Distribution of random effects

Question

Why do we usually assume that random effects come from a normal distribution? Can we assume another distribution? Or maybe because the CLT indicates that a random effect is normally distributed?

Answer 1

Assuming a normal distribution for the random effects is convenient from a computational point of view. However, it could be rather restrictive. In general, incorrect distribution assumption for the random effects has unfavorable influence on statistical inferences; see e.g.

1) Agresti, A., Caffo, B. & Ohman-Strickland, P. (2004). Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies, Computational Statistics and Data Analysis, 47, 639-653.

2) Heagerty, P. J. & Kurland, B. F. (2001). Misspecified maximum likelihood estimates and generalised linear mixed models, Biometrika, 88, 973-985.

3) Litiere, S., Alonso, A. & Molenberghs, G. (2007). Type I and type II error under random-effects misspecification in generalized linear mixed models, Biometrics, 63, 1038-1044.

This motivates the search for mixed models with more flexible distributions for the random effects. For instance, there has been much literature on non-parametric modeling, such as Dirichlet process models and stick-breaking processes.

Komarek, A. & Lesaffre, E. (2008), CSDA, 52, 3441-3458, have proposed a parametric extension, replacing the normal distribution in generalized linear mixed models with a penalized Gaussian mixture distribution.

Some others have proposed to use skew normal distribution as extensions to the normal. For example, see Hosseini, F., Eidsvik, J. and Mohammadzadeh, M. (2011). Approximate Bayesian inference in spatial generalized linear mixed models with skew normal latent variables, Computational Statistics and Data Analysis, 55, 1791-1806.