An examination of consumer heterogeneity in a stochastic model of consumer purchase dynamics with explanatory variables

European Journal of Operational Research76 (1994) 259-272 North-Holland

259

An examination of consumer heterogeneity in a stochastic model of consumer purchase dynamics with explanatory variables J a m e s H. P e d r i c k a n d F r e d S. Z u f r y d e n

School of Business Administration, Department of Marketing, University of Southern California, Los Angeles, CA 90089-1421, USA Received May 1990; revised March 1991 Abstract: A stochastic model of brand choice and purchase incidence is proposed that relates marketing

variables, including advertising exposures, to brand performance measures such as market share, penetration and depth of repeat purchase dynamics over time. The approach considers various aspects of consumer heterogeneity including individual differences in brand loyalty levels, advertising exposures, and purchase rates for a population of consumers. A major focus of this study is on the evaluation of the impact of consumer heterogeneity in alternative models on the basis of single-source scanner panel data for an actual market situation. In particular, we demonstrate the critical importance of considering heterogeneity over a consumer population as a means of enhancing descriptive and predictive model fit as well as for allowing the analysis of advertising media schedules. Keywords: Stochastic models; Purchase behavior; Consumer heterogeneity; Marketing

1. Introduction

In this paper, we develop and empirically evaluate a composite non-stationary stochastic model framework that integrates brand choice and purchase incidence components as well as explanatory variables. Our main focus is on studying the impact of alternative methods of modeling household heterogeneity using scanner panel data as well as on the analysis of advertising media schedules. Most past stochastic models of purchase incidence account for heterogeneity with a mixing distribution on key model parameters over the consumer population (e.g., Ehrenberg, 1959; Chatfield and Goodhardt, 1973; Jeuland et al., 1980). This allows the model to account for differences in household behavior that may otherwise be unobservable to the researcher. However, they ignore information about past purchasing behavior and causal marketing variables which may be obtained from recently available 'single-source' scanner panel databases. A number of papers in the choice model literature have shown that accounting for observable household heterogeneity in the form of brand loyalty variables dramatically improves model fit (e.g. Guadagni and Little, 1983; Gupta, 1988; Tellis, 1988; Lattin and Bucklin, 1989). However, applications using these models for aggregate market-level predictions ignore any form of heterogeneity by holding the brand loyalty variables fixed at some constant value.

Correspondence to: Dr. J.H. Pedrick, School of Business Administration, Department of Marketing, University of Southern California, Los Angeles,CA 90089-1421, USA. 0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0377-2217(93)E0299-D

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In our approach, we attempt to overcome some of the limitations of these previous models by including both methods of accounting for heterogeneity. In particular, we include a mixing distribution to account for individual differences in category purchase rates as well as observable heterogeneity of marketing and brand loyalty variables. This leads to a general integrated model framework which relates marketing variables, including advertising exposures, to brand performance measures such as market share, penetration and depth of repeat purchase dynamics over time. To empirically test our approach, we consider heterogeneity assumptions at different levels of complexity. This demonstrates that the consideration of observable brand loyalty and advertising exposure heterogeneity over a population significantly enhances both the descriptive and predictive model fit. Moreover, our analyses support the use of a binary model version in practical applications as it is shown to yield similar empirical results to those from a more complex multi-brand model version. 1.1. Overview o f related literature

Over the past thirty years, there has been a considerable evolution in the development of stochastic models of consumer purchase behavior. ~ Here, we provide a brief overview of literature that relates to our study. Ehrenberg (1959) was first to show the empirical viability of accounting for unobservable household heterogeneity by placing a Gamma mixing distribution on individual Poisson purchase rates to develop aggregate purchase distributions. Studies that have followed focused on the integration of brand choice and purchase incidence behavior within composite model structures (e.g., Herniter, 1971; Zufryden, 1977; Jeuland et al., 1980). However, despite their predictive ability, the prior models' managerial usefulness has been limited by their ignorance of the specific impact of marketing mix variables. A recent emphasis has been to develop more managerially-relevant models that permit general specifications of explanatory variables (e.g., Jones and Zufryden, 1980; Hauser and Wisniewski, 1982; Wagner and Taudes, 1986; Gupta, 1988; Rosenqvist and Strandvik, 1989). However, these models ignore the heterogeneity of explanatory variables. Our model attempts to overcome this problem by specifying distributions for household brand loyalty and advertising exposure levels across the consumer population. The latter feature is of particular significance for advertising planning purposes as it permits the aggregation of advertising exposures over a population. Thus, in contrast to Wagner and Taudes (1986), our approach provides a way to study the specific influence of media schedule characteristics such as reach and frequency rather than merely aggregate advertising expenditures. It also offers advantages over the experimentally-based Zufryden (1987) model which considers media schedule effects but ignores the specific influence of marketing variables.

