controlling confounding variables vs. excluding confounding variables

Question

I'm working on a meta-analysis project that looks at the effect of "pure" depression (i.e., depression with no anxiety) on mortality. For studies that looked at the effect of pure depression on mortality, they either controlled comorbid anxiety (by adjusting it as a covariate) or excluded individuals with comorbid anxiety. I was wondering if those two procedures could be considered to be statistically similar or different. That being said, I'm wondering if I could include both types of studies to investigates the association between pure depression and mortality.

I'd appreciate any answers or advice!

Answer 1

Running a regression model that included depression, anxiety, and their interaction could yield the same results as excluding those with anxiety if the coefficients of the model were coded correctly. If $X_d$ represents depression (0/1) and $X_a$ represents anxiety (0/1), in the model $$Y=\beta_0 + \beta_1 X_d + \beta_2 X_a + \beta_3 X_d X_a + \epsilon$$ $\beta_1$ represents the effect of depression on $Y$ for those with no anxiety, which would be equal to the slope on depression in a model fit to participants who had no anxiety.

A problem is that "controlling for" is vague, and includes running the above model without including the interaction term. This will not fully control for anxiety, and the coefficient on depression will not be the isolated effect of depression in those without anxiety unless the effect of depression on mortality is the same in anxious and non-anxious people.

If anxiety were controlled for using propensity score methods (e.g., matching or weighting), the estimated depression effect would be the marginal effect of depression in the population, i.e., averaged across levels of anxiety, so this also would probably not be the quantity you want to estimate.

Answer 2

I would add two points to @Noah's answer:

1. Ceteris paribus, the precision of beta estimates will differ between a controlled regression and a model run only on one subgroup because of different sample size, i.e., standard errors and p-values would be larger in a model run only on one subpopulation.
2. Splitting a sample by level of confounding variable (anxiety) is not recommended if the researcher is interested in showing heterogeneous treatment effect for different levels of anxiety. This is because there can be another confounding variable in the relationship between depression and mortality that is only correlated with depression for people with anxiety (or without anxiety), e.g., some other condition or use of a particular medication. That means that one may end up with an insignificant effect of depression on mortality when estimating on the sample where depression is correlated with that other confounding variable.