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Can we interact the treatment variables with independent control variables that have been used for matching?

I have a binary treatment variable called 'medicine'. It indicates whether a person received a medicine.

I also have 5 control variables and 1 dependent variable 'days' - counting the days survived.

Using the MatchIt package in R, I match observations in the treatment group (medicine = 1) to similar observations in the control group (medicine=0) using CEM.

After matching I run a Poisson regression to estimate the effect of the medicine on days survived, while controlling for 5 variables.

Question 1: Can I interact the treatment variable (medicine) with the control variables in this matched sample?

Question 2: Assuming the treatment and control group have similar distributions across the control variables (well balanced), what does an interaction between a control variable and treatment tell me?

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You can interact treatment and control variables in the matched sample. There are a few reasons one might do this. One would be simply to adjust for the control variables beyond the adjustment afforded by the matching (i.e., as Ho, Imai, King, & Stuart, 2007, recommend). To do this, you would center you control variables before interacting them with the treatment. Then the coefficient on the treatment variable can be interpreted as a causal effect.

Another would be to examine treatment effect heterogeneity, or effect modification (i.e., does the treatment effect differ based on other covariates?). For this, interacting the treatment and the covariates only allows you to assess this if you have balance between the treated and control groups within levels of the interacting covariates. If you do not have this, the observed treatment effects within each level of the other covariates may be due to confounding.

If you do have balance within levels of the covariates, then an interaction between the covariates and the treatment tells you that the treatment effect varies along the covariate values. It doesn't tell you that the covariate is causing the treatment effect to vary; you would need to randomly assign the covariate to assess this phenomenon, which is called interaction (i.e., in contrast to effect modification). The interpretation of the coefficient for the treatment when an interaction is present in the model is the effect of treatment when the covariate takes a value of 0. The coefficient on the interaction term is the change in the treatment effect for a 1-unit increase in the covariate.

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