Market shares and brand loyalty

Sometimes measurements of substances in food are available at a more detailed food coding level than consumption data. For example, measurements may have been made for specific brands of a food whereas the consumption survey did not record the brand. MCRA allows to specify market share data for subtypes of a food (e.g. A$1, A$2, A$3 are three brands of food A), and to calculate acute exposure based on such market shares.

For chronic assessments brand loyalty should be specified according to a simple Dirichlet model (Goodhardt et al. (1984)). Technically, the Dirichlet model for brand choice needs nbrand parameters \(\alpha_{i}\) (which should be positive real numbers). The average brand choice probability for each brand is

\[\alpha_{i}/S\]

where

\[S = \sum \alpha_{i}\]

By definition, the market shares \(m_{i}\) should be proportional to the brand choice probabilities, and thus to the parameters \(\alpha_{i}\). Thus means that \(S\), the sum of the alphas, is the only additional parameter that should be specified, and indeed this is the parameter that determines brand loyalty. \(S=0\) corresponds to absolute brand loyalty, and brand loyalty decreases with increasing \(S\). We define \(L = (1+S)^{-1}\) as an interpretable brand loyalty parameter, where now \(L = 0\) and \(L = 1\) correspond to the situations of no brand loyalty and absolute brand loyalty, respectively.

The multinomial distribution models the probability of counts and is a generalization of the binomial distribution. The Dirichlet does the opposite, it models for a number of counts the distribution of probabilities, so for numbers \(\alpha_{1} = x_{1},...,\alpha_{k} = x_{k}\) the distribution of \(m_{1},...,m_{k}\) is modelled with \(m_{k}\) the marketshare for brand k.

Given empirical or parametric distributions of consumption and concentration values, the algorithm for chronic exposure assessment now operates as follows:

  1. Simulate consumptions for \(n\) individual(day)s.

  2. Simulate \(n\) selection probabilities from the Dirichlet distribution.

  3. For each individual, simulate \(d\) brand choices from a multinomial distribution using the individual specific selection probabilities from step 2.

  4. For all individuals and days simulate values from the appropriate concentration distribution.

  5. Multiply consumption with concentration to obtain exposure.