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A Model of Heterogeneous Multicategory Choice for Market Basket Analysis

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Published/Copyright: September 28, 2013
Review of Marketing Science
From the journal Review of Marketing Science Volume 11 Issue 1

Abstract

Based on market basket data, using multicategory purchase incidence models, we analyze demand interdependencies between product categories. We propose a finite mixture multivariate logit model to derive segment-specific intercategory effects of market basket purchase. Under the assumption that only a fraction of intercategory effects are significant, we exclude irrelevant effects by variable selection. This leads to a detailed description of consumers’ shopping behavior that varies over segments not only with respect to (w.r.t.) parameters’ values but also w.r.t. included interaction effects. As the high number of product categories in the model prohibits exact maximum likelihood estimation, we adopt pseudo-likelihood estimation. We apply our model to a data set with 31 product categories and 1,794 households purchasing 17,280 baskets in one store. The best fitting model is determined by predictive model selection. We find that a homogeneous model would overestimate the intensity of interaction between product categories.

Keywords: finite mixture model; market basket analysis; variable selection; multivariate logistic regression; pseudo-likelihood estimation

Appendix: pseudo-likelihood estimation

The FM-MVL model is estimated in iterations each consisting of three sampling steps. First, the membership probability of each household h for segment k is sampled. Given an assignment of households to components based on the probabilities in step 1, components are independent (Shi, Murray-Smith, and Titterington 2005). As mentioned before, we assume that not all interaction effects contribute to the explanation of purchase behavior and include a variable selection step that determines which effects are not significantly different from zero, that is, which categories are pairwise independent (step 2). This step is located between segmentation and parameter estimation. Afterward, component-specific parameters are drawn by a hybrid Monte Carlo step (Duane et al. 1987; Horowitz 1991; Neal 1996) that simulates a dynamical (Hamiltonian) system (step 3). The advantage of a hybrid MC algorithm is the possibility to suppress random walks for a sequence of steps L thus enabling a guided search for new parameter values (Neal 1996). In each iteration, step 2 and step 3 are performed for every component k. If possible, we suppress the iteration index t in the following for ease of exposition.

Sampling details

Step 1: The membership probability of household h for component k given the component assignments z of all other households is evaluated by a Gibbs sampling step according to

[15]
[15]

with and (Shi, Murray-Smith, and Titterington 2005). With these probabilities, the complete household assignment to segments z is sampled for one iteration resulting in K subsets of households with a total of market baskets that are assigned to a segment k. The intractable joint basket probability is replaced with the respective PL value . is calculated as in Shi, Murray-Smith, and Titterington (2005).

Step 2: We make use of an algorithm explained in Geweke (2005) for linear regression which Dippold and Hruschka (2013) applied to a homogeneous logit model. Given its contribution to the explanation of purchase behavior, the probability of being zero is determined for every single interaction18 parameter conditioned on the other parameters determined as significant within the component as

[16]
[16]

being the stochastic utility of a purchase in category j for a market basket i given segment k with random uniform numbers , as proposed in Tüchler (2008). The precision is distributed as

[17]
[17]

with the sum of squared errors and the number of baskets assigned to category k, , as well as prior values for variance and free parameters .

Only if this probability is smaller than a random uniform number , the respective interaction parameter is sampled in the next step.

Step 3: Assuming an a priori independence of parameters across segments, we can sample segment-specific significant parameters independently for each segment with

[18]
[18]

where serves as prior. Again, is replaced with . In a hybrid MC, the conditional parameter sampling density is conceived to be as proportional to a fictitious potential energy , that is,

[19]
[19]

treating the parameters like position parameters in a dynamical system. For every parameter within , a respective momentum vector with npar elements is defined. Together with the associated kinetic energy (mass as suggested by Rasmussen (1996)), we get the total energy of the system . This total energy H facilitates draws from the joint distribution of the parameters in the dynamical system, because

[20]
[20]

This relation is used for updating position and momentum parameters in a stepwise process. Holding the total energy H constant within one iteration allows for a sequence of L directed improvements of values. These substeps L are called leap-frog steps. With one leap-frog step and step size , the parameter update for one parameter of , for example, with respective momentum variable , reads as (Shi, Murray-Smith, and Titterington 2005)

[21]
[21]

We use and for burn-in, afterward and . After one iteration, the value for the kinetic and consequently also the total system energy are slightly changed by perturbating thus exploring the whole phase space (Neal 1996). In a Metropolis–Hastings step, the new estimates are proposed as new parameters with probability . Further details on these sampling steps can be found in Neal (1996) and Shi, Murray-Smith, and Titterington (2005).

