U-statistics-based Pairwise Discordance Regression for Joint Mean-Variability Inference in Longitudinal Studies with Missing Data

Longitudinal studies are central to scientific inquiry across disciplines; yet, statistical analyses have predominantly focused on mean trajectories while treating variability as a secondary or nuisance component. Increasing evidence suggests that temporal variability and heterogeneity themselves encode critical scientific information, particularly in studies with repeated measurements, heterogeneous populations, or rich covariate structures. The classical mixed-effects and location-scale model family represents variability through latent variable components but relies on strong parametric assumptions and often requires numerical approximation or computationally intensive estimation. Here we introduce the semiparametric pairwise discordance regression, a unified framework for longitudinal data that directly models pairwise contrasts to simultaneously characterize mean trajectories, between-subject heterogeneity, and within-subject temporal variability. By targeting pairwise discordances rather than latent variance components, the proposed approach enables robust inference without specifying full parametric distributions for latent components or error terms, provided the specified conditional moment restrictions hold. Pair-level covariate encodings further accommodate flexible covariate structures and allow variability parameters to be interpreted through pairwise contrasts or dissimilarities. Estimation and inference are carried out via (weighted) U-statistic-based generalized estimating equations (UGEE) with a valid sandwich variance estimator, and we extend the fitting procedure to accommodate missing-at-random dropout. We establish the asymptotic distributions of the resulting estimators under mild regularity conditions. Simulation studies demonstrate reliable finite-sample performance in the settings considered, including non-Gaussian errors and misspecified working correlation structures, whereas likelihood-based parametric fits can be sensitive to distributional assumptions or numerically unstable. An application to longitudinal blood pressure during pregnancy illustrates the framework on real data.