Choice Modeling: Marketing Engineering Technical Note 1

Transcription

1 Choce Modelng: Maretng Engneerng Techncal Note 1 Table of Contents Introducton Descrpton of the Multnomal Logt (MNL) Model Propertes of the MNL Model S-shaped response functon Inverted U Margnal response Elastcty of response Proportonal Draw Logt Model Estmaton va Maxmum Lelhood Usng Logt Models for Customer Targetng Usng Logt Models for Customer Segmentaton Determnng the number of latent segments n MNL models Summary References Introducton Frms today have access to ncreasng amounts of maret response data at the level of ndvdual customers, ncludng data from scanner panels, drect maretng efforts, onlne retalng, loyalty programs, and the le. These data nclude both the maretng effort drected at a customer (e.g., prce dscount, or specfc emal sent to that customer) and the assocated specfc behavors (e.g., purchase, customer support) of that customer. Consequently, there s also ncreasng nterest among mareters n developng and usng response models specfed at the ndvdual level. Analyses of ndvdual-level data are useful for frms even for mang decsons about aggregate maretng actons, such as TV advertsng. After all, marets are composed of ndvduals, and acnowledgng 1 Ths techncal note s a supplement to the materals n Chapter 1,2, and 7 of Prncples of Maretng Engneerng, by Gary L. Llen, Arvnd Rangaswamy, and Arnaud De Bruyn (2007). (All rghts reserved) Gary L. Llen, Arvnd Rangaswamy, and Arnaud De Bruyn. Not to be re-produced wthout permsson.vst for addtonal nformaton.

2 and ncorporatng customer heterogenety can be benefcal n a wde varety of maretng decson contexts. One of the most wdely used approaches for modelng ndvdual customer behavor s the multnomal logt model (MNL), whch can be used to explan and predct the choces that customers mae (e.g., choosng brands, respondng to an emal, upgradng product). Other methods n Maretng for modelng behavor nclude Regresson Analyss, Neural Networs, and Dscrmnant Analyss. In ths note we only descrbe the MNL model, and descrbe how t can be used for customer targetng and for customer segmentaton. Descrpton of the Multnomal Logt (MNL) Model The theory of ratonal choce underles much of modern Economcs. Accordng to ths theory, ndvduals have well-ordered preferences for any set of choce alternatves (e.g., products, brands, canddates n an electon), and they choose that alternatve that maxmzes ther preferences. The MNL model offers a way to operatonalze the theory of ratonal choce wthn a probablstc framewor. The obectve of the MNL model n Maretng s to predct the probabltes that a customer would choose each of several alternatves whch are avalable on a partcular purchase/choce occason. The MNL model s based on several core concepts: (1) The customer has an unobservable (at least to the modeler) preference or utlty for each of the choce alternatves, (2) the utlty of each choce alternatve s composed of two addtve terms, namely, a determnstc component (the ntrnsc value or attractveness of the choce alternatve), and a random component that vares randomly across choce alternatves, customers, and purchase occasons, (3) the dstrbuton of the random component can be specfed, and (4) on each choce occason, the customer chooses the alternatve that provdes hm or her the hghest utlty. Below, we elaborate on these core concepts: On each choce occason, the (unobserved) utlty that customer gets from choce alternatve s gven by: U = A + ε (1) where ε s the random component of the customer s utlty. We assume that ε s are dstrbuted ndependent Gumbel (.e., type 1 extreme value) dstrbuton. 2

