Incorporating Time-Invariant Covariates into the Pareto/NBD and BG/NBD Models

Transcription

1 Incorporating Time-Invariant Covariates into the Pareto/NBD and BG/NBD Models Peter S Fader wwwpetefadercom Bruce G S Hardie wwwbrucehardiecom August Introduction This note documents how to incorporate the effects of time-invariant covariates into the Pareto/NBD and BG/NBD models It is assumed that the reader is familiar with the derivations of these models, as presented in Fader and Hardie 2005 and Fader et al 2005 Let z 1 be the vector of time-invariant covariates that are assumed to explain some of the cross-sectional heterogeneity in the purchasing process, and z 2 be the vector of time-invariant covariates that are assumed to explain some of the cross-sectional heterogeneity in the dropout process While it will typically be the case that z 1 = z 2, this equality is not assumed in these derivations Note that since these covariates are specific to each individual, both z 1 and z 2 should have an individual-specific subscript i; this has been suppressed for notational convenience The basic results are as follows: For the Pareto/NBD model, we simply replace α and β with α = α 0 exp γ 1z 1 β = β 0 exp γ 2z 2 where γ 1 and γ 2 capture the effects of these two vectors of covariates; r and s remain unchanged c 2007 Peter S Fader and Bruce G S Hardie This document can be found at < 1

2 For the BG/NBD model, we simply replace α, a, and b with α = α 0 exp γ 1z 1 a = a 0 exp γ 2z 2 b = b 0 exp γ 3z 2 where γ 1, γ 2 and γ 3 capture the effects of these two vectors of covariates; r remains unchanged 2 Preliminaries A popular, easily interpretable method for incorporating the effects of exogenous covariates in event-time models is the proportional hazards approach In this framework, the covariates have a multiplicative effect on the hazard rate More specifically, let F 0 t θ be the so-called baseline cdf for the distribution of an individual s interpurchase times, and f 0 t θ and h 0 t θ the associated pdf and hazard rate function The most common formulation of the proportional hazards specification states that ht θ, γ, z i =h 0 t θ expγ z where z denotes the vector of time-invariant covariates and γ denotes the effects of these covariates Note that z must not include an intercept term When the baseline is distributed exponential with rate parameter θ, ht θ, γ, z =θ expγ z It follows from the definition of the hazard rate function, ht = that, since F 0=0, F t =1 exp Therefore, and ft 1 F t t F t θ, γ, z =1 e θ expγ zt 0 hudu ft θ, γ, z =θ expγ ze θ expγ zt 2

3 3 The Case of the Pareto/NBD Model The Pareto/NBD model is based on the following assumptions: i Customers go through two stages in their lifetime with a specific firm: they are alive for some period of time, then become permanently inactive ii While alive, the number of transactions made by a customer follows a Poisson process with transaction rate λ This is equivalent to assuming that the time between transactions is distributed exponential with transaction rate λ, ft j t j 1 λ =λe λt j t j 1, t j >t j 1 > 0, where t j is the time of the jth purchase iii Heterogeneity in transaction rates across customers follows a gamma distribution with shape parameter r and scale parameter α: gλ r, α = αr λ r 1 e λα iv A customer s unobserved lifetime of length ω after which he is viewed as being inactive is exponentially distributed with dropout rate μ: fω μ =μe μω Note: Previous discussions of the Pareto/NBD have used τ to denote the time at which the customer becomes inactive Because of notational conflicts, we now use ω to denote this quantity v Heterogeneity in dropout rates across customers follows a gamma distribution with shape parameter s and scale parameter β gμ s, β = βs μ s 1 e μβ Γs vi The transaction rate λ and the dropout rate μ vary independently across customers We now assume that interpurchase times are distributed according to the with-covariates exponential distribution ft j t j 1 λ 0, γ 1, z 1 =λ 0 expγ 1z 1 e λ 0 expγ 1 z 1t j t j 1 and F t j t j 1 λ 0, γ 1, z 1 =1 e λ 0 expγ 1 z 1t j t j 1, 3

