Chapter 21 HAZARD/SURVIVAL MODELS IN MARKETING

Transcription

1 Introducton Chapter 21 HAZARD/SURVIVAL MODELS IN MARKETING Pradeep K. Chntagunta, Unversty of Chcago Xaojng Dong, Northwestern Unversty A varable of consderable nterest to marketng researchers s tme. Tme can take on the role of an explanatory varable n many contexts, for example when one s dealng wth data that reflect a temporal trend. At the same tme, there are several stuatons n whch tme s the dependent varable that the researcher would lke to focus on. For example, one mght be nterested n the duraton of a relatonshp between a servce provder (such as an advertsng agency) and ts clent (the frm employng the ad agency), and n studyng the effects of varous factors that mght nfluence the duraton of the relatonshp. Survval models also referred to as duraton models, hazard models or falure tme models are unquely suted to addressng such ssues. Before proceedng wth a descrpton of the model tself, t may be worthwhle askng: snce tme s a contnuous varable and snce regresson methods are very well developed, why not smply use a regresson model to study the nfluence of explanatory varables on the duratons of nterest? There are two man reasons. The frst s referred to as censorng or more precsely, for a vast majorty of topcs of nterest, rght censorng. The second s the ssue of handlng explanatory varables or factors that vary over tme also known as tme varyng covarates.

2 The censorng ssue can be explaned as follows. Consder agan, the example of advertsng agency relatonshp duratons. Suppose we have a sample of such relatonshps that were establshed n the year 1990 and we observed these relatonshps tll the year For each such relatonshp, the data would ndcate two possbltes ) where the relatonshp termnates between the years 1990 and 2000 n whch case we have a duraton observaton; or ) where the relatonshp s stll ongong as of 2000 n whch case, snce the relatonshp has not ended, we do not have a duraton observaton except that we know the duraton exceeds 10 years. The data n the latter case s referred to as a rght censored observaton snce the end of the relatonshp (as opposed to the begnnng of the relatonshp whch s referred to as left censorng ) s not observed. Now f one used a regresson framework, we can easly deal wth the data correspondng to ) above. However, when the data are rght censored, we can ether use 10 years as the duraton to approxmate ts true value or we can gnore the observaton. As the number of censored observatons ncreases, so wll the bas assocated wth both these approaches. Consequently, the regresson model to study duratons wll not be approprate when dealng wth censored data. Note that left censorng s also an ssue wth regresson models but s less prevalent n marketng data. The tme varyng covarates ssues can be thought of as follows. Gong back to our llustraton on advertsng agency relatonshps, one of the key drvers of the relatonshp s lkely to be the talent pool at the advertsng agency. If ths talent pool s declnng over tme, then t would have an mpact on the duraton of the relatonshp. Now, n a regresson model the dependent varable wll contnue to be the duraton, but what value of the explanatory varable sze of talent pool should one use n the analyss? Would t be the pool n 1990, the pool when the relatonshp termnated or some average sze of the pool over the duraton of the relatonshp? 2

3 Hence there s no natural way for the regresson model to deal wth explanatory varables whose values change over the duraton. Ths nablty to handle tme varyng covarates necesstates an alternatve approach that can deal wth such data. Havng descrbed why standard regresson based models mght be unsutable for studyng the effects of explanatory varables or covarates on duratons, we now descrbe the methodology of hazard models. Frst, why do we refer to these models as hazard models? The ntuton for ths s as follows. What we are nterested at any gven pont n tme s the lkelhood that an event wll occur gven that the event has not occurred thus far. Specfcally, n the advertsng agency relatonshp case, one can thnk of ths as studyng the probablty that the relatonshp wll end at some pont n tme, gven that t has not ended thus far. In other words, we are attemptng to assess the rsk or hazard of termnaton of the relatonshp. Hence, the term hazard models. Equvalently, one can also recast the event of nterest n terms of ts non-occurrence tll ths pont n tme,.e., the lkelhood of the event occurrng after ths tme pont. Snce the relatonshp (n the advertsng agency case) has survved tll a gven pont,.e., the relatonshp has not yet termnated, these models are also referred to as survval models. And snce they deal wth duratons as the dependent varable, they are also called duraton models. Followng the same logc, t s easy to see why these models are also referred to as falure tme models snce f a relatonshp between an ad agency and ts clent dd termnate then t can be thought of as havng faled or not havng survved. The notons of hazard and survval descrbed n the prevous paragraph must shed some lght on why ths class of models s able to accommodate rght censorng and tme varyng covarates. Recall that the ssue of rght censorng was assocated wth falure not havng occurred by the end of the data collecton duraton (.e., the year 2000). What ths mples s that 3

