Wheat and Morrison (1990) and Gupta(1988). Second, random buyers appear to buy consistentlyless frequently than regular buyers, a finding we expectedby comparing the size of the scale parameters of the randomand regular segments in Table 2.Also as expected by their characterization based on the

shape of their corresponding hazard rates, random buyersexhibit a much higher coefficient of variation than theirregular counterparts with the margarine random segmentexhibiting the highest variation (0.91) implying an almost-exponential behavior. The time elasticities allow us to in-terpret the propensity of random and regular segments interms of the acceleration (percentage reduction of interpur-chase times) caused by prices and promotions. Promotionalsignal (FoD) elasticities appear to be higher (wheneversignificant) than price cuts confirming previous conclusionson the importance of advertised price reductions (Totten andBlock, 1987).Another important issue worth pursuing is whether the

purchase regularity exhibited by a household depends on theproduct category. In other words, can a random buyer ofsugar be a regular buyer of ketchup? By looking at thedifferences in random segment sizes across the four cate-gories one would tentatively answer yes to such question,since we studied the same households in all four categories.We further pursued the issue by calculating the correlationsamong the probabilities of belonging to the random segmentfor all four categories, since the assignment of households toeither segment has been done probabilistically rather than ina dichotomous manner. The corresponding correlations ap-pear in Table 5 and they are all statistically significant at the

1% level. Despite their statistical significance, no correla-tion is greater than 0.45 revealing a relative loose rela-tionship of random purchasing across the four product cat-egories. One possible interpretation of the numbers of Table5 is that a households purchase regularity varies acrosscategories.

The relationship between purchase regularity andpropensity to accelerate

Random buyers appear to accelerate their purchases onlyin the two occasionally purchased product categories ofketchup and sugar. Regular buyers on the other hand, ex-hibit propensity to accelerate in the frequently purchasedcategories of bathroom tissue and margarine and only onsugar from the occasionally purchased categories. In otherwords, for categories purchased on a regular basis (frequent-ly), regular buyers are more sensitive to marketing mixactivities, whereas for categories purchased on a less regularbasis (necessity) random buyers exhibit equal or highersensitivity to marketing mix than regular buyers. Althoughtheir low inventory sensitivity and the lack of regularity intheir purchasing pattern may suggest that random buyersexhibit a higher flexibility in their purchasing schedule,such flexibility does not always translate into higher pro-pensity to accelerate. Purchase flexibility therefore cannotfully account for the response of random buyers to pro-motional events.Our interpretation of the results regarding the response of

random and regular buyers to prices and promotions reliesheavily on another significant empirical finding, consistentacross all four categories, that random buyers are less fre-quent (lighter) buyers than regular buyers. Regular buyersbeing high-demanders (Diamond, 1987), frequent users of

Table 4Purchase behavior statistics for random and regular buyers

Segment Size Coefficient of Variation Average InterpurchaseTime

Price Cut Timeelasticity

Feature or DisplayTime elasticity

Random Regular Random Regular Random Regular Random Regular Random RegularKetchup 0.57 0.43 78.7 (18.1) 57.1 (10.4) 83.2 (25.1) 54.3 (19.0) )2.4* )0.3 )3.1* )3.8Sugar 0.63 0.37 88.9 (18.2) 70.3 (8.7) 56.0 (23.0) 29.1 (9.5) )1.1* )0.9* )10.7* )9.0*Bathroom Tissue 0.67 0.33 82.3 (17.5) 53.0 (5.4) 24.3 (10.2) 17.1 (5.5) 0.1 )0.7* )0.7 )3.7*Margarine 0.54 0.46 91.4 (18.0) 62.4 (7.1) 29.8 (16.0) 15.7 (4.2) 0.6 )0.7* 2.5 )3.4** Denotes that the corresponding coefficients are significant at least at the 10% level.

Table 5Correlation Matrix for Probabilities of Belonging to Random Segment

Ketchup Sugar BathroomTissue

Margarine

Ketchup 1 0.32 0.44 0.36Sugar 1 0.24 0.28Bathroom Tissue 1 0.31Margarine 1


all four categories, are more likely to be informed about thedistribution of prices than random buyers and should thus beable to respond more efficiently to prices and promotions(Kim and Rossi, 1994). Because they buy more frequently,they also have more at stake in each of the four categoriesthan random buyers do, but more so for the frequentlypurchased categories. Regular buyers therefore choose tofocus their response to marketing mix activities on thecategories of bathroom tissue and margarine where there ismore at stake than in the occasionally purchased categoriesof ketchup and sugar. For the frequently purchased catego-ries therefore, regular buyers exhibit higher sensitivity thanrandom buyers do.Random buyers on the other hand, being less frequent

users of all four categories, are less informed about pricesand have less at stake than regular buyers. Thus for pur-chases of frequently bought categories, where demand hasto be satisfied fairly regularly and consumers cannot skippurchases random buyers, with more limited informationthan regular buyers, cannot respond efficiently to prices andpromotions. In the occasionally purchased categories on theother hand, where demand is more flexible, random buyersuse price cuts and promotions as opportunities to buy intosuch categories. Our interpretation therefore of the resultsregarding the relationship between random buying and pro-pensity to accelerate is based on information-theoretic ar-guments suggesting that such propensity depends on thefrequency at which a category is purchased. Extensive rep-lication can potentially strengthen the support for our argu-ment.

