McCullagh and Nelder's book on glm suggest to plot standardized deviance residuals against either the linear predictor ($\hat{\eta}$) or the fitted values ($\hat{\mu}$) transformed to the constant information scale which for poisson errors is $2\sqrt{\hat{\mu}}$, for model checking. Why most of the examples show a plot of deviance residuals vs untransformed fitted values $(\hat{\mu})$?

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    $\begingroup$ Some guesses: People haven't read the book. The kind of plot you see seems simpler or is more obviously available in software. $\endgroup$ – Nick Cox Apr 11 '16 at 13:32
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    $\begingroup$ The aim of this recipe is to back-transform the Poisson to a homogeneous scale, but this approach will not work well for small counts - the simulation-based approach in the DHARMa R package provides an exact solution to the problem, see cran.r-project.org/web/packages/DHARMa/vignettes/DHARMa.html $\endgroup$ – Florian Hartig Jan 5 '17 at 12:15

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