# Causal inference and Propensity score

I am trying to understand Rubin's causal model but I can not make the connection between certain notions. The problem of causal inference lies in calculating the counterfactual, i.e. knowing what the outcome would have been in the absence/with treatment.

The causal effect is individual (and unobservable), so we are more interested in two aggregate effects: ATE and ATT. If treatment allocation is independent, then selection bias is eliminated, so ATT=ATE (randomization). In causal inference methods, we try to reproduce randomization (control selection bias as much as possible).

To do this, we can calculate a propensity score and apply various methods (matching, IPTW, stratification).

I can not make the connection between ATT/ATE and propensity score method:

1. Why logistic regression (then matching, IPTW or stratification) reduce this bias?
2. How does this relate to the counterfactual? Should not we try to estimate it?
3. Why are there different methods depending on whether you want to estimate ATT or ATE? Because if you want to remove the bias, then ATT=ATE.

In my mind, it is difficult to make the link between these methods and Rubin's general model.

Thank you for enlightening me on the subject.

• Can you clarify your first question? Are you asking why propensity scores reduce bias? Or why logistic regression reduces bias? And do you mean logistic regression for the propensity score or for the outcome? Also, how technical do you want your answer to be (i.e., do you want derivations and proofs or just intuitive answers)? Have you read Rosenbaum & Rubin (1983), which explains all this in a technical way? Have you read Hernán & Robins (2006), which explains this in an intuitive way?
– Noah
Mar 11 at 16:49
• Hi Noah, To be more precise, why does a method like the propensity score reduce bias in a technical way? And above all, what is the link between this method and the notions of ATE and ATT? I've read a lot on the subject but I don't really understand. What's the link between the propensity score method and Rubin's conceptual model? Mar 14 at 13:03
• Do you understand the technicalities of the Horvitz Thompson estimator in survey statistics?
– Kuku
Apr 5 at 10:23