Improving customer profit predictions with customer mindset metrics through multiple overimputation

Abstract

Research and practice have called for the incorporation of customer mindset metrics (CMMs) to improve the accuracy of models that predict individual customer profits. However, as CMMs are self-reported data, collected through customer surveys, they are seldom available for a firm’s entire customer database and in addition always measured with some degree of error. Their usage in models for individual-level predictions of customer profit has therefore proven challenging. We offer a solution through a new method called multiple overimputation (MO). MO treats missing data as an extreme form of measurement error and imputes the CMMs for both customers with observed, albeit with measurement error, as well as missing values, that are then included as predictors in a model of individual customer profits. Through a simulation study, empirical application in the pharmaceutical industry, and a customer selection exercise, we demonstrate the predictive and economic value of applying MO in the context of CRM.

Appendices

Appendix A: Details on model specification for sales and retention

Zero-inflated Poisson (ZIP) model

In each month t during period 3 (i.e., t = 11-45), we observe for each customer i (i = 1 to N) the level of sales ($y_{it}^{p3}$) and sales calls ($Det_{it}^{p3}$) directed toward that customer. We assume that sales from customer i in month t follow a ZIP model (Lambert 1992), such that at any time t customer i can belong to either of two latent states, dormant, or inactive, (B_it = 1) versus active (B_it = 0). Market forces, marketing, and other influences likely affect customer i’s switching between states. We assume that customers never quit a relationship, such that there is always a finite probability (1 − π_it) that they will prescribe the firm’s drugs, in line with extant research (Kumar et al. 2008). Under the ZIP model, the probability that sales (y_it) from customer i in time t equals k is;

$$ \begin{array}{@{}rcl@{}} p(y_{it}^{p3}&=&0|\lambda_{it},\pi_{it})=\pi_{it}+(1-\pi_{it})\exp(-\lambda_{it})\\ p(y_{it}^{p3}&=&k|\lambda_{it},\pi_{it})=(1-\pi_{it})\frac{\lambda^{k}_{it}\exp(-\lambda_{it})}{k!},\\ k&=&1,2,..., \end{array} $$

(A.1a)

where λ_it > 0. As per Eq. A.1a, customer i is active (π_it = 0) when sales reach at least one new prescription in time t (i.e., $y_{it}^{p3} > 0$). When we do not observe sales in time t, customer i could either belong to the dormant state with probability π_it or the active state with probability 1 − π_it , or y_it = 0. We therefore include the term (1 − π_it)exp(−λ_it) when modeling the probability that sales equal 0, or $p (y_{it}^{p3} = 0)$. Both, λ_it and π_it are unknown customer-specific parameters, modeled as functions of observed covariates (Ghosh et al. 2006). We rewrite Eq. A.1a as a mixture model of latent random variables $V_{it}^{p3}$ and $B_{it}^{p3}$;

$$ \begin{array}{@{}rcl@{}} y_{it}^{p3} &=& V_{it}(1 - B_{it}),\\ V_{it}&\sim&Poisson(\lambda_{it}), {\text{and}}\\ B_{it}&\sim& Bernoulli(\pi_{it}). \end{array} $$

(A.1b)

The expected number of new prescriptions from physician i in time t, which represents the Poisson mean λ_it, is modeled as;

$$ ln(\lambda_{it})=\beta^{\lambda}_{i}X^{\lambda p3}_{it}, $$

(A.2a)

where $\beta ^{\lambda }_{i}$ represents the customer-specific coefficients and $X^{\lambda p3}_{it}$ refers to the corresponding covariates that capture customer past purchase behavior, i.e., lagged sales ($y_{it-1}^{p3}$) and firm actions, i.e., sales calls ($Det_{it}^{p3}$). We model the Bernoulli random variable B_it (Eq. A.1b), which represents the probability that customer i is inactive in time t, as;

$$ logit(\pi_{it})=\beta^{\pi}_{i}X^{\pi p3}_{it} $$

(A.2b)

Similar to Eq. A.2a, $\beta ^{\pi }_{i}$ represents the customer-specific coefficients and $X^{\pi p3}_{it}$ refers to the covariates that capture firm actions and customer past purchase behavior.¹²

Hierarchical model

With the following hierarchical model of customer-specific coefficients, $\beta _{i} = (\beta ^{\lambda }_{i},\beta ^{\pi }_{i})$, we can assess the influence of CMMs and their behavioral predictors on sales and retention;

$$ \beta_{i}=\gamma Z_{i}^{p2}+\nu_{i} $$

(A.3)

