COMMON GROUND FOR BASS AND NBD/DIRICHLET: IS CONDITIONAL TREND ANALYSIS THE BRIDGE?

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1 COMMON GROUND FOR BASS AND NBD/DIRICHLET: IS CONDITIONAL TREND ANALYSIS THE BRIDGE? Abstract Cullen Habel, Cam Rungie, Gus Geursen and John Kweh University of South Australia Track: Conceptual Papers / Marketing Theory Two models are constantly referred to in the discussion of marketing science. These are Ehrenberg s Dirichlet model of brand choice (and its foundation application of the Negative Binomial Distribution) and the Bass Model of the Diffusion of Innovation. Both models have been used to foster discussion about marketing science. Both models are at the core of market behavioral discussion. However, despite considerable research extending their individual scopes and applications they have remained as singular models. The Dirichlet provides understanding of frequently purchased products and services in stable market conditions and the Bass model providing the adoption mathematics of infrequently purchased products and services. The purpose of this paper is to suggest there is common ground between these models and to explore the implications of this observation. In this first instance an NBD based model has been shown to replicate some (but not all) of the Bass curve shapes and there are grounds to believe that, with further development an NBD/Dirichlet based penetrations model may also be able to account for brand choice behaviour in a growing category. Introduction This paper aims to demonstrate that, under certain conditions, a model based on conditional trend analysis can generate a curve of cumulative period-to-period penetrations (a purchase frequency 0). A curve such as this appears similar to the Bass curve under certain conditions. Generating these curves is a relatively well-trodden path; to our knowledge Fourt & Woodlock took the earliest step in 960. Our paper is the first (to our knowledge) that involves an attempt to integrate Bass and Dirichlet. It proposes a new functional form, in addition to those reviewed by Hardie, Fader and Wisniewski (998). These authors compared eight models of new product trial, which included the Bass Model (969). A major finding of their paper was that forecasting accuracy of models such as ours (incorporating heterogeneity, simplicity and likelihood estimation) appeared to be beyond that of many other classes of models. A brief review of that finding in included in our discussion. Another aim of our paper is to define the strict limitations of the CTA approach that Bass is a far more robust model that addresses the issues of non-stationarity and customer commitment in a straightforward and intuitively appealing manner. Further research directions are suggested to address in particular the need for data, non-stationarity issue and the possible inclusion of brand choice behaviour into new category growth forecasts. The Bass Model The Bass model (969) models the process of the diffusion of an innovation throughout a population of potential adopters. Drawing on roots in epidemiology (Bailey, 957 cited in (Mahajan and Muller 979), the model requires an independent estimate of three parameters. These parameters are the coefficients of innovation p and imitation q; the third being an estimate of the total number of potential adopters (or Nbar). The number of adoptions in a given time t is given by the parsimonious algebraic expression: Conceptual Papers / Marketing Theory Track 520

