  Linear regression adjustment is commonly used to analyse randomised
controlled experiments due to its efficiency and robustness against model
misspecification. Current testing and interval estimation procedures leverage
the asymptotic distribution of such estimators to provide Type-I error and
coverage guarantees that hold only at a single sample size. Here, we develop
the theory for the anytime-valid analogues of such procedures, enabling linear
regression adjustment in the sequential analysis of randomised experiments. We
first provide sequential $F$-tests and confidence sequences for the parametric
linear model, which provide time-uniform Type-I error and coverage guarantees
that hold for all sample sizes. We then relax all linear model parametric
assumptions in randomised designs and provide nonparametric model-free
sequential tests and confidence sequences for treatment effects. This formally
allows experiments to be continuously monitored for significance, stopped
early, and safeguards against statistical malpractices in data collection. A
particular feature of our results is their simplicity. Our test statistics and
confidence sequences all emit closed-form expressions, which are functions of
statistics directly available from a standard linear regression table. We
illustrate our methodology with the sequential analysis of software A/B
experiments at Netflix, performing regression adjustment with pre-treatment
outcomes.
