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?

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    $\begingroup$ Related (if not duplicate): Are group effects in a mixed effects model assumed to have been picked from a normal distribution? $\endgroup$ – Firebug Apr 25 '17 at 18:30
  • $\begingroup$ @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. $\endgroup$ – Haitao Du Apr 25 '17 at 18:50
  • $\begingroup$ 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. $\endgroup$ – Firebug Apr 25 '17 at 19:10
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    $\begingroup$ @Firebug I think Generalized Linear Mixed Effects means we use link function to predict binary count etc. variables, instead of changing the $\gamma$ $\endgroup$ – Haitao Du Apr 25 '17 at 19:12
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    $\begingroup$ 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 $\endgroup$ – amoeba May 2 '17 at 10:08

It's because of the mathematics.

Multivariate normality gives some really slick results. These were the first ones worked out theoretically. When you have normal or gaussian variables then maximum likelihood methods turn out to use least squares methods and only means and covariance matrices are needed to make all the estimates.

The methods of computation worked well when our forebears had only electomechanical desk calculators to work with. There were some shortcut methods to compute covariances and to invert matrices, particularly those that had standard patterns, eg, diagonal or block-diagonal.

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  • $\begingroup$ thanks and I agree to your points. But could be more specific? what exactly are nice properties, could you describe it in formula? $\endgroup$ – Haitao Du Apr 26 '17 at 14:29
  • $\begingroup$ I don't think I can add anything. I recommend standard texts on linear models. $\endgroup$ – David Smith Apr 26 '17 at 16:38

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