[R] Predicted values based on fixed effects do not correspond with actual data in cross-classified generalized linear mixed model (lmer)

[Gert]

Dear R-Users,

I have fitted a cross-classified generalized linear mixed model using the
lmer package with the following code. 

Mod<-lmer(y~x+(1|a)+(1|b)+ (1|c), family=binomial)

In this case, only including a covariate (x) as a fixed effect.

The fitted values, using fitted(mod), correspond to the raw data nicely, and
the mean of the fitted values is equal to the mean of the raw data. In
addition, the parameter estimate for the fixed effect (x) corresponds to the
data as well (the slope ‘seems’ right). So far so good. 

The problem arises when I calculate the predicted values based on the
intercept and the parameter estimate of the fixed effect, using the formula
exp(X)/(1+EXP(X)), where X=intercept + par. Est. * x. 

When I use calculate the mean of these predicted values, this mean is much
lower than the mean of the actual data. The shape of the predicted curve
fits nicely to the data, but the predicted lines is always ‘below’ the
actual data. Apparently, the intercept of the curve is not predicted
correctly.

Does anyone know why this is? 

I guess it has something to do with the fact that the intercept for the
fixed effects is estimated for a certain value of the random effects?
According to the R documentation on fitted values; ‘the fitted values at
level i are obtained by adding together the contributions from the estimated
fixed effects and the estimated random effects…’. But is there an 'average
contribution' of the random effects?

Is there a way to evaluate the fixed effects at the ‘average level’ of the
random effects? Do I need to adjust the formula for the predictions to take
into account the random effects?

Many thanks,
Gert Stulp


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