The most important practical difference between the two is this: Let's say you have a model with a categorical predictor, which divides your observations into groups according to the category values.* The model coefficients, or "effects", associated to that predictor can be either fixed or random. "Mixed" just means the model has both fixed and random effects, so let's focus on the difference between fixed and random. That is especially true for random and mixed effects models. In statistics, jargon should never be used as a substitute for a mathematical understanding of the models themselves. What follows is essentially a summary of their perspective.įirst of all, you should not get too caught up in the terminology. There are good books on this such as Gelman and Hill. This definition is standard in the multilevel modeling literature (see, for example, Snijders and Bosker, 1999, Section 4.2) and in econometrics. “If an effect is assumed to be a realized value of a random variable, it is called a random effect.” (LaMotte, 1983)įixed 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).
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“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) Searle, Casella, and McCulloch (1992, Section 1.4) explore this distinction in depth. Kreft and De Leeuw (1998) thus distinguish between fixed and random coefficients.Įffects are fixed if they are interesting in themselves or random if there is interest in the underlying population. For example, in a growth study, a model with random intercepts $a_i$ and fixed slope $b$ corresponds to parallel lines for different individuals $i$, or the model $y_ = a_i + b t$. Here we outline five definitions that we have seen:įixed effects are constant across individuals, and random effects vary.
#Stats modeling the world chapter 13 answers full#
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). Perhaps you can pick out which one of the 5 definitions applies to your case.
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Statistician Andrew Gelman says that the terms 'fixed effect' and 'random effect' have variable meanings depending on who uses them.