This is a post-print version of an article accepted for publication in the journal Marketing Letters

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1 This is a post-print version of an article accepted for publication in the journal Marketing Letters The final publication is available at Springer via

2 1 Predicting future purchases with the Poisson lognormal model Giang Trinh (corresponding author) Ehrenberg-Bass Institute, University of South Australia Postal address: School of Marketing, GPO Box 2471, Adelaide SA 5001, Australia Phone: Fax: Cam Rungie Ehrenberg-Bass Institute, University of South Australia cam.rungie@unisa.edu.au Malcolm Wright School of Communication Journalism and Marketing, Massey University m.j.wright@massey.ac.nz Carl Driesener Ehrenberg-Bass Institute, University of South Australia Carl.Driesener@unisa.edu.au John Dawes Ehrenberg-Bass Institute, University of South Australia John.Dawes@marketingscience.info

3 2 Abstract The negative binomial distribution (NBD) has been widely used in marketing for modeling purchase frequency counts, particularly in packaged goods contexts. A key managerially relevant use of this model is Conditional Trend Analysis (CTA) - a method of benchmarking future sales utilizing the NBD conditional expectation. CTA allows brand managers to identify whether the sales change in a second period is accounted for by previous non; light; or heavy buyers of the brand. Although it is a useful tool, the conditional prediction of the NBD suffers from a bias: it under predicts what the period one non-buyer class will do in period two; and over predicts the sales contribution of existing buyers. Also, the NBD s assumption of a gamma-distributed mean purchase rate lacks theoretical support - it is not possible to explain why a gamma distribution should hold. This paper therefore proposes an alternative model using a lognormal distribution in place of the gamma distribution, hence creating a Poisson-lognormal distribution (PLN). The PLN has a stronger theoretical grounding than the NBD as it has a natural interpretation relying on the central limit theorem. Empirical analysis of brands in multiple categories shows that the PLN gives better predictions than the NBD. Keywords Probability models; purchase frequency counts; conditional predictions; NBD; Poisson lognormal

4 3 1 Introduction Goodhardt and Ehrenberg (1967) introduced to marketing science a method to benchmark future sales change based on past performance, which they termed conditional trend analysis (CTA). CTA is based on the conditional expectation of the negative binomial distribution (NBD) for modeling consumer behavior in two sequential time periods under stationary conditions. It is regarded as a very important method with significant managerial implications (Morrison and Schmittlein 1988). The conditional expectation is crucial for brand managers who wish to identify whether a change in overall brand sales is accounted for by previous non-buyers, light buyers or heavy buyers (Schmittlein et al. 1985). Suppose a manager for a repetitively purchased brand reports an increase in sales compared to last year. The manager would perhaps ask: Where do our increased sales come from? Is it because there are more new buyers or because the existing buyers purchase more? The reason she needs to know the answer to such a question is that it will help her plan future marketing activity and assess past marketing activity. For example, if she finds that the source of sales growth comes mainly from new buyers, she would carry out marketing activity with a focus on customer acquisition. On the other hand, if the brand sales growth comes mainly from existing buyers, she would concentrate on loyalty marketing activity to retain and bolster current customers. CTA is a method to answer these questions. By comparing the actual sales during the growth period with the expected sales predicted by the conditional expectation of the NBD model under the stationary condition (i.e. the no change condition), the manager is able to identify the sources of growth. Due to its ability to determine the source of sales gains or losses, conditional expectation is regarded as one of the most managerially useful constructs in the stochastic modeling of brand choice (Schmittlein et al. 1985).

