A Case For Replication: Fitting Product Variants Data to the Dirichlet Model

ABSTRACT - To-date replication has been undervalued in marketing resulting in a focus on single studies. Non-replicated results have thus become disseminated as established knowledge. This paper discusses replication and why it is important. It looks at datasets from different countries and time periods as an example of how easy it can be to conduct differentiated replications. The model tested is the well-established NBD Dirichlet model of buyer behavior. Here it is extended to look at its application to purchasing of product variants (e.g. purchasing across different pack sizes). The results show the usefulness of replication studies and open up further areas of investigation.


Rachel Kennedy and Jaywant Singh (2002) ,"A Case For Replication: Fitting Product Variants Data to the Dirichlet Model", in AP - Asia Pacific Advances in Consumer Research Volume 5, eds. Ramizwick and Tu Ping, Valdosta, GA : Association for Consumer Research, Pages: 374-378.

Asia Pacific Advances in Consumer Research Volume 5, 2002      Pages 374-378


Rachel Kennedy, University of South Australia, Australia

Jaywant Singh, South Bank University, U.K.


To-date replication has been undervalued in marketing resulting in a focus on single studies. Non-replicated results have thus become disseminated as established knowledge. This paper discusses replication and why it is important. It looks at datasets from different countries and time periods as an example of how easy it can be to conduct differentiated replications. The model tested is the well-established NBD Dirichlet model of buyer behavior. Here it is extended to look at its application to purchasing of product variants (e.g. purchasing across different pack sizes). The results show the usefulness of replication studies and open up further areas of investigation.


Replication in social sciences is rare, even though its importance is widely recognised (Burgsthaler and Sundem 1989, Campbell 1969, Carver 1978). It is clearly undervalued in marketing. So undervalued that in an audit of three major marketing journals, only 2% of the published studies were found to be replications, and yet of these only 15% fully confirmed the original findings(Hubbard and Armstrong 1994). Given this lack of confirmation of results, the focus on single studies is concerning, especially as many findings have become disseminated as established knowledge.

If we are to advance marketing knowledge, it is important we adopt good scientific practice, pay more attention to empirical foundations and thus replicate more and across a wide variety of conditions. This paper outlines what replication is, why it is important to knowledge development, especially with extensions, and then presents a summary of a replication with extensions. [It extends ideas presented in Kennedy (2001) with a different example.] We extend the replication to nine different product categories taking a few product variants and demonstrate how market-performance can systematically be analysed using the generalisable results.


A replication study is simply the repeat of a piece of research to ascertain whether the results hold in different conditions. The aim is to see if the same result is found, or equally important, if a different result is found. Replications determine whether we have a result at all (Lindsay and Ehrenberg 1993) rather than a spurious finding, and they also lead to generalisable results.

They are central to the scientific process as they overcome uncertainty and remove biases. Specifically they reduce the problem of sampling error as further samples are tested, different data collection methods, measures and researchers limit different biases that can be inherent is a single study, and different times or places increase generalisability of the particular replicated finding. Thus successful replications produce re-useable knowledge rather than just a finding, that may or may not hold at a later point in time. This allows for prediction of the outcomes from marketing activity (Stern and Ehrenberg 1997). If we know what to expect as normal, then we have a benchmark to compare results of marketing activities.

In practice, replications are not mere repetitions. As a minimum they will vary in time (i.e. being conducted in the year 2000 or 2001) and/or the researchers undertaking the work. They can, however, also be designed to purposefully extend the scope of a previous result, so as to lead to more powerful empirical generalisations. Such replications if successful deepen the explanatory findings and any theory developed.

When one deliberately repeats a study under some new condition it is often referred to as a replication with extension, or a differentiated replication. Extensions are often intentionally designed to extend the result across different product or service categories, populations (i.e. experts, general public, students), countries, situations, using variations in measures and procedures, or any other dimensions of interest. The more conditions that differ the better (Lindsay and Ehrenberg 1993), and the stronger the test of the generalisability of the finding when consistent findings are generated.

Caution however is required in designing a study with too many conditions changed. If multiple extensions are made (i.e. the study is conducted in a different country, using different measures, looking at behaviour across different categories etc) and the result is not consistent, the exact boundary conditions will not be known. That is, the limitsBoutside of which the result does not hold, will not be clear. It will not necessarily be known if the difference relates to the different sample or different measure, for example or an interaction between the two.