4 the relatonshp has survved from the start of the relatonshp n 1990 to the end of data collecton n year So the nformaton contaned n ths observaton wll be the lkelhood of the relatonshp havng survved at the end of the data collecton perod. In ths way the methodology can naturally explot the nformaton contaned n rght censored or survved observatons. In the same way, one can also account for left censorng f t exsts n the data on hand. What then about tme varyng covarates? To go back to our example, we have annual measures of the sze of the talent pool at the ad agency for the years 1990 to 2000 and the pool dd declne from one year to the next. Now suppose the relatonshp ended n the year Snce the explanatory varables only change from year to year, t s straghtforward to dvde the duraton from 1990 to 1997 nto sub-duratons each beng a year n length. For the frst sub-sx duratons, what we observe s a survval of the relatonshp gven the talent pool n that year or sub-duraton. For the seventh year, we observe a falure (snce the relatonshp termnates) gven the talent pool n year Hence the overall lkelhood of the data wll be the product of the lkelhoods of the seven sub-duratons. In ths way, the modelng approach accounts for tme varyng covarates. The descrpton n the prevous paragraph mght prompt the reader to thnk of a duraton model as smply a bnary outcome probablstc model (such as the logt or the probt) where the two outcomes are falure and survval. Such a characterzaton s accurate except that n a bnary outcome model, the ntrnsc probablty of falure or survval (.e., the probablty wthout any explanatory varables or covarates) s constant over tme. By contrast, n a duraton model, ths ntrnsc probablty s allowed to change over tme even f the explanatory varables reman unchanged. Loosely speakng then, the duraton model as descrbed n the paragraph on tme 4

5 varyng covarates can be construed as a bnary outcome probablstc model wth a tme-varyng ntercepts. The specfcaton of duraton or hazard models (the hazard functon ) has three man buldng blocks () the baselne hazard functon; () a functon that reflects the effects of explanatory varables on duratons; and () an approach to accountng for heterogenety n the baselne hazard and the covarate functon among the varous cross-sectonal elements whose behavor s beng analyzed. Block () s mostly relevant only n stuatons where multple observatons are avalable for each cross-sectonal unt under consderaton. Each of the three blocks s dscussed n detals n the followng. Baselne Hazard Functon Specfyng the baselne hazard s akn to specfyng a probablty dstrbuton on the duraton tmes n the absence of explanatory varables (see the paragraph on the bnary outcome probablstc model). Snce duraton tmes are postve numbers, the standard dstrbutons used for the purpose are the exponental, Webull, the log-logstc and the expo-power. Followng are the functonal forms for these four baselne hazard functons. ht () denotes the hazard functon, S( t ) denotes the survvor functon. Usng the dervaton from Appendx A, one can obtan the relatonshp between the hazard functon ht ( ) t hudu ( ) 0 and the survvor functon S() t, as S() t = e. Exponental () = γ t () = e γ ht S t 5

6 where γ > 0. One of the nterestng propertes of the exponental dstrbuton s that the hazard functon correspondng to ths dstrbuton s a constant and s the mean parameter of that dstrbuton. Ths property s referred to as the property of no duraton dependence. In many marketng applcatons, ths property could be unappealng. Consder the case of the advertsng agency relatonshp. What ths means s that condtonal on the relatonshp not havng termnated tll the year 1995, the probablty of the relatonshp termnatng s the same as the probablty of the relatonshp termnatng condtonal on t not havng termnated tll the year In other words, the hazard functon does not depend upon how long the relatonshp has lasted. In realty, one mght thnk that the hazard functon ether decreases over tme (as the relatonshp strengthens) or ncreases over tme (as the relatonshp deterorates). Ths would requre the dstrbuton to have the property of duraton dependence, such as the followng baselne functons. Webull () = γα ( γ t) α ( γ t) () = e ht S t α 1 where γ, α > 0. The Webull dstrbuton has a hazard that can be ncreasng n duraton f α > 1, decreasng over tme f 0< α < 1, and constant f α = 1, whch wll be the same as the exponental hazard. The shapes of the baselne hazard for exponental and Webull are shown n Fgure

7 2 1.5 Exponental: γ=1.0 Webull: α=1.5, γ=0.6 Webull: α=0.5, γ= Duraton Fgure 21.1 Hazard functons for Exponental and Webull Log-logstc () ht () S t ( t) ( γ t) γα γ = = 1 + ( γ t) α 1 α α where γ, α > 0. The log-logstc n addton, allows for non-monotonc hazards,.e., those that can ncrease ntally and then decrease. Specfcally, for α > 1, the hazard frst ncreases wth duraton, then decreases. Such a specfcaton would be approprate for data wth perodcty where the lkelhood of an event occurrng ncreases over tme but f the event gets delayed beyond a pont, t becomes less lkely to occur. If 0< α 1, the hazard functon decreases wth duraton. The shapes of the functons under dfferent values of α are shown n Fgure

