Data analysis methods for the study of treatments or exposures in relation to a clinical outcome in the presence of competing risks have a long history, often with inference targets that are hypothetical, thereby requiring strong assumptions for identifiability with available data. Here data analysis methods are considered that are based on single and higher dimensional marginal hazard rates, quantities that are identifiable under standard independent censoring assumptions. These lead naturally to joint survival function estimators for outcomes of interest, including competing risk outcomes, and provide the basis for addressing a variety of data analysis questions. These methods will be illustrated using simulations and Women\'s Health Initiative cohort and clinical trial data sets, and additional research needs will be described.

There are several different topics that can be addressed with multivariate failure time regression data. Data analysis methods are needed that are suited to each such topic. Specifically, marginal hazard rate models are well suited to the analysis of exposures or treatments in relation to individual failure time outcomes, when failure time dependencies are themselves of little or no interest. On the other hand semiparametric copula models are well suited to analyses where interest focuses primarily on the magnitude of dependencies between failure times. These models overlap with frailty models, that seem best suited to exploring the details of failure time clustering. Recently proposed multivariate marginal hazard methods, on the other hand, are well suited to the exploration of exposures or treatments in relation to single, pairwise, and higher dimensional hazard rates. Here these methods will be briefly described, and the final method will be illustrated using the Women\'s Health Initiative hormone therapy trial data.

In this article we are interested in a general class of distributions for independent not necessarily identically distributed random variables, closed under minima, that includes a number of discrete and continuous distributions like the Geometric, Exponential, Weibull or Pareto. The main parameter involved in this class of distributions is assumed to be time varying with several possible modeling options. This is of particular interest in reliability and survival analysis for describing the time to event or failure. The maximum likelihood estimation of the parameters is addressed and the asymptotic properties of the estimators are discussed. We provide real and simulated examples and we explore the accuracy of the estimating procedure as well as the performance of classical model selection criteria in choosing the correct model among a number of competing models for the time-varying parameters of interest.

With the aim to improve the communication of trial results, we introduce a novel graphical approach that complements the analysis of time to event outcomes in two-arm randomized trials. We define the so-called two-sample survival probability curve and propose a nonparametric estimator of the curve based on a random walk using Kaplan-Meier survival estimates for the two arms. We then use the estimated curve to visualize treatment effect as well as potential effect modification of factors of interest. We also propose to estimate two-sample survival probability curves within the framework of the Cox model to graphically assess model fit. The proposed two-sample survival probability plot puts trials in a standardized [0,1] × [0,1] space, allowing for a simple visualization of the main effect, effect modification, and the adequacy of a model fit.

The Dabrowska (Ann Stat 16:1475-1489, 1988) product integral representation of the multivariate survivor function is extended, leading to a nonparametric survivor function estimator for an arbitrary number of failure time variates that has a simple recursive formula for its calculation. Empirical process methods are used to sketch proofs for this estimator\'s strong consistency and weak convergence properties. Summary measures of pairwise and higher-order dependencies are also defined and nonparametrically estimated. Simulation evaluation is given for the special case of three failure time variates.