Multi-category purchase incidences with marketing cross effects

Abstract

We focus on cross effects of marketing variables and cross category dependences for multi-category decisions which households take during a shopping trip to a retail store. A cross effect is defined as the effect which a marketing variable used for a certain product category exerts on purchases of another category. Using Dirichlet process mixture models with multivariate probit components we analyze purchase incidences of 24,047 shopping visits of a random sample of 1500 households. Independent variables of these models encompass marketing variables for 25 product categories and household attributes. We discuss differences between the two best performing models, a full model which includes both cross effects and cross category dependences, and a related restricted model which ignores cross effects. We obtain several high and significant differences with respect to category constants and cross category dependences between these two models. We also present explanations for the larger (in absolute terms) cross effects of features or displays. We demonstrate that by ignoring cross effects management runs the risk to obtain in many product categories too optimistic forecasts of sales revenue changes due to promotions. In contrast to previous related work suggesting not to use promotions which are not tailored to individual households in any of the investigated categories, we obtain support for such promotions in at least 48 % of the 25 product categories. In addition, based on the full model we demonstrate that often different categories are appropriate for promotions which are targeted at household clusters.

Appendices

Appendix 1: Overview of MCMC Simulation

Clusters are indexed as $c = 1, \ldots , C$ with C as the number of clusters currently formed. I and $I_{-i,c}$ denote the number of households and the number of households of cluster c without household i, respectively. Following existing practice we set $\alpha =1$ (Neal 2000). One MCMC iteration consists of the following steps which correspond to algorithm 7 of Neal (2000):

• 1. For each cluster:

  • (a) Draw new samples of coefficients from their respective prior distributions (see “Appendix 2”) and calculate the corresponding likelihood.

  • (b) Decide whether to form a new segment $c_i^*$ with probability

    $$\begin{aligned} min \left[ 1,\frac{\alpha L_i^*}{(I-1) L_i}\right] \end{aligned}$$

• 2. For each singleton:

  • (a) Decide on allocating the singleton to one of the clusters with probability proportional to

    $$\begin{aligned} \frac{I_{-i,c}}{I-1} \end{aligned}$$

  • (b) Calculate the corresponding likelihood and place the singleton into the new cluster $c_i^*$ with probability proportional to

    $$\begin{aligned} min \left[ 1,\frac{(I-1) L_i^*}{\alpha L_i} \right] \end{aligned}$$

• 3. Decide on allocating each customer belonging to a cluster to one of the other clusters with probabilities proportional to

  $$\begin{aligned} g \frac{I_{-i,c}}{I-1} L_i \end{aligned}$$

  where g serves as normalizing constant.

• 4. Estimate parameters for each cluster as described in “Appendix 2”.

On the level of stochastic utilities the multivariate probit model corresponds to a seemingly unrelated regression model (Chib and Greenberg 1998). That is why the likelihoods of steps 1–3 are based on latent residuals of the multivariate probit model which are defined as $e_{jit} = U_{jit} - V_{jit}$ and depend on parameter values assumed. The residual vector of household i for basket t is $e_{it} = (e_{1it}, e_{2i1t},\ldots , e_{Jit})^{\prime}$. (J, J) matrix $A_{i}$ of residual cross products across all baskets of household i is given by:

$$\begin{aligned} A_{i} = \sum _{t=1}^{T_i} e_{it} e_{it}^{\prime} \end{aligned}$$

(3)

Finally we obtain as likelihood $L_i$ for household i (see, e.g., the analysis of the seemingly unrelated regression model in Zellner 1971 or Greene 2003):

$$\begin{aligned} L_i = |\varSigma |^{-T_i/2} \exp {\left( -\frac{1}{2} tr (A_{i} \varSigma ^{-1}) \right) } \end{aligned}$$

(4)

Appendix 2: Clusterwise parameter sampling

Estimation steps described here concern all households which in the current iteration are assigned to segment c, i.e., all i $\in$ c. Note that we suppress the suffix c in this appendix to simplify notation.

Estimation of the multivariate probit model is based on the Bayesian analysis of Zellner (1971) for the seemingly unrelated regression 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 seemingly unrelated regression model.

Stochastic utilities are multivariate normally distributed with mean vector $V_{it}$ containing deterministic utilities $V_{1it}, \ldots V_{Jit},$ and covariance matrix $\varSigma$. 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:

$$\begin{aligned} V_{jit|-j} = V_{jit} + \sigma _{s -k} \varSigma _{s -k}^{-1} (V_{-kit} - \mu _{-jit}), \quad \sigma _{-j} = \sigma _{j,j} - \sigma _{-j} \varSigma _{-j}^{-1} \sigma _{-j}^{\prime} \end{aligned}$$

(5)

$V_{-kit}$ 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. $\varSigma _{-j}$ corresponds to $\varSigma$ without row and column j.

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

Coefficients are collected in a (P, J) matrix $\beta$ with $P=1+S+2J$ denoting the number of coefficients, i.e., a constant, S coefficients for sociodemographic variables, and 2J coefficients if own effects and cross effects of marketing variables as given in Eq. (1). T symbolizes the total number of baskets of all the households currently assigned to the cluster. (T, P) matrix X holds the P predictors including the constant, (T, J) matrix V the stochastic utilities of all baskets and categories.

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

• 1. For all baskets $t=1,\ldots ,T$ and categories $j=1,\ldots ,J$

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

• 2. Compute matrix of cross products of latent residuals $A=\sum _i A_{i}$

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

• 4. Compute proposal correlation matrix $\tilde{R} = D \varSigma D$ 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 $\varSigma$ in row and column k)

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

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

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

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

  $h_{jj}$ 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.

Appendix: 3 Significant differences of category constants and coefficients

The following table contains all the differences of category constants and coefficients between the full and the restricted model whose respective 95 % probability intervals only include either positive or negative values (Table 11).

Table 11 Significant differences of category constants and coefficients