8 2 1.5 Log-logstc: α=1.5, γ=1.2 Log-logstc: α=0.5, γ= Duraton Fgure 21.2 Hazard functons for log-logstc wth dfferent parameter values Expo-power () α α 1 θt ht = γαt e () S t = e γ θt α 1 e θ where γ, α > 0. The expo-power s a bt more flexble than the log-logstc and also allows for U-shaped hazard functons. It also nests the Webull functon, as well as the exponental functon. The dfferent shapes of expo-power wth dfferent parameter values are llustrated n Fgure Expo-power: θ=0.1, γ=1, α=0.5 Expo-power: θ=-0.1, γ=1, α=1.5 Expo-power: θ=0.1, γ=0.5, α= Duraton 8

9 Fgure 21.3 Hazard functons for expo-power wth dfferent parameter values Besdes these baselne hazard specfcatons, there are other more flexble forms such as the Box-Cox or the sem-parametrc that nvolve the estmaton of a larger number of parameters. The tradeoff for the researcher s the addtonal flexblty at the cost of ncreased computatonal burden. In most practcal applcatons, the parametrc forms descrbed above appear to suffce ncely. Specfcaton of Explanatory Varables Effects on the Hazard Wth the baselne hazard specfed, one needs to decde on how to ntroduce explanatory varables nto the specfcaton. There are three popular approaches to dong ths, referred to as the proportonal hazards model (PHM), the addtve rsks model (ARM), and the accelerated falure tme model (AFTM). 1. Proportonal hazard model (PHM), proposed by Cox (1972) ( ) ( ) h t X h t e β t, X t = Where (, ) h t X t stands for household s hazard functon at tme t, t X s a vector of explanatory varables, β s a vector of parameters for household. In other words, the PHM says that the effect of the functon for explanatory varables on the baselne hazard s multplcatve. 2. Addtve rsks model (ARM), proposed by Aalen (1980) ( ) ( ) Xt, t = + h t X h t e β ARM specfes that the effect of functon for explanatory varables on the baselne hazard s addtve. 9

10 3. Accelerated falure tme model (AFTM), proposed by Prentce and Kalbflesch (1970) AFTM specfes one of the parameters n the baselne hazard functon to be a functon of explanatory varables. For example, Chntagunta (1998) specfes parameter γ n the log-logstc hazard functon to be a lnear functon of the explanatory varables, that s γ = γ0 + X t γ1. Essentally these three specfcatons dffer n the way n whch the dervatve of the hazard functon wth respect to the explanatory varable (when t s contnuous),.e., the margnal hazard, depends upon the baselne hazard. In the PHM case the dervatve s proportonal to the baselne hazard, n the ARM case t s ndependent of the baselne hazard and n the AFTM case t can be a complcated functon of the baselne hazard, typcally non-proportonal. To understand the ntuton underlyng these alternatve approaches, recall the example of the advertsng agency relatonshps. One mght be nterested n answerng the queston what happens to the hazard functon f I am able to margnally ncrease the talent pool n a gven tme perod. The PHM model would predct an effect sze proportonal to the value of the hazard at that tme pont. The ARM specfcaton would predct that the correspondng change n the hazard functon wll not depend on the current duraton of the relatonshp, whereas the nature of mpact under the AFTM s not easy to characterze. A pror there s no theory that can gude the researcher on whch approach s best suted n an emprcal context. Hence the choce wll have to based on (a) whether the resultng estmates from the model are nterpretable as beng ntutvely plausble; (b) statstcal crtera such as ft and ablty to predct n a hold-out sample. Unobserved Heterogenety Once the baselne hazard and the effects of explanatory varables have been accounted for, the last ssue to address s that of accountng for unobserved heterogenety. In other words, 10

11 the parameters of the baselne hazard and / or the effects of the explanatory varables need not be the same across all the cross-sectonal unts n the analyss. By cross-sectonal unts we mean for example, the agency-clent par n our example of advertsng agency relatonshps; households when we are nterested n the nter-purchase tme behavor of consumers; segments f we nterested n dfferences n behavor across large, medum and small-szed busnesses, etc. If such heterogenety exsts but s not accounted for n the estmaton, t wll result n based estmated effects for the explanatory varables. One example s to study household s purchase behavor, among whch households responses to promoton (or prce cut) are beleved to dffer across households. Two popular approaches can be employed to account for ths heterogenety, as shown n the followng. Dscrete Heterogenety Ths approach also has other names, such as latent class, semparametrc or fnte mxture model approach. The underlyng assumpton here s that there are dscrete segments n the market and each segment has a unque set of parameters characterzng the baselne hazard functon and / or the effects of explanatory varables. In the example of households purchase behavor analyss, one can beleve that there are dstnct groups of households n ther responses to promoton. The number of groups can be decded usng statstcal- crtera. More detals of estmaton process are dscussed n Appendx B. Contnuous Heterogenety The second s the contnuous dstrbuton approach where each unt s parameters are assumed to be a draw from some assumed contnuous dstrbuton (such as the Normal dstrbuton). The estmaton process for ths approach s a lttle more complcated, because of ts 11

