If the Hosmer-Lemeshow indicates a lack of fit but the AIC is the lowest among all the models....should you still use the model?
If I delete a variable, the Hosmer-Lemeshow statistic is not significant (which means there is no gross lack of fit). But the AIC increases.
Edit: I think in general, if the AIC's of different models are close (i.e. $<2$) to each other then they are basically the same. But the AIC's are much different. This seems to indicate that the one with the lowest AIC is the one I should use even though the Hosmer-Lemeshow test indicates otherwise.
Also maybe the H-L test only applies for large samples? It has low power for small sample sizes (my sample size is ~300). But if I am getting a significant result... This means that even with low power I am getting a rejection.
Would it make a difference if I used AICc versus AIC? How do you get AICc's in SAS? I know there could be problems with multiplicity. But a priori I hypothesize that the variables have an effect on the outcome.
Edit2: I think I should use the model with one fewer variable and the higher AIC with non-significant H-L. The reason is because two of the variables are correlated with each other. So getting rid of one makes sense.