Overdispersion

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A common task in applied statistics is coming up with a parametric model of a given set of empirical observations. This necessitates an assessment of the fit of the chosen model. It is usually possible to choose the model parameters in such a way that the theoretical population mean of the model is approximately equal to the sample mean. However, especially for simple models with few parameters, theoretical predictions may not match empirical observations for higher moments. When the observed variance is higher than the variance of a theoretical model, overdispersion has occurred. Conversely, underdispersion means that there was less variation in the data than predicted.

Overdispersion is often encountered when fitting very simple parametric models, such as those based on the Poisson distribution. The Poisson distribution has one free parameter and does not allow for the variance to be adjusted independently of the mean. The choice of a distribution from the Poisson family is often dictated by the nature of the empirical data. For example, Poisson regression analysis is commonly used to model count data. If overdispersion is a problem, an alternative model with additional free parameters may provide a better fit. In the case of the Poisson distribution, a Poisson mixture model like the negative binomial distribution can be used instead.