12 need for smulaton. Appendx C shows more detals regardng the estmaton process of ths approach. Whle the choce among these approaches s to some extent, a matter of taste, the mportant emprcal fndng s that heterogenety when present and gnored can severely bas estmated parameters from these models. One can also use statstcal approaches to determne whch type of heterogenety assumpton s more approprate for a set of data. A Numercal Example Up to now, the three buldng man blocks for a duraton or hazard model have all been dscussed; a numercal example from the paper by Seetharaman and Chntagunta (2003) s presented here to llustrate how hazard models are appled emprcally. The data are panel data, recordng the purchases of 300 panelsts n the detergents category over a perod of two years from June 1991 to June The emprcal dstrbuton of nter-purchase tme s plotted n Fgure Interestngly, ths plot shows peaks at multple of seven days. Ths s consstent wth the emprcal observatons that households makng shoppng trps n weekly ntervals (e.g., Kahn & Schmttlen, 1989; Dunn, Reader, & Wrgley, 1983). The modelng process can be consdered as tryng to use the three buldng blocks to separate out ther effects on nter-purchase tmes. 12

13 Fgure 21.4 Observed Inter-Purchase Tmes The nter-purchase tme s nfluenced by the followng marketng varables, prce, dsplay, feature, and a household-specfc varable: product nventory. Usng the expo-power baselne hazard wth dscrete tme PHM, the parameter estmates for these varables are lsted n Table 21.1, for both homogeneous and heterogeneous models. Note that, n the homogeneous model, the effect of prce s under-estmated (closer to zero), relatve to prce estmates for any segment 13

14 obtaned from the heterogeneous model. Ths ndcates that the model assumng no taste varatons among households does not gve the average effect on a marketng varable, as one mght expect. Ths can be also verfed by the fact that the weghted average of the three estmates from the heterogeneous model (wth the probabltes as the weghts) s not the same as the parameters from the homogeneous model. The results from the heterogeneous model dentfy three groups of people. The largest group accounts for 46% among the panelsts, who are most senstve to prce and feature. Ths group could be those planned shoppers, who have seen the features from ther newspapers or onlne, and learnt about prces, based on whch they decde whether to buy detergents, or make a shoppng lst. The second group accounts for 44% of the panelsts, who are least senstve to prce, but most senstve to dsplay. They mght be those unplanned shoppers, who go to the store wthout a shoppng lst, but when they see a dsplay and realze that ther nventory s low (note that ths groups shows the hghest mpact from nventory, comparng to the other two groups), they wll buy t. The thrd group accounts for only 10% of the panelsts, who are least senstve to feature or dsplay, but only care about prce. Table 21.1 Results Usng Expo-Power Baselne Hazard Functon Parameter Homogeneous Heterogeneous Prce (0.06) Dsplay 1.41 (0.07) Feature 1.40 (0.07) Inventory (0.00) Probablty Other Types of Hazard Models Encountered n the Marketng Lterature 14

15 So far, our dscusson has focused on a sngle event that s of nterest to marketers, for example, the duraton of a relatonshp between a clent frm and an advertsng agency. Two other types of duratons that have been studed n the marketng lterature are as follows. The frst stuaton s one n whch, at the end of the duraton.e., at the tme of falure, a number of alternatve outcomes are possble. Ths stuaton s very commonly encountered wth scanner panel data. Consder a household makng purchases of nstant coffee. When the marketer records that household s purchase (or falure ), the household could have purchased from among a set of dfferent brands of such coffee for example, Maxwell House, Folgers or Taster s Choce. Indeed, the household could have chosen to purchase Maxwell House n the current week because that brand was promoted ths week. In the absence of the promoton, the household mght have wated an addtonal week to purchase coffee and even then mght have purchased a brand other than Maxwell House. In essence then, each of the brands s competng for the household s purchase. Such a stuaton n whch duraton s defned not as the tme to purchase of nstant coffee but rather as the tme to purchase of the ndvdual coffee brand s referred to as the competng rsks hazard model. More generally, n the coffee example, f there are J brands the household can choose from, the household at the tme of a purchase can be classfed as beng n one of J states. Followng that purchase, (s)he s at rsk for agan purchasng one of the J brands at the next purchase. In essence the, there are J*J possble duratons of nterest to the researcher. Each of these duratons (also called transtons snce they are based on movement from one state to another) can then be modeled exactly as descrbed for the sngle duraton context above. An mportant consderaton when modelng competng rsks s the potentally large number of parameters that need to be estmated when the number of states (or brands n ths case), J gets large. Further, snce the transton from state j to state k requres a suffcent 15

