# CUPED

## CUPED: Controlled-experiment using pre-existing data

CUPED (Controlled-experiment Using Pre-Existing Data) is a technique that uses user information from before an experiment to reduce variance and increase confidence in experimental metrics. At Statsig, this pre-experiment data is the 7 days before each user's exposure, rather than a fixed window before the experiment starts for all users. This approach helps debias experiments that have meaningful pre-exposure bias (for example, groups that were randomly different before any treatment).

The Cloud product uses a 7-day window for CUPED calculation. For Warehouse Native customers, Statsig recommends a 7-day window, but you can customize it to any length.

For more information, go to the [Variance Reduction](https://docs.statsig.com/experiments/statistical-methods/variance-reduction) page.

For an in-depth look at the methodology, refer to <a href="https://drive.google.com/file/d/1cLXDhwK4jcu5Oss2rVGHtp22LxVLLzBC/view?usp=sharing" target="_blank" rel="noreferrer">CURE by Statsig</a>.

## CUPED for simple aggregations

The original [Microsoft paper](https://www.exp-platform.com/Documents/2013-02-CUPED-ImprovingSensitivityOfControlledExperiments.pdf) and Statsig's in-depth [article](https://www.statsig.com/blog/cuped) on the technique describe the methodology for simple aggregations.

The Cloud product uses stratification alongside CUPED to account for users who may not have pre-experiment data. Statsig groups users into strata based on available pre-experimentation information. Statsig first estimates treatment and control effects within each stratum, then aggregates them to produce an overall result. Statsig then applies the standard difference-in-means and variance estimation. This approach retains users with missing pre-data while still benefiting from variance reduction where applicable.

## CUPED for ratio metrics

The Microsoft paper also gives details on how to implement CUPED for a different analysis unit (Appendix B). Statsig extends this to work for ratio metrics, where each experiment unit has a numerator and a denominator. The variance reduction process works 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 of a unit as $Y$, $N$, $X$, and $M$, respectively. Using the CUPED-reduced variance formula,

$$
Var(\frac\{Y\_\{cv}}\{N\_\{cv}})=Var(\frac\{Y}\{N})+\theta^2 Var(\frac\{X}\{M})-2\theta Cov(\frac\{Y}\{N}, \frac\{X}\{M})
$$

where Statsig finds the optimal $\theta$ as

$$
\frac\{Cov(\frac\{Y}\{N}, \frac\{X}\{M})}\{Var(\frac\{X}\{M})}
$$

expanded to
\\

$$
\frac\{Cov(\frac\{Y}\{\mu\_N}-\frac\{\mu\_Y N}\{\mu^2\_N}, \frac\{X}\{\mu\_M}-\frac\{\mu\_X M}\{\mu^2\_M})}\{Var(\frac\{X}\{\mu\_M}-\frac\{\mu\_X M}\{\mu^2\_M})}
$$

From this:

$$
\frac\{\hat\{Y\_\{c}}}\{\hat\{N\_\{c}}}=\frac\{Y\_\{c}}\{N\_\{c}}-\theta( \frac\{X\_\{c}}\{M\_\{c}} - \mathbb\{E}\[R])
$$

$$
\frac\{\hat\{Y\_\{t}}}\{\hat\{N\_\{t}}}=\frac\{Y\_\{t}}\{N\_\{t}}-\theta( \frac\{X\_\{t}}\{M\_\{t}} - \mathbb\{E}\[R])
$$

Because $\mathbb\{E}\[R]$ is difficult to derive and the expectation term is the same for both groups, Statsig substitutes $\mathbb\{E}\[R]$ with $\frac\{X\_\{c}}\{M\_\{c}}$, transforming the formulas above to:

$$
\frac\{Y\_\{cv}(control)}\{N\_\{cv}(control)}=\frac\{Y(control)}\{N(control)}
$$

$$
\frac\{Y\_\{cv}(test)}\{N\_\{cv}(test)} \\\\
:=\frac\{Y(control)}\{N(control)} - (\frac\{Y(control)}\{N(control)} - \theta \frac\{X(control)}\{M(control)}) + (\frac\{Y(test)}\{N(test)} - \theta\frac\{X(test)}\{M(test)}) \\\\
:=\frac\{Y(test)}\{N(test)} - \theta\frac\{X(test)}\{M(test)} + \theta \frac\{X(control)}\{M(control)}
$$

Using the optimal $\theta$, Statsig reduces group-level variance by plugging the parameter back in to calculate the adjustment. Across-group $\theta$ doesn't necessarily reduce variance for one group, or the sum of variances of all groups, but in most cases it does. Simulation shows that 98.3% of metrics saw a decrease with CUPED.

Statsig uses CUPED variance when all of the following conditions hold:

* Core assumptions of the CUPED model hold; rounding error or other data artifacts can violate this
  * E(X\_hat) = E(X)
  * The pooled variance of the adjusted population across groups is \< the variance of the unadjusted population
* Enough units have pre-experiment values (> 100)
* Enough percentage of units have pre-experiment values (> 5%)