Marketing Bulletin, 2004, 15, Technical Note 2

Calculation of Theoretical Brand Performance Measuresfrom the Parameters of the Dirichlet Model
Cam Rungie and Gerald Goodhardt
The Dirichlet Model is used in marketing to provide a probability density function for the repeatedpurchases, by shoppers over a period of time, of the competing brands within a product category. The
model is used to analyze a range of measures which report on the performance and competitive statusof the brands. This Research Note provides an overview of the algebra and statistical theory of aspectsof the Dirichlet Model. Specifically, it provides the algebra for using the parameter of the Dirichlet
Model (or estimates) to generate the theoretical brand performance measures. These are marketing
metrics such as market share, penetration, 100% loyals, share of category requirements and others.
This algebra, which is critical to the further development of the Dirichlet Model, has not beenpreviously documented. Program files to calculate the formulas are provided separately for both Excel
and MATLAB.
Keywords: Brand Choice, Buyer Behavior, Choice Models, Data Mining, Marketing Metrics, Dirichlet

Introduction
The Dirichlet Model was developed by Goodhardt, Ehrenberg and Chatfield (Ehrenberg 1959;
Goodhardt, Ehrenberg & Chatfield 1984). It is a model for the patterns of repeat purchases ofthe brands within a product category. It models simultaneously the counts of the number ofpurchases of each brand, over a period of time. It has been shown to be applicable to many
product categories and to have substantial uses, particularly with regards to the analysis ofwhat are known as the ‘brand performance measures’ (Ehrenberg 1988; Ehrenberg & Uncles
1999; Ehrenberg, Uncles & Goodhardt 2003; Uncles, Ehrenberg & Hammond 1995).
Dirichlet Modelling usually involves applying the model to generate estimates of the brandperformance measures including market share, average purchase rate, penetration, purchasefrequency, share of category requirements,100% loyals, average portfolio size and repeat rate.
These estimates are known as the ‘theoreticals’ and are compared to the ‘observeds’ which
are calculated directly from the data (Ehrenberg, Uncles & Goodhardt 2003). The proceduresfor calculating the theoreticals from the parameters of the Dirichlet Model traditionally havebeen considered to be “fairy complex”, which is true (Ehrenberg, Uncles & Goodhardt 2003)
and are not well documented. Consequently, the Dirichlet Model is usually applied throughpackaged software (Hewitt 1990; Kearns 2000; Uncles 1989). However, as the use anddevelopment of the model expands, there is a requirement to document the functional forms,
and their derivations, for calculating the brand performance measures from the parameters of
the Dirichlet Model. This has not been previously done and is presented in this paper.
This Technical Note presents the algebra but not the underlying theoretical structure of themodel. In particular, there is no discussion here of the underlying gamma distributions(Ehrenberg, Uncles & Goodhardt 2003; Goodhardt, Ehrenberg & Chatfield 1984; Rungie &Page 1 of 19

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Marketing Bulletin, 2004, 15, Technical Note 2

Laurent 2003b; Rungie, Laurent & Stern 2003). Also, this Research Note does not discuss
methods for estimating the parameters of the Dirichlet Model (Ehrenberg 1988; Goodhardt,
Ehrenberg & Chatfield 1984; Rungie 2004).
The notation used in this Research Note differs slightly from the notation used in the original
documentation of the model. This has been to move the notation closer to statistical tradition
where parameters are given Greek symbols and to avoid duplication in the use of symbols.
The variations in the notation, where they occur, are noted.
Many of the expressions in this research note use the mathematical function gamma. This isgiven the symbol Γ. It is a function available in Excel as the log of gamma. The Excel
function is GAMMALN. Where it is necessary to convert this to the gamma function thentake the exponential. Thus the Excel gamma function is Γ=EXP(GAMMALN). A propertyof the gamma function is that rΓ(r ) = Γ(r + 1) and r!= Γ(r + 1) . Do not confuse the gammafunction, Γ, with the parameter, γ, nor with the gamma distribution.

The purpose of this Research Note is to document the algebra of the model in a form available
to researchers working with computer programs such as Excel. Many researchers fit themodel to data by using existing packaged software. This is certainly appropriate. However,should the researcher wish to explore the model further then, with this Research Note, it ispossible to access, understand and program the formulas for the theoretical brand performancemeasures.

