Catching Copulas in a Lie: Goodness-of-Fit Testing for Covariate-Dependent Copulas under Censoring

Many important questions in medicine, engineering, and public health involve the timing of several related events, such as when a tumor progresses and when a patient dies, or when different components of a machine fail in sequence. Understanding how these events are related, and how those relationships change with factors such as age, treatment, or operating conditions, is essential for making reliable predictions and informed decisions in machine learning and artificial intelligence. Researchers often study these relationships using copula models, which describe how the timing of one event is linked to another. However, a major challenge is that complete event histories are often unavailable because studies may end before all events occur or because some events are observed only between scheduled assessments. Despite their widespread use, researchers currently lack reliable tools for determining whether copula models accurately represent the observed data. This project develops new statistical methods that allow scientists and engineers to validate these models before drawing scientific conclusions that may ultimately inform patient care, product safety, and public policy. By improving the reliability of quantitative analyses across medicine, engineering, and public health, the project advances the national interest in scientific discovery, health, and economic well-being. The project will also train graduate and undergraduate students in modern statistical methodology, integrate its findings into graduate education, and produce freely available open-source software that makes these methods broadly accessible to researchers and practitioners. This project develops the first unified statistical framework for assessing the goodness-of-fit of covariate-dependent copula models when event times are subject to censoring. Copula models are widely used in multivariate survival analysis because they separate the marginal behavior of individual event times from the dependence among events, while conditional copulas allow that dependence to vary with covariates. Existing methods cannot determine whether the assumed copula family is appropriate when dependence varies with covariates and event times are censored, including in semi-competing risks settings. The research addresses this gap through three methodological advances. First, it establishes conditional information-matrix equivalence for covariate-dependent copulas under right- and interval-censoring and extends this identity to semi-competing risks, where the dependence structure is only partially identifiable. These theoretical results yield conditional moment restrictions that hold only under correct copula specification and provide the analytical foundation for the goodness-of-fit tests developed in this project. Second, the project constructs test statistics using orthogonalized influence functions and cross-fitting to remove first-order sensitivity to slowly converging nuisance estimators, thereby enabling statistically valid and efficient inference under flexible semiparametric modeling. Third, it combines single-index dimension reduction with sieve approximation to achieve flexible nonlinear modeling, built-in regularization, and stable estimation in moderate-dimensional covariate settings. The resulting methodology provides rigorous diagnostic tools for multivariate event-time data under independent censoring and semi-competing risks, together with complementary diagnostics based on martingale residuals. These methods will be disseminated through peer-reviewed publications and an open-source R package with diagnostic visualizations and interactive tools for researchers in biomedicine, engineering, and public health. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.