Comparison between Nested and Non-nested Model: BIC and LRT 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.

by definition, BMI and weight are related ($BMI=w/h^2$). Taking them together does not add a significant amount of information to your model, hence the unimpressive BIC difference. In large sample sizes, you would almost always find the additional variable significant - that's a key problem of the traditional significance testing and specifically looking only at the p-value.
Recall that p-value is not "the probability that $H_0$ is wrong" but rather "the probability for rejecting $H_0$ in favor of a powerful alternative $H_1$", and then think of how to compute the power of $H_1$.
Nah, I'm kidding. You don't really need to compute it. Just keep in mind that a p-value does not say anything by itself and that with great sample sizes comes great responsibility for being cautios in interpreting the results.