Probability Density Function
The Dirichlet Model is a probability density function. It specifies the distribution ofpurchases by a population of shoppers of each of the brands within a product category over a
specified period of time. The category might be, for example, detergents, the period of time ayear, the population all households (called shoppers) in France, and the brands the set of about
twenty brands of detergents available in supermarkets in that country. The data in thisexample would record for each household in the sample the purchases of each of the brands
over the specified year. The data is multivariate in that there are several brands. It models thecount of the purchases where one purchase of one SKU counts as one purchase of the relevant
brand, regardless of other characteristics of the SKU which might be, for example, pack size,
product formulation, flavor etc. It is discrete, integer and nonnegative because purchases arecounts which are whole numbers and, by definition, can’t be negative.
The Dirichlet Model is the combination of two probability density functions, the negative
binomial distribution (NBD) and the Dirichlet multinomial distribution (DMD). In the
Dirichlet Model the category purchase rate is assumed to have a NBD over the population of
shoppers. The NBD is derived through specifying that each shopper’s purchases for the
category follow a Poisson process based on a category propensity. Over the population of
shoppers these propensities have a gamma distribution. The purchases of the individual
brands are assumed to have a DMD which is conditional on the category purchase rate. The
DMD is derived through specifying that each shopper’s purchases of the brands (conditionalon the category purchase rate) follow a multinomial process based on brand choiceprobabilities. Over the population of shoppers, these probabilities have a multivariate
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Marketing Bulletin, 2004, 15, Technical Note 2

Dirichlet distribution. The two distributions the NBD and the DMD are assumed to be
independent. In addition their parameters are assumed to have no associations.
Over the population of shoppers let the category purchase rate be a random variable K. (Thisnotation differs from the original notation developed for the Dirichlet Model in which thesymbol K was used for another purpose.) The Dirichlet Model assumes that the categorypurchase rate has a NBD; i.e. K has a NBD. Over the population of shoppers K is a randomvariable for which an individual shopper has an observation category purchase rate of k.
The NBD has two parameters which are both positive:••

the shape1 parameter γ (In the original notation of the Dirichlet Model this parameter
was given the Arabic symbol K. This should not be confused with the randomvariable referred to above as K.)
the scale2 parameter (which also influences the shape) β .

The probability density function for the NBD is (Johnson, Kotz & Kemp 1993):
Equation 1

f γ , β (k ) =Γ(γ + k )βkΓ(γ )k! (1 + β )(γ + k )for k = 0,1,2,...

Let there be h brands. Over the population of shoppers let the purchase rate of each brand bea set of random variables R1,R2,...,Rh. Then, the sum of these purchase rates is the categorypurchase rate; R1+R2+...+Rh = K. The Dirichlet Model assumes that the purchases of eachbrand, conditional on the category purchase rate, have a DMD; i.e. R1,R2,...,Rh , conditional on
K, has a DMD. Over the population of shoppers R1,R2,...,Rh are random variables for which
an individual shopper has observed purchase rates for the brands of r1,r2,...,rh.
The DMD has h parameters, one for each brand. These are α1,α2,...,αh where each is positive.
The probability density function for the DMD is (Johnson, Kotz & Balakrishnan 1997)

Equation 2 hΓ ∑ α j k!
h Γ (α + r )
j =1
j
j

f α1 ,α 2 ,...,α h (r1 , r2 ,..., rh | r1 + r2 + ... + rh = k ) = ∏h j =1 r j !Γ(α j )Γ ∑ α j + k 
 j =1

The Dirichlet Model is a probability density function for the purchases of all brands in a
product category over a period of time. The model combines these two distributions; theNBD and the DMD. The probability density function for the Dirichlet model is given by:
1

Referred to as the K parameter in the original Goodhardt Ehrenberg and Chatfield 1984 paper and often
referred to as the α parameter in the statistical literature.
2
Referred to as the A parameter in the original Goodhardt Ehrenberg and Chatfield 1984 paper and often
referred to as the β or λ parameter in the statistical literature β=1/λ.
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Marketing Bulletin, 2004, 15, Technical Note 2

Equation 3

f γ , β ,α1 ,α 2 ,...,α h (r1 , r2 ,..., rh ) = f γ , β (k ) f α1 ,α 2 ,...,α h (r1 , r2 ,..., rh | r1 + r2 + ... + rh = k )The number of brands in a product category, h, can typically be of the order of 20 to 40 but it
can be as low as two or as high as 150 or more. It is rare for the individual brand alphas to bemuch greater than one and they are often in the range of 0.7 to 0.01 and less. The value of γ isof the order of 0.5 to 1 but it can be higher or lower. It is greater for product categories which
are purchased by many shoppers. The value of β is proportional to the length of the time
period over which the data is collected and can be of the order of 0.1 to 50.

