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I do not have enough points to comment so... Thank you Ben, you did interpret my question correctly. There are three treatment categories: control, A, and B. Thank you for clarifying that from Y=treatment+covariates, the coefficients of each A and B are the treatment effects relative to control. On the other hand if I wanted to measure the treatment effect of A relative to B, how would I do it? Lastly, does the statistical significance of either or both treatments imply the statistical insignificance of the control (baseline)? I am guessing not, but then how would I test whether control is significant if the intercept can be significant from accounting for the variations from missing variables? Thank you!

Original:

If there are two treatments (control, let's say A, and B) and I want to measure the effects of A relative to control, and A relative to B, am I to run regression:

Y=INTERCEPT+A, with subsetting data with Treatment B, and

Y=INTERCEPT+A, with subsetting data with Control?

If so, if I were to include all conditions in one regression (Y=INTERCEPT+A+B), what would be the interpretation of coefficients?

As you can probably tell, I am very new to RCT evaluation. Are there any nearly-mandatory robustness check that I should run?

Thank you so much!

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Without knowing more about your control and treatments, it sounds like you are dealing with an applied treatment giving rise to a categorical variable. When putting a categorical variable into a regression model you have an intercept term to deal with the base category and you have indicator variables to deal with the other categories. Typically the control group is set as the base category and the treatment groups are then given indicator variables. So, if we let $A_i$ and $B_i$ denote the treatment indicators for Treatments A and B for the $i$th data point, your model would look like this:

$$Y_i = \beta_0 + \beta_A A_i + \beta_B B_i + \text{other model terms} + \varepsilon_i.$$

When implementing regression analysis in most statistical software, there is usually an automated way to deal with categorical variables (also called "factor" variables in some programs). For example, if you were to implement this regression model in R you could do this with a single treatment variable (usually set as a character variable describing the treatment category) and your regression formula would look like this:

Y ~ factor(treatment) + other model terms

You should bear in mind that the regression analysis for an RCT is not fundamentally different to a regression analysis with any other categorical variable. In all these cases we would include all possible categories in the analysis and make comparisons between categories in the usual way. The main difference in an RCT is that we interpret the statistical results in a causal manner, based on randomisation of the applied treatment. The most important thing here is to ensure that your treatments were applied randomly, so that they are not affected by other covariates in the model. (There are a number of questions on this site on causal inference for RCTs, so I'd encourage you to search and read some of them.)

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  • $\begingroup$ Thank you so much for your answer! If I may ask a few follow up questions: Including both treatment A and B, the coefficient of A would not be relative to control but to control and A right? So if I wanted to coefficient of A to be the ATE solely with respect to control then should I exclude the sample that received treatment B? Also, to see the coefficient A relative to B, I should not exclude the control unit but run the regression you suggested? I am sorry if my questions are confusing... $\endgroup$
    – Mike J
    Commented Sep 16, 2022 at 4:13
  • $\begingroup$ I'm afraid I don't understand your follow-up question. I'm interpreting your question as having three treatment categories: Control, Treatment A, and Treatment B. Assuming that this interpretation is correct, the coefficients $\beta_A$ and $\beta_B$ in the above would both be effects relative to Control (i.e., relative to the baseline category in the regression). $\endgroup$
    – Ben
    Commented Sep 16, 2022 at 6:07

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