For multivariate functional data, a functional latent factor model is proposed, extending the traditional latent factor model for multivariate data. The proposed model uses unobserved stochastic processes to induce the dependence among the different functions, and thus, for a large number of functions, may provide a more parsimonious and interpretable characterization of the otherwise complex dependencies between the functions. Sufficient conditions are provided to establish the identifiability of the proposed model. The performance of the proposed model is assessed through simulation studies and an application to electroencephalography data.

Covariate-adaptive randomization methods are widely used in clinical trials to balance baseline covariates. Recent studies have shown the validity of using regression-based estimators for treatment effects without imposing functional form requirements on the true data generation model. These studies have had limitations in certain scenarios; for example, in the case of multiple treatment groups, these studies did not consider additional covariates or assumed that the allocation ratios were the same across strata. To address these limitations, we develop a stratum-common estimator and a stratum-specific estimator under multiple treatments. We derive the asymptotic behaviors of these estimators and propose consistent nonparametric estimators for asymptotic variances. To determine their efficiency, we compare the estimators with the stratified difference-in-means estimator as the benchmark. We find that the stratum-specific estimator guarantees efficiency gains, regardless of whether the allocation ratios across strata are the same or different. Our conclusions were also validated by simulation studies and a real clinical trial example.