My regression is of the form:

$$y = \beta_{0} + \beta_{1} Treat + \beta_{2} X + u$$

The $y$ variable is normalized. The $Treat$ variable is a binary dummy.

How should I interpret $\beta_{1}$ or the effect of the treatment? Is it something like, "getting the treatment changes $y$ by t standard deviations" where $t=\beta_{1} / \sigma_{y}$?



The model predicts that the treatment group scores $\beta_1$ standard deviations higher on the (unstandardized) $y$ variable than the non-treatment group, controlling for the $X$ variable. No need to divide $\beta_1$ by $\sigma_y$ if $y$ is already standardized and no need to divide $\beta_1$ by $\sigma_{treat}$ if $treat$ is a dummy code.

  • $\begingroup$ Thanks so much, @Jeffrey! Can I write is as "getting the treatment changes y by β1 standard deviations"? $\endgroup$
    – Jerry
    Apr 28 '21 at 4:05
  • $\begingroup$ The word "changes" implies a causal effect, which you may or may not have evidence of (I would need more information.) Safer to just describe the group difference as I did. $\endgroup$ Apr 28 '21 at 4:07
  • $\begingroup$ Ahh, thanks! Yes, the model is written in a way that it's to explain the causality between Treat and y. Treat is exogeneous and most variables affected y are controlled for. Thanks again! $\endgroup$
    – Jerry
    Apr 28 '21 at 4:09
  • $\begingroup$ Also note that my answer assumes that by "normalized" you mean standardized or z-scored by subtracting the sample mean and then dividing by the sample SD. $\endgroup$ Apr 28 '21 at 4:16

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