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  1. 1

    Examples for this approach which deal with another multicategory problem by focusing on the covariance of coefficients across categories are the probit models of Ainslie and Rossi (1998) and Seetharaman, Ainslie, and Chintagunta (1999) or the logit models of Hansen, Singh, and Chintagunta (2006) and Singh, Hansen, and Gupta (2005). In the latter two studies, a factor analytic structure is imposed on the covariance matrix of coefficients.

  2. 2

    There has been a long discussion on the performance of finite vs continuous mixture models (e.g. Wedel et al. 1999; for a short summary, see Varki and Chintagunta 2004). Whereas Allenby, Arora, and Ginter (1998) and Allenby and Rossi (1999) argue that FM models do not sufficiently represent consumer heterogeneity, especially when the number of segments is small and complete homogeneity within a segment is an unrealistic assumption, Andrews, Ainslie, and Currim (2002) do not find any performance superiority of the continuous over the FM models. Even for a very limited number of segments (one to three), the continuous and the discrete model recover parameters and forecast holdout data equally well. Additionally, Wedel and Kamakura (1998) and Wedel et al. (1999) stress the consistency of the FM model with the way management thinks about consumers in segments.

  3. 3

    Segment is the marketing interpretation of a component in a FM model. Therefore, the terms segment and component are used interchangeably in the text.

  4. 4

    In contrast to probit models, there is no biasing effect of joint non-purchase that would be the most frequently occurring event. We also remark that we follow the cross-category effect definition by Hruschka, Lukanowicz, and Buchta (1999). In contrast to Russell and Petersen (2000), cross-category effects do not depend on a household’s typical basket size. This modeling decision is justified, because (1) the inclusion of basket size resulted only in a weak improvement of the LL value for holdout data in the RP model and (2) our model already accounts for interaction effect variability by estimating different effects for different segments.

  5. 5

    We thank one anonymous reviewer who suggested to discuss this issue and drew our attention to the latent variable interpretation of co-incidence. We also thank the editor for suggesting relevant references.

  6. 6

    With , Z has elements, that is, all possible market baskets. Huang and Ogata (2002) observe exponents between 9 and 15 to be the limit of computation.

  7. 7

    The general disadvantage of wrong standard errors can be easily adjusted for, as correct standard errors can be computed with bootstrapping (e.g. Efron and Tibshirani 1998).

  8. 8

    This two-step approach is conventionally used in FM models for multicategory choice (e.g. Song and Chintagunta 2007).

  9. 9

    See, for example, Andrews and Currim (2002) for a complete tabulation including formulas for calculations.

  10. 10

    We argue that the homogeneous model smoothes interaction effects. The lower number of interaction effects included for the heterogeneous model might also contribute to such results. Boztuğ and Reutterer (2008) formulate a similar hypothesis, though they motivate it differently. Chib, Seetharaman, and Strijnev (2002) present the opposite effect.

  11. 11

    Our model specification does not include RP’s category-specific household variables time since last category purchase (TIME) and loyalty (LOYAL). As this model is estimated over the purchases within one shop only neglecting purchases in other stores, we do not have complete information on a consumer’s shopping history. Therefore, the values of TIME and LOYAL would not be meaningful. Besides, we already account for heterogeneity with the FM model and do not need auxiliary measures of consumer diversity.

  12. 12

    Test runs showed that the model parameters and especially the household assignment to segments stabilize quickly.

  13. 13

    For reasons of comparability, PLL is also calculated for the independence model whose parameters are estimated by ML.

  14. 14

    We only consider interaction effects larger than 0.001 in absolute size.

  15. 15

    We thank one anonymous reviewer who recommended to compare the results of our model to those obtained by a correlational analysis.

  16. 16

    Household income in thousand US$.

  17. 17

    HS: high school; C: college; TS: technical school; PG: postgraduate work. SC: some college, what means that the person left college without a degree.

  18. 18

    For model stability, variable selection is not applied to category constants or marketing-mix coefficients.

Published Online: 2013-09-28

©2013 by Walter de Gruyter Berlin / Boston

Articles in the same Issue

  1. Article
  2. A Model of Heterogeneous Multicategory Choice for Market Basket Analysis
  3. Examining the Dynamics of Consumer Interest and Live Performance Event TicketSales in the Presence of a Critical Industry-Wide Event
https://doi.org/10.1515/roms-2012-0001
Keywords for this article
finite mixture model; market basket analysis; variable selection; multivariate logistic regression; pseudo-likelihood estimation

Articles in the same Issue

  1. Article
  2. A Model of Heterogeneous Multicategory Choice for Market Basket Analysis
  3. Examining the Dynamics of Consumer Interest and Live Performance Event TicketSales in the Presence of a Critical Industry-Wide Event
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