Jose Pedro Gala sends in some quesetions about statistical methods for causal inferences. I’ll give his questions, then my responses. Jose writes:

1. Type of U in Sensitivity Analyses: We’ve now seen various different methods proposed for sensitivity analysis for causal inference. For many of these methods, a decision has to be made about the type of variable U is–i.e., whether it is binary, continuous, etc. This decision is often arbitrary (in particular when one does not really have a specific unmeasured confounder in mind), although a binary U seems to be the most common choice probably for mathematical convenience! My question is whether it really matters? I.e., if we assume U is binary when in fact the unmeasured confounder needed to really achieve ignorability is in fact continuous, does it matter?? Any thoughts on this? I cannot find appropriate articles in the lit that shed light on this. What might be missed in the subsequent sensitivity analysis if we pretend to have the wrong U? Or does assuming binary U suffice? (I thought this question would be particularly interesting since there is currently very little on your blog concerning sensitivity analyses.)

2. I have heard time and time again, folx say that propensity score methods (and others) rely on the “constant treatment effect assumption” that E [Y(1) – Y(0)] = ATE = Y_i(1) – Y_i(0) for all i. Y(z) here is the potential outcome at the index z \in {0,1}. I disagree with this. If one wants to make this assumption, that is fine, but I don’t see the _need_ to. In fact, in general I simply don’t believe that the average treatment effect is equal to the individual treatment effects for everyone in the population. I may, even, later want to study how the individual causal effects vary (summarized by a covariate, say), such as for example, look at the effect E [Y(1) – Y(0) | Gender]. What are your thoughts?

3. Finally, what work has been done that discusses the propogation of errors in propensity score methods? Here’s what I mean: propensity scores are typically estimated, yet I don’t see users of the method ever state how they adjust their standard errors for the subsequent \hat{ATE} to account for the “error” in the estimation of the propensity score. I should note that I see this happening less with folx that use propensity score weighting (they often use the robust or sandwich standard errors which account for the estimation of the weights (prop scores)).

My responses:

1. In thie sorts of problems I work on, I think it makes sense to think of U as continuous (e.g., there is a continuous range of people, rather than 2 types). But if you work in the Rubin causal framework (which I’m still trying to understand), it might be possible to avoid these latent U’s entirely and just work with principal strata defined by the distribution of potential outcomes.

2. I don’t see why propensity scores would rely on a constant treatment effect. Often if’s of interest when treatment effects vary. See the work of Rajeev Dehejia on propensity scores and on varying treatment effects.

3. This is a tough problem. I think the right way to do it is through hierarchical Bayes, allowing treatment effects to vary by categories, but in practice I don’t know that it’s been done. You could try bootstrap or jackknife, maybe.