In short, my question is as follows:

  • Why is it common to assume normally distributed random effects (especially in generalized linear mixed models)?

A longer version:

Under some circumstances, an approximately normally distributed random effect makes sense. For example, say we measure individuals' weight ($y$) depending on the type of diet ($x$) they were on, once before and once a month after dieting. If individuals ($\upsilon$) are measured twice, then the following LMM:

$$y_{ij} = \beta_0+ \beta_1 x + \upsilon_i + \epsilon_{ij} \\ \upsilon \sim \mathcal{N}(0,\,\sigma_\upsilon^2), \; \epsilon \sim \mathcal{N}(0,\,\sigma_\epsilon^2)$$

basically assumes that individuals ($\upsilon$) come from some larger population, which causes a random, normally distributed offset in their initial weight. One could argue that whatever (unknown) differences are present between individuals (genetic, environmental, lifestyle), might sum up to a normal distribution as many sums of independent random variables would. In fact, we could use almost the same argument for the errors of the outcome variable ($\epsilon$).

However, say we count birds ($y$) in different terrain types ($x$) on different islands ($\upsilon$) and use a Poisson GLMM, why, if at all, is the normality assumption still defensible? Surely the sum of random variables differing between two islands can cause a normally distributed offset for an outcome with normally distributed errors, but how can we justify this for a non-normal error structure?

I understand that a GLMM models the random effect in the linear part, but isn't this linear part still not assumed to have a normal error structure? (Sorry for the double negative.)

Bonus question:

  • Are there any simple examples of non-normal random effects (e.g. Poisson distributed)?
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    $\begingroup$ don't have time to answer properly, but basically: the random effects are always imposed on the linear predictor scale, which means that we're looking for a continuous variable without constraints (can take any real value); Normality is defensible on the same central-limit grounds. Also, it's analytically and computationally convenient. $\endgroup$ – Ben Bolker Sep 12 '18 at 1:49
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    $\begingroup$ Another small point others haven't mentioned is that it's much simpler with normal distributions to specify certain covariance structures in the random effects which allow you to generalize an iid random effect (as in your linear model example) to non-iid scenarios, i.e. temporal or spatial random effects. $\endgroup$ – aleshing Sep 12 '18 at 6:22
  • $\begingroup$ @BenBolker Thank you for your comment (and for lme4!). I can imagine the computational convenience, but I still don't quite understand why normality on the linear predictor scale is defensible on the same grounds. Perhaps I'm thinking in the wrong direction, but aren't the 'residuals' of a GLMM non-normal like described here: stats.stackexchange.com/a/139624/176202)? Even after applying the link function, I can't see how the normal distribution arises. $\endgroup$ – Frans Rodenburg Sep 13 '18 at 3:13
  • $\begingroup$ @marmle That is a very good point. I'm still curious if for some cases there is a theoretical justification though. $\endgroup$ – Frans Rodenburg Sep 13 '18 at 3:15

Some points:

  1. The choice of a normal distribution for the random effects in linear mixed models (i.e., normally distributed) outcome is typically done for mathematical convenience. That is, the normal distribution of $[Y \mid b]$ works nicely with the normal distribution for the random effects $[b]$, and you get a marginal distribution that for the outcome $[Y]$ that is multivariate normal.

  2. In that regard it helps to see a mixed model as a hierarchical Bayesian model. Namely, in the linear mixed model assuming a normal distribution for the random effects is a conjugate prior that gives you back a posterior in closed-form. Hence, you can do the same for other distributions. If you have Binomial outcome data, the conjugate prior for the random effects is a Beta distribution, and you get the Beta-Binomial model. Likewise if you have Poisson outcome data, the conjugate prior for the random effects is a Gamma distribution, and you get the Gamma-Poisson model. Just to make clear here that in the previously mentioned examples, the distribution of the random effects was on the scale of the mean of the outcome conditional on the random effects not on the scale of the linear predictor (e.g., in the Gamma-Poisson example, on the linear predictor scale the assumed distribution would be a log-Gamma distribution).

  3. There is nothing stoping you changing the distribution. For example, in the linear mixed model you could use a Student’s t distribution for the random effects, or in categorical outcomes to use a normal distribution. But then you lose the computational advantage of having a closed-form posterior. There is considerable literature looking into the impact of changing the random-effects distribution. Many people have proposed flexible models for it; for example, using splines or mixtures to be able to capture random-effects distributions that are multi-modal. However, the general consensus has been that the normal distribution works pretty well. Namely, even if you simulate data from a bimodal or skewed distribution for the random effects, and you assume in your mixed model that it is normal, the results (i.e., parameter estimates and standard errors) are almost identical to when you fit a flexible model that captures this distribution more appropriately.

  4. Hence, the choice of the normal distribution has dominated, even though other options do exist. With regard to your point on whether the choice of a normal distribution is defensible for categorical data, as Ben mentioned, note that the distribution of the random effects is placed not on the outcome but rather on the transformed mean of the outcome. For example, for Poisson data you assume a normal distribution for the random effects for $\log(\mu)$ where $\mu$ denotes the expected counts of the outcome variable $Y$ which is the observed counts.

  • $\begingroup$ Thank you for your answer @DimitrisRizopoulos, but do I understand from (2) that you are saying the normal distribution is not a logical prior choice for a GLMM (e.g. binomial) in the context of BHM because it wouldn't be a conjugate prior? $\endgroup$ – Frans Rodenburg Sep 13 '18 at 4:09
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    $\begingroup$ @FransRodenburg No, I’m not saying this. Conjugacy is just mathematical/computional convenience (actually before the advancement of MCMC the Bayesian approach has been criticized of only being able to practically work with conjugate priors). The normal distribution is a more natural choice because the random effects are in a sense regression coefficients, and for the coefficients of regression models we know that their posterior (Bayesian) or their sampling distribution (maximum likelihood) is approximately normal. $\endgroup$ – Dimitris Rizopoulos Sep 13 '18 at 6:46
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    $\begingroup$ I'd add to that that the standard LMMs / GLMMs actually use multivariate normal REs. For example, when you fit a random slope / intercept, lme4 will also fit the correlation between the two. It would be very difficult (though of course not impossible) to keep up this principle with other distributions. $\endgroup$ – Florian Hartig Oct 9 '18 at 8:50

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