Assumptions in the use of propensity scores

Question

A researcher performed a propensity score matching analysis where the exposure was sex (i.e. males vs females). This was not intended to be a causal analysis and the results were described only in terms of 'associations' - no mention of causal inference was made in the manuscript.

It was suggested in review that the positivity assumption was violated as females have a zero chance of being males and vice versa. I am wondering what people's opinions are on this as my thought is that would be technically true if in fact the propensity score matching was being used as a step in a causal inference analysis. That is not the case here - instead the intent was more as a data reduction step.

Am I correct in thinking that assumptions such as positivity, exchangeability, etc are assumptions for causal inference, not for the use of propensity scores per se? Or am I missing the point? - which if I am I can't help but think is a semantic one (i.e. a propensity score is really just a predicted probability, and people often just assume that when propensity scores are calculated, a causal analysis follows).

Answer 1

I agree with you that the reviewer misunderstood how propensity score can be applied; your final parenthetical sums it up pretty well.

Propensity score methods are not causal inference methods; they are methods of estimating the contrast between expectations of conditional expectations, i.e.,

$$\tau_0 \equiv E[E[Y|A=1,X]] - E[E[Y|A=0,X]]$$

This is a purely statistical estimand and does not need to have a causal interpretation. $Y$ does not need to be an "outcome", $A$ does not need to be a treatment, and $X$ does not need to be a set of confounder sufficient to eliminate confounding.

Propensity score methods are one of many methods that allow you to estimate this quantity. They are often not the best methods, but they do allow for data reduction and can have some robustness to certain modeling assumptions. The reason they can be used to estimate this statistical quantity quantity is that the propensity score is a balancing score. It has nothing to do with the causal status or interpretation of any of the variables involved; it ability to estimate $\tau_0$ is purely statistical fact that can be derived from statistical principles. This purely statistical perspective on the propensity score is well articulated in section 11.3.5 of Judea Pearl's book Causality: Models, Reasoning and Inference.

Under the assumptions you mention (i.e., positivity, exchangeability, etc.), $\tau_0$ can be equal to the causal estimand $$ \tau \equiv E[Y^1]-E[Y^0] $$ where $Y^a$ is the potential outcome under treatment $A=a$. This connection between $\tau_0$ and $\tau$ is why propensity score methods are often used to estimate causal effects. But if one is not willing to invoke those assumptions, then $\tau_0$ can be interpreted on its own terms without any reference to causality as a purely statistical estimand.