# Overdispersion

A common task in applied statistics is choosing a parametric model to fit 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 a very common feature in applied data analysis because in practice, populations are frequently heterogeneous.

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 feature, an alternative model with additional free parameters may provide a better fit. In the case of the count data, a Poisson mixture model like the negative binomial distribution can be used instead where the mean of the Poisson distribution can itself be thought of as a random variable drawn - in this case - from the gamma distribution thereby introducing an additional free parameter (note the resulting negative binomial distribution has 2 parameters).

As a concrete example, it has been observed that the random number of boys born to each family do not - as might be expected - conform faithfully to a Binomial distribution. Instead, each family seems to skew the sex ratio of their children in favor of either boys or girls (see, for example the Trivers-Willard hypothesis for one possible explanation) i.e. there are too many all boy families, too many all girls families, and not enough families close to the population 51:49 boy-to-girl mean ratio thereby yielding an estimated variance that is larger than predicted by the binomial model. In this case, the beta-binomial model is a popular and perhaps theoretically defensible alternative to the binomial that captures the overdispersion absent from the binomial model thereby providing a better fit to the observed data. To capture the heterogeneity of the families, one can think of the p parameter (proportion of boys) in the binomial model as itself a random variable (i.e. random effects model) drawn for each family from a beta distribution as the mixing distribution. The resulting compound distribution (Beta-Binomial) has an additional free parameter.

Over- and underdispersion are terms which have been adopted in branches of the biological sciences. In parasitology, the term 'overdispersion' is used as defined here - meaning a distribution with a higher than expected variance. In some areas of ecology, however, meanings have been transposed, so that overdispersion is actually taken to mean more even (lower variance) than expected. This confusion has caused some ecologists (e.g. Greig-Smith, Quantitative Plant Ecology, editions 1957, 1964, 1983) to suggest that the terms 'aggregated', or 'contagious', would be better used in ecology for 'overdispersed'. This suggestion has not been adopted, and confusion persists in the literature. Furthermore in demography overdispersion is often evident in the analysis of death count data, in this case demographers refer to it with the term of unobserved heterogeneity.

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