[R-sig-ME] In simple terms, how is the estimated variance of higher-level effects calculated?

[Jeremy Koster]

I'm teaching some grad students about mixed-effects modeling.  To their credit, they're paying close attention and asking good questions.

Today, we were talking about variance components in a basic two-level binomial glmer with no fixed effects.  The output includes these estimates:


Random effects:
 Groups  Name        Variance Std.Dev.
 Subject (Intercept) 0.93537  0.96715 

They know about standard deviations and variances from their intro class, so when I showed them the dotplot (caterpillar plot) of the estimated intercepts for each of the higher-level subjects, they wondered if the standard deviation above was simply the sd of those varying intercepts.  Well, I acknowledged that the estimated variance was going to reflect the dispersion in those estimates, but that there is no doubt some extra stuff going on behind the scenes, particularly since the estimated ranef intercepts themselves vary in their precision, often relating to different numbers of observations for each subject (they didn't fully believe me and tried it anyway, only to confirm my suspicion).

So if one were to describe in simple terms how lme4 generates a number for the estimated variance of the random effects, what might be said?