New Perspectives on Customer “Death” Using a Generalization of the Pareto/NBD Model

Several researchers have proposed models of buyer behavior in noncontractual settings that assume that customers are “alive” for some period of time and then become permanently inactive. The best-known such model is the Pareto/NBD, which assumes that customer attrition (dropout or “death”) can occur at any point in calendar time. A recent alternative model, the BG/NBD, assumes that customer attrition follows a Bernoulli “coin-flipping” process that occurs in “transaction time” (i.e., after every purchase occasion). Although the modification results in a model that is much easier to implement, it means that heavy buyers have more opportunities to “die.” In this paper, we develop a model with a discrete-time dropout process tied to calendar time. Specifically, we assume that every customer periodically “flips a coin” to determine whether she “drops out” or continues as a customer. For the component of purchasing while alive, we maintain the assumptions of the Pareto/NBD and BG/NBD models. This periodic death opportunity (PDO) model allows us to take a closer look at how assumptions about customer death influence model fit and various metrics typically used by managers to characterize a cohort of customers. When the time period after which each customer makes her dropout decision (which we call period length) is very small, we show analytically that the PDO model reduces to the Pareto/NBD. When the period length is longer than the calibration period, the dropout process is “shut off,” and the PDO model collapses to the negative binomial distribution (NBD) model. By systematically varying the period length between these limits, we can explore the full spectrum of models between the “continuous-time-death” Pareto/NBD and the naïve “no-death” NBD. In covering this spectrum, the PDO model performs at least as well as either of these models; our empirical analysis demonstrates the superior performance of the PDO model on two data sets. We also show that the different models provide significantly different estimates of both purchasing-related and death-related metrics for both data sets, and these differences can be quite dramatic for the death-related metrics. As more researchers and managers make managerial judgments that directly relate to the death process, we assert that the model employed to generate these metrics should be chosen carefully

Customer-Base Analysis using Repeated Cross-Sectional Summary (RCSS) Data

We address a critical question that many firms are facing today: Can customer data be stored and analyzed in an easy-to-manage and scalable manner without significantly compromising the inferences that can be made about the customers’ transaction activity? We address this question in the context of customer-base analysis. A number of researchers have developed customer-base analysis models that perform very well given detailed individual-level data. We explore the possibility of estimating these models using aggregated data summaries alone, namely repeated cross-sectional summaries (RCSS) of the transaction data. Such summaries are easy to create, visualize, and distribute, irrespective of the size of the customer base. An added advantage of the RCSS data structure is that individual customers cannot be identified, which makes it desirable from a data privacy and security viewpoint as well. We focus on the widely used Pareto/NBD model and carry out a comprehensive simulation study covering a vast spectrum of market scenarios. We find that the RCSS format of four quarterly histograms serves as a suitable substitute for individual-level data. We confirm the results of the simulations on a real dataset of purchasing from an online fashion retailer

Estimating CLV Using Aggregated Data: The Tuscan Lifestyles Case Revisited

The Tuscan Lifestyles case (Mason, 2003) offers a simple twist on the standard view of how to value a newly acquired customer, highlighting how standard retention-based approaches to the calculation of expected customer lifetime value (CLV) are not applicable in a noncontractual setting. Using the data presented in the case (a series of annual histograms showing the aggregate distribution of purchases for two different cohorts of customers newly “acquired” by a catalog marketer), it is a simple exercise to compute an estimate of “expected 5 year CLV.” If we wish to arrive at an estimate of CLV that includes the customer\u27s “life” beyond five years or are interested in, say, sorting out the purchasing process (while “alive”) from the attrition process, we need to use a formal model of buying behavior that can be applied on such coarse data. To tackle this problem, we utilize the Pareto/NBD model developed by Schmittlein, Morrison, and Colombo (1987). However, existing analytical results do not allow us to estimate the model parameters using the data summaries presented in the case. We therefore derive an expression that enables us to do this. The resulting parameter estimates and subsequent calculations offer useful insights that could not have been obtained without the formal model. For instance, we were able to decompose the lifetime value into four factors, namely purchasing while active, dropout, surge in sales in the first year and monetary value of the average purchase. We observed a kind of “triple jeopardy” in that the more valuable cohort proved to be better on the three most critical factors

Counting Your Customers the Easy Way: An Alternative to the Pareto/NBD Model

Today’s managers are very interested in predicting the future purchasing patterns of their customers, which can then serve as an input into “lifetime value” calculations. Among the models that provide such capabilities, the Pareto/NBD “counting your customers” framework proposed by Schmittlein et al. (1987) is highly regarded. However, despite the respect it has earned, it has proven to be a difficult model to implement, particularly because of computational challenges associated with parameter estimation. We develop a new model, the beta-geometric/NBD (BG/NBD), which represents a slight variation in the behavioral “story” associated with the Pareto/NBD but is vastly easier to implement. We show, for instance, how its parameters can be obtained quite easily in Microsoft Excel. The two models yield very similar results in a wide variety of purchasing environments, leading us to suggest that the BG/NBD could be viewed as an attractive alternative to the Pareto/NBD in most applications

