The raw residual is defined as $r_i = y_i - \hat{y}_i$. Why raw residual does not make sense in GLM? Why do we have to standardize it to Pearson residuals?
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1$\begingroup$ It always makes sense as being what it is. Whether it is the best residual for model assessment depends on which GLM you are using. Other kinds of residuals can be better behaved and so more useful. Pearson residuals are one such example, but not always the best and certainly not the only possibility. Substantively, how far the model is off in its predictions on the original scale of the response should be of interest and even important. $\endgroup$– Nick CoxSep 7, 2020 at 8:34
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$\begingroup$ So you mean using raw residuals, how far the model is off the response is not very informative, because of the variance function $V(\mu)$? $\endgroup$– WCMCSep 7, 2020 at 15:59
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$\begingroup$ If anything I mean the opposite. If my model predicts 42 frogs and there were 666, that kind of detail is substantive. Using say residuals that scale for heteroscedasticity as appropriate for a Poisson family and log link. is not in conflict with that simple idea. $\endgroup$– Nick CoxSep 7, 2020 at 16:02
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$\begingroup$ Some related posts: stats.stackexchange.com/questions/283674/…, stats.stackexchange.com/questions/29271/…, stats.stackexchange.com/questions/307221/…, stats.stackexchange.com/questions/374452/… $\endgroup$– kjetil b halvorsen ♦Sep 7, 2020 at 17:07
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$\begingroup$ see also discussion in cran.r-project.org/web/packages/DHARMa/vignettes/DHARMa.html $\endgroup$– Florian HartigSep 8, 2020 at 20:58
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