# Do propensity scores reflect the probability of treatment or outcome?

I am a non-statistician PhD student working on a project that has involved some propensity score matching (PSM). I had initially assumed that propensity scores would represent the probability of each patient receiving the outcome of interest given their baseline characteristics. However, everything I read subsequently suggested that propensity scores represent the probability of each patient being allocated to a treatment group.

This was my new understanding of PSM until I completed the analysis and a senior statistician on the project commented that I have misunderstood PSM altogether as propensity scores should be calculated to represent the probability of each patient receiving the outcome of interest.

I am not sure how to square this with my reading around PSM, although I am also conscious that I cannot easily read many of the technical papers in the field. Is anyone here able to help me understand whether propensity scores should represent the probability of treatment or outcome, and how confident I should be when replying to this statistician?

• It's supposed to be the treatment. In a true experiment, you know why a patient got the active treatment--it's because you flipped a coin & it came up heads. The idea is that you can replicate this w/ observational data: if you have propensity scores for the probability someone will be on the treatment, & they are right, then you can form matched sets & w/i those sets you know why some people were on the treatment--nature flipped a coin & it came up heads. This knowledge of the true causal structure leading to the treatment can be leveraged to allow for valid causal inference. – gung - Reinstate Monica Apr 6 '18 at 19:00
• A similarly named value, the "prognostic score", is the probability of experience the outcome of interest. There has been some research into using the prognostic score in a similar way that you might use the propensity score. – Noah Apr 11 '18 at 19:13

The propensity is for treatment assigned, not outcome.

While there are natural situations where propensity strongly mimics randomization, there are more scenarios where treatment is determined in the most non-random ways possible. Given a sufficiently large sample, searches for the probability of treatment assignment will be successful. If treatment assignment can be perfectly determined from the data, those variables should be scrutinized as likely representing treatment bias (guilty until proven otherwise). If the propensity is measuring the latent variable of disease severity, the underlying estimates obtained from propensity matching or regression are likely biased.

As both others have said, propensity scores represent the probability of receiving treatment. From the Stata manual for its native propensity score matching command (emphasis mine):

Propensity-score matching uses an average of the outcomes of similar subjects who get the other treatment level to impute the missing potential outcome for each subject. The ATE is computed by taking the average of the difference between the observed and potential outcomes for each subject. teffects psmatch determines how near subjects are to each other by using estimated treatment probabilities, known as propensity scores. This type of matching is known as propensity-score matching (PSM).

So, propensity score matching is used to calculate the average treatment effect or the average treatment effect among the treated, but it does so by matching individual observations on the propensity score. Which, as you see above, is the probability of receiving treatment.

Now, do note that you can use propensity scoring with a continuous or binary outcome of interest (or count, or whatever else you can imagine). Maybe the outcome in your case is binary and this is the source of the misunderstanding? Either way, the propensity score itself is, as has been said ad nauseam, the probability of receiving treatment, and if the senior statistician seriously thinks that it is the probability of receiving the outcome, then this person is not qualified to be a senior statistician. I'm betting on a misunderstanding.

The propensity score was developed for the most part by Donald Rubin. Here's the abstract to his 1983 paper with Rosenbaum from Biometrika. You don't need a PhD to understand it.

The propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates. Both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates. Applications include: (i) matched sampling on the univariate propensity score, which is a generalization of discriminant matching, (ii) multivariate adjustment by subclassification on the propensity score where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations, and (iii) visual representation of multivariate covariance adjustment by a two- dimensional plot.

PAUL R. ROSENBAUM, DONALD B. RUBIN; The central role of the propensity score in observational studies for causal effects, Biometrika, Volume 70, Issue 1, 1 April 1983, Pages 41–55, https://doi.org/10.1093/biomet/70.1.41

There is a strong connection between propensity scores and confounding adjustment. Confounders predict the outcome and receipt of treatment$^1$, so candidate factors which are confounders are subsets of candidates propensity factors. Thus, when you select covariates for developing a propensity score, it is often the case that they also predict the outcome. That's not surprising. Compare cancer treatments vs. survival. People with advanced cancers may opt for more aggressive treatment, so when you compare survival, cancer stage at diagnosis is a very important confounder.

$^1$ they're a bit more subtle than that, see Pearl, Causality 2nd edition.