16 number of observatons on that transton, we need a large number of parameters across all transtons to be able to estmate all the duratons. Nevertheless, the approach allows for an analyss of a household s transtons from one state (brand) to another over tme. Consequently, a key beneft to ths approach s that one can study not just the nfluence of marketers actons on the tmng of coffee purchases but also the effects of these actons on brand choce behavor. The other type of hazard model that has appeared n the marketng lterature s referred to as the splt hazard model. Consder an example where we mght be nterested n characterzng the effects of varous factors on tme to adopton of frms of a new technology. Gven a sample of frms and ther duratons, we wll observe ether falures (.e., adoptons) or survvals (.e., no adoptons) durng the data gatherng perod. However, t s possble that some of the survvors are unlkely to ever adopt the new technology n whch case, those frms need to be treated dfferently than those who are observed not to have made a purchase durng the data collecton perod but who could potentally adopt the technology at a later pont n tme. The splt hazard model formalzes ths stuaton by frst assgnng a probablty to each frm of ever adoptng the technology. Then for those who are classfed as potental adopters, the duraton s characterzed very much as descrbed prevously (.e., we can observe ether falures or survvors durng the data observaton perod). Marketng Lterature: Some Illustratve Examples In the marketng lterature, there have been two broad classes of applcatons of duraton models. The frst set of studes can be thought of beng at the mcro or ndvdual level. The dependent varable of nterest here s the tme between successve purchases n a frequently purchased product category such as coffee or detergents. Several papers have been publshed n ths area wth a varety of dfferent specfcatons for the three buldng blocks descrbed above. 16

17 The second set of studes tends to be more macro n nature and looks at ssues pertnent to marketng strategy what determnes the falure of new ventures? When does a new technology takeoff or when does a domnant desgn emerge n a technology product market. These applcatons have by and large been very successful snce researchers have consderable flexblty n specfcaton and estmaton of the three key buldng blocks descrbed prevously. The classc study of hazard functon or duraton models n marketng s that by Jan and Vlcassm (1991). Ths s the frst study n the marketng lterature to study households category purchase tmng behavor n the context of the ground coffee category. So the duraton of nterest was the tme to next purchase of the coffee category and the nfluence of marketng actvtes such as prce and promoton on ths duraton. It was also the frst study to formally decompose the hazard nto the baselne hazard, the effects of explanatory varables and the effects of unobserved heterogenety as descrbed earler n ths chapter. The authors also tested a number of dfferent parametrc specfcatons for the baselne hazard and concluded that the coffee data, the baselne hazard had a non-monotonc pattern. Prma face, ths was a curous fndng as one would thnk that a monotoncally ncreasng hazard s more lkely snce the probablty of makng a purchase condtonal on not havng made a purchase should ncrease due to the depleton of one s nventory n the pantry. The explanaton offered by the authors for the nonmonotonc pattern s based on the regularty of purchases made n ths category. In other words, say that a household n ths category typcally makes a coffee purchase every 4 weeks. Then, t s reasonable (n the absence of any other explanatory varables) to expect the hazard to ncrease for 4 weeks after a purchase. However, f the regular duraton passes for that household then t becomes less lkely to observe the household makng a purchase at a later data. Hence, the estmated non-monotonc nature of the baselne hazard seems reasonable n ths case. In terms of 17

18 accountng for explanatory varables, the authors used the PHM approach. The ablty of Jan and Vlcassm (1991) to account for unobserved heterogenety stemmed from the panel nature of the data wth repeated observatons (also referred to as multple spells ) on the same household. Indeed, ths was also one of the frst studes to hghlght the mportance of accountng for unobserved heterogenety when nvestgatng household purchase behavor. Smlar models publshed n the marketng nclude those by Gupta (1991), Gonul and Srnvasan (1993), Helsen and Schmttlen (1993) and Wedel et al. (1995). For a revew of PHM models publshed n the marketng lterature, the reader s referred to Seetharaman and Chntagunta (2003). For a comparson of some baselne hazard comparsons, see Grover and Tadkamalla (1997). Wedel et al. (1995) also demonstrate how to account for tme-varyng covarates wth these data by dscretzng the duraton nto sub-duratons as prevously descrbed. A recent applcaton of that approach to onlne purchasng s n Manchanda et al. (2005). All the above studes can handle rght censorng of the data as well. Whle the Jan and Vlcassm (1991) and the Seetharaman and Chntagunta (2003) studes compare a number of alternatve specfcatons of the baselne hazard model, few studes have compared the dfferent approaches to accountng for explanatory varables n hazard functon analyses n marketng. A notable excepton s Seetharaman (2004) who compares the performance of the ARM wth those of the PHM and AFTM. Usng household scanner panel data on three product categores laundry detergents, paper towels and tolet tssue, Seetharaman fnds that across all product categores, ARM performs best followed by PHM and fnally, AFTM. Two crtera are used for ths comparson n-sample ft and predcton n a hold-out sample of households. Further, smlar to Seetharaman and Chntagunta, he fnds that the more 18