4 with the unobserved heterogeneity in λ 0 captured by a gamma distribution with shape parameter r and scale parameter α 0 : gλ 0 r, α 0 = αr 0 λr 1 0 e λ 0α 0 We also assume that lifetimes are distributed according to the withcovariates exponential distribution fω μ 0, γ 2, z 2 =μ 0 expγ 2z 2 e μ 0 expγ 2 z2ω and F ω μ 0, γ 2, z 2 =1 e μ 0 expγ 2 z2ω, with the unobserved heterogeneity in μ 0 captured by a gamma distribution with shape parameter s and scale parameter β 0 : gμ 0 s, β 0 = βs 0 μs 1 0 e μ 0β 0 Γs Following the logic of the derivation presented in Fader and Hardie 2005, it follows that and Lλ 0, γ 1 z 1,t 1,,t x,t, ω > T =λ x 0 expγ 1z 1 x e λ 0 expγ 1 z 1T Lλ 0, γ 1 z 1,t 1,,t x,t, inactive at ω t x,t] = λ x 0 expγ 1z 1 x e λ 0 expγ 1 z1ω Removing the conditioning on ω yields the following expression for the individual-level likelihood function: Lλ 0,μ 0, γ 1, γ 2 z 1, z 2,x,t x,t = Lλ 0, γ 1 z 1,x,T,ω >TP ω >T μ 0, γ 2, z 2 T { + Lλ 0, γ 1 z 1,x,t x,t,inactive at ω t x,t] t x } fω μ 0, γ 2, z 2 dω = λx 0 expγ 1 z 1 x μ 0 expγ 2 z 2 λ 0 expγ 1 z 1+μ 0 expγ 2 z 2 e λ 0 expγ 1 z 1+μ 0 expγ 2 z 2t x + λ x+1 0 expγ 1 z 1 x+1 λ 0 expγ 1 z 1+μ 0 expγ 2 z 2 e λ 0 expγ 1 z 1+μ 0 expγ 2 z 2T 1 We remove the conditioning on unobserved latent variables λ 0 and μ 0 by taking the expectation of 1 over the distributions of λ 0 and μ 0 To facilitate 4

5 this calculation, we perform the change of variables λ = λ 0 expγ 1 z 1 and μ = μ 0 expγ 2 z 2, which implies gλ r, α = αr λ r 1 e λα gμ s, β = βs μ s 1 e μβ Γs, where α = α 0 exp γ 1z 1, and, where β = β 0 exp γ 2z 2 Note the addition of the addition of minus signs in the exponential terms as we go from multipliers of λ 0,μ 0 to multipliers of α 0,β 0 This gives us Lr, α 0,s, β 0, γ 1, γ 2 z 1, z 2,x,t x,t { λ x μ = 0 0 λ + μ e λ+μtx + λx+1 λ + μ e λ+μt } gλ r, αgμ s, β dλdμ We know from Fader and Hardie 2005 that the solution to this is where Lr, α 0,s, β 0, γ 1, γ 2 z 1, z 2,x,t x,t = Γr + xαr β s { } s r + x A 1 + A 2 r + s + x r + s + x A 1 = 2F 1 r + s + x, s +1;r + s + x +1; α β α+t x α + t x r+s+x if α β 2F 1 r + s + x, r + x; r + s + x +1; β α β+t x β + t x r+s+x if α β 2 3 and A 2 = 2F 1 r + s + x, s; r + s + x +1; α β α+t α + T r+s+x if α β 2F 1 r + s + x, r + x +1;r + s + x +1; β α β+t β + T r+s+x if α β 4 In other words, the likelihood function for the time-invariant-covariates version of the Pareto/NBD model is the likelihood function associated with the basic model where α and β are replaced by α = α 0 exp γ 1 z 1 and β = β 0 exp γ 2 z 2; r and s remain unchanged Suppose we have transaction data for a sample of N customers, where customer i made x i purchases in the period 0,T i ] with the last transaction occurring at t xi, and covariate vectors z 1i and z 2i The sample 5

6 log-likelihood function is given by LLr, α 0,s,β 0, γ 1, γ 2 = N ln [ Lr, α 0,s,β 0, γ 1, γ 2 z 1i, z 2i,x i,t xi,t i ] i=1 In numerically evaluating this function, we note that the α / β condition associated with 2 4 must now be checked for each i For any given α 0,β 0, it may be the case that α>βfor person i while α<βfor person j because of the values of z 1i, z 2i and z 1j, z 2j and γ 1, γ 2 For all related expressions eg, E[Xt],EY t x, t x,t, it is easy to show that we simply replace α and β with α = α 0 exp γ 1 z 1 and β = β 0 exp γ 2 z 2 to arrive at their with-time-invariant-covariates equivalents 4 The Case of the BG/NBD Model The BG/NBD model is based on the following assumptions the first three of which are identical to the corresponding Pareto/NBD assumptions: i Customers go through two stages in their lifetime with a specific firm: they are alive for some period of time, then become permanently inactive ii While alive, the number of transactions made by a customer follows a Poisson process with transaction rate λ This is equivalent to assuming that the time between transactions is distributed exponential with transaction rate λ, ft j t j 1 λ =λe λt j t j 1, t j >t j 1 > 0, where t j is the time of the jth purchase iii Heterogeneity in transaction rates across customers follows a gamma distribution with shape parameter r and scale parameter α: gλ r, α = αr λ r 1 e λα iv After any transaction, a customer becomes inactive with probability p Therefore the point at which the customer drops out is distributed across transactions according to a shifted geometric distribution with pmf P inactive immediately after jth transaction = p1 p j 1, j =1, 2, 3, 6

