1 Motivating GLMMs
I briefly summarize the motivations for GLMMs (in linguistic modeling):
• The Language-as-fixed-effect-fallacy (Clark 1973 following Coleman 1964). If you want to make state- ments about a population but you are presenting a study of a fixed sample of items, then you cannot legitimately treat the items as a fixed effect (regardless of whether the identity of the item is a factor in the model or not) unless they are the whole population.
– Extension: Your sample of items should be a random sample from the population about which claims are to be made. (Often, in practice, there are sampling biases, as Bresnan has discussed for linguistics in some of her recent work. This can invalidate any results.)
• Ignoring the random effect (as is traditional in psycholinguistics) is wrong. Because the often significant correlation between data coming from one speaker or experimental item is not modeled, the standard error estimates, and hence significances are invalid. Any conclusion may only be true of your random sample of items, and not of another random sample.
• Modeling random effects as fixed effects is not only conceptually wrong, but often makes it impossible to derive conclusions about fixed effects because (without regularization) unlimited variation can be attributed to a subject or item. Modeling these variables as random effects effectively limits how much variation is attributed to them (there is an assumed normal distribution on random effects).
• For categorical response variables in experimental situations with random effects, you would like to have the best of both worlds: the random effects modeling of ANOVA and the appropriate modeling of categorical response variables that you get from logistic regression. GLMMs let you have both simultaneously (Jaeger 2007). More specifically:
– A plain ANOVA is inappropriate with a categorical response variable. The model assumptions are violated (variance is heteroscedastic, whereas ANOVA assumes homoscedasticity). This leads to invalid results (spurious null results and significances).
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– An ANOVA can perform poorly even if transformations of the response are performed. At any rate, there is no reason to use this technique: cheap computing makes use of a transformed
ANOVA unnecessary.
– A GLMM gives you all the advantages of a logistic regression model:1
 Handles a multinomial response variable.
 Handles unbalanced data
 Gives more information on the size and direction of effects
 Has an explicit model structure, adaptable post hoc for different analyses (rather than re- quiring different experimental designs)
 Can do just one combined analysis with all random effects in it at once.
• Technical statistical advantages (Baayen, Davidson, and Bates). Maybe mainly incomprehensible, but you can trust that worthy people think the enterprise worthy.
– Traditional methods have deficiencies in power (you fail to demonstrate a result that you should be able to demonstrate)
– GLMMs can robustly handle missing data, while traditional methods cannot.
– ?? GLMMs improve on disparate methods for treating continuous and categorical responses ??.
[I never quite figured out what this one meant – maybe that working out ANOVA models and tractable approximations for different cases is tricky, difficult stuff?]
– You can avoid unprincipled methods of modeling heteroscedasticity and non-spherical error vari- ance. – It is practical to use crossed rather than nested random effects designs, which are usually more appropriate – You can actually empirically test whether a model requires random effects or not.
 But in practice the answer is usually yes, so the traditional ANOVA practice of assuming yes is not really wrong.
– GLMMs are parsimonious in using parameters, allowing you to keep degrees of freedom (giving some of the good effects listed above). The model only estimates a variance for each random
effect.