19 flexble functonal forms for the baselne hazard ft the data much better than those that mpose monotoncty n that hazard, even after adjustng for the larger number of estmated parameters. In terms of accountng for the effects of unobserved heterogenety, the early studes, e.g., Jan and Vlcassm (1991) found that dscrete / latent-class approach to accountng for heterogenety appeared to ft the data better than usng specfc parametrc functonal forms for the dstrbuton of ths heterogenety (e.g., the normal dstrbuton). More recently however, the advent of Bayesan methods (Allenby, Leone, & Jen, 1999; Manchanda et al., 2005), and the ablty to obtan household level parameter estmates has re-nvgorated proponents of the contnuous mxture approach to accountng for heterogenety. A few studes n marketng have looked at competng rsks models. Agan, the frst study was that by Vlcassm and Jan (1991). That study nvestgated the effects of marketng actvtes on the purchases of saltne crackers. As n the case of coffee category purchases, the fndng of non-monotonc hazards was repeated albet at the level of brand to brand transtons. The authors agan used the PHM framework to account for the effects of the explanatory varables. And they contnue to fnd that t s crtcal to account for unobserved heterogenety n order to obtan the correct pattern for the transtons as well as for the effects of the covarates. An mportant emprcal fndng here s the ablty to compare swtchng as well as repeat purchase duratons to nfer not only the brands that seem to show the hghest loyalty (after controllng for the effects of marketng actvtes) but also to dentfy the ablty of brands to ether draw customers from competng brands or to lose customers to those brands. As dentfed prevously, ths methodology gets cumbersome wth a large number of brands and also requres a large amount of data to estmate all the transtons. To overcome these problems, Chntagunta (1998) proposes an approach to reduce the computatonal burden of the competng rsks model. The competng 19

20 rsks model has also been appled n the context of a household s personal nvestment data va a Bayesan framework by Allenby, Leone and Jen (1999). The splt hazard model has also seen a few applcatons n marketng. Early studes n ths area were by Snha and Chandrashekaran (1992) who nvestgated the drvers of bank adopton of ATM machnes. More recently the approach has been appled to study the adopton behavor of physcans of new drugs. The popular press (Wall Street Journal July 2005) has recently emphaszed the mportance of early adopter physcans to pharmaceutcal companes. By focusng ther attenton on these physcans, the frm then expects these physcans to act as nfluencers n gettng other physcans to adopt the drug. The queston then s: how does one dentfy the early adopters? In a recent study Kamakura, Kossar and Wedel (2004) use tme-tofrst-prescrpton data for a large number of physcans across several drugs and estmate a splt hazard model on these data. After controllng for the effects of explanatory varables, they are then able to dentfy physcans who are more nclned to prescrbe a drug early. The splt hazard formulaton s useful n ths case to account for the possblty that a physcan mght never prescrbe a partcular drug. Future Drectons Gong forward, one can see several drectons n whch researchers can use and enhance hazard models to better understand marketng phenomena. Very often, marketers have to deal wth multple dependent varables n some cases these varables could all be duratons and n other cases t could be a combnaton of a duraton and a dependent varable wth dfferent propertes (.e., not a postve valued varable, but a dscrete varable or a regular contnuous varable). A small number of studes n marketng have looked at the ssue of multple dependent varables. For example, when studyng the mpact of marketng actvtes on the duraton of 20

21 purchases n two dfferent categores by a household, one needs a bvarate hazard functon that s correlated across categores. Chntagunta and Haldar (1998) and more recently Park and Fader (2004) have addressed ths ssue. Researchers have also combned the hazard model wth a logt model to study category purchase duraton and brand choce behavor of households wthout havng to estmate a competng rsks model prevously descrbed (see, e.g., Chntagunta & Prasad, 1998). Further researchers have also combned a duraton model wth a regresson model to study the effects of marketng actvtes on the jont purchase tmng and expendture decsons of households. An llustraton of ths approach s n Manchanda et al. (2002). Gong forward one expects to see more work n ths area. Substantvely, an area where hazard models are just beng appled s n measurng the lfetme value of a frm s customers. As the topc of customer relatonshp management becomes more and more central to the workngs of an organzaton, t s reasonable to expect ncreased applcaton of hazard / duraton models n ths area. Indeed, studyng customer lfetme value wthn the context of hazard models poses an nterestng methodologcal challenge. Marketers are nterested n nfluencng two types of duratons as they relate to ther customer pool. The frst s the duraton of the relatonshp the tme elapsed snce the ncepton of the relatonshp. The second duraton nvolves varous events that occur durng the span of the relatonshp such as the dfferent purchases that the customer mght make from the frm. Hence the second duraton s embedded wthn or s nested wthn the frst duraton. One potental framework for addressng ths ssue s the so called hazard n hazard framework (Lllard, 1993). That framework was developed to account for the tmng of brths of chldren wthn the duraton of a couple s marrage. Another potental methodologcal area of potental nvestgaton (not specfc to customer lfetme value) s the area of excess hazards. The dea s to combne the advantages 21

