Validating the way of creating a composite score using elastic net

Question

We have a dataset where we want to see the association of different personal/demographic characteristics and biomarkers with post-injury depression. For that, we first want to create a composite score with important biomarkers (step-1) and use it in a logistic regression of depression to show the associations in terms of adjusted odds ratios and p-values (step-2).

We are using an elastic net to select these biomarkers as the biomarkers are highly correlated. The idea is to use the $\beta$ weights obtained from this elastic net to create a composite score of selected biomarkers.

I'm confused about if I should use the personal/demographic variables in the elastic net too. Adding some of them may change the coefficients/beta weights by a good margin and possibly make the coefficients of the biomarkers more "adjusted". We want to use these personal/demographic variables along with the created composite in the (step-2) logistic regression of depression anyway, even if some of these do not get selected by the elastic net (for example, gender/premorbid depression). The selection of biomarkers and getting stable coefficients for them are the main goals in step-1. As a result, I'm planning to make the composite like: $\hat{\beta_1}*Biomarker_1 + ... + \hat{\beta_n}*Biomarker_n$.

If we only used biomarkers in the elastic net (and not the other demographic variables), I'd probably think about finding a predicted logit ($X\hat{\beta}$) which would also include $\hat{\beta_0}$ (intercept) and use that as a composite.

Could you let me know which one seems more correct to you?

Answer 1

What I think you're asking is whether you should adjust for demographic variables in your step 1, and, if so, how? Is my understanding of your question correct?

I agree with your intuition that

Adding some of [the personal/demographic variables] may change the coefficients/beta weights by a good margin and possibly make the coefficients of the biomarkers more "adjusted"

Furthermore, I think that if this were to happen it would be a good thing. You don't want to select biomarkers the associations of which are merely surrogates for the patient/demographic variables that you plan to include in your model anyways.

I think you should try to fit a one-step model by applying your elastic-net type shrinkage but decreasing or even zeroing out the penalty on the patient/demographic coefficients that you definitely want included in your model. In doing so, you would only penalize the biomarker coefficients, which you are less certain about. If you are using glmnet in R, you can achieve this differential penalization of coefficients with the penalty.factor argument.

I am generally not in favor of a two-step process such as what you propose because in step 2 you are not accounting for the underlying uncertainty in your derived predictor.