I am fitting gams (in R using mgcv) to count data and am uncertain about how to select from among competing models. In my model specification I am using "ts" as my smoothing basis and a negative binomial distribution. As I understand it this essentially adds an extra penalization to the the smooth and thereby has the potential to help exclude some terms completely during the model fitting.
The modeling strategy I employed is a follows
I first fit a global model (i.e. with all my independent variables of interest) e.g. (N~s(x1)+s(x2)+s(x3)+s(x4)). When I find that the terms x3 has an edf that is practically equal to 0 and that x4 is not significant at the 0.05 level (but with edf>1) I subsequently fitted a model by eliminating these two terms i.e. N~s(x1)+s(x2).
What does it mean when I compare the the two models and find that the nested model (now with all terms significant and non zero edfs) has a lower deviance explained? Also is this a defensible approach to identify important predictor variables?
Any advice will be greatly predicated.