In summary, replications can help develop re-useable knowledge and give marketers confidence in research findings. They enable us to know the range of conditions under which a finding is so far known to hold, under which it is therefore becoming routinely predictable, and under which it can be applied to practical problems (Lindsay and Ehrenberg 1993). Extensions help us to know the boundary condition on a finding, generalisation or theory. Knowing when a generalisation does not hold can be as important as knowing when it does hold.

We now move on to present a replication with extension of a buyer behaviour model as a case to demonstrate how easily it can be done, especially given the growing availability of large scale marketing data sets such as buyer behaviour tracking across countries and categories. We conclude with a summary of what the replication tells us that a single study can not.


The particular model that we will use as a case study to demonstrate how to conduct replications is the NBD Dirichlet model or theory of market structure and buyer behaviour (for full details see Ehrenberg and Uncles 2001, Ehrenberg 1988). It is a descriptive, empirical model that was developed when patterns were identified in tracking consumer panels that monitored how households bought fast moving consumer goods. The core inputs to this theoretical model are how many people buy the category and each brand, and how often. The outputs include benchmarks of many brand performance measures plus a framework for understanding generalisable patterns of brand buying including:

1. Penetrations (i.e. number of customers) are much lower for small brands than bigger brands;

2. Brand purchase rates are very similar across competitive brands (i.e. people buy coffee 5 times a year on average, whatever brand they buy);

3. But there is a Double Jeopardy trend where small brands get hit twice with less customers who buy less on average;

4. 100%-loyalty in a year is relatively rare (as most buyers have a repertoire of brands they will buy); and

5. Loyalty measures (i.e. 100% loyalty, Share of Category requirements, etc) are higher in shorter periods.

The model (the above and other generalisable findings) has been successfully fitted to many fmcg categories including coffee, washing powders, pet food and more (e.g. Ehrenberg and Uncles 2001, Ehrenberg 1988). Each new category study is basically a replication of the model with simple extension, whether seen in this light or not.



As summarised in Table 1, the same "Dirichlet-type" patterns hold in over 50 varied product and service categoriesBsome quite different to the original setting where the patterns were found. For example one very different setting tested included doctors prescribing of drugs (Stern 1995, Stern and Ehrenberg 1995). This could be expected to be quite different to household purchasing of grocery categories for many reasons including that doctors are experts, trained in prescribing drugs, they are not the end users of the product, they do not pay for it and so on. Similarly one may have expected different patterns for car purchasing given the amount of money spent compared to a can of beans.

Testing the model in these more extreme conditions were more radical replications / extensions. But because the model successfully holds and predicts market performance in these varied conditions, marketers can have greater confidence that the model will also hold in other situations. This generalisability of the patterns is why they provide usable (and re-usable) norms or benchmarks.

Due to such extensions Dirichlet-type findings are now well established, but more can still be learnt from seeing if we can extend the model further, or equally importantBnot. In this paper we demonstrate how the model fits across nine different product categories to demonstrate how it is possible to study the patterns in numbers and, in this case, establish a predictable regularity of buyer behaviour. The most important extension that we will present however is more differentiate than simply to new data sets and categories. Specifically we extend it from brand buying to look at loyalty and buying of different product variants. Different variants can include different pack sizes, different flavours, products with different features, fragrances and so on. The idea of fitting this type of data to this established model is new.

To simplify this paper we present only top-line outputs of the model and compare them to our observed results. Specifically that is the key loyalty-related performance measures (i.e. the penetration figures, the number of 100% loyal customers, and the Share of Category requirements) as predicted by the model for demonstration purposes. The model has however been fitted in its entirety including the levels of switching across the competitive brands, product variants and so forth. The observed replication results are quantitatively validated and compared with the theoretical model predictions.


Most products have a range of variants. Unlike brands, which are either big or small, product variants are highly differentiated. There are small vs large pack sizes, tablet-powder-liquid formats, etc., each of these representing a particular characteristic or attribute. Table 2 summarizes the loyalty-related performance measures for four size-related product variants in the USA. The table gives both the observed (O) figures and those predicted by the model and known as Theoretical (T) norms. The Observed figures in all the tables are sourced from scanner panel data consisting of 872 households in the examples from IRI, Philadelphia (1991) and 10,000 households in the examples from TNSofres UK (1999).

It demonstrates that we have a successful extension through a differentiated replication of the results. Specifically the replication showed us the model extends from the buying of brands to also hold for the buying of different product variants (in this case pack sizes). With this observed data we see the generalisable patterns that we would expect to find, including;

1. Double JeopardyBwhere the variants bought least often (i.e. Extra large packs) get hit twice. Not only do they have less people buying them (with a penetration of only 13% compared with 58% of households buying the medium pack size) but they buy less on average (i.e. 2.9 times in the period compared with 4.4 for the medium packs).

