Chapter 30
Over- and Underdispersion Models
30.1 Introduction
This chapter explores some count statistical models, which are tied to the phenomenon of over-/equi-/underdispersion. This concerns many fields in applied statistics, including public health, medicine, and epidemiology.
In general, one says overdispersion if the observed variability exceeds the expected variability and underdispersion if it is lower than expected; sometimes, equidispersion means there is no discrepancy between both variabilities. For example, when applying generalized linear models (e.g., [69,73]) with a known scale or dispersion parameter, there are two possible scenarios to examine the quality of the fitting model through the similarity of residual deviance and residual degrees of freedom: standard model diagnostics (e.g., outliers, omitted terms or variables in the linear predictor, incorrect relationship between mean and explanatory variables) and the phenomenon of over-underdispersion (i.e., the empirical variance or variation may be greater/smaller than that predicted by model). The second scenario is of interest to us and it was observed a long time ago in the particular case of the Poisson model [103; see also 32]. Overdispersion considered as a burden by statisticians is most frequent than underdispersion hardly noticed, in several domains of statistics for which we can be inspired by it for clinical trials; see, for example, [2–23, 28–32, 36–40, 65–122].
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