Information Matrix

I wrote earlier that deviance residuals were better than Pearson residuals when examining a model fit for Poisson counts. It is worth expanding on why this is, since it also neatly illustrates why there are limits to models based on grouped counts.

When fitting a model for Poisson counts, an important step is to check the goodness of fit using the following statistic:

\[\tilde{\chi}^2 = \sum_{i=1}^n r_i^2\]

One of the most important means of checking a model's fit is to look at the residuals, i.e. the standardised differences between the actual data observed and what the model predicts.  One common definition, known as the Pearson residual, is as follows: