From: Thomas Lumley <tlumley_at_u.washington.edu>

Date: Tue 20 Jun 2006 - 04:33:26 EST

R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Tue Jun 20 04:37:34 2006

Date: Tue 20 Jun 2006 - 04:33:26 EST

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

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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Tue Jun 20 04:37:34 2006

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