Anytime-Valid Inference in Linear Models with Clustered Outcomes: Exact Multivariate Tests and Cluster-Robust Confidence Sequences

Abstract: We extend anytime-valid inference for linear models from independent scalar outcomes to units that contribute a vector of dependent outcomes. We develop a parametric and a nonparametric treatment. For the Gaussian multivariate linear model with an unstructured error covariance, we derive a closed-form Bayes factor under the right-Haar prior and a matrix-normal mixture alternative and prove that it is an exact e-value at every sample size and a test martingale in the invariant filtration. To our knowledge, this Bayes factor is new in the generality considered here: a d x q coefficient matrix, nuisance regressors and an arbitrary fixed positive-definite prior precision under the right-Haar nuisance prior. It yields sequential Hotelling and MANOVA tests with exact confidence sequences. Under a common within-cluster design, the cluster-robust Wald statistic of a pooled regression is equal to Hotelling's T^2 statistic computed from the per-cluster least-squares summaries, giving an exact Gaussian benchmark with G-1 degrees of freedom. For general clustered population least-squares projections, the corresponding sandwich statistic is an asymptotic e-process, without Gaussian, common-design or linearity assumptions. Each treatment includes growth and first-order expected stopping-time results; under stronger regularity, direct comparisons of expected log evidence identify a fixed mixture precision that maximizes expected log growth through constant order within a specified family, for a specified frequentist alternative. Applied to randomized experiments with several outcomes, the robust procedure gives asymptotically anytime-valid joint tests, confidence ellipsoids and simultaneous confidence sequences for the vector of average treatment effects from ordinary least-squares estimates and cluster-robust standard errors.