2. Model development We now describe the underlying behavioral assumptions of the proposed model. We then develop four alternative model versions that include both binary and multi-brand formulations (see Table 1). 2.1. Brand choice model

A logit brand choice model is proposed to describe individual household brand purchase probability as a function of explanatory variables (e.g. Jones and Zufryden, 1980; Guadagni and Little, 1983). For expository purposes, we begin by considering a binomial logit model which examines a brand of interest 1 Refer to a comprehensivesurveyby Wagnerand Taudes(1987) for a detailed accountof the literature in this area.

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Table 1 Summary of four alternative WNBD models and components Model component Brand choice Category incidence Exposure heterogeneity Loyalty heterogeneity

Binary

Multi-brand

Binomial logit Negative Binomial (Poisson-Gamma) Beta-Binomial Beta-Binomial-NBD or Bivariate NBD (Polya)

Multinomial logit Negative Binomial (Poisson-Gamma) Beta-Binomial Dirichlet-Multinomial-NBD or Multivariate NBD (Polya)

(Brand 1) versus all other brands (Brand 0). The extension of the model to a multi-brand case is straight-forward as will be discussed in Section 2.4. Thus we express P(x.,)

= 1/[1 + exp(flx.,)],

(1)

where: P ( x , t ) = Household Brand 1 purchase probability given an environment x , t , at purchase occasion n during period t. = (fl0' fl01' "', flu ) = A vector of constant p a r a m e t e r s to be estimated. We specify Xnt = (Znt, iXnt, iOnt, b t_ 1, K t - 1 - b t - 1), with components defined as follows: z,, = Vector of point-of-purchase marketing variables at purchase occassion n during period t. ikm = Brand k (k = 0,1) advertising exposures received by a household prior to the current purchase occasion n during period t. b,-1 = Household short-term loyalty for Brand 1, expressed as the number of household Brand 1 purchases made during period t - 1. gt-1 = N u m b e r of category purchases made by a household during period t - 1. Kt_ 1 - b t _ 1 = Household short-term loyalty for 'other' brands, expressed as the number of household Brand 0 purchases made during period t - 1. Here, we note certain advantages of our logit model specification. First, the 'lagged' loyalty variables (i.e., bt_ 1 and Kt_ 1 - b t _ 1) capture the short-run carry-over effects due to past marketing, including advertising, effects (e.g., as in Clarke, 1972, Bass and Clarke, 1972, and Rao, 1986). These variables also allow for differential brand choice probability response to advertising and other marketing variables with changes in the level of a household's brand loyalty. 2.2. Purchase incidence model

We now consider the development of the brand purchase incidence model that will be related to the logit brand choice model and explanatory variables. We first consider purchase incidence behavior conditional on a fixed purchase environment vector Xnt during period t. Thus, dropping the subscript n for notational convenience, we posit that purchases of Brand 1, as a function of x t during t, are Poisson with mean uB(xt). 2 Now if the individual product category purchase rate is defined as u , . S t where u c is a constant across periods, and S, is a seasonal index, then the brand purchase rate may be stated as uB(x,) = ucS, P(x,).

(2)

To account for the heterogeneity of individual consumer purchase rates during period t, it is assumed that u c is distributed as G a m m a over the population with parameters a, fl > 0 (e.g., Ehrenberg, 1959). 2 An alternative assumption that may be used is the Condensed Poisson (e.g., see Chattfield and Goodhardt, 1975, and Zufryden, 1977).

