I would like to select the best model for predicting breast cancer risk, specifically, it is the comparisons between weight/BMI/height, as other covariates remain the same in all the models. But I got opposite results for nested model selection by using BIC and LRT. Say that the P-value of LRT is <0.05, but the BIC of the richer model is larger than the less rich model, and the delta BIC is larger than 2, which suggests positive evidence.
Q1: Should I use BIC for nested model comparison? Can BIC penalize the overfitting issue adequately for the richer models?
Q2: Can I use BIC for non-nested model selection and LRT for nested model selection? For example, using BIC to select the best model among weight-only/BMI-only/height-only models; using LRT to select the best model between weight+BMI model Vs BMI-only model to see if weight added additional significant information to the BMI-only model.
I know there are many posts about model selections using BIC/AIC/LRT. But none of them really solve my question.