2 n q N ( t) = pn + ( q p) N( t) [ N( t) ] 2 equation Where N(t) is the cumulative number of adoptions at the beginning of time t. The model can be calibrated in a number of ways. One manner is calibration/parameter estimation by analogy. Using this technique, one may consider the nature of the new product, the consumer behaviour that is likely to accompany it, and look for previous adoptions of a similar nature. In this manner a marketer wishing to forecast, say the adoption pattern of DVD s may see some parallels between the nature of the compact disc, introduced in 983 and their product. As the CD (Bass 995) is an adoption that has run its course, the parameters from that earlier adoption may be used in the Bass model to develop a forecast. A second method may be to estimate the parameters based on the first few periods of adoptions. For instance, the DVD player has been available for about three years. A marketer wishing to model DVD introduction may also choose to consider those first three periods to estimate the model s parameters. Given an independent estimate of the total size of the market the value of the innovation and imitation parameters can be estimated for the first three periods then extrapolated to forecast subsequent periods. It is important to stress the value of an independent estimate of the total number of adopters. Bass is a robust, powerful model, but much of its strength relies on solid inputs. A marketer is well advised to spend significant resources on determining the size of the potential market in order to give the model a chance of delivering reasonable forecasts. Bass has the ability to describe many diffusion phenomena. At the one extreme is an innovation where the population is high on imitation but low on innovation, for example, mobile phones. At the other extreme is an innovation where the population is low on imitation but high on innovation, for example, blockbuster movies (Lilien and Rangaswamy 2003). The differing pattern of adoptions per period is intuitively appealing, empirically well substantiated and modeled reliably by Bass. The NBD/Dirichlet In 959 Andrew Ehrenberg demonstrated that the repeat purchase rate of a category across a population of consumer could be described by the Negative Binomial Distribution. At the heart of NBD theory is the concept that category purchase propensity is distributed gamma across a population of consumers; a Poisson process is applied to the gamma to create the discrete NBD for the category. The NBD retains two parameters, A (the scale parameter) and K (the shape parameter). As a shape parameter K controls the balance of light and heavy users within the distribution. As a scale parameter A is directly proportional to the volume of purchases. We use the technique of likelihood maximization (Rungie 2003) to estimate the parameters. A key property of the A parameter is that it is proportional to the length of the time period of observation of the consumer repurchase behaviour (Goodhardt and Ehrenberg 967, Morrison 969, Schmittlein, Cooper and Morrison 993). This is a property of the NBD that we will make full use of in building the bridge to Bass. Whilst even the initial fitting of the NBD to category purchases was empirically well substantiated (Ehrenberg 959) it has even further empirical support through the development of The Dirichlet (Goodhardt et al. 984, Ehrenberg 2000, Uncles et al. 995). The Dirichlet builds on the foundation of the NBD as it accounts for brand choice behaviour within a category. Again, at the heart of The Dirichlet is the gamma distribution; the latent propensity to purchase each particular brand in the category is said to be distributed gamma across the population of consumers. Each brand within that category then has its own ANZMAC 2003 Conference Proceedings Adelaide -3 December

3 parameter (its brand alpha) that defines the shape of the distribution of the population s propensities to purchase it. The higher a brand s alpha, the more likely there are to be high volume users within the population. Dirichlet then models the competition between the brands each brand takes its alpha parameter into the mixing distribution, generating estimates of the purchase amounts per brand, per customer. Various brand performance measures can be calculated from the model, for instance market share is based on the value of its brand alpha relative to the sum of all of the brand s alphas in the category. α MS = Equation 2 α + α 2 + α α n For n brands in the category. Whilst the deeper mathematics of The Dirichlet is beyond the scope of this paper, the model consistently predicts many of the observations of marketing. Double Jeopardy (McPhee 963), Duplication of Purchase (e.g. Goodhardt and Ehrenberg 969) and correlation of penetration with market share are all incorporated into the structure of Dirichlet, rendering it, too, a robust model that also reflects observed regularities. It is important to note that a cornerstone of The Dirichlet is the Negative Binomial Distribution, as it describes the population s purchase rate of a category. It is with the NBD that we consider we can make the link to the Bass Model. Two Separate Worlds? The two models are similar in many ways; their broad applicability, their theoretical rigor and their empirical substantiation being the most obvious. In some ways, however, they occupy two separate worlds. Consumer Durables vs Frequently Purchased Items Whilst diffusion of innovation applies to all product classes Bass has found its greatest application in the area of consumer durables (Bass 969; Dodds; 973, Kalish and Lilien 986). In these markets the incidence of repeat purchase is low a family buys a dishwasher or a DVD player only once. The repurchase cycle of these products suits the Bass model, which has only ever purported to model adoptions each customer s first purchase. The customer who purchases a second dishwasher within the modeling timeframe has the effect of generating noise. That said, there can still be common ground for the two models; this was identified by Hardie, Fader and Wisniewski (998) in their justification for including Bass (969) in their review of product trial forecasting models. Adoption vs Penetration Bass is a model of consumer behaviour that proposes to describe the rate of adoption by a population. In a simplistic sense this can be seen the first purchase by a consumer but this is subtly different to adoption. Adoption implies the commitment of a consumer to continue the use of the innovation (Rogers 983) With expensive consumer durables it appears a reasonable assumption that first purchase approximates adoption the cost of the item means that a prepurchase evaluation and consumer commitment has taken place. The lower risk associated with the first purchase of, say, a new ready to drink alcohol beverage means this commitment cannot be assumed. This estimating of commitment is provided by most penetration models in their repeat component. (Hardie, Fader and Wisniewski 998). Thus a period to period pure penetration model based on NBD/Dirichlet may lack some of the richness of the Bass Model when Bass is used in its preferred context. Dynamic Markets vs Static Markets A core assumption of NBD/Dirichlet is one of a static market. Goodhardt et al. (984) mentioned that the underlying purchasing behaviour analysed by the Dirichlet model must Conceptual Papers / Marketing Theory Track 522

