# Why are random effects coefficients assumed to come from a Gaussian distribution?

I am learning about mixed effects models from here. The linear mixed effects model can be represented as

$$\mathbf{y} = \boldsymbol{X\beta} + \boldsymbol{Z\gamma} + \boldsymbol{\varepsilon}$$

Here is my understanding:

Intuitively, I understand the random effect part is some "fine tuning" on fixed effect, i.e., change the intercept and other coefficients by certain amount according the grouping options.

In addition, $\boldsymbol{\gamma}$ is a long vector and has many degrees of freedom, so in order to avoid over fitting we have constraints on $\boldsymbol{\gamma}$, where as stated in the web page:

We nearly always assume that: $\boldsymbol{\gamma} \sim \mathcal{N}(\mathbf{0}, \mathbf{G})$

Why we have the Gaussian constraint? Do other constraints on $\boldsymbol{\gamma}$ exist in mixed effects models?

• Related (if not duplicate): Are group effects in a mixed effects model assumed to have been picked from a normal distribution? – Firebug Apr 25 '17 at 18:30
• @Firebug thanks for the link!! the answer helped me a lot. The only reminding part is do we have other constraints on $\gamma$ in addition to normal distribution. – Haitao Du Apr 25 '17 at 18:50
• True, it's an interesting question and one I wouldn't know how to answer if asked. I'm expecting Generalized Linear Mixed Effects modelling to come into play. – Firebug Apr 25 '17 at 19:10
• @Firebug I think Generalized Linear Mixed Effects means we use link function to predict binary count etc. variables, instead of changing the $\gamma$ – Haitao Du Apr 25 '17 at 19:12
• cran.r-project.org/web/packages/hglm/vignettes/hglm.pdf "The package fits generalized linear models with random effects, where the random effect may come from a conjugate exponential-family distribution (Gaussian, Gamma, Beta or inverse-Gamma)." @Firebug – amoeba May 2 '17 at 10:08