While conducting a Monte Carlo study to explore the empirical rejection rates of the output null probabilities associated with the hypothesis of zero slopes in the generalized linear model formulation using the gam function of the mgcv package, I have come across an unexpected result. When using the gaussian or binomial families, I found the expected result: as sample size increases (from 10 to about 200 in this study), the reported asymptotic null probabilities converge nicely. However, when I use the poisson family, the empirical rejection rates do not converge to the alpha rates, even when sample size is large (200). The models converge normally without any error or warning messages. For example, at a sample size of 200, the empirical quantiles of p-values (1000 Monte Carlo simulations) using the poisson family are 0.0117 for p=0.05 and 0.0246 for p=0.10. Using the binomial family the empirical quantiles are 0.0463 for p=0.05 and 0.0906 for p=0.10.
I suspect that the problem is that gam is having trouble with the log link using the default choices for convergence and estimation while these choices are fine for the identity link (gaussian) or the logit link (binomial).
Question 1: Has this already been reported?
Question 2: What would be better choices for controlling estimation and convergence?