Conclusions

Although much important research has been conductedon purchase regularity, a study directly relating purchaseregularity to marketing mix sensitivities had been missingfrom the marketing literature. In this paper we empiricallyinvestigated the relationship between purchase regularityand propensity to accelerate, by formulating a Weibull mix-ture model of purchase timing and applying it to scannerdata on four product categories. We found that randombuyers exhibit propensity to accelerate in the occasionallypurchased categories of sugar and ketchup but do not re-spond to prices and promotions in the frequently purchasedcategories of bathroom tissue and margarine. Having alsofound that in all product categories random buyers are lessfrequent (lighter) buyers than regular buyers, we basedour explanation of these results on the frequency at which aproduct category is purchased.More specifically we believe that in the occasionally

purchased categories, random buyers, who are less frequentusers, take advantage of promotions and price cuts in orderto buy into those categories exhibiting opportunistic buy-ing behavior. Regular buyers on the other hand, prefer toconcentrate their response to marketing mix activities on the

more frequently purchased categories where they have a lotmore at stake. In such categories regular buyers being morefrequent users and thus better informed about prices andpromotions than random purchasing households are theones who exhibit propensity to accelerate.At least two other findings that merit discussion were

obtained. First random buyers exhibit lower inventory sen-sitivity than regular buyers with the exception of the sugarcategory where both segments are inventory insensitive.This, along with the pattern of a random households pur-chases, suggests that random buyers should be more flexiblein terms of their purchasing schedules. Such flexibility how-ever, as evidenced from the empirical results, does notalways translate into a higher propensity to accelerate. Sec-ond we consistently found that in all categories householdsare more random than regular. This finding, while in agree-ment with Dunn Reader and Wrigley (1983) contradicts theresults obtained by Wheat and Morrison (1990) and Gupta(1988) on coffee purchases.We believe that our results concerning the purchasing

behavior, and in particular the propensity to accelerate, ofrandom and regular buyers call for extensive replication inorder to further confirm or challenge them. We also thinkthat the relationship between purchase frequency and pur-chase regularity should be further investigated. Our study,and past on the subject (Dunn, Reader and Wrigley, 1983;Bawa and Ghosh, 1990), suggested that random buyerspurchase less frequently than regular buyers do. Are there-fore random purchasing behavior and purchase frequencytwo sides of the same coin? In other words do they representthe same aspect of purchasing behavior or are there statis-tical reasons that infrequent buyers appear random in theirpurchasing behavior? For example, a household that pur-chases infrequently may be more likely characterized asrandom since there are fewer observations available forsuch household and thus its purchasing pattern will be moresensitive to an abnormally long or short interpurchaseinterval (i.e., an outlier). A careful examination, includingsimulations, of the effects of purchase frequency on theparameter estimates characterizing households as random orregular (i.e., shape of hazard rate) will provide an excellentopportunity for research.Another potentially insightful future study could com-

bine household and store-level data to investigate the effectof the size of random and regular segments on the spikesand dips of store sales data during and after promotionalevents (Neslin and Schneider-Stone, 1996; Van Heerde,Leeflang, and Wittink, 2000). Such research would furthersolidify the practical dimension of a segmentation schemebased on purchase regularity.Finally a considerable amount of purchase occasions in

our sample (39%) involve simultaneous purchases of atleast two of the four items studied. This suggests that theremay be dependencies in the interpurchase times of thesefour items (Chintagunta and Haldar, 1998). The issue ofcross-category dependencies is however broader as inter-


purchase times of multiple items should be conditional onthe shopping interval in which case a larger number ofcategories, those that constitute a typical shopping basket,need to be studied.3Thus a thorough investigation of the issue of dependence

of interpurchase times of multiple items should require dataon a sufficiently large basket of goods (more than the fouritems studied here) and a computationally demanding modelthat captures such interdependencies. Hopefully, the sub-stantive implications from such complex investigation willbe worth the effort.

Notes1. The range of days used for plotting the hazard ratesdoes not extend beyond two standard deviations fromthe mean interpurchase time for that category.

2. We thank an anonymous reviewer for suggesting thismethod of calculating segment-level statistics.

3. We would like to thank the editor, L. P. Bucklin, andan anonymous reviewer for their suggestions on thisissue.

Acknowledgments

This paper has benefited from comments of participantsin the Marketing Science Conference at the University ofCalifornia at Berkeley, and marketing seminars at the Uni-versity of Southern California and Santa Clara University.The authors would like to thank in particular Don Morrisonand Ram Rao for their helpful insights and suggestions.

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