22 of PHM and ARM by specfyng the hazard as the sum of the PHM (that ncludes the baselne hazard and covarates) and another baselne hazard. Such a hazard functon would be more flexble than ether the ARM or PHM. A thrd area of research could be n the area of fralty models. The dea here s that a group of ndvduals share a common characterstc that nfluences all ther duratons. Identfyng and estmatng ths (unobserved) common characterstc can help shed lght on the relatonshp between duratons of multple ndvduals. ENDNOTE 1 We can thnk of tme as ether a contnuous varable ( tme to an event ) or as a dscrete varable n terms of sub-duratons (as n our dscusson of tme-varyng covarates). Here, we wll restrct ourselves to the contnuous case. Estmaton of the dscrete-tme verson s n Appendx B. 22

23 REFERENCES Aalen, O. O. (1980). A model of nonparametrc regresson analyss of countng processes. New York: Sprnger-Verlag. Allenby, G. M., Leone, R. P., & Jen, L. (1999), A dynamc model of purchase tmng wth applcaton to drect marketng. Journal of the Amercan Statstcal Assocaton, 94(446), Chntagunta, P. K. (1998). Inerta and varety seekng n a model of brand-purchase tmng. Marketng Scence, 17(3), , & Haldar, S. (1998). Investgatng purchase tmng behavor n two relates product categores. Journal of Marketng Research, 35(1), , & Prasad, A. R. (1998). An emprcal nvestgaton of the "dynamc McFadden" model of purchase tmng and brand choce: Implcatons for market structure. Journal of Busness and Economc Statstcs, 16(1), Cox, D. R. (1972). Regresson models and lfe tables. Journal of the Royal Statstcal Socety, B, 34,

24 Dunn, R., Reader, S., & Wrgley, N. (1983). An Investgaton of the Assumptons of the NBD Model Appled to Purchasng at Indvdual Stores. Appled Statstcs, 32(3), Gonul, F., & Srnvasan, K. (1993). Consumer purchase behavor n a frequently bought product ca. Journal of the Amercan Statstcal Assocaton, 88(424), Grover, R., & Tadkamalla, P. (1997). A comparson of some stochastc models of nterpurchase ntervals and of aggregate purchase ncdence, Amercan Journal of Mathematcal and Management Scences, 17, Gupta, S. (1991). Stochastc models of nterpurchase tme wth tme-dependent. Journal of Marketng Research, 28, Helsen, K., & Schmttlen, D. C. (1993). Analyzng duraton tmes n marketng: Evdence for the effectveness of hazard rate models.. Marketng Scence, 12(4), Jan, D. C., & Vlcassm, N. J. (1991). Investgatng household purchase tmng decsons: A condtonal hazard functon approach. Marketng Scence, 10(1), Kahn, B. E., & Schmttlen, D. C. (1989). Shoppng trp behavor: An emprcal nvestgaton. Marketng Letters, 1(1), Kamakura, W. A, Kossar, B. S., & Wedel, M. (2004). Identfyng nnovators for the cross-sellng of new products. Management Scence, 50(8), Lllard, L. (1993). Smultaneous equatons for hazards: Marrage duraton and fertlty tmng. Journal of Econometrcs, 56, Manchanda, P., Dube, J.-P., Goh, K., & Chntagunta, P. (2002). The effect of banner advertsng on consumer nter-purchase tmes and expendtures n dgtal envronment. Workng paper, Unversty of Chcago. 24

25 -----, -----, -----, & (2005). The effect of banner advertsng on nternet purchasng. Journal of Marketng Research, forthcomng. Park, Y.-H., & Fader, P. S. (2004). Modelng browsng behavor at multple webstes. Marketng Scence, 23(3), Prentce, R. L., & Kalbflesch, J. D. (1970). Hazard rate models wth covarates. Bometrcs, 35(1), Ross, P., & Allenby, G. M. (2003). Bayesan statstcs and marketng. Marketng Scence, 22, Schwarz, G. (1978), Estmatng the dmenson of a model. The Annals of Statstcs, 6(2), Seetharaman, P. B. (2004). Modelng multple sources of state dependence n random utlty models: A dstrbuted lag approach. Marketng Scence, 23(2), , & Chntagunta, P. K. (2003). The proportonal hazard model for purchase tmng: A comparson of alternatve specfcatons. Journal of Busness and Economc Statstcs, 21(3), Snha, R. K., & Chandrashekaran, M. (1992). A splt hazard model for analyzng the dffuson of nnovatons. Journal of Marketng Research, 29, Vlcassm, N. J., & Jan, D. C. (1991). Modelng purchase-tmng and brand-swtchng behavor ncorporatng explanatory varables and unobserved heterogenety. Journal of Marketng Research, 28, Wedel, M., Kamakura, W. A., Desarbo, W. S., & ter Hofstede, F. (1995). Implcatons for asymmetry, nonproportonalty, and heterogenety n brand swtchng from pece-wse exponental mxture hazard models. Journal of Marketng Research, 32,