4 show no overall trend from one period to the next. It is assumed that both the category parameters and the distribution of purchase propensities for most brands of frequently purchased consumer goods are approximately stationary most of the time. Bass (995) expressed a desire for a generalised model that reduced as a special case to the Dirichlet model, explaining nonstationary markets as well as stationary markets. This process has already been followed in the development of the Generalised Bass Model, (Bass et al., 994) which then reduces to the special case of the Bass Model. Science is full of examples of special cases that are later generalised. A key example of this is that of Newtonian Mechanics, applying as a special case of Einstein s Theory of Relativity under the special condition of sub-lightspeed environments. (Bass 995) It may be, indeed, this limiting condition of Dirichlet that limits an NBD based model to approximate a greater range of adoption curves. The Bass Model implies dynamic market conditions. The (p) coefficient of innovation relates to the external influence on the population of customers, such as advertising, awareness campaigns, trial offers and other inducements. Even more importantly the (q) coefficient of imitation relates to internal influence the dynamics within the customer population such as word of mouth, opinion leaders, group influences and other internal effects. Bass accounts for both of these effects. Free of the assumption of a static market one may expect Bass to describe a greater range of adoption patterns. Method We propose that a plot of period to period category penetrations can be generated from the NBD parameters extracted from the disaggregated data of first period s purchases. The technique we use is that of likelihood maximization (Rungie 2003) although others such as method of moments (eg: Ehrenberg 959, Goodhardt, Ehrenberg and Chatfield 984) are also used. Given the assumption that the A and K parameters for the category will not vary from one period to the next the procedure will be: i/ to estimate the first period s penetration from the NBD parameters ii/ create an estimate for a two period penetration and iii/ subtract a period penetration from a 2 period penetration to extract the second period s penetration figure. A generalised form of this process is proposed, and then becomes the means whereby we produce curves for period-to-period penetrations. Estimating Penetration from NBD Parameters The Negative Binomial Distribution is a probability density function for the number of purchases r of a category. It is given by the expression: r Γ () ( K + r) A f r = Equation 3 ( K ) r ( A) ( K + Γ! + r ) Where K and A are parameters, and Γ is the gamma distribution for the bracketed expression following it. f (r) is a non-negative number less than and represents the proportion of the sample that is purchasing () r times. Thus f (0) is the proportion of the sample that does not purchase and can be calculated by: 0 Γ ( ) ( K ) A f 0 = or f () 0 = Equation 4 Γ( K ) 0! ( + A) ( K ) ( + A) ( K ) Penetration is therefore the entire sample (00% or ) minus the proportion of the sample that purchases none at all in the time period. Using equation 5 as the expression for the proportion of the sample purchasing none at all, penetration can be expressed as: PEN = Equation 5 ( + A) ( K ) ANZMAC 2003 Conference Proceedings Adelaide -3 December

5 The A parameter of the NBD has shown to be proportional to the time period of observation (Schmittlein et al. 993), (Goodhardt and Ehrenberg 967)) By doubling the A parameter it is therefore possible to calculate the penetration for a period of twice the length, thus if we consider monthly data the penetrations for these periods to be: PENmonth = ( + A ) ( K ) and PEN2month = month ( + 2A ) ( K ) Equations 6 & 7 month Subtraction of equation 7 from equation 8 would give the penetration for that second time period. This process could be repeated as necessary to give incremental penetration increases for as many time periods as necessary. A Generalised Form Given periods of equal time, a generalised form for period-to-period penetration can be generated by considering that a period s new penetrations are equal to that period s penetration minus the penetrations of the previous period. Thus for a given A and K parameter, the number of new penetrations for time t (or n(t)) can be expressed as: () n t ( ) or K + ta ( + ( t ) A) n( t) = Equation 8 & 9 K K ( + ( t ) A ( + ta) Where the curves are similar = K Thus period to curves can be generated using the negative binomial distribution and conditional trend analysis. Manipulation of the NBD parameters does not give a broad range of shapes they are all minor variations on the convex penetration pattern that is generated by Bass under conditions of high innovation and low imitation. Figure : An NBD period to period penetrations curve exhibits a convex saturation pattern A Static Market Cumulative Penetration Figure 2: The Bass Curve Exhibits a greater variation of shapes, convex and s-shaped. A High p Low q Innovation Time A Low p High q Innovation Cumulative New Adoptions Cumulative New Adoptions We can see that the NBD Time based curve approximates the Bass curve under some Timeconditions. These conditions are when the innovation parameter (p) is high and the imitation parameter (q) is low. These conditions would occur where the product is simply an idea whose time Conceptual Papers / Marketing Theory Track 524

