lme4, cloglog vs. binomial link

[...snippage...]
My questions are as follows:

1. Is it correct to assume that given a cloglog link, the less frequent response should be considered the success?

No, cloglog is asymmetric, so it will make a difference which outcome is considered success, but there is no mathematical reason to choose between them. In survival data, the cloglog comes out of the proportional hazards model when you have death within a fixed time period as the response (exact date of death not recorded). In that case, death is "success" (!); hopefully, it is the least likely outcome, but it might not be. If cloglog is just used as a generic link function, then no such logic applies.

2. Is it correct to conclude that the changes in the model have led to less influence of the random factor?

No. The scales are different. At the very least, you need to somehow compare it to the fixed effects on the same scale.

3. How shall I react to the increase in AIC?

(Or, equivalently, the deviance). The cloglog link model seems to give the worse fit to data.

A final question, which may not have an answer at all: I am most curious to learn about possible modifications of the model so that an observed random effect can be minimized (while its presence cannot be denied). 

First, is that desirable, and why? The only logic, that I can think of, is that you want to get the fixed-effect part of the model right, so that the error is not mistakenly taken as part of the random variation.

Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com