Überdispersion

Überdispersion, also called overdispersion, is a statistical phenomenon in count data where the observed variance exceeds the mean, violating the Poisson assumption that Var(Y) = E[Y]. In practice, data are more variable than the Poisson model allows, leading to unreliable inferences if standard Poisson models are used.

Causes of überdispersion include unobserved heterogeneity (differences between subjects not captured by the model), clustering or grouping effects, varying exposure or at-risk time, and general model misspecification. Zero inflation (an excess of zero counts) can also contribute to apparent overdispersion. The effect is common in fields such as epidemiology, ecology, and social sciences.

Implications and diagnostics: under the Poisson framework, overdispersion causes standard errors and test statistics to be biased, typically yielding overly optimistic p-values. Diagnostics often use a dispersion parameter φ in the variance function, or assess the ratio of the Pearson chi-square statistic to its degrees of freedom. Other approaches include deviance-based measures and tests such as the Cameron–Trivedi test for overdispersion.

Modeling remedies: several approaches address überdispersion. The quasi-Poisson model uses Var(Y) = φ μ to scale the variance. The negative binomial model specifies Var(Y) = μ + α μ^2, capturing extra-Poisson variation through a dispersion parameter. Zero-inflated or hurdle models handle excess zeros, while mixed-effects or random-effects models address clustering. Robust or sandwich standard errors provide protection when variance is misspecified. Model comparison and validation should rely on information criteria and, when possible, likelihood-based tests.

Terminology: the term is widely used in German-speaking literature and among practitioners; in English, overdispersion is the common label, with überdispersion recognizing the same phenomenon.