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Given this assumption, it is easily shown that the aggregate distribution of brand purchase counts j during a period t of length T, becomes a Negative Binomial Distribution (NBD): +J') P ( j l x t, T; a, /3) = F(ot F(a,)j!

Ot(xt)J[ 1 Ot(xt)]r ~ _

(3)

for j = 0, 1, 2, ....

with O,(x t) = T / [ T +/3/(StP(x,)]. Formulation (3) provides an alternative to the model of Rosenqvist and Strandvik (1989). The latter defines the individual mean brand purchase rate as an exponential function of explanatory variables with an error term that is Gamma-distributed over the consumer population. However, unlike our model (2), the mean brand purchase rate function does not reflect diminishing returns and saturation effects as the level of marketing inputs, such as advertising, increases (e.g., see Simon, 1965, and Rohloff, 1968).

2.3. Aggregate market model We now turn to the development of the aggregate brand purchase distribution model. First, to derive the marginal aggregate brand purchase distribution during t, we take a weighted average of the P ( j l xt,T;a,/3) probabilities, from the NBD. This is accomplished by defining weights w(x t) that reflect the probability distribution on the purchase environment variables xt, at product category purchase occasions n during t. Thus, we obtain a Weighted Negative Binomial Distribution (WNBD) model:

P(j[T;cr,/3)=

Y'~P(Jlxt;a,/3).w(xt)

for j = 0 , 1 , 2 , . . . ,

(4)

with mean

E,(j) = [aTS,//3].

EP(x,)

(5)

.w(x,).

Jgt

In order to obtain the probabilities P ( j l T;a,13) and utilize (4) for forecasting purposes, w(x,) must be derived in terms of the distributions of the random variables of vector x, (i.e., i~,, iot , bt_l, and Kt_l). In past stochastic models (e.g. Ehrenberg, 1959; Jeuland et al., 1980; Wagner and Taudes, 1986), heterogeneity is typically considered by defining probability distributions on specific model parameters as we did for u c. In addition to this approach, we model loyalty and advertising exposure counts heterogeneity by placing a distribution on the corresponding variables. As a result of this, the brand choice model parameters can be estimated directly using disaggregated panel data, thus simplifing estimation and avoiding possible aggregation bias.

2.3.1. Exposure heterogeneity We now consider advertising exposure heterogeneity over a target population. Suppose that a total of

N t T V spots have been scheduled for Brand 1 during period t. We assume that individual exposures given N t follow a binomial distribution with Pt, the average individual probability of viewing any one of the spots during t, Beta-distributed over the population. Then, it is easily shown (e.g., see Zufryden, 1987) that the aggregate population exposure distribution during t is a Beta-Binomial distribution (BBD): N,!

r(m, +n,)

P ( i l N t ; mr, n , ) = ( N , - i ) !

i! r ( m , ) r ( n , )

r(m,+i)r(nt+N,-i) r(m,+n,+N,)

f o r i = O , 1,2 .... , N t. (6)

This model (6), first proposed by Metheringham (1964), has commonly been applied to the analysis of media exposure distributions (e.g., Headen et al., 1977; Zufryden, 1987). Here, we assume that each

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263

brand can be represented by an independent BBD. Thus, the BBDs provide a useful way to evaluate the impact of corresponding media schedules during particular time periods t. 2.3.2 Loyalty heterogeneity A model approach that enables the consideration of the joint behavior of the loyalty variables b t_ 1 and K t_ 1 is the B B D - N B D model. 3 This model has been used to describe brand choice and purchase incidence behavior in several previous studies (e.g., see Morrison and Schmittlein, 1988, and Zufryden, 1984). Here, it is assumed that category purchase incidence behavior in a population is NBD. Thus, following the form of (3), the probability that K t_ 1 category purchases are observed during period t - 1 is given by P ( g t _ l , o~, fl), an NBD with parameters a, /3, for Kt_ 1 = 0, 1, 2 , . . . . Furthermore, the number of brand purchases conditional on the number of product category purchases during period t - 1 is expressed as a BBD model. Then, following the functional form of (6), the conditional probability of bt-1 brand purchases given that K t_ 1 category purchases have been made during period t - 1 may also be stated as a BBD,

