fixed effects estimates too small in binomial GLMM with low "success"-rate

Dear list,

I would really appreciate your input on an issue in which the fixed
effects estimates of a binomial GLMM with relatively low rate of
"successes" are too low (compared to the observed effect) by around .5%.

Don't know, but one comment and one thought:

  (1) the convergence warning goes away with the most recent version
of lme4 (1.1-7)

  (2) I'm _guessing_ that you will find that this phenomenon is
due to Jensen's effect (i.e., a nonlinear averaging effect) --
that would mean its magnitude should be proportional to the
random-effects variances and to the strength of the nonlinear
(inverse link) relationship.
  (I think there was a similar, possibly neglected, question along
these lines on the mailing list earlier ...)

  Ben Bolker