# Anytime-Valid Linear Models and Regression Adjusted Causal Inference in Randomized Experiments

arXiv: https://arxiv.org/abs/2210.08589 · HTML: https://mlsanity.com/paper/2210.08589/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2022.

## Built on by (most cited first, 4 of 4)

- [Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance](https://mlsanity.com/paper/2310.03722/) (2023)
- [Anytime-valid Optimal Policy Identification](https://mlsanity.com/paper/2606.17515/) (2026)
- [GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function](https://mlsanity.com/paper/2602.08096/) (2026)
- [Agentic RCA for Internet-Scale Services Using Constrained Creativity](https://mlsanity.com/paper/2610.08622/) (2026)

## Abstract (arXiv)

Linear models are foundational tools in statistics and ubiquitous across the applied sciences. However, conventional statistical inference -- such as t-tests and F-tests -- are only valid at fixed sample sizes, making them unsuitable for sequential settings such as online A/B testing. We develop an anytime-valid theory of inference for the linear model, introducing sequential analogues of classical tests and confidence sets that provide Type-I error control and coverage guarantees uniformly over all sample sizes. Our construction is based on likelihood ratios of invariantly sufficient statistics, yielding simple closed-form expressions of ordinary least squares estimators and standard errors. The resulting tests are optimal in the GROW/REGROW sense for both frequentist and Bayesian alternative hypotheses. We then relax the linear model assumptions to provide heteroskedasticity-robust asymptotic sequential tests and confidence sequences, which enable sequential regression-adjusted inference for causal estimands in randomized controlled experiments. This formally allows experiments to be continuously monitored for significance, stopped early, and safeguards against statistical malpractices in data collection. We demonstrate the practical utility of our approach through simulations and applications to real A/B test data from Netflix.