Estimating the Parameters of the Dirichlet Model
The data should be aggregated such that for each shopper and for each brand there is a countwhich is the purchase rate. Shoppers who do not purchase (non buyers) should be included.
If there are h brands and n shoppers there will be nh counts. It is usual to set this data up in atable with n rows (one per shopper) and h columns (one per brand).
There exists a range of methods for estimating the parameters of the Dirichlet Model. The
developers of the model used a combination of the method of zeros and means and themethod of moments (Ehrenberg 1988; Goodhardt, Ehrenberg & Chatfield 1984). This isquick and relatively simple. Likelihood theory provides an alternate method which is also
now relatively easy to apply (Rungie 2004). The original method has the excellent property
that it generates estimates of the market shares for the brands which exactly match theobserved market shares. In addition the original method is well matched to the traditional useof the Dirichlet Model in estimating the brand performance measures. The likelihood method
is more efficient (less sampling variation) and is well matched to extensions of the Dirichlet
Model such as its development into a generalised model (Rungie, Laurent & Stern 2003).
The methods documented and demonstrated in this Research Note do not rely on any one
method of estimation for the parameters of the Dirichlet Model. The model has a set of h+2
parameters which are γ, β, α1, α2,…, αh . In order to use the methods presented here theseparameters can be estimated using the traditional approach, likelihood theory or any otherapproach.

Brand Performance Measures
One of the substantive applications of the Dirichlet Model is the estimation of the brandperformance measures. These are a series of measures which relate to the characteristics ofindividual brands. It is customary to calculate the brand performance measures from the raw
data, these are known as ‘observed’, and then to also estimate the measures from the Dirichlet
Model, these are known as ‘theoretical’ (Ehrenberg, Uncles, & Goodhardt 2003). Eachtheoretical value is a probability statement.

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Marketing Bulletin, 2004, 15, Technical Note 2

Definitions

The bulk of the brand performance measures relate to purchases over a fixed period of timesuch as a month or a year.
Shopper (respondent, panellist, household etc). The unit for which the purchase data isrecorded. The shoppers are all the potential buyers regardless of whether or notthey buy. A shopper can have a purchase rate of zero.
Buyer.

A shopper who makes at least on purchase. Their purchase rate is greater thanzero. An individual shopper may be a buyer for one brand but and not another.

Purchase Rate. The count of the quantity purchased by the shopper over the specified time
period. Each shopper has a separate purchase rate for the category and for eachbrand. The category purchase rate is the sum of the brand purchase rates.
Average Purchase Rate. The purchase rate averaged over all shoppers. There is a separate
average purchase rate for the category and for each brand. The category averagepurchase rate is the sum of the brand average purchase rates.
Penetration3. The proportion of shoppers who are buyers. There is a separate penetration forthe category and for each brand.
Purchase Frequency4. The purchase rate averaged over buyers. The average purchase rate forthose shoppers who buy the brand. There is a separate purchase frequency for the
category and for each brand.
100% Loyals (Sole Buyers)5. For each brand, the proportion of buyers who only buy thebrand.
Share of Category Requirements (SCR). In Dirichlet modelling SCR is defined as:
Total Purchases of brand j
Total Purchases of the category by buyers of brand j
Average Portfolio Size.

Average number of brands purchased by buyers of the category.

3

Referred to as the B, for the category, and b, for a specified brand, in the original Goodhardt Ehrenberg andChatfield 1984 paper.
4
Referred to as the W, for the category, and w, for a specified brand, in the original Goodhardt Ehrenberg andChatfield 1984 paper.
5
The definitions for measures such as 100% loyals and SCR include the shopper who only buys once and so is100% loyal.
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Marketing Bulletin, 2004, 15, Technical Note 2

All of the above definitions refer to purchases over a fixed period of time. Different shoppers
make different numbers of purchases. This is not a characteristic used in the specification of
the remaining brand performance measures.
Repeat Rate. The proportion of buyers of a brand at the last purchase occasion who then buy
the same brand again at the next purchase occasion. A purchase occasion is theevent where the shopper makes a purchase from the category. This is an intuitive
measure of loyalty. It records how much a brand hangs onto its buyers.
Dirichlet S and Category Polarization. Two statistical measure of loyalty which are discussed
below.

