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I am analyzing data that are originating from an randomized experiment and I am new to this. There was one control groups and three different treatment groups. The treatment groups are discrete and unordered (you can think of them as of three completely different drugs).

I want to analyze the data employing an regression.

At first I subseted the data (so that only one treatment group and the control group were left) and analyzed the following:

$y_i=\alpha+\beta T_i + error_i$

where $T_i$ is a dummy indicating whether an individual received the treatment. In this case, $\beta$ corresponds to the average causal effect of the treatment.

I have a question: Would, instead of subsetting, also the following expression feasible?

$y_i=\alpha + \beta_1 T_{1,i} + \beta_2 T_{2,i} + \beta_3 T_{3,i} + error_i$

where $T_{1,i}$, $T_{2,i}$, $T_{3,i}$ are dummies indicating whether an individual received the first, second or third treatment.

Best regards

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  • $\begingroup$ Yes, the model with 3 dummy variables is correct. $\endgroup$ – user158565 Nov 5 '18 at 14:36
  • $\begingroup$ However, the dummies are correlated right? This might lead to bigger standard errors? However, due to the greater subset, the estimate might get more precise. I don't know which of these effects might be stronger :O $\endgroup$ – FeldO Nov 6 '18 at 13:08
  • $\begingroup$ You have 4 groups (one control groups and three different treatment groups.), so 3 dummy variables will not correct collinearity (correlation). If you create 4 dummy variables, collinearity will appear. After you fit the model, check the 3 estimated 3 regression coefficients to see which drug is the best. $\endgroup$ – user158565 Nov 6 '18 at 18:44
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Oneway ANOVA was designed specifically for this task, so you might want to look there as well. Essentially this involves running an omnibus test of whether all the treatment group means are the same, and, if this hypothesis is rejected, performing pairwise comparisons with a control for alpha (e.g., Scheffe's test) to determine which groups differ from each other.

If you are determined to use regression, you should include all the treatment dummies and all the data points. The reason for doing this is to increase the precision of your estimate of the mean squared error, which is part of the test statistic for each group comparison. One problem with regression is that the dummy codes only provide tests of each treatment against the reference (i.e., control) group. If you want to make other comparisons, you need to perform additional Wald tests for the differences in coefficients (this is straightforward in Stata using test). If you perform many Wald tests, your Type I error rate increases unless you correct for it. Many ANOVA procedures have this built in, which is why I recommend it over regression unless your only question is whether each treatment group differs from control.

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