[R] lmer binomial model overestimating data?

[Thomas Lumley]

On Wed, 14 Jun 2006, Martin Henry H. Stevens wrote:

> Hi folks,
> Warning: I don't know if the result I am getting makes sense, so this
> may be a statistics question.
>
> The fitted values from my binomial lmer mixed model seem to
> consistently overestimate the cell means, and I don't know why. I
> assume I am doing something stupid.

Not really, there is something subtle going on.  The model says that

  logit E[Y|x, random effects] = x*beta+random effects

Now, when you compute the observed values you are averaging over the 
random effects to get

    E[E[Y|x, random effects]]= E[ invlogit(x*beta +random effects)]

where invlogit is the inverse of logit.

When you compute the fitted values you are also averaging, but on the 
linear predictor scale to get
    E[logit(E[Y|x, random effects])]= invlogit(x*beta)

The logit/unlogit operation is not linear, so these are not the same. In 
fact, invlogit(x*beta) is always further from 1/2 than E[Y|X].


With linear regression it is useful and fairly standard to think of the 
random effects part of a mixed model as giving a model for the covariance 
of Y, seperate from the fixed-effects model for the mean of Y.  With 
generalized linear models these can no longer be separated: adding random 
effects changes the values and the meaning of the fixed effects 
parameters.


 	-thomas