Summary of the Formulas for the Theoretical Brand Performance Measures
The derivation the formulas for the theoretical brand performance measures are given in the
appendix
Average Purchase Rate for brand j=

α j γβS

Market Share for brand j

µj =

αj
S

Penetration for brand j∞ Γ(S )Γ(S − α j + k ) = ∑ f γ , β (k )1 − Γ(S + k )Γ(S − α ) 
k =1
j 


Purchase Frequency for brand j=

Average Purchase Rate for brand j
Penetration for brand j

100% Loyals for brand j Γ(S )Γ(α + k ) ∑ f γ β (k ) Γ(S + k )Γ(α ) ∞=

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j

,

j 

Penetration for brand j

k =1

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Marketing Bulletin, 2004, 15, Technical Note 2

Share of Category Requirements for brand j=

Average Purchase Rate for brand j∞ Γ(S )Γ(S − α j + k ) 

1 −
(
)
f
k
k
∑
γ ,β


(
)
(
)
S
k
S
α
Γ
+
Γ
−
k =1
j 


Average Portfolios Sizeh=∑ Penetration for brand j
j =1

Penetration for the category

Repeat Rate for brand jρj =α j +1S +1

Dirichlet Sh

S = ∑α jj =1Category Polarization,ϕ

ϕ = 1/(S+1)
And, where market share = µ, and repeat rate = ρ for any one brand, then

ρ = µ +ϕ - µϕSome of the formulas above specify infinite summations. These can be numericallyapproximated with finite summations. This is discussed in the appendix.

Program Files
Two program files accompany this paper. They will be of assistance to researchers wanting to
explore the brand performance measures further. These are for the advanced user. Those
researchers simply wanting to fit the Dirichlet Model to data would be far better using one of
the established software packages.
Excel

An Excel workbook accompanies this research note. It demonstrates each of the steps inusing estimates of the parameters of the Model to calculate the theoretical brand performancemeasures. The name of the workbook is dirichletbpm.xls.
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Marketing Bulletin, 2004, 15, Technical Note 2

The Excel workbook has three sections:
SectionInputsResults
Detailed working

Work Sheets
Inputs and Results
Inputs and Results
All other work sheets

The workbook contains five worksheets. The function of each sheet is described below.
Introduction
Documentation
Inputs and Results
Enter the estimates of the parameter values in this work sheet. This work sheet
presents the brand performance measures. An example is given in Table 1. Those
measures which have simple closed forms are calculated in this work sheet. Thosethat have more complex forms requiring numerical approximations draw interim
calculations from the three supporting work sheets (1) NA Brand A, (2) NA Brand B
and (3) NA Brand C.
Table 1.

An example of the results provided by the attached Excel file.
Category

INPUTS
Gamma (also known as K)
Beta (also known as A)
Brand Alphas
RESULTS
Market Share
Average Purchase Rate
Penetration
Purchase Frequency
100% Loyals
Share of Category Requirements
Average Portfolio Size
Repeat Rate
Category S (sum of brand alphas)
Category Polarization

BrandsBrand A

Brand B

Brand C

1.10000.50000.4000

55%
0.81
46%
1.77
61%
73%

25%
0.37
24%
1.52
44%
58%

20%
0.29
20%
1.48
42%
56%

70%

50%

47%

2.10000.7000

100%
1.47
67%
2.19

1.34
2.00
0.33

NA Brand A
Numerical Approximations for Brand A. This work sheet generates the interim
calculations for the penetration, 100% loyals and the share of category requirements.
This involves a sum over many values of the category purchase rate, k. Each row is avalue of k starting at k = 1. There must be enough rows to ensure that the summations
are sufficiently accurate. It is clear that successive rows contribute less and less to thesummations and it is, in this case, acceptable to stop the summation after about 100
rows.
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Marketing Bulletin, 2004, 15, Technical Note 2

