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From my understanding, the deviance residual of a GLM model, when plotted against the fitted values, should give a scatterplot distributed with mean 0 and constant variance? Does this hold for any GLM family with any link function? I am mostly interested in Gamma GLM with the identity link function right now.

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Deviance residuals will not in general have 0 mean; they don't for Gamma models.

However the mean deviance residual tends to be reasonably close to 0.

Here's an example of a residual plot from a simple identity link gamma fit (to simulated data for which the model was appropriate; in this case the shape parameter of the gamma was 3):

![enter image description here

The plot on the left is a typical deviance residuals vs fitted type plot. The one on the right splits the fitted values into bins so we can use boxplots to help judge whether the spread is near constant; the 0 line is marked in red.

As you can see from the boxplots, judging from the IQR, the spread is pretty much constant (with some random variation at the right where there are few values), but the medians there are consistently below 0. We can see that (in this case) the deviance residuals appear to be close to symmetric.

The mean deviance residual for this model is -0.1126, (marked in blue) which is very close to where those marked medians are sitting. With such a big sample, this mean is many standard errors from 0, but the mean is still "near" 0 (in the sense that the standard deviation of the residuals is more than 5 times larger than 0.1126).

Based on simulations, it looks like (as long as n is large and the shape parameter is not too small) the average deviance residual for a Gamma will be about $-\frac{1}{3\alpha}$, where $\alpha$ is the common shape parameter for the gamma-distributed response. The relationship comes in fairly well by about $\alpha=2$, but much below that it tends to overestimate.

In summary: the mean deviance residual should be close to constant, with close to constant variance, but the mean of the deviance residuals should be "near" 0 rather than 0.

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I am going to preface this statement with I am no statistician (I can understand and apply statistical concepts) and I an no GLM expert. From my understanding, GLMs follow the same assumptions of linear models. If the residuals deviate from the fitted values in an uniform way it would indicate that the model is either biased (or unbiased) and heteroscedastic(or homoscedastic). Therefore Yes, these should apply to all GLM link functions (I think).

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  • $\begingroup$ Can you explain what you mean by residuals deviating from fitted values? $\endgroup$ – Glen_b Aug 16 '15 at 23:44
  • $\begingroup$ One of the best graphic I have seen was this Residual Plots. Also while this focuses on linear models I think the same can apply for GLMs R-Blog. Hope this helps. $\endgroup$ – Paul Julian Aug 17 '15 at 0:00
  • $\begingroup$ I'm afraid that doesn't really answer the question. $\endgroup$ – Glen_b Aug 17 '15 at 0:07
  • $\begingroup$ @Glen_b --sorry I just realised the link I provided to the residual plots didn't have any axis labels. These plots are of residual versus fitted values. Ideally the residuals would be spaced somewhat uniformly along the fitted values (i.e. top- left plot of the link above and here). As residual values deviate from the fitted the points can go all different directions (i.e. all other plots). ...hope this was closer to what you were expecting. $\endgroup$ – Paul Julian Aug 17 '15 at 0:16
  • $\begingroup$ I still don't see what you mean by "residual values deviate from the fitted" in that phrase. How are you measuring "deviation" between residuals and fitted (and why?). Do you actually mean something like "residual values deviate from 0 as fitted values change"? $\endgroup$ – Glen_b Aug 17 '15 at 0:31

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