THE DIRICHLET‐MULTINOMIAL MODEL: ACCOUNTING FOR INTER‐TRIAL VARIATION IN REPLICATED RATINGS

Differences in sensory acuity and hedonic reactions to products lead to latent groups in pooled ratings data. Manufacturing locations and time differences also are sources of rating heterogeneity. Intensity and hedonic ratings are ordered categorical data. Categorical responses follow a multinomial distribution and this distribution can be applied to pooled data over trials if the multinomial probabilities are constant from trial to trial. The common test statistic used for comparing vectors of proportions or frequencies is the Pearson chi-square statistic. When ratings data are obtained from repeated ratings experiments or from a cluster sampling procedure, the covariance matrix for the vector of category proportions can differ dramatically from the one assumed for the multinomial model because of inter-trial. This effect is referred to as overdispersion. The standard multinomial model does not fit overdispersed multinomial data. The practical implication of this is that an inflated Type I error can result in a seriously erroneous conclusion. Another implication is that overdispersion is a measurable quantity that may be of interest because it can be used to signal the presence of latent segments. The Dirichlet-Multinomial (DM) model is introduced in this paper to fit overdispersed intensity and hedonic ratings data. Methods for estimating the parameters of the DM model and the test statistics based on them to test against a specified vector or compare vectors of proportions are given. A novel theoretical contribution of this paper is a method for calculating the power of the tests. This method is useful both in evaluating the tests and determining sample size and the number of trials. A test for goodness of fit of the multinomial model against the DM model is also given. The DM model can be extended further to the Generalized Dirichlet-Multinomial (GDM) model, in which multiple sources of variation are considered. The GDM model and its applications are discussed in this paper. Applications of the DM and GDM models in sensory and consumer research are illustrated using numerical examples.

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