Anytime-Valid F-Tests for Faster Sequential Experimentation Through Covariate Adjustment

We introduce sequential $F$-tests and confidence sequences for subsets of coefficients of a linear model. This generalizes standard univariate Gaussian confidence sequences that are often used to perform sequential A/B tests. When performing inference on treatment effects, the ability to include covariate and treatment-covariate interaction terms reduces the stopping time of the sequential test and the width of the confidence sequences. Our sequential $F$-tests also have other practical applications concerning sequential tests of treatment effect heterogeneity and model selection. Our approach is based on a mixture martingale, using a Gaussian mixture over the coefficients of interest and the right-Haar mixture over the remaining model parameters. This exploits group invariance properties of the linear model to ensure that our procedure possesses a time-uniform Type I error probability of $\alpha$ and time-uniform $1-\alpha$ coverage for all values of the nuisance parameters. This allows experiments to be continuously monitored and stopped using data dependant stopping rules. While our contributions are to provide anytime-valid guarantees, that is, time-uniform frequentist guarantees in terms of Type I error and coverage, our approach is motivated through a Bayesian framework. More specifically, our test statistic can be interpreted as a Bayes factor, which can be readily used in a traditional Bayesian analysis. Our contributions can also be viewed as providing frequentist guarantees to the Bayesian sequential test.