3 The ndependence assumpton mples that nowledge of the value of the random component for any customer, choce alternatve, or purchase occason does not provde any nformaton about the value of the random component for another customer, choce alternatve, or purchase occason. Notce that utlty, ( U ), s the sum of an observable component ( A ) and an unobservable component ( ε ), mang t unobservable, or latent. A s the overall attractveness (vew t as nferred preference or utlty value) of alternatve to customer A = β X. (2) X s the value (observed or measured) of a contextual varable (e.g., Color of product; prce of product, whether product was on a specal promoton on that purchase occason) for product alternatve on a gven purchase occason. β s the mportance weght assocated wth varable (estmated by the model ths s smlar to regresson coeffcents). We assume that customer chooses the product whch offers hm or her the hghest utlty. Then, the probablty that the customer wll choose alternatve s gven by: P = P{ U U m; for all m n the choce set} (3) That s P s the probablty that the utlty of product wll be at least as hgh as the utltes of any other product on that purchase occason. Then, t can be shown that ndvdual s probablty of choosng product 1 or choce alternatve 1( P ) s gven by: P 1 e = e A A for =. 1, 2, K (4) Thus the logt model s a sequence of K equatons (where K s the number of alternatves). When appled to a typcal brand choce problem, the model components have the followng nterpretatons: 3

4 X = customer s evaluaton of brand on product attrbute (brand qualty, for example), where the summaton s over all brands that ndvdual s consderng purchasng; β = revealed mportance weght" showng the degree to whch attrbute nfluences brand preferences (apples to all brands). These parameter estmates are revealed by an analyss of the past behavor (e.g., choce) of customers rather than by drectly asng consumers. They can be broadly nterpreted n much the same way as regresson coeffcents; β X. = Overall attractveness (utlty) of brand for customer In the aggregate Logt model, gven n Eq. (4), β s the same for all ndvduals n a target maret. Propertes of the MNL Model What s the value of MNL models n Maretng? The answer, brefly, s that the structure of logt mrrors the dfferental senstvtes we expect n actual choce behavor. To see how ths wors, consder Eq. (4). The exponentaton n Eq. (4) ensures that the probabltes are always postve, snce the exponentaton of any real number s always postve. Exponentaton also ensures that the probabltes do not change f all the measures of attractveness are ncreased by a constant. Thus the measures of attractveness need only form nterval scales, somethng qute useful snce most customer-based measures only acheve nterval-scale qualty. S-shaped response functon: An mportant characterstc of logt s that t produces an S-shaped curve, tracng the expected relatonshp between attractveness and choce. Graphng Eq. (4) as a functon of A produces an S- shaped curve that asymptotes to zero (no chance of beng chosen) for very unattractve brands and to one for very attractve ones (almost certan to be chosen). In most applcatons of the logt model, the attractveness of a brand (or, more generally, a choce alternatve) s assumed to be a functon of ts characterstcs. Ths attractveness functon s typcally lnear as n Eq. (3). Inverted "U" Margnal response: The margnal mpact of a change n an 4

5 attrbute of an alternatve X ι taes a partcularly smple form. For example, consderng Product 1, the dervatve of P 1 as a functon of X 1 s dp dx 1 1 * 1 * 1 = w P (1 P ) (5) where * P1 s the predcted probablty (as predcted by the model) that consumer wll choose product 1 from the current choce set (Analogous expressons apply for other products n the choce set). Thus the margnal change n the probablty that consumer wll choose product 1, for a unt change n varable, turns out to be a functon of the predcted probablty of choosng product 1 ( P ). A graph of Eq. (5) s gven n Exhbt 1. The margnal mpact of a gven maretng effort s maxmzed when the probablty of choosng the product s equal to.5, but the margnal mpact approaches zero when the probablty of choosng that product s near zero or close to one. Thus the logt model has a nce behavoral property: t ensures that the ncremental mpact of maretng effort drected at a product s at ts pea when the consumer s on the fence about choosng t. * 1 EXHIBIT 1 The margnal mpact of maretng effort depends on the probablty of choce. Elastcty of response: Lewse, we can compute the elastcty of choce 5

