Poisson and binomial submodels: `non-integer` issue and lme4 issue #682

While working on a Poisson example for the revised vignette(s), I realized that lme4 indeed runs through for a Poisson submodel, but the lme4 run is slow and throws a lot of warnings concerning non-integer values (the warnings being suppressed by projpred, though). The reason is probably that in the submodels, the response values resulting from the reference model's predictions are not integers (in contrast to the original response).

So my first question is: Does lme4 return valid results in this case, despite the (suppressed) warnings? And my second question is: If yes, shouldn't projpred throw a warning when creating a Poisson reference model, so the user gets notified of the increased runtime?

A similar issue might exist for GAMs and GAMMs. I haven't looked into that.

For GLMs: I don't know how the projpred-internal GLM submodel fitter handles this. Perhaps this was exactly the reason why projpred has an internal GLM submodel fitter instead of using glm() (or lm(), for the Gaussian family)? If not, then the projpred-internal GLM submodel fitter might exhibit a similar issue.

In an example run with non-suppressed warnings, I did not encounter this issue for the binomial family. In principle, the binomial family could have the same issue because the reference model's predictions are probabilities there, but I guess the issue does not occur there since the usual R convention for binomial fitters is to also accept proportions for the response.

Start by reproducing the Poisson example described in the issue with warnings unsuppressed, then inspect the projpred submodel fitting path for Poisson and binomial responses. Determine whether the non-integer-response warnings indicate invalid results and whether projpred should notify users about the runtime or fitting behavior; also check whether the concern applies to GAMs, GAMMs, or the internal GLM fitter.