7 v Heterogeneity in dropout probabilities follows a beta distribution with parameters a and b: fp a, b = pa 1 1 p b 1 Ba, b, 0 p 1 vi The transaction rate λ and the dropout probability p vary independently across customers We now assume that interpurchase times are distributed according to the with-covariates exponential distribution ft j t j 1 λ 0, γ 1, z 1 =λ 0 expγ 1z 1 e λ 0 expγ 1 z 1t j t j 1 and that the unobserved heterogeneity in λ 0 is distributed gamma with shape parameter r and scale parameter α 0 : gλ 0 r, α 0 = αr 0 λr 1 0 e λ 0α 0 Following the logic of the derivation presented in Fader et al 2005, Lλ 0,p,γ 1 z 1,x,t x,t=1 p x λ x 0 expγ 1z 1 x e λ 0 expγ 1 z 1T + δ x>0 p1 p x 1 λ x 0 expγ 1z 1 x e λ 0 expγ 1 z 1t x Taking the expectation of this over the distribution of λ 0 gives us Lr, α 0,p,γ 1 z 1,x,t x,t =1 p x Γr + xα0 r expγ 1 z 1 x α 0 + expγ 1 z 1T r+x + δ x>0 p1 p x 1 Γr + xα0 r expγ 1 z 1 x α 0 + expγ 1 z 1t x r+x =1 p x Γr + x[α 0 exp γ 1 z 1] r α 0 exp γ 1 z 1+T r+x + δ x>0 p1 p x 1 Γr + x[α 0 exp γ 1 z 1] r α 0 exp γ 1 z 5 1+t x r+x Note that we have yet to include the effects of z 2, the vector of timeinvariant covariates that are assumed to explain some of the cross-sectional heterogeneity in the dropout process It is not possible to include the effects of covariates into p and then allow for unobserved heterogeneity using the beta distribution Using the logic of the beta-logistic model Heckman and Willis 1977, Rao and Steckel 1995, we incorporate the effects of time-invariant covariates on the dropout process via the parameters of the beta mixing distribution 7

8 by making a and b functions of z 2 That is, we assume that heterogeneity in the dropout probabilities follows a beta distribution with parameters a 0 exp γ 2 z 2 and b 0 exp γ 3 z 2: fp a 0,b 0,γ 2,γ 3 ; z 2 = pa 0 exp γ 2 z p b 0 exp γ 3 z 2 1 Ba 0 exp γ 2 z 2,b 0 exp γ 3 z 2 Taking the expectation of 5 over this distribution of p gives us Lr, α 0,a 0,b 0,γ 1, γ 2, γ 3 z 1, z 2,x,t x,t { Ba0 exp γ 2 = z 2,b 0 exp γ 3 z 2+x Ba 0 exp γ 2 z 2,b 0 exp γ 3 z 2 Γr + x[α 0 exp γ 1 z 1] r } α 0 exp γ 1 z 1+T r+x { Ba0 exp γ 2 + δ z 2+1,b 0 exp γ 3 z 2+x 1 x>0 Ba 0 exp γ 2 z 2,b 0 exp γ 3 z 2 Γr + x[α 0 exp γ 1 z 1] r } α 0 exp γ 1 z 1+t x r+x In other words, the likelihood function for the time-invariant-covariates version of the BG/NBD model is the likelihood function associated with the basic model where α, a, and b are replaced by α = α 0 exp γ 1 z 1, a = a 0 exp γ 2 z 2, and b = b 0 exp γ 3 z 2; r remains unchanged 1 For all related expressions eg, E[Xt],EY t x, t x,t, it is easy to show that we simply replace α, a, and b with α = α 0 exp γ 1 z 1, a = a 0 exp γ 2 z 2, and b = b 0 exp γ 3 z 2 to arrive at their with-time-invariantcovariates equivalents 1 We note that the BG/NBD has 50% more covariate parameters than the Pareto/NBD While this difference could be removed by making only a a function of covariates ie, set γ 3 to 0, doing so would constrain the way in which the covariates could influence the shape eg, variance of the beta distribution Furthermore, it would be inconsistent with the basic beta-logistic formulation 8

9 References Fader, Peter S and Bruce G S Hardie 2005, A Note on Deriving the Pareto/NBD Model and Related Expressions < Fader, Peter S, Bruce G S Hardie, and Ka Lok Lee 2005, "Counting Your Customers" the Easy Way: An Alternative to the Pareto/NBD Model, Marketing Science, 24 Spring, Heckman, James J and Robert J Willis 1977, A Beta-logistic Model for the Analysis of Sequential Labor Force Participation by Married Women, Journal of Political Economy, 85 February, Rao, Vithala R and Joel H Steckel 1995, Selecting, Evaluating, and Updating Prospects in Direct Marketing, Journal of Direct Marketing, 9 Spring,