P(bt_ 1, Kt_ 1, mr_l, n t _ l )

for bt_ 1 = 0, 1, 2 . . . . , K t _ l ,

with parameters mt_ 1 and n t _ 1. Now that the components of w ( x t) have been described, the WNBD in (4) may be evaluated by appropriately summing over the random variables of x, as

e(jlT;a,[3)=~e(jlxt;a,/3)'w(xt)

for j = 0 , 1 , 2

.... ,

(7)

X t

where 1 w ( x t ) = e ( g t _ 1 Io~,/3)e(bt_ 11 g t _ l , rot_l, nt_l) H P(ik, I Nit; mk,, n~,). k=0

(8)

2.4. Multi-brand model extensions The theoretical model framework that has been developed can readily be extended to a multi-brand situation. This may be acomplished by first extending (1) to a multinomial logit model defined for a set S of brand alternatives (e.g., see McFadden, 1974, and Guadagni and Little, 1983). In the multi-brand case, a key consideration in the development of aggregate purchase behavior becomes that of modeling the joint distribution of the loyalty variables bk, t_ 1, for brands k ~ S, of the weights w(xt). In this study, we examined two alternative ways of modeling these weights. 4 Our first method examined the heterogeneity of the loyalty variables bk, t_ 1 by using a Dirichlet-Multinomial distribution which is multivariate analogue of the BBD (e.g., Goodhardt et al., 1984; Wagner and Taudes, 1986). We also considered the multivariate NBD, or Polya, model as a joint distribution of the loyalty variables over the population (e.g., see Chattfield and Goodhardt, 1975; Wagner and Taudes, 1986). In our application, a potential advantage of the Polya model is that it is easier to estimate than the Dirichlet-based model. Also, as it directly models the joint marginal brand purchase distribution, it tends to reduce computational requirements for the multi-brand case.

3. Empirical analysis

We used a database provided by the A.C. Nielsen Marketing Research Division to estimate our alternative models. This data source is particularly appropriate for our purposes as it combines both 3 Based on supporting evidence from Currim and Shoemaker (1989), we assume that the loyalty and exposure distributions are independent. 4 See Appendix A for details concerning related mathematical developments.

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individual brand purchase as well as advertising exposure information over time. The information pertains to the yogurt product category and comes from a US city in which all major supermarkets were equipped with electronic scanning equipment to record purchase transactions. In addition, the database contains detailed records of causal data such as displays, feature ads, price promotions, and coupon activity on a weekly basis at point-of-purchase. The dataset contains three major national brands of yogurt along with a number of small market share regional and store private label brands. In the binary logit case, we focused on the leading brand in the market relative to all other brands. In the multinomial logit analysis, we focused on the three national brands since they accounted for most of the media advertising in the category and classified the remaining brands into a fourth all 'other' brand grouping. 5 The data was divided into three periods. The first 12 months were used as an initialization period in the computation of long-term loyalty for each brand. The second 8 months were used to estimate our models, while the last 5 months of data was used as a hold-out sample for predictive model validation.

3.1. Brand choice model variable definitions In this section, we specify the variables used in the brand choice analyses. The variables are similarly defined for both the binary and multi-brand cases. Advertising effects are specified as a 'share of exposure voice' by dividing the advertising exposures a household receives for a brand of interest by those of all brands prior to each purchase occasion in a given period. This formulation has the advantage of including competitive advertising effects in the binary choice model. In addition to the short-term loyalty variables, we included long-term loyalty variables. The latter measure the heterogeneity of preferences across individuals, while the short-term loyalty variables measure changes in an individual's preferences over time periods. We operationalized long-term loyalty as the monthly rate of brand k purchases made by a household during the initialization period of our sample. We now turn to the description of the marketing mix variables zt. The latter include shelf price variables for each national brand k. These are calculated as a market share weighted average of prices for available UPC's for brand k in the store visited by a household at a purchase occasion. 6 For the 'other' brand, because a weighted average price across UPC's smoothes out most of the relevant information, we focused instead on a market share weighted average of available promotional percentage price discounts. Moreover, we estimated our model with separate coefficients to allow for different response rates for major versus minor promotions. Measures of both store and manufacturer couponing activity are also included in the choice model. The store coupon variable records the value of store coupons offered for brand k, while the manufacturer coupon variable measures the availability of brand k coupons in the market.

