# CUPED - estimation of correlation with pre treatment covariates

I came across this variance reduction technique https://www.exp-platform.com/Documents/2013-02-CUPED-ImprovingSensitivityOfControlledExperiments.pdf

the main idea is using correlation with a pretreatment covariate

$$\hat{Y}cv = \bar{Y} - \theta (\bar{X} - E(X))$$

my confusion is on how to estimate below

$$\theta = \frac{\text{cov}(X, Y)}{\text{var}(X)}$$

$$\rho = Corr(Y,X)$$

if we use Y (post treatment) ,X (pre treatment) from experiment results, they are a random variable so $$\theta$$ would also be a random variable. moreover, the treatment and control populations could have different correlations.

what makes sense to me is to get the correlation from pre experiment data where we use consecutive time periods before the experiment and assume it will hold in the experiment period too.

• why 𝜃 is a random variable is a problem? we can use the estimated value from the pooled population of control and treatment.
– wei
Commented Mar 14, 2022 at 7:55
• Take an example of ecommerce where exposure to some change(treatment) causes some users to increase spend compared to earlier and others to decrease to spend compared to earlier. the pooled estimate is influenced by the treatment. pooling reminds me of the 2 sample t test we check if the variances are similar before taking a pooled variance. in that case we estimate them separately. Commented Mar 14, 2022 at 9:51
• in this case, it seems only influences the variance of 𝜃 , not the point estimate
– wei
Commented Mar 14, 2022 at 10:20