NA Brand B Numerical Approximations for Brand B.
NA Brand C Numerical Approximations for Brand C.
For extreme values of the parameters it is possible that the numerical approximations in the
Excel file will not be sufficiently accurate. This is because an infinite summation is replacedwith a summation which has 1000 terms. Under these conditions a warning is automaticallygiven.
MATLABThe Excel file provided with this research note does not require the user to have access to
MATLAB. A researcher who wishes to work extensively with the theoretical brandperformance measures will find Excel too limiting. MATLAB is one of many programming
platforms which are more suitable to extensive automation of calculations. A MATLAB filealso accompanies this research note (known as an m file). Unlike the Excel file the MATLAB
file can process any number of brands. Copy the file into the ‘work’ directory within the
MATLAB program directory on your computer. Then, for help on using this file, in
MATLAB, type ‘help dirichletbpm’.
Figure 1.

Help Information for the MATLAB m Filedirichletbpm(parcat,alpha)e.g.
dirichletbpm([1 1],[1 .5 .5])
Function
Calculates
theoretical
brand
performancemeasuresfromtheparameters of the Dirichlet Model.
Inputsparcat[gamma, beta] parameters for the NBD; ie [K A]
alpha[brandalpha1 brandalpha2 ....]
Outputs
Columns
Brands and then category is the final column.
Rows
1Market Share2
Average Purchase Rate
3
Penetration
4
Purchase Frequency
5
100% Loyals
6
Share of Category Requirements
7
Average Portfolio Size (category only)
8
Repeat Rate
9
Brand alphas (i.e. input) and category S
10Category polarization (category only).
Author:Cam
Rungie2003.
UniversityofSouth
Australia.
cam.rungie@unisa.edu.au

The m file as two inputs. Both are vectors. The first, parcat, has two elements which are theparameters of the NBD, γ and β. The second, alpha, contains α1, α1,… αh and has as many
elements as there are brands.
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Marketing Bulletin, 2004, 15, Technical Note 2

In MATLAB type in:dirichletbpm([.21 .7],[1.1 .5 .4])

In this case parcat = [0.21 0.7] and alpha = [1.1 0.5 0.4]. The result is shown in Figure 2. The
output has no print statements because the m file has been written to be a subroutine for otherprogram files. The columns and rows of the results are explained in Figure 1.
Figure 2.

Output from the MATLAB m fileans =0.55000.08090.06351.27330.82500.846000.70001.100000.25000.03670.03081.19390.72600.750900.50000.500000.20000.02940.02491.18070.71070.735800.46670.400001.00000.14700.1054
1.39411.00001.00001.13021.00002.00000.3333For extreme values of the parameters the numerical approximations in the Excel fileautomatically adjust. The infinite summations are estimated using finite summations. The
number of terms in the finite summation will be increased automatically. For extreme valuesof the parameters the execution will be slower.

Conclusions
This Research Note has documented the array of formulas used in the Dirichlet Model; thefundamental probability density functions and the expressions for the brand performancemeasures. The formulas presented here will be of use to researchers who wish to experimentwith and get to know the model by programming it themselves in software such as Excel and
MATLAB.

References
Bass FM, Jeuland A & Wright GP (1976). Equilibrium stochastic choice and market
penetration theories: Derivations and comparisons. Management Science, 22 (10),1051-63.
Brown B, Rungie CM, Laurent G & Rudrapatna S (2003). A Standard Error Estimator for the
Polarization Index: Assessing the Uncertainty in Loyalty. Adelaide, Australia: Schoolof Marketing, University of South Australia.
Chatfield C & Goodhardt GJ (1970). The Beta-Binomial Model for consumer purchasing
behaviour. Journal of the Royal Statistical Society: Series C (Applied Statistics), 19(3), 240-50.
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Marketing Bulletin, 2004, 15, Technical Note 2

Ehrenberg ASC (1959). The pattern of consumer purchases. Applied Statistics, 8, 26-41.
Ehrenberg ASC (1988). Repeat-buying, Facts, Theory and Applications. New York: New
York: Oxford University Press.
Ehrenberg ASC & Uncles MD (1999). Understanding Dirichlet-Type Markets. London:
Centre for Research in Marketing, South Bank University.
Ehrenberg ASC, Uncles MD & Goodhardt GG (2003). Understanding brand performancemeasures: Using Dirichlet benchmarks. Journal of Business Research (forthcoming).
Fader PS & Schmittlein DC (1993). Excess behavioral loyalty for high-share brands:
Deviations from the Dirichlet Model for repeat purchasing. Journal of Marketing
Research, 30 (4 November).
Goodhardt GJ, Ehrenberg ASC & Chatfield C (1984). The Dirichlet: A comprehensive modelof buying behaviour. Journal of the Royal Statistical Society, Series A (General), 147(Part 5), 621-55.
Hewitt I (1990). DIPRED: A Program for Predicting Buyer Behavior from the Dirichlet
Model (Version 2.6). Port Sunlight, UK: Unilever Research Laboratory.
Johnson NL & Kotz S (1972). Distributions in Statistics: Continuous Multivariate
Distributions. New York: John Wiley & Sons, Inc.
Johnson NL, Kotz S & Balakrishnan N (1994a). Continuous Univariate Distributions, Volume
1 (Second ed.). New York: John Wiley & Sons, Inc.
Johnson NL, Kotz S & Balakrishnan N (1994b). Continuous Univariate Distributions,
Volume 2 (Second ed.). New York: John Wiley & Sons.
Johnson NL, Kotz S & Balakrishnan N (1997). Discrete Multivariate Distributions. New
York: John Wiley & Sons, Inc.
Johnson NL, Kotz S & Kemp AW (1993). Univariate Discrete Distributions (Second ed.).
New York: John Wiley & Sons, Inc.
Kalwani MU (1980). Maximum Likelihood Estimation of zero-order models given variable
numbers of purchases per household. Journal of Marketing Research, XVII, 547-51.
Kalwani MU & Morrison DG (1977). A parsimonious description of the Hendry System.
Management Science, 23 (January), 467-77.
Kalwani MU & Morrison DG (1980). Sample size requirements for zero-order models.
Journal of Marketing Research, XVII, 221-27.
Kearns, Zane (2000). Dirichlet No Solver Software. New Zealand: Massey University.
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Marketing Bulletin, 2004, 15, Technical Note 2

Rungie CM (2003). How to estimate the parameters of the Dirichlet Model using Likelihood
Theory in Excel. Marketing Bulletin, 14, Technical Note 1.
http://marketing-bulletin.massey.ac.nz.
Rungie CM & Laurent G (2003a). Repeated Binary Logit (RBL): Fitting Explanatory
Variables to the Beta Binomial Distribution. Adelaide: School of Marketing,
University of South Australia.
Rungie CM & Laurent G (2003b). Repeated Revealed Preference Methods: Dirichlet
Multinomial Regression. Adelaide: School of Marketing, University of South
Australia.
Rungie CM, Laurent G & Stern P (2003)., Modeling Long-Run Dynamic Markets. Adelaide:
School of Marketing, University of South Australia.
Sabavala, Darius J. and Donald G. Morrison (1977), "A Model of TV Show Loyalty," Journalof Advertising Research, 17 (6), 35-43.
Uncles MD (1989). BUYER: Buyer Behavior Software. London: London Business School(refer to web site).
Uncles MD, Ehrenberg ASC & Hammond K (1995). Patterns of buyer behavior: Regularities,
models and extensions. Marketing Science, 14 (3, 2 of 2), G71-G78.
Appreciation
The authors wish to thank Helen Monroe and Carl Driesener for their assistance in thedevelopment of this paper.
Cam Rungie is in the School of Marketing, University of South Australia, and Gerald Goodhardt is in theCentre for Research in Marketing, South Bank University

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Marketing Bulletin, 2004, 15, Technical Note 2

Appendix A: Derivation of the Formulas for Theoretical Brand
Performance Measures
Statistical Properties of the Dirichlet Model

The properties of the NBD and DMD are useful in developing applications for the Dirichlet
Model (Ehrenberg 1959; Ehrenberg 1988; Goodhardt, Ehrenberg & Chatfield 1984; Johnson& Kotz 1972; Johnson, Kotz & Balakrishnan 1994a; Johnson, Kotz & Balakrishnan 1994b;
Johnson, Kotz & Balakrishnan 1997; Johnson, Kotz & Kemp 1993).
NBD

E [K ] = γβ = average purchase rate for the category1
Pr (K = 0) =(1 + β )γDMD[] αS k = average purchase rate for brand j

