The Beta‐binomial Model: Accounting for Inter‐trial Variation in Replicated Difference and Preference Tests

Abstract

Binomial tests are commonly used in sensory difference and preference testing under the assumptions that choices are independent and choice probabilities do not vary from trial to trial. This paper addresses violations of the latter assumption (often referred to as overdispersion) and accounts for variation in inter‐trial choice probabilities following the Beta distribution. Such variation could arise as a result of differences in test substrate from trial to trial, differences in sensory acuity among subjects or the existence of latent preference segments. In fact, it is likely that overdispersion occurs ubiquitously in product testing. Using the Binomial model for data in which there is inter‐trial variation may lead to seriously misleading conclusions from a sensory difference or preference test. A simulation study in this paper based on product testing experience showed that when using a Binomial model for overdispersed Binomial data, Type I error may be 0.44 for a Binomial test specification corresponding to a level of 0.05. Underestimation of Type I error using the Binomial model may seriously undermine legal claims of product superiority in situations where overdispersion occurs. The Beta‐Binomial (BB) model, an extension of the Binomial distribution, was developed to fit overdispersed Binomial data. Procedures for estimating and testing the parameters as well as testing for goodness of fit are discussed. Procedures for determining sample size and for calculating estimate precision and test power based on the BB model are given. Numerical examples and simulation results are also given in the paper. The BB model should improve the validity of sensory difference and preference testing.

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The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion

“…In general, random generation from the Beta-Binomial distribution is used as a standard method to simulate overdispersed Binomial variables (See for example: Ennis & Bi, 1998). However, since our present study focuses on comparing three Binomial mixture distributions including the Beta-Binomial distribution in handling overdispersed Binomial data, there is a suspicion that perhaps the results may be influenced towards Beta-Binomial distribution.…”

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“…The responses of one subject are not independent from each other. If subjects are not acting in an identical fashion, the variance due to differences in between-person and within-person is explained by the beta distribution (Ennis & Bi, 1998). This variability is known as overdispersion, which is measured by gamma (c), a value that ranges from 0 to 1.…”

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