# Interactions and Regressors Conditional on Other Regressors

Short version: I am trying to build a model in which some binary regressors (whether or not a specific medical procedure took place) are conditional on other regressors (whether or not a specific medical procedure was clinically indicated for a given person), and I'm not sure of the best way to go about it. The regressand is the difference in total medical expenditures between a base year (when the medical procedures do or do not occur) and a later year, to see the impact of those procedures on subsequent costs with the ultimate goal of building a predictive model. My instinct is to include, for each rule, both binary regressors in the model individually along with an interaction, but I'm not certain how sound that is or if it's enough to capture the relationship between the "procedure applies" indicator and the "procedure was done" indicator. So:

1. Is that a sound approach?
2. If (1), are there any major pitfalls I should be aware of?
3. If not (1), what other approaches might be suitable instead?

Additional details: I'm working on a project attempting to relate certain medical "best practices" in a base year to medical expenditures in following years. I have a list of best practices (from a vendor) as well as whether or not they applied to specific participants and were followed with a particular in a given year. Not all of the best practice rules apply to all members, and I need my model to reflect a difference between a rule not applying to a participant versus a rule applying and the procedure not taking place. Some rules apply to nearly everyone, while others apply to as few as seven participants. The conditions I am working with are chronic and so are assumed to have some (fairly) regular, nonzero cost from year to year.

I have around 60 rules to consider for inclusion. Because every rule would be represented by three terms in my planned model, the model itself will be pretty big and I am concerned about overfitting. I've thought a little bit about variable selection but haven't settled on an approach yet and may try including every rule.

I appreciate any advice, and am eager to review any referenced material related to my situation. I have not had much luck finding anything addressing a strongly similar situation, but what I can find seems to suggest that my intended approach is acceptable.

To 1. and some of 2. but not 3.:

You don't need to model with an interaction. Binary predictors for indicated and happened are enough.

An interaction would allow you to model a separate effect for all the combinations of indicated and happened. Since having the procedure without being indicated the procedure is not possible (i.e. indicated = 0; happened = 1), you can achieve all the necessary freedom with main effects. If B_* indicates the coefficient for *:

indicated  happened                          effect
0         0                             B_0
1         0               B_0 + B_indicated
1         1  B_0 + B_happened + B_indicated


Every group can be given a unique prediction and the model is adequately specified. Using an interaction will lead to undefined coefficients in a standard linear model. Excluding the interaction means fewer coefficients for the same amount of freedom.

If you regularise then you will get slightly different results by using an interaction, but only because the model can use two coefficients to affect the same group and cheat the regularisation penalty (depending on its nature). So again, no interaction is better.

Given the nature of your problem it's definitely worth trying a regularized model. Including every rule in an L1 and tuning $\lambda$ by cross-validation is a standard approach.

• If it is true that happened => indicated, (happened =1 => indicated =1) then happened*indicated = happened. – Filipe Sep 18 '16 at 17:50