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I would like to apply Coarsened Exact Matching (CEM) on my data to reduce the differences between the characteristics of the treated and those of the untreated. I was advised to do matching to reduce these differences and then to apply some econometric method (OLS for example) on the matched dataset.

I am quite confused about the variables I can include or not in the CEM and in the post-matching estimated equation:

  • can I include some variables that were used in matching in the post-matching estimated equation?

  • at the contrary, do I have to include all variables that may influence the outcome in the CEM?

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You can include variables that were used in the matching in the outcome regression model used to estimate the treatment effect. Ho, Imai, King & Stuart (2007) argue that you should perform in the matched dataset whatever analysis you would have performed in the original dataset, now with some added robustness to model misspecification. If you exactly matched on any of these variables, including the variables in the outcome regression will not change the effect estimate or standard error. It's a good idea to include covariates in the outcome model after using coarsened exact matching (especially continuous variables or collapsed categorical variables) because the matching will not completely eliminate the imbalance in these variables.

You do not have to match on all variables that affect the outcome to arrive at an unbiased or low-error estimate of the treatment effect. To address confounding, you need to adjust for (either through matching or regression) a sufficient set of variables, which can be determined using graphical criteria. See Elwert (2013) for a nice introduction. I also explain confounders in this post. As long as your conditioning strategies together adjust for the required variables (and you have got the set of required variables correct), you can have some confidence in the validty of your effect estimate as unbiased. You can match on some variables and then use regression on the others. This is generally not a recommended practice, though; you should attempt to use all methods available to adjust for variables you need to adjust for. Sometimes, however, you can incidentally achieve balance on some variables when matching on others. Some research has shown that there isn't much value in including covariates in a regression when the standardized mean difference is below 0.1 (Nguyen et al., 2017), though I wouldn't take that result too seriously.

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  • $\begingroup$ "It's a good idea to include covariates in the outcome model after using coarsened exact matching [...] because the matching will not completely eliminate the imbalance in these variables." I thought that the aim of the CEM was to completely eliminate the imbalance whatever the number of observations left (contrary to the PSM). I've done some matching with {cem} R package and I had no imbalance at all after matching but what you said is making me doubt, is it possible to have no imbalance left after CEM? If so, I guess that using matching variables in the outcome regression model is useless? $\endgroup$
    – bretauv
    Commented May 1, 2020 at 8:17
  • $\begingroup$ @bretauv My understanding is there will usually be some residual post-matching imbalance if you're grouping a continuous variable into bins. For example, if you've created age categories of 18-29, 30-39, 40-49, etc. years of age in CEM, you will group subjects of 18 and 29 years of age into the same age category for matching purposes. Without post-matching regression you're averaging any differences in the outcome variable due to the age difference of 11 years. $\endgroup$
    – RobertF
    Commented Jul 22, 2020 at 18:55
  • $\begingroup$ In order to avoid the Table 2 fallacy, I'm assuming the post-matching regression coefficients for secondary variables cannot be interpreted as direct effects on the outcome variable after controlling for other variables, is that correct? $\endgroup$
    – RobertF
    Commented Sep 5, 2020 at 19:31
  • $\begingroup$ That's correct, because they may still be seriously confounded. $\endgroup$
    – Noah
    Commented Sep 5, 2020 at 19:54

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