Linear models are commonly used in causal inference for the analysis of experimental data. This is motivated by the ability to adjust for confounding variables and to obtain treatment effect estimators of increased precision through variance reduction. There is, however, a replicability crisis in applied research through unknown reporting of the data collection process. In modern A/B tests, there is a demand to perform regression-adjusted inference on experimental data in real-time. Linear models are a viable solution because they can be computed online over streams of data. Together, these motivate modernizing linear model theory by providing ``Anytime-Valid'' inference. These replace classical fixed-n Type I error and coverage guarantees with time-uniform guarantees, safeguarding applied researchers from p-hacking, allowing experiments to be continuously monitored and stopped using data-dependent rules. Our contributions leverage group invariance principles and modern martingale techniques. We provide sequential $t$-tests and confidence sequences for regression coefficients of a linear model, in addition to sequential $F$-tests and confidence sequences for collections of regression coefficients. With an emphasis on experimental data, we are able to relax the linear model assumption in randomized designs. In particular, we provide completely nonparametric confidence sequences for the average treatment effect in randomized experiments, without assuming linearity or Gaussianity. A particular feature of our contributions is their simplicity. Our test statistics and confidence sequences have closed-form expressions of the original classical statistics, meaning they are no harder to use in practice. This means that published results can be revisited and reevaluated, and software libraries which implement linear regression can be easily wrapped.

翻译：线性模型在因果推断中常用于实验数据的分析，其动机在于能够调整混杂变量，并通过方差缩减提高处理效应估计的精度。然而，由于数据收集过程的不透明报告，应用研究中存在可重复性危机。在现代A/B测试中，需要实时对实验数据进行回归调整推断。线性模型是一种可行的解决方案，因为它可以在数据流上在线计算。这些因素共同推动了线性模型理论的现代化，即提供“任意有效”推断。这用时间一致保证取代了经典固定样本的第一类错误率和覆盖保证，从而保护应用研究者免受p值操纵的影响，允许实验使用数据依赖规则进行持续监控和终止。我们的贡献利用群不变性原理和现代鞅技术。我们提供了线性模型回归系数的序贯t检验和置信序列，以及回归系数集合的序贯F检验和置信序列。针对实验数据的特殊性，我们能够在随机化设计中放宽线性模型假设。特别地，我们为随机实验中的平均处理效应提供了完全非参数的置信序列，无需假设线性或高斯性。我们贡献的一个显著特点是其简洁性。我们的检验统计量和置信序列具有原始经典统计量的闭式表达式，这意味着它们在实践中同样易于使用。这也意味着已发表的结果可以被重新审视和评估，并且实现线性回归的软件库可以轻松封装。