LongiBand: Covariance-Robust Confidence and Conformal Prediction Bands for Longitudinal Treatment Effects

Repeatedly randomized longitudinal studies pose two distinct uniform-inference tasks: estimating a population treatment-effect surface and predicting a new subject’s latent conditional-mean treatment-effect curve. We introduce LongiBand, a framework that explicitly separates the targets, sampling units, and inferential guarantees associated with these tasks. For population inference, LongiBand combines treatment-specific series regressions fitted under working independence with a subject-block CR3 sandwich estimator and subject-level Gaussian-multiplier calibration. Under explicit approximation, leverage, moment, and grid conditions, the resulting simultaneous confidence band is asymptotically valid even when the within-subject covariance is misspecified. For personalized prediction, pooled functional principal component analysis estimates a joint two-arm eigensystem and uses the new subject’s observed history to predict the latent scores. Because the calibration curves are unobserved, subject-level recovery sets yield feasible conformal scores, while a truncation adjustment accounts for omitted components and extends protection to the full latent curve. Under explicit recovery-event conditions, the resulting band achieves marginal simultaneous coverage for an exchangeable future subject. For covariate-dependent FPCA, we further establish an exact-label finite-grid guarantee and show that recovered proxy labels require an additional recovery-gap condition. Simulations show that longer histories reduce prediction error and model-based band width. They also reveal substantial undercoverage when omitted components are ignored. In one representative setting, coverage increases from 0.776 under naive truncation to 0.940 after truncation adjustment. End-to-end recovery-set conformal bands achieve conservative coverage under Gaussian innovations and either Gaussian or heavy-tailed random effects, although they are substantially wider than infeasible latent-label bands, quantifying the cost of recovering unobserved calibration curves. An analysis of HeartSteps illustrates the population-level procedure.