marginal predictions on response scale

[mebrooks]

Currently, it seems that predict.merMod does not give predictions on the response scale averaged over possible realizations of the random effects, i.e. marginalized with respect to the random effects. I expected that a new level of the random effect would be given a marginalized prediction. It would also be very useful in general. Maybe this could be clearer in the documentation, since I think many people don't realize it.

MCMCglmm uses a couple of approximations to do this for log or logit links (described in course notes page 46-47 from 20 June 2015).

@bbolker's initial thoughts were

Thinking about marginalizing predictions opens up a lot of questions:

• marginalize by taking the mean of random samples?
• by doing numerical integration?
• by using the delta method? (one might be able to use higher-order terms to reduce error ...)

This is related to #37 and #281 and was discussed in this thread https://stat.ethz.ch/pipermail/r-sig-mixed-models/2016q1/024543.html