5 4 Although it is a very useful tool, the NBD conditional expectation is often biased. A typical bias is that the NBD conditional expectation under-predicts what the period one nonbuyer class will do in period two. Similarly, it over predicts the sales contributions of existing buyers (Lenk et al. 1993; Morrison and Schmittlein 1988; Morrison and Schmittlein 1981). In explaining the poor predictions of the NBD model, the Poisson assumption of the model has been questioned by several authors. For example, Chatfield and Goodhardt (1973) assumed that interpurchase times follow the Erlang 2 distribution (more regular than the exponential distribution implied by Poisson purchasing behaviour), which leads to the condensed negative binomial distribution (CNBD). However, while comparing the NBD model with the CNBD model, they found that the Poisson assumption is more robust than the Erlang 2 assumption. As such they concluded In those cases where the NBD does not give a good fit (essentially for large variances), it is therefore likely to be mainly due to a failure in the gamma assumption. Ehrenberg (1988, p. 63) also noted that the gamma distribution is not a particularly precise assumption, it is not possible to adduce any strong reasons why a gamma distribution should hold. Similarly, Brockett et al. (1996, p.96) suggested when the NBD model fails, the most lucrative avenue for deriving a model that does fit involves relaxing some assumption other than the Poisson individual purchase assumption. The conditional prediction of the CNBD has also been examined by Schmittlein and Morrison (1983). They found that the NBD predictions are slightly superior to the CNBD predictions. They also concluded that in regard to the typical bias, if we assume individual purchases are more regular than the Poisson process and use an Erlang 2 distribution of the CNBD, this only makes the bias worse, as existing buyers purchases in period two are even more over predicted (Morrison and Schmittlein 1988).

6 5 In light of this criticism, we propose an alternative to the gamma distribution for the mean purchase rate, namely the lognormal distribution. This approach creates the Poisson lognormal distribution (PLN) for modeling purchase frequency counts and predicting future purchases based on past performance. The PLN has not previously been fitted to purchase frequently count data. Also, our paper focuses on the conditional prediction of the PLN model rather than in-sample fitting, and this has never been studied. This conditional prediction is an important and managerially relevant topic that requires comprehensive examination. There are other distributions that can also replace the gamma distribution such as the Pareto, inverse Gaussian, Weibull and Lomax distribution, yet, they lack development of possible causal relationships, and their parameters are less well understood and do not directly relate to consumer variability. They also lack possible elaborations (e.g. to multivariate models) (Stewart 1994). We choose the lognormal distribution because it has an attractive theoretical grounding and its parameters are easily understood and interpreted. Additionally, the multivariate variant of the PLN model is ready available for future elaborations (Aitchison and Ho, 1989). The next section reviews the applications of the lognormal distribution in the literature and proposes a theoretical interpretation to justify the lognormal distribution of purchase rates. 2 Poisson lognormal model The log-normal distribution has been used in many disciplines such as geology, economics, telecommunication, biochemistry, demography, health, and risk analysis (Aitchison and Brown 1969; Cassie 1962; Crow and Shimizu 1988; Johnson et al. 1994). Yet surprisingly few applications have been demonstrated in marketing. The earliest application of the lognormal distribution in marketing science seems to be reported in Lawrence (1980), in which it was used to model purchase frequency rates. However, Lawrence (1980) s model is not a complete model of purchase frequency as no individual distribution is chosen (no mixing

7 6 model) and it consequently suffers from several shortcomings. It assumes that an individual consumer always purchases at the average purchase rate; as such, Lawrence s model has not addressed the question of within-consumer variability (Morrison 1981). In addition, the lognormal model used by Lawrence (1980) is a continuous distribution whereas purchase frequency counts are integers, which should be described by a discrete distribution. Finally, that approach also creates a problem of estimating the non-buyers as the log of zero is negative infinity. With conditional trend analysis, it is crucial to estimate what the non-buyers in period one will do in period two. More recently, Abe (2009) proposed the multivariate lognormal distribution to model the relationship between purchase rates and dropout rates in customer base analysis. The author compared the Pareto/NBD model with a multivariate lognormal based model and found that the modified model perform as well as the Pareto/NBD model for individual level predictions. Rungie and Laurent (2010) compared the multivariate log-normal model with the Dirichlet multinomial model of brand choice. The authors showed that the multivariate lognormal model gives a better fit than the Dirichlet multinomial model if the number of draws is large (e.g. 10,000 or larger number of draws). Yet, there is no study that has applied the univariate mixture Poisson lognormal distribution to model purchase frequency counts and predict future purchases at an aggregate level, for example predicting purchase quantities for a brand. This is an important area that has not been examined. Jerath et al. (2011) point out that many companies face difficulties in accessing individual level data and even when they can get access to this data, the data format is often unfamiliar, or there is possible data loss. These problems potentially create barriers to implement individual level data models. As such, aggregate count models are crucial for such situations. The Poisson log-normal model (PLN) proposed in this study overcomes the disadvantages of Lawrence s model by allowing variability within a given individual