3.2. Brand choice model estimation results In order to test the validity of our choice model specification, we first estimated four nested binary logit models for Brand A, the leading brand. We focus on Brand A since it has the dominant market share and the results for the other brands are similar. Estimation results for the nested models are presented in Table 2. The first model includes only an individual brand constant. The next model adds short and long-term brand loyalty variables to account for consumer heterogeneity. Model 3 then includes point-of-purchase marketing variables price, promotions and store and manufacturer coupon usage. 7 Finally, model 4 adds the advertising share of voice variable. 5 See Pedrick and Zufryden (1990) for a more detailed description of these data, the corresponding market situation and the empirical data reduction procedures used in the study. 6 Data analysis showed that displays and feature ads for the national brands did not improve the model's predictive fit and were thus omitted. 7 Competitive prices are included in the model only for competitor Brand B, since data analysis showed that prices for the other brands did not significantly affect Brand A purchase probabilities.

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Table 2 Parameter estimates for nested Br and A binary logit choice models Parameter

Model 1

Brand constant

2 - 1.32 ( - 39.40) a

Short term loyalty: Brand A

3 - 2.05 ( - 32.56)

4 - 2.26

- 2.33

( - 2.09)

( - 2.16)

0.77

A l l other brands

Long term loyalty: Brand A All other brands

0.81

0.82

(16.09) - 0.41

(13.52) - 0.40

(13.62) - 0.40

(8.19)

( - 7.89)

( - 7.81)

0.04

0.05

0.05

(6.38) - 0.02

(7.76) - 0.02

(8.30) - 0.02

(- 3.22)

( - 3.14)

( - 3.19)

National brand price: Brand A Brand B

- 2.31

- 2.45

( - 3.48) 3.38

( - 3.66) 3.52

(1.93)

(1.94)

Other brand minor deal

- 1.26

- 1.26

Other brand major deal

(-2.19) - 2.20

(-2.18) - 2.33

( - 1.90)

( - 1.90)

Brand A store coupon size Brand A manufacturer coupon index

1.18

1.17

(3.00) 0.01

(2.93) 0.01

(3.48)

(3.35) 0.39

Brand A share of ad exposure voice L o g likelihood ( N = 2 6 1 5 ) U 2

Hold-out sample weekly market share R M S E

- 1356 6.35

- 1022 0.246 5.31

-944 0.304 4.84

(2.26) -941 0.306 2.62

a Estimated parameters with asymptotic t-statistics in parentheses.

Within sample Likelihood ratio tests, individual t-statistics and parameter stability support the addition of each successive model component. Hold-out sample root mean square error (RMSE) prediction results for Brand A weekly market shares, shown in Table 2, also support the addition of each model component. The full model weekly market share estimation and validation sample predictions for Brand A are shown in Figure 1. The plot shows that the model reproduces market share fluctuations reasonably well during both periods. We next consider the impact of simplifying the brand choice model by using a binomial instead of a multinomial logit choice model as a way of reducing the computations when using the WNBD model for market predictions. To investigate this, we compare weekly Brand A market share forecasts from a binomial versus multinomial logit model in Figure 2. 8 The plot shows that predictions for the two models are almost identical. Given this close coincidence, we use the binary logit model for most of our subsequent empirical work with the WNBD model. This is done in the interest of tractability, since as was noted above the binary logit model is easier to work with. A further analysis of the implications of this decision will be made in Section 4.1.

