Analyzing the dependences of multi-category purchases on interactions of marketing variables

Abstract

We extend the usual specification of the multivariate probit model frequently used to analyze multi-category purchase incidence data by introducing interaction effects between marketing variables. Models are estimated by a Markov Chain Monte Carlo simulation method using 24,047 shopping visits made by a random sample of 1500 households in one specific grocery store over a one year period. Our data refer to a total of 25 food and non-food product categories and include socio-demographic household attributes in addition to purchases and marketing variables. Information criteria agree on the superiority of the extended specification. Estimation results demonstrate that many interaction effects are erroneously attributed to the main effects of marketing variables if one applies the usual specification instead. We derive managerial implications with respect to sales revenue by stochastic simulation. If managers base decisions on the usual specification in spite of its worse statistical performance, they run the risk to overestimate sales revenue increases due to sales promotion activities.

Appendix: Overview of MCMC simulation

Estimation of the MVP model is based on the Bayesian analysis of Zellner (1971) for the seemingly unrelated regression (SUR) model and the sampling of latent variables due to Albert and Chib (1993) as well as Chib and Greenberg (1998). Sampling of the correlation matrix of latent residuals in steps 3–5 draws upon Liu and Daniels (2006). Sampling of individual coefficients works similar to the approach developed by Banerjee et al. (2008) for the SUR model.

Stochastic utilities are multivariate normally distributed with mean vector $V_{it}$ containing deterministic utilities $V_{1it}, \ldots V_{Jit},$ and covariance matrix $\Sigma$. Therefore the conditional distribution of the stochastic utility of category j given the remaining $J-1$ categories is univariate normal with conditional mean $V_{jit|-j}$ and conditional variance $\sigma _{-j}$ as parameters:

$$\begin{aligned} V_{jit|-j}= & V_{jit} + \sigma_{s-k} {\mathop \Sigma \limits_{s-k}^{-1}} (V_{-kit} - \mu _{-jit}),\nonumber \\ \sigma _{-j}= & {} \sigma _{j,j} - \sigma _{-j} {\mathop \Sigma \limits _{-j}^{-1}} \sigma _{-j}^{'} \end{aligned}$$

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$V_{-kit}$ corresponds to vector $V_{it}$ without element k. $\sigma _{-j}$ denotes the vector containing the residual covariances of category j with the other $J-1$ categories. $\Sigma _{-j}$ corresponds to $\Sigma$ without row and column j.

$\mathcal{N}(\mu , \sigma )$ denotes the univariate normal distribution with expectation $\mu$ and variance $\sigma$. $\mathcal{TN}^{+}(\mu, \sigma )$ and $\mathcal{TN}^{-}(\mu, \sigma )$ denote univariate normal distributions truncated to positive and negative values, respectively. $\mathcal{IW} (\nu,A)$ denotes the inverse Wishart distribution with degrees of freedom $\nu$ and symmetric, nonsingular matrix A.

Coefficients including category constants are collected in a (P, J) matrix $\beta$ with $P=1+S+M$ and $P=1+S+M+0.5 \, M \, (M-1)$ for the specifications main and inter, respectively. S and M denote the number of socio-demographic and marketing variables. T symbolizes the total number of baskets across all households. (T, P) matrix X holds the P predictors, (T, J) matrix V the stochastic utilities of all baskets and categories.

Latent residuals are defined as differences between stochastic and deterministic utilities $e_{jit} = U_{jit} - V_{jit}$. The residual vector of household i for basket t is $e_{it} = (e_{1it}, e_{2i1t},\ldots, e_{Jit})^{'}$.

Initially all coefficients are set to zero and $\Sigma$ to a diagonal correlation matrix. Each coefficient has the prior $\mathcal N(0,\underline{v})$ with $\underline{v}= 4$. Estimation steps of each iteration are as follows:

• 1. For all categories $j=1,\ldots,J$, households $i=1,\ldots,I$ and their baskets $t=1,\ldots,T_i$

  $$U_{jit} \sim \left\{ \begin{array}{ll} {\mathcal{TN}}^{+}(V_{jit|-j}, \sigma _{-k}) & \text{ if } \quad y_{jit}=1 \\ {\mathcal{TN}}^{-}(V_{jit|-j}, \sigma _{-k}) & \text{ else } \end{array} \right.$$

• 2. Compute matrix of cross products of latent residuals across households $A= \sum _{i=1}^{I} \sum _{t=1}^{T_i} e_{it} e_{it}^{'}$.

• 3. Sample covariance matrix $\Sigma$ from $\mathcal IW (T-J-1,A)$.

• 4. Compute proposal correlation matrix $\tilde{R} = D \Sigma D$ where D is a (J, J) diagonal matrix with elements $d_{j,j} = \sigma _{k,k}^{-1/2}$ ($\sigma _{k,k}$ denotes the element of $\Sigma$ in row and column k).

• 5. Accept $\tilde{R}$ with probability $min(1, \exp {(0.5 (J+1) (\log {|\tilde{R}|}-\log {|\Sigma _{0}|})}))$ with $\Sigma _{0}$ as correlation matrix of the previous iteration.

• 6. Set $\Sigma = R$ and compute its inverse $H=\Sigma ^{-1}$

• 7. Sample coefficients $\beta _{pj}$ for predictors $p = 1,\dots,P$ and categories $j =1, \dots,J$:

  $$\begin{aligned} \beta _{pj} \sim \mathcal N (m,v) \quad \text{ with } \quad v = 1/(1/\underline{v} + h_{jj} x_p' x_p) \quad \text{ and } \quad m = h_j' (V - X \beta _0)' \, x_p \, v \\ \end{aligned}$$

  $h_{jj}$ is the element in row j and column j of H. $h_j$ and $x_p$ denote columns j and p of H and X, respectively. $\beta _0$ equals $\beta$ except for a zero value in row p and column j.