Minding our P's, Q's and R's

(Mar 22, 2016)

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:

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

where \(r_i\) represents the residual for the \(i^{\rm th}\) Poisson count, usually a cell in a contingency table with \(n\) such cells.  If the model is correct, the residuals \(\{r_i\}\) are usually assumed to be values drawn from the N(0,1) distribution.  This in turn means that \(\{r_i^2\}\) are values…

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Tags: Pearson residuals, deviance residuals, Poisson, quantile-quantile plot

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