E Rj | k =jMarginal DMD: If R ′j = K − R j then R j and R ′j jointly also have a DMD withparameters α j and S − α j .
Thus
Γ(S )Γ(S − α j + k )Pr (R j = 0 | k ) =Γ(S + k )Γ(S − α j )Pr (R j = k | k ) =Γ(S )Γ(α j + k )Γ(S + k )Γ(α j )Performance Measures
Category Purchase Rate

Average Purchase Rate for the category = E[K ] = γβPenetration for the category = Pr (K > 0) = 1 − Pr (K = 0) = 1 −Page 13 of 19

1(1 + β )γ

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Marketing Bulletin, 2004, 15, Technical Note 2

Purchase Frequency for the category=

Average Purchase Rate for the category
Penetration for the category

=

E[K ]
Pr (K > 0)

Restatements of the Definitions of the Brand performance Measures

The definitions for the brand performance measures can be rewritten as probability
statements:[ ]Average Purchase Rate for brand j = E R j
Penetration for brand j = Pr (R j > 0 )Purchase Frequency for brand j==

Average Purchase Rate for brand j
Penetration for brand j

[ ]

E Rj

Pr (R j > 0 )100% Loyals for brand j==

Probability of a shopper buying brand j only
Probability of a shopper buying brand j
Pr (R j = K > 0 )Pr (R j > 0 )Define a new variable:
K ′ = K if R j > 0otherwise K ′ = 0

Share of Category Requirements for brand j=

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[ ]

E Rj

E[K ′]

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Marketing Bulletin, 2004, 15, Technical Note 2

Results Conditional on Category Purchase Rate

In order to examine the brand performance measures for a single brand it is useful to firstconsider the case where the category purchase rate is fixed; K = k. These are the theoreticalbrand performance measures for brand j conditional on the category purchase rate being k.

] αSk

[

Average Purchase Rate = E R j | k =[E Rj | k]=αMarket Share=

Penetration= Pr(R j > 0 | k )

k

j

j

S

= 1 − Pr(R j = 0 | k )= 1−

Γ(S )Γ(S − α j + k )Γ(S + k )Γ(S − α j )100% Loyal interim calculation
Pr (R j = k | k ) =Γ(S )Γ(α j + k )Γ(S + k )Γ(α j )Share of Category Requirements interim calculation
E [K | k ] = k
Therefore Γ(S )Γ(S − α j + k ) E [K ′ | k ] = k Pr (R j > 0 | k ) = k 1 −
 Γ(S + k )Γ(S − α ) j Results Incorporating the Distribution for the Category Purchase Rate

These conditional results can be converted to results which are not conditional on k by mixing
over the probability density function for the category purchase rate; i.e. by mixing over the
NBD. These are the unconditional results. These are the theoretical brand performancemeasures for brand j.

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Marketing Bulletin, 2004, 15, Technical Note 2

In some cases mixing over the NBD simplifies to a closed form; e.g.

[ ]

[

∞

] α SγβAverage Purchase Rate for brand j = E R j = ∑ f γ , β (k )E R j | k =∞Market Share for brand j = ∑ f γ , β (k )k =1[

E Rj | k
k

k =1

]=α

Alternatively, Market Share = Pr (R j = 1 | k = 1) =

j

j

S

αj
S

However, for the remaining cases the expressions are infinite sums which can be numericallyestimated.
Penetration for brand j= Pr (R j > 0)∞

= ∑ f γ ,β (k ) Pr (R j > 0 | k )k =1∞
 Γ(S )Γ(S − α j + k ) = ∑ f γ ,β (k )1 − Γ(S + k )Γ(S − α j ) 
k =1



Also
Purchase Frequency for brand j
=

[ ]

E Rj

Pr (R j > 0 )100% Loyals for brand j=

Pr (R j = K > 0)
Pr (R j > 0)∞

∑ f γ β (k ) Pr (R = k | k )=Pr (R > 0 ) Γ(S )Γ(α + k ) ∑ f γ β (k ) Γ(S + k )Γ(α ) =
Pr (R > 0 )
k =1

j

,

j

∞

k =1

j

,

j

j

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Marketing Bulletin, 2004, 15, Technical Note 2

Share of Category Requirements for brand j
=

[ ]

E Rj

E [K ′]

[ ]