6 probablty, namely, the percentage change n the probablty of choce for a 1% change n ndependent varable, whch s gven by: dp dx X 1 1 * w ( 1 P 1) X 1 1 P 1 = (6) Other thngs equal, the response s more elastc when * P 1 s lower,.e., when product 1 has a lower probablty of beng chosen. In other words, low-share choce brands can gan proportonately more for ther maretng efforts, as compared to hgh-share choce brands. The above propertes of the logt model are more credble than the propertes of a lnear probablty model, whch smply predcts P l as a functon of a lnear combnaton of the X l s. The lnear probablty model assumes a constant probablstc mpact of any change n the X l s. That s counter to our deas of what the mpact of maretng and contextual factors on choce ought to be and can result n predcted probabltes that are less than zero or greater than one! Proportonal draw: Ths s another property of the logt model, whch we llustrate wth an example: EXAMPLE Suppose that someone performed a survey of shoppers n an area to understand ther shoppng habts and to determne the share of shoppers that a new store mght attract. The respondents rated three exstng stores and one proposed store (descrbed by a wrtten concept statement) on a number of dmensons: (1) varety, (2) qualty, (3) parng, and (4) value for the money (Exhbt 2.). By fttng shoppers choces of exstng stores to ther ratngs through the logt model, we can estmate the coeffcents [b ]: A = b X + b X b X J J (7) where A = attractveness of store (for customer ); X = customer 's ratng or evaluaton of store on dmenson, = 1,..., J; and 6

7 b = mportance weght for dmenson. The data n Exhbt 2 come from a group of smlar customers. Exhbt 3 gves the share of the old stores wth and wthout the new store, the potental share of the new store, and the draw estmated from ths group. EXHIBIT 2 Ratngs and mportance data for the store-selecton example. EXHIBIT 3 Logt model analyss of new store share example. In column e of Exhbt 3, the draw s proportonal to maret share (column c). In other words, ths model assumes that all ndvduals consder all brands n ther choce process, that they do not go through any prescreenng or elmnate some brands. (Ths prescreenng s often referred to as a consderaton process.) The proportonal draw property mples, for example, that f a new lght beer s ntroduced nto the maret, t wll draw share from every product n the maret (ncludng regular beers), n proporton to the current maret shares of the exstng products. However, t s lely that the lght beer wll draw a dsproportonate share from other lght beers, rather than from regular beers. To 7

8 mnmze the effects of such dscrepances, t s mportant that n applcatons of the logt model, we carefully specfy the actual set of choces avalable to customers, or customer segments, based on maret realtes. Researchers have also developed several ways to deal wth the proportonal draw problem. One way s a pror segmentaton; the researcher segments the maret nto groups that do consder (dfferent) sets of brands dfferently. Another alternatve s to group products (rather than customers) nto groups that more drectly compete wth one another. If we vew the choce process as a herarchy, we can then assume that consumers select among branches of a tree at each level of the herarchy (Exhbt 4). The consumer mght frst choose the form of the deodorant and then, condtonal on that choce, choose the brand. The form of the logt model that apples here s called the nested logt, and t ncorporates an equaton le Eq. (4) for the selecton of product form (the upper level of the herarchy) and a separate logt model for brand (condtonal on the selecton of form) at the lowest level of the herarchy. EXHIBIT 4 Consumer decson herarchy for deodorant purchase. Source: Urban and Hauser 1980, p. 92. where The nested logt model can be represented as P P = P (8) P = probablty that customer chooses brand and product form P = probablty that customer frst chooses product form 8

9 P = probablty that customer chooses brand gven he or she has chosen product form (We drop the superscrpt n the dscusson below for smplcty.) If we assume attractveness s separable, we get A = A + A (9) where A A = attractveness of product form and brand = attractveness of product form A = attractveness of brand (when n product form ) The brand choce (bottom level of the herarchy n Exhbt 4) can be represented as a multnomal logt model as before: (10) Under sutable assumptons, the product form probablty has a smlar structure: (11) where s a normalzng constant to ensure that the sum of all choce probabltes add to 1. Substtutng Eq. (11) and Eq. (10) n Eq. (8) gves the full equaton for the nested logt model. (See Roberts and Llen, 1993, for a more complete dscusson). Logt Model Estmaton va Maxmum Lelhood Indvdual choce models of all sorts are dffcult to estmate. We outlne here the general approach to estmatng the smple MNL model (Eq. 4). Let, 9