8 F o r a complete description of the multinomial brand choice model results see Pedrick and Zufryden (1990).

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J.H. Pedrick, F.S. Zufryden / A n examination of consumer heterogeneity 50-

ESTIMATION SAMPLE

45-

40

35

M

~ 30 T

~

25

~ ~o Z

15

~0

5

0 27AU087

OSOEC~7

14r"Aa EE

22JUN88

5CSEaE£

DATE

Fig. 1. Brand A weekly market share predictions. A = actual brand a market shares; P = binary logit predictions

5.0

[SfIMATION SAMPLE 45

40

35

3O

25

20

I

i./

"L~l

:5

.$

5

0 27AC3~7

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Fig. 2. Binary vs. mult. logit Brand A percent market share predictions. B = binary logit predicted % market shares; M = mult. logit predicted % market shares

J.H. Pedrick, F.S. Zufryden / A n examination of consumer heterogeneity

267

3.3. Aggregate WNBD model estimation and empirical results An attractive feature of the proposed model is that its parameters can be easily obtained using a multi-stage estimation procedure which first separately estimates the household brand choice model parameters along with the advertising exposure and brand loyalty distributions. Then, the aggregate NBD purchase incidence model parameters are estimated conditionally using the estimated parameters from the other three model components. Estimation of the brand choice model parameters was done using the method of maximum likelihood. The estimation and corresponding empirical results for advertising exposure and brand loyalty heterogeneity distributions as well as the product category purchase incidence model are described below. The parameters of the BBD advertising exposure heterogeneity distributions, for each brand, were estimated by the method of zeros and means using observed advertising reach, frequency and number of scheduled spots (e.g., see Zufryden, 1987). Here, the BBD provided an excellent fit to the exposure data of each advertised brand (see Pedrick and Zufryden, 1990). Alternative techniques were used to estimate the loyalty heterogeneity parameters. The BBD-NBD model version was estimated from brand market shares and penetrations using a method of means and zeros as in Chattfield and Goodhardt (1973). The Dirichlet-Multinomial-NBD version was estimated as in Goodhardt et al. (1984). The bivariate and multivariate NBD (Polya) model versions were estimated by extending the means and zeros technique as in Goodhardt et al. (1984). At the final step, we extended the means and zero procedure in Zufryden (1988) to estimate the NBD purchase incidence parameters conditionally using the previously estimated parameters (See Appendix B for details). The fit of the overall WNBD model for aggregate market level purchase distributions is demonstrated in Table 3. Here, actual versus predicted Brand A depth of repeat purchase distributions for the five 4-week periods in the validation sample are shown. The Chi-square goodness of fit statistics given in the table provide support for the model, as none indicate that a significant deviation exists between actual and predicted household brand purchase frequencies.

4. Comparison of competing aggregate brand purchase incidence models We begin our comparison of competing models by empirically contrasting the predictive performance of alternative specifications of household heterogeneity for a set of nested aggregate purchase incidence models. We also evaluate the impact of using a binary versus multinomial logit brand choice formulation

Table 3 Brand A validation sample purchase distributions for binary logit W N B D model with N B D - B B D brand loyalty heterogeneity Period

9 10 11 12 13

N u m b e r of purchases

Actual Predicted Actual Predicted Actual Predicted Actual Predicted Actual Predicted

Validation sample total:

Chi-square

0

1

2

3

4+

531 534 535 539 542 536 551 545 576 576

33 32 32 30 25 32 23 28 8 7

14 10 10 9 11 10 3 7 0 1

5 4 4 3 3 4 4 2 0 0

1 4 3 3 3 2 3 2 0 0

4.15 0.61 2.45 5.74 1.14 14.09

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J.H. Pedrick, F.S. Zufryden / An examination of consumer heterogeneity

Table 4 Evaluation of the impact of modeling heterogeneity on prediction results for market performance measures Market performance measure

Purchase distribution Chi-square a

Market share RMSE

Average purchase rate RMSE

Brand penetration RMSE

6.33 4.59

1.96 2.16

2.50 1.86

288.13 311.19

6.33 4.59

2.43 2.43

1.29 1.29

23.25 46.46

3.07 3.10

1.16 1.10

0.79 0.79

14.09 14.22

3.41 3.16

1.79 1.63

0.74 0.76

26.22 23.02

No heterogeneity (Poisson model): Binary logit Multinomial logit

Add category purchase rate heterogenity (NBD model): Binary logit Multinomial logit

Add brand loyalty and exposure heterogenity (WNBD model): Binary logit: NBD-BBD loyalty Polya loyalty Multinomial logit: Dirichlet loyalty Polya loyalty a

The Chi-square statistics shown are the sum of the five 4-week period Chi-squares defined in Table 3. Thus, they reflect the goodness of fit of the depth of repeat distributions over the entire validation sample.

in our model. Finally, we highlight the managerial benefits of our model over existing approaches with an application to media analysis.