Bayesian Inference for the Negative Binomial Distribution via Polynomial Expansions

To date, Bayesian inferences for the negative binomial distribution (NBD) have relied on computationally intensive numerical methods (e.g., Markov chain Monte Carlo) as it is thought that the posterior densities of interest are not amenable to closed-form integration. In this article, we present a “closed-form” solution to the Bayesian inference problem for the NBD that can be written as a sum of polynomial terms. The key insight is to approximate the ratio of two gamma functions using a polynomial expansion, which then allows for the use of a conjugate prior. Given this approximation, we arrive at closed-form expressions for the moments of both the marginal posterior densities and the predictive distribution by integrating the terms of the polynomial expansion in turn (now feasible due to conjugacy). We demonstrate via a large-scale simulation that this approach is very accurate and that the corresponding gains in computing time are quite substantial. Furthermore, even in cases where the computing gains are more modest our approach provides a method for obtaining starting values for other algorithms, and a method for data exploration

A joint model of usage and churn in contractual settings. Marketing Sci

A s firms become more customer-centric, concepts such as customer equity come to the fore. Any serious attempt to quantify customer equity requires modeling techniques that can provide accurate multiperiod forecasts of customer behavior. Although a number of researchers have explored the problem of modeling customer churn in contractual settings, there is surprisingly limited research on the modeling of usage while under contract. The present work contributes to the existing literature by developing an integrated model of usage and retention in contractual settings. The proposed method fully leverages the interdependencies between these two behaviors even when they occur on different time scales (or "clocks"), as is typically the case in most contractual/subscription-based business settings. We propose a model in which usage and renewal are modeled simultaneously by assuming that both behaviors reflect a common latent variable that evolves over time. We capture the dynamics in the latent variable using a hidden Markov model with a heterogeneous transition matrix and allow for unobserved heterogeneity in the associated usage process to capture time-invariant differences across customers. The model is validated using data from an organization in which an annual membership is required to gain the right to buy its products and services. We show that the proposed model outperforms a set of benchmark models on several important dimensions. Furthermore, the model provides several insights that can be useful for managers. For example, we show how our model can be used to dynamically segment the customer base and identify the most common "paths to death" (i.e., stages that customers go through before churn)

Customer-Base Analysis Using Repeated Cross-Sectional Summary (RCSS) Data

Abstract We address a critical question that many firms are facing today: Can customer data be stored and analyzed in an easy-to-manage and scalable manner without significantly compromising the inferences that can be made about the customers' transaction activity? We address this question in the context of customer-base analysis. A number of researchers have developed customerbase analysis models that perform very well given detailed individual-level data. We explore the possibility of estimating these models using aggregated data summaries alone, namely repeated cross-sectional summaries (RCSS) of the transaction data (e.g., four quarterly histograms). Such summaries are easy to create, visualize, and distribute, irrespective of the size of the customer base. An added advantage of the RCSS data structure is that individual customers cannot be identified, which makes it desirable from a privacy viewpoint as well. We focus on the widely used Pareto/NBD model and carry out a comprehensive simulation study covering a vast spectrum of market scenarios. We find that the RCSS format of four quarterly histograms * Corresponding author Email addresses: [email protected] (Kinshuk Jerath), [email protected] (Peter S. Fader), [email protected] (Bruce G.S. Hardie) URL: www.petefader.com (Peter S. Fader), http://www.brucehardie.com (Bruce G.S. Hardie) 1 The authors thank David Bell for providing the Bonobos data used in this paper. 2 The second author acknowledges the support of the Wharton Customer Analytics Initiative. serves as an suitable substitute for individual-level data. We confirm the results of the simulations on a real dataset of purchasing from an online fashion retailer

Stuck in the Adoption Funnel: The Effect of Interruptions in the Adoption Process on Usage

Many firms have introduced Internet-based customer self-service applications such as online payments or brokerage services. Despite high initial sign-up rates, not all customers actually shift their dealings online. We investigate whether the multistage nature of the adoption process (an “adoption funnel”) for such technologies can explain this low take-up. We use exogenous variation in events that possibly interrupt adoption, in the form of vacations and public holidays in different German states, to identify the effect on regular usage of being interrupted earlier in the adoption process. We find that interruptions in the early stages of the adoption process reduce a customer's probability of using the technology regularly. Our results suggest significant cost-saving opportunities from eliminating interruptions in the adoption funnel.NET InstituteLondon Business School. Centre for MarketingMack Center for Managing Technological InnovationDeutsche Forschungsgemeinschaf