In simple terms, how would you explain (perhaps with simple examples) the difference between fixed effect, random effect and mixed effect models?

Alek RichterEnlightened

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Statistician Andrew Gelman says that the terms ‘fixed effect’ and ‘random effect’ have variable meanings depending on who uses them. Perhaps you can pick out which one of the 5 definitions applies to your case. In general it may be better to either look for equations which describe the probability model the authors are using (when reading) or write out the full probability model you want to use (when writing).

Here we outline five definitions that we have seen:

Fixed effects are constant across individuals, and random effects vary. For example, in a growth study, a model with random intercepts ai

and fixed slope b corresponds to parallel lines for different individuals i, or the model yit=ai+bt

. Kreft and De Leeuw (1998) thus distinguish between fixed and random coefficients.

Effects are fixed if they are interesting in themselves or random if there is interest in the underlying population. Searle, Casella, and McCulloch (1992, Section 1.4) explore this distinction in depth.

“When a sample exhausts the population, the corresponding variable is fixed; when the sample is a small (i.e., negligible) part of the population the corresponding variable is random.” (Green and Tukey, 1960)

“If an effect is assumed to be a realized value of a random variable, it is called a random effect.” (LaMotte, 1983)

Fixed effects are estimated using least squares (or, more generally, maximum likelihood) and random effects are estimated with shrinkage (“linear unbiased prediction” in the terminology of Robinson, 1991). This definition is standard in the multilevel modeling literature (see, for example, Snijders and Bosker, 1999, Section 4.2) and in econometrics.