8 7 consumer. Also, by combining the log-normal distribution with the Poisson distribution, the continuous log-normal distribution is converted to a discrete distribution, which is more appropriate when modeling counts. Finally, estimation of the non-buyers class is not a problem as the model includes the zero counts. While the proposed PLN model has not been fitted to purchase frequency counts previously, it has been shown that the model gives a better fit for count data compared to the negative binomial model (e.g. Connolly et al. 2009; Tsionas 2010; Winkelmann 2008). It is evident that the log-normal distribution s tails are heavier than that of the gamma distribution (Sohn 1994; Kaas and Heseelager 1995; Miranda-Moreno et al. 2005). Previous research has shown that for data with outliers, the PLN model gives a better fit than the NBD model (Connolly et al. 2009; Sohn 1994; Miranda-Moreno et al. 2005). Thus the PLN model may be more suitable for purchase frequency of heavily brought brands, or where there are outliers such as excessively heavy buyers, where the NBD shows a lack of fit (Ehrenberg 1959; Chatfield et al. 1966; Ehrenberg 1988). Not only does the PLN model appeal in fitting to empirical data, but also the lognormal distribution has an attractive theoretical interpretation (Cassie 1962; Winkelmann 2008). It is quite reasonable to assume that many independent unobserved factors contribute to the variance in the mean purchase rate across consumers. These factors could include different needs and different exposures to marketing activity such as advertising, promotion and word of mouth. If these many independent unobserved factors have proportional but different impacts on each consumer in a multiplicative process, then central limit theorem suggests the geometric mean purchase rate across consumers converges to a log-normal distribution (Aitchison and Brown 1969; Johnson et al. 1994; Winkelmann 2008).

9 8 In light of the empirical results and the theoretical advantages of the PLN model, Winkelmann (2008, p.134) suggests, the previous neglect of the Poisson lognormal model in the literature should be reconsidered in future applied work. 3 Mathematical expressions Let mean rates of purchasing of different consumers in the long run be lognormal distribution 1 log f ( ;, ) exp where is the mean and is the standard deviation of the normal distribution Y (1) where Y log( ). Then probability density function of x purchases is f (x) PLN f (x) poisson f ( ;, )d 0 1 x 1 exp( )exp log x! (2) d The mean of the PLN distribution is E[x] exp( 2 /2) (3) and the variance is (4) var[x] exp( 2 /2) 1 (exp( 2 /2)(exp( 2 ) 1) Crow and Shimizu (1988) As this is a compound Poisson distribution, the conditional expectation of the PLN can be estimated as follow

10 9 E[X 2 X 1 x] (x 1) f (x 1) f (x) (5) That is, the number of purchases in period two made by the buyers who bought x purchases in period one is the same as the number of purchases made by the buyers who bought (x + 1) purchases in period one (Robbins 1977). Consequently, we will test the PLN model for predicting future purchases of multiples brands and product categories. Then we will compare this model with the well-known NBD model. We first compare the models using simulated data; then apply them to household panel data. 4 A simulation study to compare NBD and PLN models This section shows the comparison between the theoretical conditional predictions of the NBD model and the PLN model. We first specify a population mean purchase rate, population variance and sample size for the simulated data. We use three values of population mean (0.25; 0.5; 1) and three values of population variance (1.5; 2.5; 5) similar to those in empirical data analysed later. This gives a total of nine combinations. The sample size is As the simulation uses assumed means and variances, the parameters can be estimated by the method of moments. The NBD frequency distribution can be calculated directly from its parameters. However, the PLN model does not have a closed form, hence we apply numerical estimation based on 1000 random draws and average the results to estimate the frequency distribution. Finally we use equation (5) to calculate the conditional predictions. Figures 1-3 show examples of the theoretical conditional predictions of the PLN and the NBD model. Results are similar for all nine scenarios. The purchases in period two of the period one non buyer class are greater for the PLN model than the NBD model. Conversely the purchases in period two of the period one existing buyer class are less for the PLN model

