# Propensity Score Matching

What is the right approach when there is a variable that does not affect participation but effects the outcome measure? I have data set of the health outcome of a treatment and control group. I have data on 3 covariates. 2 demographic and one is a baselines score. I believe that base line score did not affect the participation decision. Hence including it estimating the propensity score will be wrong. But measuring ATT by ignoring it will be ignoring data.

• 'baseline' means an individual outcome measurement that is taken before the beginning of the experiment? Jun 7, 2015 at 10:56
• Keep in mind you can do a lot of things with propensity scores, e.g. compute differences, or use them as weights in a ordinary regression that has other variables in them, etc. What ATT estimation strategy are you expecting to use? Jun 7, 2015 at 11:04
• Hi ,i wanted to use kernel matching for ATT estimation. Can you please provide a link to where i can learn about using pscores as weights. Jun 7, 2015 at 14:26
• There's a discussion in section 5.3.3 of Morgan and Winship (2007), among other places. Jun 7, 2015 at 17:11

## 2 Answers

You do propensity score matching on the covaritates to create a quasi-randomized study. You only need to match on the covariates that are different in the study groups (treatment vs. control) and affect the treatment decision (as the study is NOT randomized which is pracitically done by a fair coin toss or binary algorithm (0 or 1)). As a matter of fact the propensity score is defined as the conditional probability of treatment given confounding covariates:

propscore(x) = Pr(T=1 | X=x) (1)

Let O(C) and O(T) represent the potential outcomes under control and treatment. Then treatment assignment is conditionally unconfounded if potential outcomes are statistically independent of treatment conditional on confounding covariates X. This can be written as

O(C), O(T) ⊥ T│X (2)

where ⊥ denotes statistical independence. My question to you is: How do you know that the baseline covariate did not affect treatment decision? I don't understand how you can be sure that this is the case. I also don't think that estimating the propensity score using the baseline covariate would be wrong. So my recommendation is: Use the baseline covariate in the propensity score matching process. As you already mentioned, there's a general principle in statistics: The more data the better (I call it the TMDTB axiom). It would be interesting to know on which scale the covariates are and what exactly they are. Then I could give you some more tips. I hope this helps a bit.

Philipp

• The 'TMDTB axiom' is inapplicable here. Whether to put a variable in a propensity score mode is a causal identification question, not a statistical estimation question. Here, if the questioner's assumption is correct that a variable causally affects the outcome but not the participation then it will be useful for making the ATT more precise (a statistical issue) but not relevant for removing confounding (a causal inference issue). Hence there is no reason to put it in the propensity score model, but some reason to use it elsewhere. Jun 7, 2015 at 10:54
• To be clear, your suggestion is that it might be a confounder so to put it in the p-score model. This is very reasonable if it does indeed block a backdoor path to the outcome. The point is simply that the axiom is generally false because such variables may also open such a path or make one misidentify a total effect, depending on the graph structure being assumed. Jun 7, 2015 at 11:02
• I am sure because i the person who did the allocation of treatment tried to only make sure that the means of the demographic variables are as close as possible.Hence i know this about the data generating process. Albiet, the matching was not perfect. I have been trying to find out what to do with variables that do not affect treatment decision but affects outcome. I am unable to get any conclusive advice. Thanks Jun 7, 2015 at 14:30
• Those variables will make your estimates more precise, so condition on them somehow, e.g. using a regression model. Remember that p-scores are just a conditioning tool themselves, albeit in the lower dimensional slice of variation that affects treatment assigment and without the implicit conditional variance weighting that regression models will generate. Morgan and Winship (2007) has a reasonable discussion, and there a doubtless other newer ones. Jun 7, 2015 at 17:17
• The TMDTBPIAACNACOIV axiom (The More Data The Better Provided Its About A Confounder Not A Collider Or Intervening Variable) seems somehow less pithy. Jun 7, 2015 at 20:48

According to Cuong (2013)'s "Which covariates should be controlled in propensity score matching?..." all those variables should be part of the psm calculation which affect program participation and outcome but not those which only affect outcome but not program participation.

• Could you please give a full citation for the paper? Nov 16, 2015 at 10:57
• But since you don't know and the data may be incapable of telling you, propensity scores typically include the kitchen sink. And note there are problems with matching as opposed to covariate adjustment for logit propensity. Apr 21, 2017 at 15:29