26 Appendx A Dervaton of the Survvor Functon S( t ) from the Hazard Functon ht ( ) Based on the defnton of hazard, t can be wrtten as ht () f ( t) =, where ( ) 1 F() t f t s the probablty densty functon correspondng to the hazard functon, and F( t ) s the correspondng cumulatve dstrbuton functon. t 0 ( ) () df t It can be also wrtten as htdt () =, Integrate both sdes, one can get 1 F t ( ) ln ( 1 ( )) 1 () h u du = F t F t = e t hudu ( ) 0 Based on the meanng of the survvor functon S( t) 1 F( t) =, that s S() t = e t hudu ( ) 0 Appendx B Estmaton Method for Model Accountng for Dscrete Unobserved Heterogenety (e.g., there are G unobserved segments n the market) The Lkelhood Functon Suppose household belongs to group g, and has n purchases at tme perods ( t t t ), wth the tme varant explanatory varables gven by ( ),,..., n 1 2 X1, X2,..., X n. In the analyss, 26

27 the tme varable can be treated as ether a contnuous or dscrete varable, Seetharaman and Chntagunta (2003) provdes an extensve revew. In the contnuous-tme case, the lkelhood functon for household s n ( 1, ) L = f t t X β g k k k g k = 1, where t f ( t, Xtβ) = h( t, Xtβ) S( t, Xkβg) = h( t, Xkβ) exp h( u, Xuβ) du. 0 In the dscrete-tme case, the lkelhood functon s T δ ( ) 1 k δk ( β ) ( β ) L = Pr t, X 1 Pr t, X g k k g k k g k = 1 δk, where T s the total number of shoppng trps for household, δ k denotes whether ths household made a purchase on trp number k, 1 f they do, 0 otherwse. Pr ( k, k g) t X β s the dscrete-tme hazard for household at trp k, for household, who belongs group g, whch can be computed as: t k t X X h u du tk 1 a. For PHM, Pr ( k, kβg) = 1 exp exp( kβg) ( ) t k t X X h u du tk 1 b. For ARM, Pr ( k, kβg) = 1 exp exp( kβg) + ( ) The lkelhood functon, L for household above s condtonal on belongng to segment or group g. The household s uncondtonal lkelhood s gven as: G L = L p g g g= 1 27

28 as The weghts are the probabltes of anyone fallng n each of the groups g, denoted pg and G denotes the total number of groups. Let N denote the total number of ndvduals n the sample. Then the sample lkelhood functon s gven as follows. N L= L. = 1 Decdng the Number of Groups G A most commonly used method s based on the Bayesan Informaton Crtera ntroduced by Schwarz (1978), whch s computed as BIC = 2 LL + K ln( NOBS ), where LL s the log-lkelhood value at convergence of the model estmaton, K s the number of parameters n the model and NOBS s the total number of observatons n the data. BIC can be seen as a negatve functon of log-lkelhood, wth a penalty of the number of parameters as a functon of the sample sze. Consstent wth the concept that larger log-lkelhood value s preferred, model wth smaller BIC value s preferred. Startng wth G =2, estmate the model usng maxmum lkelhood estmaton method, compute the BIC value, named t as BIC-2. Then set G=3, re-estmate the model, compute the BIC value. If ths one s larger than BIC-2, then stop and choose G=2 as the optmal number of groups; otherwse, contnue to ncrease the number of groups, untl the BIC value starts to ncrease. Parameter Estmates each group. Those parameter estmates from the optmal number of groups, ncludng β g and p g for 28

29 Appendx C Estmaton Method for Model Accountng for Contnuous Unobserved Heterogenety In ths case, the β 's across dfferent households takes dfferent values, β for household, and the β s are all from the same dstrbuton, say Normal or Multvarate normal. In order to estmate the parameters of the dstrbutons as well as all the other parameters n the model, one can follow the followng four steps. 1. For each household, make NR draws for β from the assumed dstrbuton f ( β ), compute the smulated lkelhood functon for that household L% 1 NR L r r= 1 = NR ( β ) r Where β denotes one random draw of β. 2. Repeat ths for all the households, note that β s for all are drawn from the same dstrbuton, and for the n observatons for each household, β s assumed to be constant. 3. Compute the overall lkelhood functon as a product of all the smulated lkelhood functon for each household, that s N L% = L%, where N denotes the total number of = 1 households, L % denotes the smulated overall lkelhood functon. 4. Plug n ths functon L % nto a maxmum lkelhood functon procedure, one can get the all the parameters, ncludng those for the dstrbutons can be obtaned at convergence. 29

30 Note that ths method can only obtan the dstrbuton at the populaton level, but not the parameter values for each ndvdual household, the β s. In order to obtan the β s, one way to do t s to use Herarchcal Bayesan method. For a detaled dscusson of that method, please refer to Ross and Allenby (2003). 30