6 has come. Blockbuster movies exhibit these patterns, but there better examples. A comprehensive analysis of over 40 introductions throughout the 20 th century (Lilien et al., 2000) indicates the electric frying pan, hotplates, toaster, and (to a lesser degree) the freezer as being product categories that exhibit a high p and a low q. These conditions also approximate the assumptions of NBD-Dirichlet. The lack of post launch influence reduces the adoption of the innovation to a stochastic process one which the NBD models extremely well. Where the Curves are Different Thus the models, even in their similarities, demonstrate their differences. Indeed, the greatest difference occurs around the assumption of stability. In treating period-to period penetration increases as a stochastic process, the NBD based model appears to reduce to a special case of the more versatile Bass model. The Bass model generates the well known S-Curve, which is a result of the internal influence, or the q parameter. In restricting itself to static conditions, the NBD based model cannot do this. Discussion and Further Research At the very least, the NBD based model of penetration growth is a relatively straightforward tool to estimate the level of trial during the rollout of their product. From an early snapshot of the repeat purchase profile of the customer base, an estimate can be gained for subsequent periods trial. This snapshot may be gained in a testmarketing phase (as per Fader, Hardie and Zeithammer 2003), or in the early phases of a product launch. If trial is running below expectations managerial corrective action can be taken early. There has been no empirical substantiation, as yet, of this model. The authors, however, take heart from Hardie, Fader and Wisniewski's (998) 8 model comparison where three factors were common to the more accurate models. These factors are, firstly, a simple form that does not accommodate S shaped cumulative sales curves. Secondly they explicitly accommodate heterogeneity in purchase rates and thirdly they are calibrated using Likelihood Maximisation. Our model has all three elements; simple curvilinear forms, gamma heterogeneity with A and K estimated by likelihood maximization which provides great encouragement for future empirical testing. This paper questions the need to develop a more complex version of the NBD penetrations model. As Hardie et al (998) found that simpler curves had better predictive ability, the paper suggests it might make more sense to stay with a form that describes a narrower range of phenomena but does so more accurately. We suggest, however, that the answer may be to aim for both generate a more versatile model and aim for the predictive validity of a simpler model. Other directions, on a theoretical level, involve modeling dynamic markets. Imagine an NBD model that accounted for changes in the nature of the market over a period of time. A model that could describe the drift in the A (scale) and the K (shape) parameter over the periods of observation. In effect, we could then consider the existing period-toperiod model as a special case of Bass, and indeed, a special case of the more general dynamic NBD. Bass himself foreshadowed such a development of NBD/Dirichlet in a chapter entitled Higher Level Explanations (Bass 995). Such a model has been proposed, whereby the change in the nature of the category has been given a functional form (Rungie et al. 2002); each NBD parameter and indeed Dirichlet parameters were modeled as a function ANZMAC 2003 Conference Proceedings Adelaide -3 December

7 of time. It would indeed be interesting to see if this version of the NBD could be incorporated into our model in order to generate period to period penetration estimates that cover a greater range of adoption situations, as does the Bass model. Yet a further development may involve utilizing Dirichlet s ability to model choice behaviour within a category. Once the range of adoption curves can be generated by an NBD based penetrations model a next step may be to utilize the choice modeling property of Dirichlet to describe what share of the growing category each brand is likely to have. With many new categories exhibiting brand competition almost immediately (consider DVDs and digital cameras) the ability to forecast a brand s share of a growing category may prove to be valuable to a marketing practitioner. Conclusion This paper has considered, at a theoretical level, the similarities and differences in NBD and Bass. The implications of these observations have been noted and the opportunities for further research highlighted. We have drawn parallels between two broadly substantiated models of consumer behaviour and demonstrated that while they appear to operate under different sets of assumptions there is, indeed, common ground between Bass and NBD/Dirichlet. Conceptual Papers / Marketing Theory Track 526

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