10 Y 1 f customer chooses alternatve = 0 f customer does not choose alternatve (12) Y Then P ( = 1) s the probablty that U U m for all m. Now consder the Y lelhood that P ( = 1) for a random sample of N customers whose choces we have observed. Ths sample lelhood s the product of the lelhoods that each ndvdual n the sample chose the alternatve that they actually dd, whch can be represented as: N 1 2 J = 1 C Y L( β, β,..., β ) = P( Y = 1), (13) where C s the set of alternatves (the choce set) and β s are the unnown parameters of the ndvduals utlty functon to be estmated. Substtute for Y P ( = 1) from Eq. (4) to get: Y β X N e L (.) = β C X (14) = 1 e To smplfy estmaton, we typcally consder the logarthm of L, namely, Ln(L): N β X = 1 C C Ln ( L) = Y ( β X Ln e ) The estmates for β s can then be obtaned by maxmzng the Lelhood (L), or equvalently, by maxmzng Ln(L), by settng the partal dervatves to 0: (15) Ln( L) N = ( Y P ) X = 0 for = 1, 2,...J β = 1 C (16) Ths gves a set of J equatons n J unnowns, whch can be solved usng numercal methods. It can be shown that f a soluton exsts for ths set of equatons, that soluton (.e., the maxmum lelhood estmates for β s) s unque. Further, the maxmum lelhood estmates obtaned ths way have many desrable statstcal propertes -- the estmates are consstent, asymptotcally Normal, and asymptotcally effcent. The estmated β s can be nterpreted pretty much le regresson coeffcents. 10

11 EXAMPLE Consder a stuaton where are four choce alternatves avalable to customers, and we also now the prces of the four alternatves. We can thn of the logt model for ths applcaton as generatng four equatons n four unnowns: one parameter to represent the effect of product prces (β 1 ) and three alternatvespecfc constants (α s) to represent the ntrnsc value of the four alternatves (e.g., brand mage). (One of the alternatve-specfc constants, for example, α 1, s set to 0 to ensure that the model can be estmated). Usng Logt Models for Customer Targetng Peppers and Rogers (1993) descrbe how a frm s best customers outspend ts average customers by a factor of 16:1 n the retal ndustry, 12:1 for arlnes, and 5:1 n hotels. Thus, t pays mareters to target ther maretng efforts at customers who have the hghest probabltes of purchase (or, more generally, the hghest probablty of a favorable response). An ncreasngly common approach to developng such targetng programs, especally n drect maretng (also called database maretng), s to use develop choce models to dentfy the most mportant factors drvng customer choces. Typcally, the choce model enables the frm to compute an ndvdual s lelhood of purchase, or some other behavoral response, based on varables that the frm has n ts database, such as geodemographcs, past purchase behavor for smlar products, atttudes or psychographcs. A frm can use the probablty of choce/purchase estmated from an MNL model to calculate expected customer proftablty under a partcular acton t taes. For example, a drect maretng frm can drect ts maretng campagn to those customers (or customer segments) whose expected proftablty exceeds the cost of reachng them: Expected (gross) customer proftablty = Probablty of purchase x Lely purchase volume f a purchase s made (17) x Proft margn (for ths customer). EXAMPLE Exhbt 5 shows part of a drect maretng database after the frm has 11

12 completed the choce modelng step ust dscussed. Choce modelng provded the data n column A purchase probablty. The queston, then, s whch customers should the frm target? Suppose that the total cost of reachng one of these customers s $3.50. What should the frm do? Frms commonly use several approaches to answer ths queston. Frst, f the frm loos at the average expected proft, t may decde to target all 10 groups and mae a small proft (103($3.72 $3.50) = $2.20). Or t may target customers 1, 3, 5, and 6 and mae $6.51+$3.62+$6.96+$6.20-(4 $3.50)=$9.29. Notce that by usng choce-based modelng the frm can target customers to mprove proftablty by over 400 percent. Fnally, usng a more tradtonal segmentaton by average purchase volume, the frm would target, say 30 percent, or the three largest customers n ths case 2, 4, and 9 and lose $5.02! Frms usng ths approach typcally compute the expected customer proftablty at the ndvdual level and then sort the customer database n decreasng order of expected proftablty (Exhbt 5, column D). They then target customers who exceed some threshold (a proftablty measure) or fall nto the most proftable percentage of the database. In the example below, we descrbe how ths wors. 12