E Rj

= ∞

∑ f γ β (k )E[K ′ | k ]
k =1

=

,

[ ]

E Rj

 Γ(S )Γ(S − α + k ) ∑ f γ β (k )k 1 − Γ(S + k )Γ(S − α ) 
∞

k =1

j

,





j

The above formulae require numerical approximations because it is not possible to sum toinfinity. It is necessary to sum over values for the category purchase rates starting from one
and ranging through to an upper limit ks which is large enough to generate accurate estimates.
ks

A sensible approach is to select ks such that ∑ f γ ,β (k ) > 0.99999 . This is equivalent to
k =1

keeping summing from k = 0 until a level of k is reached where the brand performancemeasures will not change.
Other Results Not Reliant on Distribution of the Category Purchase Rate

Average Portfolio Size
Define a new variable such that:R ′j = 1 if R j > 0otherwise R ′j = 0

Then, as an alternate specification,
Penetration for brand j[ ]= E R ′jAlso the average portfolio size amongst all shoppers= E [R1′ + R2′ + ... + Rh′ ]= E [R1′ ] + E[R2′ ] + ... + E [Rh′ ]h= ∑ Penetration for brand j
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Marketing Bulletin, 2004, 15, Technical Note 2

Finally, the average portfolio size for all buyers=

Average Portfolio Size for all shoppers
Category penetration

Thus, Average Portfolios size over all buyersh∑ Penetration for brand j=

j =1

Penetration for the category

Repeat Rate
Repeat Rate for brand j is defined to be=

Pr (R j = 2 | k = 2 )Pr (R j = 1 | k = 1)

Using the probability density function for the DMD:
Repeat Rate for brand j
α j +1=S +1
Polarization
Repeat rate is an intuitive measure of loyalty. It is a measure of how much a
brand hangs onto its customers. However, repeat rate is confounded withmarket share. Generally, the larger the market share the larger the repeat rate.
Polarization is a measure of loyalty which is the repeat rate standardized forthe market share.
Define three new symbols:(1) Let the repeat rate for brand j be ρj(2) Let the market share for brand j to be µj where it is shown abovethat µj =αj/S(3) Let polarization, ϕ be ϕ =1/(S+1). Then 0< ϕ <1
Then ρj = µj +ϕ - µjϕ
Without loss of generalization this can be written asρ = µ +ϕ - µϕ

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Marketing Bulletin, 2004, 15, Technical Note 2Thus, the repeat rate is a function of market share and polarization. The larger
the market share the larger the repeat rate.
Also the larger the polarization, ϕ, the greater the repeat rate and the greater
the loyalty. When ϕ = 1 there is complete loyalty. When ϕ = 0 there is no
loyalty(Brown et al. 2003; Rungie and Laurent 2003a).
Also

ϕ = (ρ – µ)/(1 - µ)Thus, Polarization is the repeat rate standardized for the market share.
Polarization φ is first described as a characteristic of the beta distribution by Sabavala andMorrison (Sabavala & Morrison 1977), and is further investigated by Kalwani and Morrison(Kalwani 1980; Kalwani & Morrison 1980). It relates to other measures of loyalty including
Hendry’s k (Kalwani & Morrison 1977), Bass’s θ (Bass, Jeuland & Wright 1976), Chatfieldand Goodhardt’s work on the beta binomial distribution (Chatfield & Goodhardt 1970) andGoodhardt and Ehrenberg’s Dirichlet S statistic (Goodhardt, Ehrenberg & Chatfield 1984).
Under Dirichlet Model conditions φ= θ = 1/(1+S)=1-k. Note that, under traditional Dirichlet
Modeling, there is one polarization for the whole category; ϕ is constant across brands.
However, in other traditions of modelling, polarization generally refers to choices between
two alternatives. In this manner it is particularly useful as it can describes loyalty in themarginal choice between one specified brand and all others in the category. Thus, there can be
one polarization for the category and one for each brand. Both can be estimated by using
likelihood maximization based on the DMD applied first to all brands and then separately toeach brand (the data is the marginal choice between the brand and all other brands combined).
This supplies a method of evaluating one condition of the Dirichlet Model, that ϕ and loyalty
is constant across brands. Also, it is an excellent method for the study of deviations from themodel in the form of variations in loyalty and of excess loyalty for big brands (Fader &Schmittlein 1993).

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