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CUPED - Controlled-experiment Using Pre-Existing Data

CUPED (short for Controlled-experiment Using Pre-Existing Data) is a technique which leverages user information from before an experiment to reduce the variance, and increase confidence in experimental metrics. This can help to debias experiments which have meaningful pre-exposure bias (e.g. the groups were randomly different before any treatment was applied).

See more at the Variance Reduction page.

CUPED for Ratio Metrics

CUPED for ratios metrics, where each experiment unit is represented by a numerator and a denominator. The variance reduction process is performed by finding the variance of experiment data, pre-experiment data, and the covariance between the two.

Denote the numerator, denominator, pre-experiment numerator, and pre-experiment denominator as YY, NN, XX, and MM, respectively. Using the CUPED-reduced variance formula,

Var(YcvNcv)=Var(YN)+θ2Var(XM)2θCov(YN,XM)\LARGE Var(\frac{Y_{cv}}{N_{cv}})=Var(\frac{Y}{N})+\theta^2 Var(\frac{X}{M})-2\theta Cov(\frac{Y}{N}, \frac{X}{M})

where optimal θ\theta is found as

Cov(YN,XM)Var(XM)\LARGE \frac{Cov(\frac{Y}{N}, \frac{X}{M})}{Var(\frac{X}{M})}

expanded to

Cov(YμNμYNμN2,XμMμXMμM2)Var(XμMμXMμM2)\LARGE \frac{Cov(\frac{Y}{\mu_N}-\frac{\mu_Y N}{\mu^2_N}, \frac{X}{\mu_M}-\frac{\mu_X M}{\mu^2_M})}{Var(\frac{X}{\mu_M}-\frac{\mu_X M}{\mu^2_M})}

Using the optimal θ\theta, we are hoping to reduce group-level variance by plugging the parameter back in to calculate the adjustment. Please note that across-group θ\theta does not necessarily reduce variance for one group, or the sum of variances of all groups, but in most cases it does. Our simulation shows that 98.3% of metrics saw a decrease by CUPED.

Statsig will use CUPED variance when all of the following are met:

  • CUPED reduces the total variance of all groups of a metric
  • Enough units have pre-experiment values (> 100)
  • Enough percentage of units have pre-experiment values (> 5%)