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With covariate adjustment. Including covariates in the outcome model is straightforward with a continuous outcome. We can include main effects for each variable and use the coefficient on the treatment as the treatment effect estimate. It can also be helpful to include interactions between the covariates and treatment if effect modification is suspected, though it is important to center the covariates at their means in the focal (i.e., treated) group if the coefficient on the treatment is to be interpreted as an average treatment effect estimate (which we demonstrate with full matching). A marginal effects procedure can also be used, which we demonstrate with binary outcomes. Below we simply include main effects of each covariate, which is the most typical way to include covariates.

#Linear model with covariates
fit3 <- lm(Y_C ~ A + X1 + X2 + X3 + X4 + X5 + 
             X6 + X7 + X8 + X9, data = md,
           weights = weights)

coeftest(fit3, vcov. = vcovCL, cluster = ~subclass)["A",,drop=FALSE]

##   Estimate Std. Error t value  Pr(>|t|)
## A    2.029     0.3202   6.338 3.735e-10

As previously mentioned, it is important to avoid interpreting coefficients in the outcome model on predictors other than the treatment, as the coefficients do no correspond to causal effects and are likely confounded. In the code above, we restricted the output to just the treatment effect. Note that sometimes the intercept of the outcome model can be useful as an estimate of the average potential outcome under control, but the covariates in the model (if any) must be centered at their means in the target population to allow for this interpretation. Without centering the covariates, the intercept is uninterpretable.

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