Assumptions of generalized linear models

Question

On page 232 of "An R companion to applied regression" Fox and Weisberg note

Only the Gaussian family has constant variance, and in all other GLMs the conditional variance of y at $\bf{x}$ depends on $\mu(x)$

Earlier, they note that the conditional variance of the Poisson is $\mu$ and that of the binomial is $\frac{\mu(1-\mu)}{N}$.

For the Gaussian, this is a familiar and often checked assumption (homoscedasticity). Similarly, I often see the conditional variance of the Poisson discussed as an assumption of Poisson regression, together with remedies for cases when it is violated (e.g. negative binomial, zero inflated, etc). Yet I never see the conditional variance for the binomial discussed as an assumption in logistic regression. A little Googling did not find any mention of it.

What am I missing here?

EDIT subsequent to @whuber 's comment:

As suggested I am looking through Hosmer & Lemeshow. It is interesting and I think it shows why I (and perhaps others) are confused. For example, the word "assumption" is not in the index to the book. In addition, we have this (p. 175)

In logistic regression we have to rely primarily on visual assessment, as the distribution of the diagnostics under the hypothesis that the model fits is known only in certain limited settings

They show quite a few plots, but concentrate on scatterplots of various residuals vs the estimated probability. These plots (even for a good model, do not have the "blobby" pattern characteristic of similar plots in OLS regression, and so are harder to judge. Further, they show nothing akin to quantile plots.

In R, plot.lm offers a nice default set of plots to assess models; I do not know of an equivalent for logistic regression, although it may be in some package. This may be because different plots would be needed for each type of model. SAS does offer some plots in PROC LOGISTIC.

This certainly seems to be an area of potential confusion!

Answer 1

These plots (even for a good model, do not have the "blobby" pattern characteristic of similar plots in OLS regression, and so are harder to judge. Further, they show nothing akin to quantile plots.

The DHARMa R package solves this problem by simulating from the fitted model to transform the residuals of any GL(M)M into a standardized space. Once this is done, all regular methods for visually and formally assessing residual problems (e.g. qq plots, overdispersion, heteroskedasticity, autocorrelation) can be applied. See the package vignette for worked-through examples.

Regarding the comment of @Otto_K: if homogenous overdispersion is the only problem, it is probably simpler to use an observational-level random effect, which can be implemented with a standard binomial GLMM. However, I think @PeterFlom was concerned also about heteroskedasticity, i.e. a change of the dispersion parameter with some predictor or model predictions. This will not be picked up / corrected by standard overdispersion checks / corrections, but you can see it in DHARMa residual plots. For correcting it, modelling the dispersion as a function of something else in JAGS or STAN is probably the only way at the moment.

Answer 2

The topic you explain is frequently called overdispersion. In my work I saw a possible solution to such topic:

Using a Bayesian approach, and estimating a Beta-Binomial distribution. This has the great advantage to other distributions (induced by other priors), to have a closed form solution.

References:

• Beta-binomial distribution
• Peter Hoff Bayes estimators notes (pdf)