13 EXHIBIT 5 Choce-based segmentaton example for database maretng: target those customers whose (expected) proftablty exceeds the cost of reachng them by comparng column D wth the cost to reach that customer. Usng Logt Models for Customer Segmentaton Choce models can also be used to segment customers on the bass of the varables that most nfluence choces n each dentfed segment. In what follows, we outlne the methods used for latent class choce segmentaton. Ths approach enables mareters to understand the unobserved (latent) choce processes drvng dfferent segments of customers to behave dfferently n mang choces. Such understandng can then be used to target dfferent groups of customers wth the approprate maretng programs. For example, customers who are more prce senstve (as evdenced by ther prevous choces) can then be dentfed and offered specal promotons not avalable to the less prce senstve customers, who can be offered products wth enhanced features or servces. In the aggregate logt model summarzed n Eq. (4), every customer has an dentcal choce process (.e., utlty functon or purchase probablty rule) although each customer maes dfferent choces because of dfferences n the determnstc or random components n ther common utlty functon. However, customers not only dffer along observed characterstcs (e.g., sex, race) but also wth respect to the unobserved, but systematc, rules that they use for mang udgments about choce alternatves. Whle we rarely have suffcent data about each ndvdual to buld separate ndvdual utlty functons, we may stll 13

14 want to segment customers accordng to ther latent choce rules to account for the heterogenety that exsts n the populaton. Customer heterogenety can be classfed nto two categores: (1) observed heterogenety (e.g., customers dffer on observable characterstcs such as gender), and (2) unobserved heterogenety (e.g., customers dffer n terms of ther prce senstvtes). Observed heterogenety can be modeled drectly by ncludng assocated ndependent varables (e.g., gender) n the choce model. However, the same dea does not wor for modelng unobserved heterogenety (e.g., we cannot construct a varable for prce senstvty because we do not observe t). A common approach for accommodatng unobserved heterogenety s to use fnte mxture modelng, n whch each segment s assumed to follow ts own choce rule. In the framewor of logt models, uncondtonal purchase probablty s then assumed to be a mxture of several condtonal purchase probabltes, where each condtonal probablty corresponds to a segment. Then, gven the actual choces people mae, we can nfer the most lely values of these segment-level parameters (e.g., prce senstvtes for dfferent segments) from the data,.e., we smultaneously form segments as well as estmate the unnown choce process wthn each segment through the maxmum lelhood estmaton method. Operatonally, ths means that the weghts (β s) n the logt model dffer across segments, but the segments are unnown (latent) and have to be nferred from the data. To accommodate ths possblty, we can specfy Eq. (4) as follows: e ( P belongs to segment s) = (18) β s X e β X s There are several methods avalable for estmatng the parameters n Eq. (7). These methods allow the estmaton of: (a) the number of segments that best ft the data, (b) the parameters (β s ) of the utlty functon of each segment, (c) the proportons of the populaton that belong to each segment, and (d) the segment to whch a partcular customer s most lely belong to, whether or not that customer s purchase behavor was used to estmate the parameters of the model. A popular method for estmatng the parameters va latent class analyss s the EM (Expectaton Maxmzaton) algorthm (Wedel and Kamaura, 2000). A specal concern n estmatng latent class models (as compared to the aggregate MNL model) s that there may ssues of dentfablty (.e., nsuffcent number of dstnct data patterns for estmatng all the model parameters), a problem that s lely to exacerbated f the predctor varables are all nomnal and there are not 14

