Solved – Why are random effects shrunk towards 0

Is there an intuitive reason for random effects to be shrunk towards their expected value in the general linear mixed model?

generally speaking, most "random effects" occur in situations where there is also a "fixed effect" or some other part of the model. The general linear mixed model looks like this:


Where $beta$ is the "fixed effects" and $u$ is the "random effects". Clearly, the distinction can only be at the conceptual level, or in the method of estimation of $u$ and $beta$. For if I define a new "fixed effect" $tilde{x}_i=(x_i^T,z_i^T)^T$ and $tilde{beta}=(beta^T,u^T)^T$ then I have an ordinary linear regression:


This is often a real practical problem when it comes to fitting mixed models when the underlying conceptual goals are not clear. I think the fact that the random effects $u$ are shrunk toward zero, and that the fixed effects $beta$ are not provides some help here. This means that we will tend to favour the model with only $beta$ included (i.e. $u=0$) when the estimates of $u$ have low precision in the OLS formulation, and tend to favour the full OLS formulation when the estimates $u$ have high precision.

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