11 Number of purchases in period 2 10 than the NBD model. These results show that the PLN predictions could reduce the well known biases of the NBD predictions. Fig. 1 Conditional prediction (mean = 0.25, variance = 1.5) Fig. 2 Conditional prediction (mean = 1; variance = 5) Buyer classes in period 1 NBD PLN Fig. 3 Conditional prediction (mean = 0.5; variance = 2.5)

12 11 5 Parameter estimation Now we fit the models to empirical data. For the NBD model, we use the maximum likelihood method to estimate the model parameters k and a. With the PLN model, probability P(x=0,1,2,3 ) no longer has a closed form. Following the numerical approximation method proposed by Train (2009), we estimate the parameters of the PLN using draws from a density function, calculating the Poisson lognormal probability for each draw, and averaging the results. We use Halton draws based on the sequence of a prime (Halton 1960). Train (2009) points out that Halton draws have several advantages compared to random draws. They provide better coverage than random draws. As a result, they tend to be self-correcting over observations. Halton draws also allow negative correlation over observations, hence reduce error in the simulated log-likelihood function. These advantages make Halton draws more effective than random draws. Indeed, Halton draws can be considered as well-placed draws from a standard uniform density (Train 2009). Previous research has shown that a small number of Halton draws (e.g. 100) provide more precise results than a large number of random draws (e.g. 1000) (Bhat 2001, Train 2000). Details on how to create Halton sequences are shown in Train (2009). 6 Measuring the fit and the accuracy of predictions To measure the fit of the models, we use the log-likelihood. The higher the log-likelihood, the better the fit. To measure the accuracy of the predictions for both the PLN and the NBD models, we use Theil s U coefficient of inequality. The U coefficient is easy to understand and interpret. It ranges from 0 to 1. The smaller the U, the better the prediction. The use of Theil s U coefficient here is appropriate as it captures the prediction of the full distribution, especially the prediction from the tail of the distribution. It is particularly useful for studies on modeling long tailed distributions (e.g. Wu and Chen 2000; Fader and

13 12 Hardie 2002). Other measures such as MAPE or MAE require one to group the tail of the distribution, and different censor points might give different results. 7 Data The empirical data used to compare the fit and accuracy of the PLN vs. NBD models is static household consumer data for a 104-week period from the Taylor Nelson Sofres (TNS) Superpanel database. The panel consists of 16,988 telephone-owning households across the UK. The panel is drawn from only full-time residents. The sample is demographically and regionally balanced in order to represent the UK population. Data is collected from panel participants twice weekly via electronic terminals in the home, with purchases being recorded via home-scanning technology (TNS 2008). We use the first 52 weeks for parameter estimation and the last 52 weeks as a test period. We analyze the top five brands in each of the following four categories: shampoo, deodorant, toilet soap, and bleach. This gives a total of 20 comparisons of the PLN and NBD models. We first examine the fit of each model to year 1 purchases, then evaluate each model s conditional predictions for year 2. 8 Results 8.1 Model fit Table 1 reports the log likelihood results of the NBD and the PLN models fitted to purchase frequency data of the twenty brands in the four categories. The log-likelihood ratios are very close between the two models, which suggests that the PLN and the NBD models are very competitive. The PLN model outperforms the NBD models in ten cases, whereas the NBD gives a better fit in the others.

14 13 Table 1 PLN and NBD fit to twenty brands in four product categories Brands Log likelihood Brands Log likelihood Models NBD PLN Models NBD PLN Toilet soap Bleach Dove Domestos Imperial leather Parozone Palmolive Toilet Duck Cussons Harpic Tesco Bloo Shampoo Deodorant Alberto Sure Head & Shoulders Lynx Pantene Sanex Soft&Gentle Herbal Essences Dove LOreal Rightguard Bold figures indicate better fits. These results are not unexpected, as the NBD model has long been reported as effective in term of modeling purchase frequency counts even though its assumptions may not strictly hold true (Ehrenberg 1959; Ehrenberg 1988; Fader and Hardie 2002; Morrison and Schmittlein 1988; Schmittlein et al. 1985). Yet Morrison and Schmittlein (1988) postulated that deviations from the NBD model show up much more in the conditional expectations than in the purchase frequency distribution. Consequently the authors suggested that conditional expectation should be used to test the NBD model even if the observed distribution looks very similar to the NBD. Indeed, Schmittlein and Morrison (1983, p.453) stated if conditional trend analysis is of primary interest, the conditional expectations are the natural quantities for model comparison. We now examine the conditional predictions of the NBD and the PLN models. 8.2 Conditional predictions Table 2 shows the accuracy of conditional predictions of the NBD and the PLN models. As assessed by U, the PLN model predicts future purchases better than the NBD model for all brands with an exception of Palmolive (toilet soap brand). On average, the U for the PLN predictions is while for the NBD predictions it is

