Partial-linear single-index Cox regression models with multiple time-dependent covariates

In cohort studies with time-to-event outcomes, covariates of interest often have values that change over time. The classical Cox regression model can handle time-dependent covariates but assumes linear effects on the log hazard function, which can be limiting in practice. Furthermore, when multiple correlated covariates are studied, it is of great interest to model their joint effects by allowing a flexible functional form and to delineate their relative contributions to survival risk.

Motivated by the World Trade Center (WTC)-exposed Fire Department of New York cohort study, we proposed a partial-linear single-index Cox (PLSI-Cox) model to investigate the effects of repeatedly measured metabolic syndrome indicators on the risk of developing WTC lung injury associated with particulate matter exposure. The PLSI-Cox model reduces the dimensionality of covariates while providing interpretable estimates of their effects. The model's flexible link function accommodates nonlinear effects on the log hazard function. We developed an iterative estimation algorithm using spline techniques to model the nonparametric single-index component for potential nonlinear effects, followed by maximum partial likelihood estimation of the parameters.

Extensive simulations showed that the proposed PLSI-Cox model outperformed the classical time-dependent Cox regression model when the true relationship was nonlinear. When the relationship was linear, both the PLSI-Cox model and classical time-dependent Cox regression model performed similarly. In the data application, we found a possible nonlinear joint effect of metabolic syndrome indicators on survival risk. Among the different indicators, BMI had the largest positive effect on the risk of developing lung injury, followed by triglycerides.

The PLSI-Cox models allow for the evaluation of nonlinear effects of covariates and offer insights into their relative importance and direction. These methods provide a powerful set of tools for analyzing data with multiple time-dependent covariates and survival outcomes, potentially offering valuable insights for both current and future studies.