15 many of them, and/or there are not suffcent number of choce alternatves for the number of parameters to be estmated. Determnng the number of latent segments n MNL models: There are several ndces to assess the goodness of ft of the estmates of the MNL model that functon smlarly to the R 2 ndex assocated wth regresson models: (1) Ht rato the proporton of out-of-sample observatons correctly classfed by the estmated model; the hgher ths rato, the hgher the predctve valdty of the model, wth a maxmum possble value of 1; and (2) AIC (Aae Informaton Crteron), BIC (Bayesan Informaton Crteron), and CAIC (Consstent AIC), all of whch ndcate superor model performance the closer they are to 0. These ndces enable the modeler to determne the number of segments n the data,.e., to choose the model for whch the number of segments results n an ndex value closest to 0. The AIC crteron enables the analyst to trade off model ft aganst model complexty. Model ft can be mproved by addng more varables, whch however may ncrease complexty, or overweght unmportant aspects that are dsproportonately present n the sample as compared to ther presence n the populaton. In addton to accountng for the number of varables n the model, the BIC crteron accounts for sample sze. We recommend the BIC crteron, unless the modeler has nowledge about the pros and cons of each ndex n a specfc applcaton. For further detals on these ndces as well as about the EM algorthm, see Jagpal (1999) and Wedel and Kamaura (2000). EXAMPLE Assume we have a two-brand maret, wth brands A and B, whose maor dfference s n prce and that each customer s attractveness for these can be assessed as Attractveness of brand A for customer s PB P A, (19) where P A and P B are the prces of the brands and s the prce senstvty parameter for customer, where the hgher the value of, the more prce senstve the customer. 15

16 Now, accordng to ths model, the probablty that customer buys brand A can be assessed as PROB ( A ) ( PA / PB ) = (20) ( PA / PB ) + ( PB / PA ) and PROB B ) = 1 PROB ( A ) ( Assume that customers are of one of two types, low prce senstvty ( l ) or hgh prce senstvty ( h ), where we now nether the level of prce senstvty () nor the proporton of the populaton wth that level of senstvty (b l, b h ),.e., the mxng dstrbuton. What can we say about the (uncondtonal) probablty of a customer buyng brand A? Usng the formula for total probablty we get PROB ( A) PROB ( A ) b + PROB ( A ) b = (21) l where PROB (A l ) and PROB (A h ) are determned from Eq. (20). The challenge here s to estmate the four parameters n Eq. (21): l, h (the levels of prce senstvty) and b l, b h (the proportons of the populaton wth those levels of prce senstvty the weghts n the mxng dstrbuton), gven observed choces that customers mae n dfferent prce stuatons. h h In the example above, we assumed two segments (hgh and low prce senstvty) and that ndvdual purchase probabltes vared only by prce senstvty. In general, response models wll have a number of parameters (le prce senstvty here) and the number of segments wll not be nown n advance. Summary In ths techncal note, we provded a broad overvew of the MNL model focusng on ts structure, propertes, estmaton, and uses. We also descrbed how choce models can be used n customer targetng and segmentaton: (1) Usng choce probabltes estmated from an aggregate choce model for purposes of selectng target customers, and (2) Segmentng customers on the bass of ther unobserved choce processes va a latent class choce model. 16

17 References Peppers, Don, and Rogers, Martha, 1993, The One to One Future, Buldng Relatonshps One Customer at a Tme, Currency Doubleday, New Yor. Roberts, John H., and Llen, Gary L., 1993, Explanatory and predctve models of consumer behavor, n Handboos n Operatons Research and Management Scence, Vol. 5, Maretng, eds. Jehoshua Elashberg and Gary L. Llen, Elsever Scence Publshers B.V., North Holland, New Yor, pp Urban, Glen L., and Hauser, John R., 1980, Desgn and Maretng of New Products, Prentce Hall, Englewood Clffs, New Jersey Wedel, Mchel, and Kamaura, Wagner A., 2000, Maret Segmentaton: Conceptual and Methodologcal Foundatons, second edton, Kluwer Academc Press, Boston, Massachusetts 17