15 14 Table 2 PLN and NBD predictions of future purchases of twenty brands Brands Theil s U Brands Theil s U Model NBD PLN Model NBD PLN Toilet soap Bleach Dove Domestos Imperial leather Parozone Palmolive Toilet Duck Cussons Harpic Tesco Bloo Average Average Shampoo Deodorant Alberto Sure Head & Shoulders Lynx Pantene Sanex Soft&Gentle Herbal Essences Dove LOreal Rightguard Average Average Figure 4 shows an example of conditional predictions of the PLN model compared to the NBD model. As we can see, the PLN model predicts better than the NBD model in most buyer classes. The sales contribution of period one non-buyers in period two as predicted by the PLN model closely matches the actual data, whereas the NBD prediction of this class largely deviates from the actual data. Again, we see that the PLN predicts closely the sales contributions of other buyer classes in period two, while the NBD predictions are biased, especially buyers who bought 1-6 purchases in period one. The 95% confidence intervals for parameters a and k of the NBD model are (3.57; 3.84) and (0.185; 0.189) respectively. The 95% confidence intervals for parameters μ and σ of the PLN model are (-2.03; -1.94) and (1.87; 1.95), respectively.

16 15 Fig. 4 Conditional predictions (Dove-Toilet Soap) Period 2 purchases NBD PLN Actual Buyer classes in period Period one zero buyer predictions Previous literature has noted The NBD tends to under predict test period purchases by the zero class, the group of customers who bought nothing in the base period. This under prediction can be a serious problem as it leads to an overstatement of one of the key goals of marketing effort - attracting previous non buyers to the brand (Lenk et al. 1993, p.289). We therefore compare the zero class s purchases in period two, predicted by the NBD and the PLN models, with the actual purchases. We calculate the ratio of difference between the theoretical predictions and the observed purchases of the zero class. A ratio closer to 1 indicates a better prediction.

17 16 Table 3 Ratios of difference between estimated and actual zero class s purchases in period 2 Models NBD PLN Models NBD PLN Toilet soap Shampoo Dove Alberto Imperial leather Head & Shoulders Palmolive Pantene Cussons Herbal Essences Tesco LOreal Average Average Deodorants Bleach Sure Domestos Lynx Parozone Sanex Soft&Gentle Toilet Duck Dove Harpic Rightguard Bloo Average Average As we can see from Table 3, the PLN model consistently outperforms the NBD model in predicting the purchases by the zero class in period two, with the exception of Palmolive. The average error rate for the PLN model is 22%, ranging from 5% to 35%, whereas that of the NBD model is 33%, ranging from 17% to 45%. The Diebold-Mariano statistical test (Diebold and Mariano 1995) between the PLN and the NBD was also calculated. The S statistic and p-value are 0.92 and for toilet soap; 3.99 and for shampoo, 2.69 and for deodorants, and 3.08 and for bleach. These results show that the PLN is a statistically significantly better predictor than the NBD model in three categories: shampoo, deodorants and bleach. 8.4 Category buying Prior studies on modeling brand buying behavior have emphasized the need to understand purchase frequency distributions of the entire product category (e.g. Ehrenberg et al. 2004; Sichel 1982). If the compound Poisson distribution gives an adequate fit to individual brand purchases, then it should do the same for the product category. We therefore present the fit and conditional prediction of the PLN and the NBD models to actual purchase count data at

