4.3 Extra-Poisson variation

Overdispersion occurs when the observed random variation is greater than the expected random variation ${\mu }_i$ under the Poisson model

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  a widely used diagnostic for the goodness of fit of a Poisson regression model is

  $$X^2=\sum _{i=1}^n\frac{{(Y_i-{\widehat{\mu }}_i)}^2}{{\widehat{\mu }}_i}$$

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    for a well-fitting model, $X^2\sim {\chi }_{n-p}^2$ where $p$ is the number of fitted regression parameters;

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    When $X^2\gg n-p$, nominal standard errors from the Poisson regression model are too small

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  Provided the $Y_i$ are independent, approximately valid inferences about $\beta$ are obtained if we multiply nominal standard errors by a factor $\sqrt{X^2/(n-p)}$.

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  But in the spatial setting, independence is not guaranteed. This is discussed further in the next section.