Variance Reduction (CUPED)

Leverage pre-exposure data to reduce the variance and increase your velocity.

The variance of your metric plays an important role when analyzing an experiment. With a higher variance, you require more samples to separate the signal from the noise. A common approach to reduce the variance of a metric is to predict the current measurement using earlier measurements. If the earlier measurements come from before the start of the experiment, they can adjust for individual variation that the treatment itself doesn’t affect. Confidence lets you use historical metric values to reduce the variance.

Statistical comparisons, like when comparing metric outcomes between two or more groups in an experiment, are uncertain. Statistical theory describes this uncertainty so that it’s possible to conclude that one treatment was superior to another. A standard comparison of means overlooks the fact that, typically, a large chunk of the variation in the means is in fact predictable. Consider an experiment on users, where the metric of interest time spent in the app. Across users, the amount spent in the app over consecutive weeks is usually fairly highly correlated. This means that a user that spends three hours a day in the app this week is likely to spend a sizable amount of time in the app also next week.

Experimental studies often apply covariate or regression adjustment to reduce the variance and increase the precision. Deng et al (2013) popularized covariate adjustment in online experimentation. Their method is commonly referred to as CUPED. For the adjustment approach to be valid, the experiment must not influence the data used for the adjustment. In online experimentation, such data is often available. Anything computed before the unit, such as a customer, entered the experiment is valid. The stronger the pre-exposure data correlates with the post-exposure outcomes, the larger the reduction in variance.

Variance Reduction for Comparisons of Means

The variance reduction method implemented in Confidence for comparisons of means is the “full regression adjustment” estimator discussed by Negi and Wooldridge (2021). The method is a better and more precise method than the original CUPED approach. Negi and Wooldridge propose to:

• Regress $Y$ on $1, X$ separately for treatment and control, where $Y$ is the outcome, and $X$ is the pre-treatment variable.
• Estimate the treatment difference by $\hat{\Delta}_{VR} = (\bar{Y}_1-\bar{Y}_0)+(\bar{X}-\bar{X}_1)\hat{\beta}_1-(\bar{X}-\bar{X}_0)\hat{\beta}_0$, where $\hat{\beta}_i$ is the estimated slope from the regressions, $\bar{X}_i$ and $\bar{Y}_i$ are the sample means for each group, and $\bar{X}$ is the overall sample mean of $X$.

Confidence reports both the adjusted and unadjusted estimates of the sample means. The variance-reduced adjusted estimate is $\bar{Y}_i-\hat{\beta}_i(\bar{X}_i-\bar{X})$ for each group $i$, and the unadjusted estimate is $\bar{Y}_i$.

In experiments with multiple treatment groups, the control group’s variance-reduced adjusted mean estimate differs between comparisons. This happens because the overall sample mean of the pre-treatment variable, $\bar{X}$, uses only the data from the groups involved in the specific comparison. Depending on which treatment group you are comparing to the control group, $\bar{X}$ changes. Regardless, the variance-reduced treatment effect estimate $\hat{\Delta}_{VR}$ is generally a more precise estimate than the unadjusted estimate, providing a more reliable measure of the treatment’s impact on the metric of interest.

Variance Reduction for Comparisons of Ratios

The approach for reducing the variance through use of pre-exposure data resembles the method described in the earlier section when the metric of interest is a ratio metric. Confidence uses the method described by Jin and Ba (2023). Let $Y_i$ and $Z_i$ be the values for the numerator and denominator for unit $i$. For example, $Y_i$ could be the total number of searches for user $i$, and $Z_i$ their number of sessions. The ratio of interest is the group-level ratio $\sum_{i=1}^n Y_i/\sum_{i=1}^nZ_i$. A difference in ratios between the groups estimates the treatment effect:

$$\hat{\Delta} = \frac{\sum_{i\text { in treatment}}Y_i}{\sum_{i\text { in treatment}}Z_i} - \frac{\sum_{i\text { in control}}Y_i}{\sum_{i\text { in control}}Z_i}$$

To reduce the variance, Confidence applies regression adjustment separately to each of the four terms in the expression. Ultimately, this leads to adjusted estimates of the ratios in the two groups and a new estimate of the treatment effect. The adjustment reduces the variance of the ratios of the two groups, and the uncertainty surrounding the treatment effect is lower.

Variance Reduction Rate

Variance reduction adjusts the comparison and reduces the variance as a result. Ultimately, the variance reduction rate summarizes the size of the reduction:

$$\text{Variance reduction rate} = 1 - \frac{\text{variance with variance reduction}}{\text{variance without variance reduction}}.$$

Relative Values

When using variance reduction, reported relative values use the unadjusted estimates in the denominator. For a comparison of means, the reported relative value is $\hat{\Delta}_{VR}/\bar{Y}_0$.

Interpret Results with Variance Reduction

Interpret the reported treatment effect for a metric that uses variance reduction in the same way as for a metric that doesn’t. The pre-exposure data helps produce a better signal of the treatment effect. Because the variance-reduced treatment effect differs from the unadjusted treatment effect, it can occasionally lead to a different conclusion than the unadjusted effect.

Use Variance Reduction

App

Variance reduction is by default enabled.

To turn on variance reduction for a metric:

1. Go to Confidence and select Metrics on the left sidebar.
2. Select the metric that you want to turn variance reduction on for and click Edit metric, or create a new metric.
3. Expand Advanced options.
4. Ensure that Variance reduction is checked.
5. Optional. Select an aggregation window for the pre-exposure data.

The metric uses the same measurement, but before exposure, to reduce the variance. Variance reduction is accounted for in the required sample size calculation by relying on historical patterns of how much the variance can be reduced.

API

Use variance reduction by populating the adjustment fields in the data objects of:

• AnalysisData in the runAnalysis endpoint
• PowerAnalysisData in the runPowerAnalysis endpoint

References

• A. Deng, Y. Xu, R. Kohavi and T. Walker (2013) “Improving the sensitivity of online controlled experiments by utilizing pre-experiment data,” WSDM ‘13: Proceedings of the sixth ACM international conference on Web search and data mining.
• Y. Jin and S. Ba (2023) “Toward Optimal Variance Reduction in Online Controlled Experiments.” Technometrics.
• A. Negi and J. M. Wooldridge (2021) “Revisiting regression adjustment in experiments with heterogeneous treatment effects”, Econometric Reviews.

Related Resources

Metrics Reference

Configure variance reduction settings

Statistical Tests

Understand test types

Statistical Settings

Configure experiment parameters