# Is matching only for treatment effects with selection bias?

Propensity score matching (and other matching techniques) are used, as far as I have seen, exclusively for identifying causal effects of a treatment (intervention) and particularly where there is a suspicion of selection bias.

However I do not see why they might not be useful for other cases of comparing two groups within a sample. E.g. I am looking at differences between rural-to-urban migrants and urban natives in education outcomes. It makes little sense to see this as a 'treatment' effect (although it might if I was comparing rural-to-urban migrants with rural natives), and causation is not exactly the issue. I want to understand whether there is a difference that can be explained by migration status separately from that explained by differences between migrants and urban natives in terms of wealth, parents' education and other observed variables.

This would usually be done using linear regression and ordinary least squares. But wouldn't there be some advantages to using a matching technique here? Specifically, doesn't OLS apply the same linear relationships among covariates to the whole sample, while matching could better allow for variation in those relationships? In this type of case, would matching perhaps be comparable to regression with lots of interaction effects allowed among the covariates?

(I am aware of this earlier question but everything I come across on matching seems to assume you are trying to evaluate causal effects of a treatment. It would be great to hear about any counter-examples to my claim that matching is only ever used for treatment effects, if there are any.)

• Given your description you are trying to evaluate causal effects, there is no difference in your situation to others you have seen referenced in literature. Commented Oct 1, 2012 at 17:37
• No, I'm comparing an individual who is born in A and migrates to B to someone born in B. Neither intuitively nor from the perspective of a causal model like Rubin's does it make sense to think of this as a causal treatment effect. Nevertheless it seems that the same methods can be useful. Commented Oct 8, 2012 at 16:52

Your presumption is not correct. In observational studies we may be looking for say differences in means between groups. There does not have to be a question of causality. the bias is simply due to non-randomness of the sample used. Propensity score matching and other match techniques are designed to eliminate that potential bias and eliminate confounding effects in a way akin to what randomization does. In case control studies the matching is used to pair cases with controls also.

• Thanks. For clarification: by my 'presumption' do you mean the claim that matching is only used for treatment effects? (To be clear, I want to use a matching technique, but am wondering why I can't find any previous studies that use it in a situation like mine.) Commented Oct 2, 2012 at 8:47
• Not exactly, you said that it is used exclusively for identifying causal effects of a treatment. My examples were meant to show that it is commonly used in studies such as case control studies where causal inference is not the goal. The idea is to correct the inference because of lack of randomization rather than to prove a cause and effect relationship. Don Rubin and Judea Pearl have formalized the concept of causal inference. In many statistical problems we have data that show that variables X and Y are correlation but that alone does not prove that X causes Y or vice versa. Commented Oct 2, 2012 at 9:58
• While I am not on top of the literature on causal inference it is my impression that Rubin developed and applied propensity scoring for applications in causal inference. But the studies I have used it for are just intended to show a difference due to a treatment and not necessarily "prove" a cause and effect relationship. Commented Oct 2, 2012 at 10:02

Having thought and read about this a bit more, here are the reasons that I think matching methods are not commonly used in studies where there is no specific treatment to be assessed and the aim is rather to compare the means of two groups differing in some characteristic while controlling for explanatory variables:

1. Most such studies are not only interested in a single comparison by one categorical variable; they are also interested in the effects of the covariates. It is therefore more appropriate to make a single model of the relationship between all of the postulated explanatory variables and the dependent variable, and test that using regression.
2. Missing explanatory variables may become more of a problem when using matching rather than fitting a specific model. In studies of treatment effects, the variation that is controlled out through matching is relatively straightforwardly interpreted as selection bias. But in studies comparing two groups differing in some characteristic there is not this interpretation, and matching might even worsen the confounding effect of some unobserved variable. In the original example, say that urban natives (A) are on average much wealthier than rural-urban migrants (B), but with some overlap, and we matched each A with a B at a similar wealth level, and didn’t have any control variables other than wealth in the data set. These ‘matched’ households would likely differ strongly in terms of missing variables, because an A at a wealth level typical for group A would have to be matched to a B that was well above average wealth for group B, and that difference might be due to an unobserved variable such as parents' education. That unobserved difference could then bias the estimated difference in terms of an outcome variable such as children's education attainment.
3. (maybe) Separate conventions and traditions used in the different types of study.

Is this about right? Maybe there are other reasons too.