During analysis of a high dimensional dataset (92 cases, 400+ variables) with the goal of statistical inference, I first used a bootstrapped LASSO (bolasso) to select predictors, and then did an OLS/logistic regression on the most often selected predictors. I proceeded by starting from the null model, and included predictors in decreasing order of selection frequency up to the first non-significant predictor.
Now, is it possible to get a better fit (as measured by AIC, for example) by including more (non-significant) predictors? And in that case, should I choose the model with only significant predictors, or the one that has the lowest AIC for best inference?
My understanding of it is that if I include a non-significant predictor in the model, it may yield a better fit only by chance, so it is not justified to include it in the model. Is that correct?