Relationship between BLUP and variance parameter When performing a mixed-effects analysis, one gets the mean (beta coefficients) of the fixed effects and the variance parameter for each of the random effects included in the model. Additionally, digging into the model, one can get the estimates for each of the levels in the grouping variable (I think these are BLUPs).
Consider a simple mixed-effects model:
$data \sim X + (1 | Group)$
where $Group$ is a factor variable with two levels. Solving this using lmer (R) or fitlme (MATLAB) will give me the $\beta$ values for the $X$ variables and the $\rho$ (standard deviation) for $Group$. If I extract these BLUPs, I get the intercepts for both the groups.

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*Is it correct to think of these two group-level intercepts as (approximately) the mean values of the two groups?


*Would it be fair to say that the $\rho$ associated with the $Group$ variable gives an estimate of how the mean varies across all levels of $Group$ (in the sense of the population of levels that might exist in $Group$)?
If both the above are (even approximately) correct, is there a mathematical relationship between the BLUPs and the overall variance parameter? In other words, if I have my raw data and the BLUP, can I calculate the $\rho$ without fitting the model again and vice-verse (having obtained the $\rho$, can I get the BLUPs without refitting the model? How would this relationship hold for more complex models?

generally, you should only calculate a mixed lm with a reasonable number of groups, though there is (as far as I know) no established Cut-Off. Calculating a mixed lm with 2 groups and a corresponding group variance makes no sense at all. I'm trying to neglect this by answering the question as if there are more groups.

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*This would a reasonable description for a model containining only the person-varying intercept, but not for model that also contain predictors. If the predictors have an influence on y and differ between the two groups, the group level intercepts will differ from the group means. Rather, the group-level intercepts could be seen as the predicted values for y if the predictors are zero.


*Assuming we have a reasonable number of groups, this would be correct.


*I'm not sure if I understand your question right. The BLUPs are inferred from the group variance but also from other parameters of the model, so I think what you want to do is not possible.
