NBDdirichlet (version 1.3)

# dirichlet: Estimation of the Dirichlet model

## Description

Given consumer purchase summary data, it estimates the parameters of the Dirichlet model, which describes the consumer repeat-buying behavior of branded products. It also returns with several probability functions for users to calculate various theoretical quantities.

## Usage

dirichlet(cat.pen, cat.buyrate, brand.share, brand.pen.obs,
brand.name = NA, cat.pur.var = NA, nstar = 50,
max.S = 30, max.K = 30, check = F)

## Arguments

cat.pen
Product category penetration, which is the observed proportion of category buyers over a specific time period.
Category buyers' average purchase rate in a given period. This is derived as the total number of category purchase occasions divided by the total number of category buyers during a time period.
brand.share
A vector of brand market share. We typically define it as the proportions of purchase occasions that belong to different brands during the time period.
brand.pen.obs
A vector of observed brand penetration, which is the proportion of buyers for each brand during the time period.
brand.name
A character vector of the brand names. If not given (default), use "B1", "B2", etc.
cat.pur.var
The observed variance of category purchase rates across individuals. It is used for the method of moment estimation of the parameter K in the Dirichlet model. If it is not given (default), then estimate K by "mean and zeros"(see reference).
nstar
Maximum number of category purchases in the time period considered in the calculation. Any number larger than nstar is assumed to have occurred with probability zero. By default, it is 50. For higher frequently purchased category and longer study time period, it is necessary to increase nstar to the level (say, 100, 300, etc.) where $\sum_{n=1}^{nstar} P(n) > 0.9999$. We did not use the truncation procedure (suggested by the reference authors) in order to simplify coding.
max.S
Upper bound for the model parameter S in the optimization procedure to solve for S. Default to 30.
max.K
Upper bound for the model parameter K in the optimization procedure to solve for K. Default to 30.
check
A logical value. If T, print some diagnostic information. Defaul to F.

## Value

M
Estimated Dirichlet model parameter: mean purchase rate of the category.
K
Estimated Dirichlet model parameter: it measures the diversity of the overal category purchase frequency (smaller K implies more diversity).
S
Estimated Dirichlet model parameter: it measures the diversity of the brand purchase propensity (smaller S implies more diversity).
nbrand
Number of brands being considered in the produt category.
nstar
Input parameter: Maximum number of category purchases considered.
cat.pen
Input parameter: Category penetration in a given time period.
Input parameter: Category buyers' average purchase rate in a given time period.
brand.share
Input parameter: A vector of brand market share.
brand.pen.obs
Input parameter: A vector of observed brand penetration.
brand.name
Input parameter: A character vector of the brand names.
check
A logical flag that indicates whether to print the intermediate information in the model estimation. Default to F.
error
A logical flag that indicates if nstar is too small to sufficently cover the support of category purchase probabilities (calculated by Pn, see below). If it is returned T, then nstar should be increased and the Dirichlet model be re-estimated.
period.set
A member function of the "dirichlet" class object with one required parameter (t), which can be any positive real number. It resets the study time period to be t times of the assumed base time period in the sample.
period.print
A member function of the "dirichlet" class object with no parameter. It indicates the current time period by printing the multiple t of the base time period.
p.rj.n
A member function of the "dirichlet" class object with three required parameters (r_j, n, j). It calculates the conditional probability of buying brand j for exactly r_j times given that the consumer has made n category purchases.
Pn
A member function of the "dirichlet" class object with one required parameter (n). It calculates the probability that a consumer has made n category purchases in the study time period.
brand.pen
A member function of the "dirichlet" class object with one required and one optional parameter (j, limit=c(0:nstar)). It calculates the probability that a consumer makes at least one purchase of the brand j (theoretical penetration) in the study time period. The optional vector limit enumerates the exact frequencies that a consumer will be buying in the category and is used to index the summation of the probabilities of not buying brand j given those category purchases in limit.
A member function of the "dirichlet" class object with one required and one optional parameter (j, limit=c(0:nstar)). It calculates the expected number of brand j purchases given that the consumer is a buyer of the brand j in the time period (theoretical brand buying rate). The limit parameter has the same meaning as that in the function brand.pen.
wp
A member function of the "dirichlet" class object with one required and one optional parameter (j, limit=c(0:nstar)). It calculates the expected number of the product category purchases given that the consumer is a buyer of the brand j in the time period (theoretical category buying rate for brand buyer). The limit parameter has the same meaning as that in the function brand.pen.

## Details

The Dirichlet model and its estimation can be found in the reference paper. It is found to fit and reproduce the patterns of repeat buying of branded products quite well. Specifically, the dirichlet model is a mixture of distributions at four levels:

1. Each consumer's purchase incidences in a product category follow the Poisson process.

• The purchase rates of the category by different consumers follow a Gamma distribution.
• Each consumer's choices among the available brands follow a multinomial distribution, and
• These brand choice probabilities follow a multivariate Beta or "Dirichlet" distribution across different consumers.
• There are three structural parameters to be estimated:

M
Mean purchase rate of the category.

K
Measures the diversity of the overal category purchase frequency across consumers (smaller K implies more diversity).

S
Measures the diversity of the brand purchase propensity across consumers (smaller S implies more diversity).

To estimate M and K, we use the observed category penetration (cat.pen) and purchase rate (cat.buyrate). To estimate S, we use additionally the observed brand penetrations (brand.pen.obs) and brand market shares (brand.share). Note however once these three parameters are estimated, only the brand market shares are needed by the Dirichlet model to compute various repeat-buying theoretical statistics.

The estimated parameters, along with several probability functions that can access the object data, are passed back in a list, which is assigned a "dirichlet" class attribute. The result can be used by the print.dirichlet, summary.dirichlet, and plot.dirichlet method.

The study period (where we report the model result) is assumed to be 4 times of the observation period (input data). So if we use quarterly data, the model output is annulized. This multiple (4) can be changed using the member function period.set.

## References

The Dirichlet: A Comprehensive Model of Buying Behavior. G.J. Goodhardt, A.S.C. Ehrenberg, C. Chatfield. Journal of the Royal Statistical Society. Series A (General), Vol. 147, No. 5 (1984), pp. 621-655

print.dirichlet, summary.dirichlet, plot.dirichlet, NBDdirichlet-package

## Examples

# The following data comes from the example in section 3 of
# the reference paper.  They are Toothpaste purchase data in UK
# in 1st quarter of 1973 from the AGB panel (5240 static panelists).

cat.pen <- 0.56 # Category Penetration
cat.buyrate <- 2.6 # Category Buyer's Average Purchase Rate in a given period.
brand.share <- c(0.25, 0.19, 0.1, 0.1, 0.09, 0.08, 0.03, 0.02) # Brands' Market Share
brand.pen.obs <- c(0.2,0.17,0.09,0.08,0.08,0.07,0.03,0.02) # Brand Penetration
brand.name <- c("Colgate DC", "Macleans","Close Up","Signal","ultrabrite",
"Gibbs SR","Boots Priv. Label","Sainsbury Priv. Lab.")

dobj <- dirichlet(cat.pen, cat.buyrate, brand.share, brand.pen.obs, brand.name)
print(dobj)