Identifying link functions for gamma glmm

Most of the time I would suggest choosing link functions on 
scientific grounds, i.e. what scale makes sense for the expected 
effects? Link functions change the expected relationship with continuous 
predictors (do I expect the effects of predictors to be linear 
(identity), exponential (log), or hyperbolic (inverse)?) and change the 
meaning of interactions (does the value of one variable change the 
expected effect of the other additively (identity), proportionally 
(log), or ?? (inverse)).

   I generally find that log links are more numerically stable (both 
identity and inverse links can sometimes lead to negative predictions). 
Logs are also nice because they essentially split the difference between 
the identity and inverse links.  If I have (say) responses that are time 
intervals, then analyzing on the identity scale describes additive 
effects on the time scale; analyzing on the inverse scale describes 
additive effects on the rate or speed (1/time) of the response; 
analyzing on the log scale describes proportional changes in *either* 
time or rate (because log(time) = -1*log(1/time)).

   My general procedure would be to use a log link and see if the 
diagnostics detected any problems.

   That said, you could use AIC or cross-validation if you are primarily 
interested in prediction (and aren't worried about snooping). 
Cross-validation will be slower but more reliable, *if* you are careful 
to maintain independence structure when you specify your training and 
testing sets (i.e., you should sample by levels of your grouping 
variable, not by individual observations)

Identifying link functions for gamma glmm

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