How to calculate confidence interval and p-value for percent change of treatment relative to control?

by Amazonian   Last Updated October 20, 2019 01:19 AM

I'm analyzing the result of an experiment where the dependent variable is a count variable (# of purchases), and the unit of observation is an individual. The way I'm calculating the treatment effect $\beta$ is

$Y_i = \alpha + \beta*1(treatment_i) + \gamma X_i+\epsilon_i$

Running this regression will give me a p-value $p$ and a 95% confidence interval $[\underline{\beta}, \bar\beta]$ for $\hat\beta$.

The interpretation of $\hat \beta$ is the average increase in # of purchases per user as the resutl of the treatment. But what I'm really interested in is how this treatment affected # of purchases in terms of percent change relative to control; i.e., I want to be able to say something like, "this treatment increased # of purchases per user by 10%".

I was thinking this is a straight forward application of Slutsky's theorem where the p-value for % change relative to control treatment effect is still the same as the p-value I got from the regression, and the confidence interval for percent change relative to control treatment effect is $[\frac{\underline{\beta}}{\bar Y^c}*100, \frac{\bar\beta}{\bar Y^c}*100]$, where $\bar Y^c$ is the average # of purchases per user from those in the control group.

Is this correct? If not, can someone explain why and the correct solution?

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