Chapter 19

Mixed-Effects Models

Up to this point, we have treated all categorical explanatory variables as if they were the same. This is certainly what R.A. Fisher had in mind when he invented the analysis of variance in the 1920s and 1930s. It was Eisenhart (1947) who realized that there were actually two fundamentally different sorts of categorical explanatory variables: he called these fixed effects and random effects. It will take a good deal of practice before you are confident in deciding whether a particular categorical explanatory variable should be treated as a fixed effect or a random effect, but in essence:

  • fixed effects influence only the mean of y;
  • random effects influence only the variance of y.

Fixed effects are unknown constants to be estimated from the data. Random effects govern the variance–covariance structure of the response variable (see p. 519). Nesting (or hierarchical structure) of random effects is a classic source of pseudoreplication, so it important that you are able to recognize it and hence not fall into its trap. Random effects that come from the same group will be correlated, and this contravenes one of the fundamental assumptions of standard statistical models: independence of errors. Random effects occur in two contrasting kinds of circumstances:

  • observational studies with hierarchical structure;
  • designed experiments with different spatial or temporal scales.

Fixed effects have informative factor levels, while random effects often have uninformative ...

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