What are the best methods for checking a generalized linear mixed model (GLMM) for proper fit?
This question comes up frequently.
Unfortunately, it isn’t as straightforward as it is for a general linear model.
In linear models the requirements are easy to outline: linear in the parameters, normally distributed and independent residuals, and homogeneity of variance (that is, similar variance at all values of all predictors).
For linear models, there are well-described and well-implemented methods for checking each of these, both visual/descriptive methods and statistical tests.
It is not nearly as easy for GLMMs.
Assumption: Random effects come from a normal distribution
Let’s start with one of the more familiar elements of GLMMs, which is related to the random effects. There is an assumption that random effects—both intercepts and slopes—are normally distributed.
These are relatively easy to export to a data set in most statistical software (including SAS and R). Personally, I much prefer visual methods of checking for normal distributions, and typically go right to making histograms or normal probability plots (Q-Q plots) of each of the random effects.
If the histograms look roughly bell-shaped and symmetric, or the Q-Q plots generally fall close to a diagonal line, I usually consider this to be good enough.
If the random effects are not reasonably normally distributed, however, there are not simple remedies. In a general linear model outcomes can be transformed. In GLMMs they cannot.
Research is currently being conducted on the consequences of mis-specifying the distribution of random effects in GLMMs. (Outliers, of course, can be handled the same way as in generalized linear models—except that an entire random subject, as opposed to a single observation, may be examined.)
Assumption: The chosen link function is appropriate
Additional assumptions of GLMMs are more related to the generalized linear model side. One of these is the relationship of the numeric predictors to the parameter of interest, which is determined by the link function.
For both generalized linear models and GLMMs, it is important to understand that the most typical link functions (e.g., the logit for binomial data, the log for Poisson data) are not guaranteed to be a good representation of the relationship of the predictors with the outcomes.
Checking this assumption can become quite complicated as models become more crowded with fixed and random effects.
One relatively simple (though not perfect) way to approach this is to compare the predicted values to the actual outcomes.
With most GLMMs, it is best to compare averages of outcomes to predicted values. For example, with binomial models, one could take all of the values with predicted values near 0.5, 0.15, 0.25, etc., and average the actual outcomes (the 0s and 1s). You can then plot these average values against the predicted values.
If the general form of the model is correct, the differences between the predicted values and the averaged actual values will be small. (Of course how small depends on the number of observations and variance function).
No “patterns” in these differences should be obvious.
This is similar to the idea of the Hosmer-Lemeshow test for logistic regression models. If you suspect that the form of the link function is not correct, there are remedies. Possibilites include changing the link function, transforming numeric predictors, or (if necessary) categorizing continuous predictors.
Assumption: Appropriate estimation of variance
Finally, it is important to check the variability of the outcomes. This is also not as easy as it is for linear models, since the variance is not constant and is a function of the parameter being estimated.
Fortunately, this is one of the easier assumptions to check. One of the fit statistics your statistical software produces is a generalized chi-square that compares the magnitude of the model residuals to the theoretical variance.
The chi-square divided by its degrees of freedom should be approximately 1. If this statistic is too large, then the variance is “overdispersed” (larger than it should be). Alternatively, if the statistic is too small, the variance is “underdispersed.”
While the best way to approach this varies by distribution, there are options to adjust models for overdispersion that result in more conservative p-values.