4.1. Empirical evaluation of alternative household heterogeneity specifications To test the need for considering household heterogeneity in our model, we start with a nonstationary Poisson model. In this base model, we do not allow for heterogeneity since all households are assumed to have the same category purchase rate u c and consequently the same brand purchase rate uB(x t) for a given market environment x r Here, nonstationarity in market conditions is modeled by setting the explanatory variables in the logit model to their period t mean values, thus excluding any corresponding heterogeneity effects. A summary of the validation sample fit for all of the competing purchase incidence models is shown in Table 4. The table shows results for aggregate incidence models using both binary and multinomial choice models. Root mean square error (RMSE) results for 4-week period market share, mean purchase rate and market penetration prediction errors as well as complete depth of repeat purchase distribution goodness of fit statistics are also shown. 9 The striking feature of the basic Poisson model results is the large purchase distribution Chi-square statistics. These indicate that a significant deviation exists from the actual hold-out sample purchase distributions as a result of the tendency of the model to dramatically underestimate the probability that a household will make multiple purchases in a period. Next, we allow for unobservable heterogeneity in individual category purchase rate by allowing it to follow a Gamma distribution over the population. This results in a nonstationary NBD model version of (3) which nests within our overall WNBD model framework. This intermediate model is similar in spirit to models of purchase incidence with explanatory variables such as those of Wagner and Taudes (1986) and Rosenqvist and Strandvik (1989). 10 9 Results are reported for 4-week periods to provide for an adequate sample size for parameter estimation of the brand loyalty and advertising heterogeneity distributions. z0 These models differ from our formulation in their specification of the brand purchase rate as a function of explanatory variables. While Wagner and Taudes (1986) use a power function and Rosenqvist and Strandvik (1989) use an exponential form, we use a logistic specification.

J.H. Pedrick,F.S. Zufryden /An examinationof consumer heterogeneity

269

PI~IZ I0.00:

/

REACH = Z 0 Z

./

8.25

REACO = 30~.

8.00 2

4

6

B

~0

Fig. 3. Binary logit WNBD Brand A percent market penetration simulation for alternative media schedule reach and frequency levels The results of the N B D versus the Poisson model in Table 4 show the dramatic improvement in purchase distribution predictions when unobservable heterogeneity is accounted for. However, the m e a n household purchase rate predictions did not improve and were actually worse for the N B D model. We note that both models yield the same market share predictions because the aggregate purchase incidence p a r a m e t e r s cancel out of their market share formulas. Model predictions from both the Poisson and NBD models were generated by setting the individual level brand choice model variables to their 4-week period means. The predictive advantage obtained from using the observable heterogeneity weighting distributions, as proposed for the W N B D models, instead of a single period m e a n value, can be seen in Table 4. The results show that the W N B D models perform significantly better in predicting all four hold-out sample market performance measures. Our final sensitivity analysis focuses on the impact of model specifications with alternative choice models and loyalty weighting distributions. The results in Table 4 show that there is little difference between predictions based on binary versus multinomial choice models. This suggests that little is lost when the simpler binary logit model is used in place of the multinomial choice model. 11 In addition, it can be seen that our alternative brand loyalty weighting distributions for the W N B D models also lead to similar results. Overall the results do however show a slight advantage for the binary logit model with N B D - B B D - b a s e d loyalty over the other three W N B D alternatives.

4.2. Managerial benefits of proposed model for media schedule analysis The proposed W N B D model provides a potentially useful tool for marketing planning. To illustrate the managerial applications of the W N B D model, a model simulation is shown which allows a m a n a g e r to assess the impact of alternative advertising media schedules. This impact can be evaluated from the 11 This conclusion is reasonable if there is interest in only one brand. The multinomial model needs to be used if multiple brands are of interest.

