# [R] Random effects and lme4

From: wmh <wmhirst_at_hotmail.com>
Date: Thu, 03 Jul 2008 10:33:27 -0700 (PDT)

I'm running some multi-level binomial models with lme4 and have a question regarding the estimated random effects.

Suppose I have nested data e.g. clinic and then patient within clinic. The standard deviations of the random effects at each level are roughly equal in a model for real life data. Attention then turns to examining the individual random effects at each level. I'm extracting these using ranef(). As I understand it one expects a certain amount of "shrinkage" here but what I'm finding is that the standard deviations of the estimated intercepts are smaller then expected and in particular are much smaller at the patient within clinic level. Is this normal or am I interpreting the estimated intercepts incorrectly?

An example from some simulated data:

Model:

Logit(p)=a+bx+c+d

Where c is the level 1 random effect - clinic - N(0,sd=sigma1) Where d is the level 2 random effect - patient within clinic - N(0,sd=sigma2).

Here a=-0.8,b=1,sigma1=0.3,sigma2=0.3

I'm simulating for 50 clinics each with 25 patients and 5 visits per patient (a total of 6250 observations per simulation). There are 200 runs.

The results - mean - from 200 runs

``` mean(a)= -0.80
mean(b)= 1.00
mean(sigma1) = 0.29
mean(sigma2) = 0.26

```

However the sd of the estimated intercepts extracted using ranef at each level are:

mean(Level 1 intercepts) = 0.23
mean(Level 2 intercepts) = 0.07

I'd expect the level 2 intercepts to be produced with more variation than the mean value of 0.07 from this simulation.

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