2. Consumers have repertoiresBon average they only give 37% of their category purchases to any one variant. Thus those people who buy the "Extra large packs" also buy, small, medium and/or large in predictable proportions.

While we can observe the patterns, knowing it we can also quantify it against the predicted figures from the model. The fit for each measure in Table 2 is close, in terms of

i. A lack of overall bias (e.g. the average Observed (O) and Theoretical (T) percentages agree within a point or two).

ii. The deviations between the individual O and T percentages have Mean Absolute Deviations (MADs) for individual brands of mostly about 1 to 3 percentages point. This measure is similar to a standard deviation but simpler.

iii. Correlations average above 0.9 suggesting a good fit between the model and what we have observed.





We now move to a further replication of the variant extension of the model, using data eight years later (1999) from the UK. The results in Table 2 support the replication and further generalise the results. Table 3 summarises the observed and theoretical values for the same key loyalty-related performance measures.

The results in Table 3 show similar patterns as in Table 2. In terms of knowledge development most noticeably we find the same exception or deviation in relation to the variant Extra Large when compared to the theoretical values. From a single study we would not know if this was a sampling issue or a real effect but having seen it twice we have confidence that something different might be going on that requires further investigation. In this instance we believe this exception is explained by a distribution effect, in that large sizes are not available in smaller outlets and/or that this very large size is functionally different in that they are too heavy for many shoppers which restrict the market for this variant. The results for product variants so far show similar generalisable patterns as brands. We are currently investigating this further.

Replication thus is a useful tool for understanding the markets and patterns of buyer behaviourBand as demonstrated by finding "perhaps" interesting exceptions, or not. Table 2 and 3 are based on data from different countries and time periods yet it is still possible to discern meaningful patterns of buyer behaviour.

We conclude with some very significant results in Table 4. It shows the averages of the top 10 brand results across 9 different product categories in the UK. That is the equivalent of the average row in Tables 2 and 3 but for brand results. The full range of observed performance measures are again closely predictable by the Dirichlet norms but are summarised here for simplicity. The 1999 results yet again demonstrate that the Dirichlet is still a robust model for predicted buyer behaviour nearly two decades after it was developed.




Stand-alone studies can be useful in marketing, especially if exploratory in nature. But replications with or without extensions which might not seem as glamorous and interesting at first sight, are likely to result in better science. Any single studyBno matter how grand, is likely to be subject to some bias or area of uncertainty. Replications can reduce or remove these and help identify boundary conditions on the knowledge we are building.

Dowling and Uncles (1997) have proposed that three exceptions exist to the Dirichlet model. Specifically these are:

1. Super-loyalty brands: some large brands that have more highly loyal users than predicted by Dirichlet;

2. Niche brands: brands that answer to very specific expectations of smaller buyer segments and realise higher purchasing frequencies in them; and

3. Variation brands: brands that are bought for specific situations or moments like alcohol-free beer. They show a much lower purchasing frequency than that predicted by Dirichlet.

It is only through empirical validation and further replications that we as scientist are truly able to determine what the boundaries on the model are. For now we have shown that the model has held in one further applicationBbuying of functionally different product variants. The appendix further extends this finding across fragrances, and flavours. From here we would like to see further replications of this model and specifically this new application including testing of the exception that was noted.

More generally we would like to see replications valued more in marketing, with more researchers conducting them and reviewers and editors accepting them as good science and a solid foundation for the discipline.







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Carver, R. P. (1978), "The case against statistical significance testing," Harvard Educational Review, 48, 378-399.

Dowling, G.R. and M. D. Uncles (1997), "Do customer loyalty programs really work?," Sloan Management Review, summer.

Ehrenberg, Andrew S.C.(1972, 1988, 2nd ed.), Repeat Buying: Facts, Theory and Applications. London: Edward Arnold; New York: Oxford University Press. Reprinted The Journal of Empirical Generalisations in Marketing Science, www.empgens.com.

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Stern, Philip and Andrew S.C. Ehrenberg (1995), "The Market Performance of Pharmaceutical Brands," Marketing and Research Today, November, 285-292.

Stern, Philip and and Andrew S.C. Ehrenberg (1997), "Replication Means Extension," in: 26th European Marketing Academy Conference vol. 4. Warwick Business School, UK: University of Warwick.



Rachel Kennedy, University of South Australia, Australia
Jaywant Singh, South Bank University, U.K.


AP - Asia Pacific Advances in Consumer Research Volume 5 | 2002

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