Learning the Covariance of Treatment Effects Across Many Weak Experiments

When primary objectives are insensitive or delayed, experimenters may instead focus on proxy metrics derived from secondary outcomes. For example, technology companies often infer the long-term impacts of product interventions from their effects on short-term user engagement signals. We consider the meta-analysis of many historical experiments to learn the covariance of treatment effects on these outcomes, which can support the construction of such proxies. Even when experiments are plentiful, if treatment effects are weak, the covariance of estimated treatment effects across experiments can be highly biased. We overcome this with techniques inspired by weak instrumental variable analysis. We show that Limited Information Maximum Likelihood (LIML) learns a parameter equivalent to fitting total least squares to a transformation of the scatterplot of treatment effects, and that Jackknife Instrumental Variables Estimation (JIVE) learns another parameter computable from the average of Jackknifed covariance matrices across experiments. We also present a total covariance estimator for the latter estimand under homoskedasticity, which is equivalent to a k-class estimator. We show how these parameters can be used to construct unbiased proxy metrics under various structural models. Lastly, we discuss the real-world application of our methods at Netflix.