18 17 the category level (using all category data). Table 4 shows the results of the NBD and the PLN models in fitting to purchase frequency data and predicting future purchases in four product categories. As we can see, across all the categories, the PLN model fits the observed data better than the NBD model. This better performance of the PLN model could be explained by the fact that at the category level, the tail of the distribution is heavier than at the brand level, which therefore fits the PLN model better than the NBD model. In terms of conditional prediction, the PLN model also outperforms the NBD model in all categories. Table 4 PLN and NBD results for four product categories Product Shampoo Toilet Soap Bleach Deodorants Model NBD PLN NBD PLN NBD PLN NBD PLN Log likelihood (model fit) Theil s U (conditional prediction) Conclusion The NBD conditional expectation is a useful tool to analyze changes in buyer behavior. Yet, it is often reported that the model is biased when being used to predict future purchases. This implies that more development of the model is needed for obtaining better predictions of consumer behavior. In this article, we propose and empirically test the PLN model and compare it with the well-known NBD model. Findings across multiple brands and product categories show that the PLN model helps reduce the well-known biases of the NBD model (to under predict purchases of the non buyer class and over predict purchases of the existing buyer class). The PLN model also has a stronger theoretical grounding than the NBD model, yet it does not have a closed form, and consequently needs to be estimated by simulation. This is a disadvantage of the PLN model compared to the simple NBD model. Yet, with

19 18 advanced statistical software packages, parameter estimation in most cases is as fast as the estimation of the NBD model. 10 Future research The NBD model of purchase frequency has been used in conjunction with the Dirichlet multinomial distribution to create a comprehensive model (known as the Dirichlet model in marketing) of buying behavior multi-brands in established competitive markets. The Dirichlet model has successfully characterized brand loyalty across a wide range of categories and conditions (Ehrenberg et al. 2004). Yet prior research has shown that the Dirichlet model has consistent deviations when predicting behavioral loyalty, e.g. excessive repeat-purchasing of high share brands (Fader and Schmittlein 1993). This shortcoming might be due to the NBD component of the Dirichlet model, as it is evident that the NBD model departs from the observed frequency when excessively heavy buyers are present (Chatfield et al. 1966; Ehrenberg 1988). Ehrenberg et al. (2004) also noted that the actual distribution of purchase frequency at the category level is a little flatter than predicted by the NBD. Future research could replace the NBD component of the Dirichlet model with the PLN distribution as the findings of this paper show that the PLN model fits purchase frequency better than the NBD model at the category level. The second direction of future research could be extending the PLN model to the bivariate PLN model for examining brand duplication of purchases or brand switching, at the disaggregate level. Goodhardt and Ehrenberg (1967) suggested that the bivariate NBD model could be used for this type of analysis. One disadvantage of using the bivariate NBD model in this case would be that the model does not allow negative correlations between the two count variables, a likely requirement for any satisfactory model for bivariate count data (Aichison and Ho 1989), whereas the Poisson lognormal model allows this to happen. The

20 19 negative correlations may be important as it is not unreasonable to think that buyers of a very expensive brand may not be generally interested in buying a very cheap private label. Although this hypothesis has not been tested, partitions between expensive brands and cheap private labels have been reported in the literature (e.g. Nenycz-Thiel et al. 2010). The NBD is also used in customer base analysis (e.g. Schmittlein et al. 1987; Fader et al. 2005). Combining the NBD with other distributions such as Pareto, Beta-geometric and gamma-gompertz enables researchers to identify inactive customers, which is crucial in customer base analysis. A further application is to media planning where the NBD is used to predict viewing behavior and benchmark advertising effectiveness (e.g. Danaher 2007). Therefore, a third direction for future research is to determine whether replacing the NBD with the PLN leads to improved forecasting in these areas. Finally, there is a growing body of recent studies on Bayesian nonparametrics in marketing to avoid misspecification of heterogeneity distributions. For example, George and Hui (2012) utilize a Polya tree prior, and Braun et al. (2006) and Kim et al. (2004) use a Dirichlet process prior. Our study generally shows that if the parametric assumption of the heterogeneity distribution is misspecified, estimation will be biased. This suggests there is a need for future work on Bayesian nonparametrics to avoid imposing restrictive assumptions, especially when knowledge of the appropriate heterogeneity distribution is not apparent.

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