We measure observed customer heterogeneity covariates (Z_i) during period 2, to control for the endogeneity among sales and CMMs (or the reinforcing effect of sales on CMMs). Our model thus captures the influence of customer i’s prior CMMs (during period 2) on his or her future behavior (during period 3).¹³ The specific heterogeneity covariates include CMMs ($\overline {CMM_{i}^{p2}}$), specialty, i.e., whether the physician is a specialist in a certain medical field (SPC_i), and logarithm of average period 2 sales, $ln(\overline {y^{p2}_{i}}$), as a proxy for the size of the customer wallet. By accounting for these measures, we can evaluate the effect of CMMs over and above commonly available measures of observed customer heterogeneity.¹⁴

Further, ν_i represents the unobserved heterogeneity component that we assume to follow a multivariate normal distribution with zero mean and a variance-covariance matrix V. Similar to Allenby and Ginter (1995), in the absence of γ’s and covariates (Z_i), Eq. A.3 represents a standard random effects distribution for β_i. Since CMMs can be considered part of the unobserved heterogeneity, ν_i allows us to assess the value of including CMM information in the hierarchical model (Eq. A.3) over and above a random effects specification of unobserved heterogeneity.

Appendix: B : Model estimation and prediction of twelve-months ahead customer profits in the holdout sample

We estimate the CMM imputation model based on behavioral predictors from period 1 using the estimation sample of 407 customers. We conduct the estimation of parameters in the ZIP and imputation models as well as the prediction of sales in the holdout sample in a fully Bayesian framework employing MCMC algorithms to enable posterior inference. We provide the prior specifications for the model parameters, estimation, and imputation algorithms in Web Appendix B in Supplementary Material. Each MCMC iteration in our model estimation proceeds in three phases. In the first phase, we simulate draws from the posterior distribution of the MO model parameters (Eq. 5) and use them to replace CMMs in the estimation dataset. In the second phase, we simulate draws from the posterior distribution of the ZIP model parameters using the multiple overimputed data from the first phase (Eqs. A.1a, A.2a, A.2b, and A.3 in Appendix A). In the third phase, we simulate the predictive posterior distribution of sales, retention, and overimputed CMMs for customer i in month T + k (where k = 1, 2, ...12; T = 10) at the end of every iteration of the MCMC algorithm as follows;

• 1. Predict CMMs ($\hat {CMM_{i}^{p2}}$) with Equation 5. Use predicted CMM values for all customers to accomplish overimputation.

• 2. Predict customer i’s hierarchical coefficients $\hat {\beta _{i}}$ using Equation A.3. Predicted CMMs ($\hat {CMM_{i}^{p2}}$) come from step 1.

• 3. Predict $\hat {\pi _{iT+k}}$ and $\hat {y_{iT+k}}$ using the predicted coefficients ($\hat {\beta _{i}}$), lagged sales ($\hat {y_{iT+k-1}^{p3}}$), and sales calls ($Det_{iT+k}^{p3}$), predicted in the holdout period.

For each iteration of the MCMC algorithm, the predicted values $\hat {\pi _{i}}=(\hat {\pi _{T+1}},\hat {\pi _{T+2}}, ...\hat {\pi _{T+12}})$ and $\hat {y_{i}}=(\hat {y_{T+1}}, \hat {y_{T+2}}, ..., \hat {y_{T+12}})$ serve to compute the profits for customer i from Eq. 4. The posterior expected profit for customer i is the Monte Carlo average;

$$ E[P_{i}(\hat{\beta_{i}}(\hat{CMM_{i}^{p2}},\hat{\pi_{i}},\hat{y_{i}})]=\sum\limits_{l=1}^{np}P_{i}(\hat{\beta_{i}}(\hat{CMM_{i}^{p2}}),\hat{\pi_{i}},\hat{y_{i}},l)/np, $$

(B.1)

where, np refers to the number of posterior iterations.

Of the 50,000 MCMC algorithm iterations, we employ the initial 30,000 as burn-in and the last 20,000 as the posterior sample to make inferences. To assess convergence, we also assess trace plots and simulate the posterior distribution using five different parallel chains. The multivariate potential scale reduction factor (MPSRF), computed using the posterior sample of five chains ranging from 1.2 to .9 (across all variables), indicates convergence in the posterior sample.¹⁵