The nonparametric Mann-Whitney-Wilcoxon (MWW) rank sum test is widely used to test treatment effect by comparing the outcome distributions between two groups, especially when there are outliers in the data. However, such statistics generally yield invalid conclusions when applied to nonrandomized studies, particularly those in epidemiologic research. Although one may control for selection bias by using available approaches of covariates adjustment such as matching, regression analysis, propensity score matching, and marginal structural models, such analyses yield results that are not only subjective based on how the outliers are handled but also often difficult to interpret. A popular alternative is a conditional permutation test based on randomization inference [Rosenbaum PR. Covariance adjustment in randomized experiments and observational studies. Statistical Science 2002; 17(3):286-327]. Because it requires strong and implausible assumptions that may not be met in most applications, this approach has limited applications in practice. In this paper, we address this gap in the literature by extending MWW and other nonparametric statistics to provide causal inference for nonrandomized study data by integrating the potential outcome paradigm with the functional response models (FRM). FRM is uniquely positioned to model dynamic relationships between subjects, rather than attributes of a single subject as in most regression models, such as the MWW test within our context. The proposed approach is illustrated with data from both real and simulated studies.

Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.