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270

aggregate WNBD model by estimating the advertising exposure heterogeneity weighting distributions to match a given schedule's reach and frequency. In contrast, models which ignore household exposure heterogeneity are not able to separate out the effects of reach and frequency from overall GRP levels. To illustrate this point, a plot of the advertising response curve for market penetration is shown in Figure 3. The simulation was generated using prevailing market conditions during a given monthly period. The plot suggests that Brand A market penetration is more sensitive to increases in media schedule reach than of frequency. This point is emphasized by the observed differences in market penetration predictions in Figure 3 for media schedules with the same overall GRP levels but different combinations of reach and frequency. For example, even though both schedules have a GRP level of 270 points, that with reach of 30% and frequency of 9 produces an 8.20% penetration, while the schedule with reach of 90% and frequency of 3 produces a larger penetration of 9.54%. Similar analyses can be done to predict a brand's market share, mean number of purchases, and depth of repeat distribution as a function of media schedule reach and frequency.

5. Summary In this study, we proposed a new stochastic model framework that integrates brand choice and purchase incidence model components as well as explanatory variables. A key feature of our model is the consideration of both observable and unobservable heterogeneity of consumers over a population. In particular, we incorporated mixing distributions to account for unobservable individual differences in category purchase and household advertising exposure rates as well as include distributions to explain the observable heterogeneity of advertising exposures and brand loyalty variables. As a result, our empirical analyses showed that the consideration of heterogeneity significantly enhances the descriptive and predictive capabilities of our model. Moreover, we illustrated the managerial relevance of considering advertising exposure heterogeneity to evaluate the market-level impact of planned advertising media schedules that are based on different media reach and frequency combinations.

Appendix A. Alternative multibrand loyalty heterogeneity models For the Dirichlet-Multinomial model, the proportion of individuals that jointly makes bk, t_ chases of brands k ~ S given Kt_ 1 = EgEsbg,t_l product category purchases during t - 1 is Kt-i~r(L,-i)

~

r(bk, t-i + Ok,t_i)

P(b'-x I K ' - l ; O t - x ) = r - ' ~ t -l~---'~t--1)-- k=l'lt .It b k , t _ l ! r ( O k , t _ l )

1

pur-

(A.1)

,

with parameters Or_ 1 = {Ok,t_i} , Ok,t_ 1 ~ 0 (k E S), and Lt_ i = EkOk,t_l . This leads to the weighting distribution B

w(x,) =P(K,_ 1 la, [3)'P(bt_I IKI_I; Ot_l) l'-I P(iktlNkt; mkt, nkt).

(A.2)

k=l

For the multivariate NBD, the proportion of individuals that jointly makes bk, t_ 1 purchases of brands k ~ S during period t - 1 of length T is stated as

]o.

e(bt-l[Z;at-l't~t-1)-~

k=lH F(g2k,t_l)bk.t_l [

(T+q~t_l)

with parameters /~t-1 = {Ok,t-l}, 12k,t-~ > 0 (k ~ S), and ~t-1 > 0.

(q~t~-I ~-T)

(A.3)

J.H. Pedrick, F.S. Zufryden / A n examination of consumer heterogeneity

This leads to an alternative formulation of

w(x t)

271

as

B

W( Xt)

= e(bt_l l Z ; l]t_t, cbt_l) I-I e( iktl gkt; mkt, nkt).

(A.4)

k=l

Appendix B. Estimation of WNBD model At the final stage of WNBD estimation, we obtain /3 by minimizing the sum of squares of the differences between the observed proportions of individuals making 0 purchases over periods t, Pot, and the corresponding theoretical values:

[Pot- Pt(0)] 2,

SSE = •

(B.1)

t

where, from (4), we obtain P,( O) = E [ [3/( S t P ( x t ) r +/3)] ~ " w ( x t ) .

(B.2)

X t

Then, a is obtained by setting the observed mean number of Brand A purchases, M t, to the theoretical mean Et(j) in (5) and averaging over time periods as

Acknowledgements This study was supported by a research grant sponsored by the A.C. Nielsen Corporation. The authors wish to acknowledge and thank J. Dennis Bender and other members of the A.C. Nielsen Marketing Research Division staff for their support. Thanks also go to Avu Sankaralingam and Susan Lawton for providing computational assistance and to Rita Wheat for her comments and suggestions during the course of the study.

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