Double robust conditional independence test for novel biomarkers given established risk factors with survival data.

Conditional independence is a foundational concept for understanding probabilistic relationships among variables, with broad applications in fields such as causal inference and machine learning. This study focuses on testing conditional independence, $T\perp X|Z$, where T represents survival data possibly subject to right censoring, Z represents established risk factors for T, and X represents potential novel biomarkers. The goal is to identify novel biomarkers that offer additional merits for further risk assessment and prediction. This can be achieved by using either the partial or parametric likelihood ratio statistic to evaluate whether the coefficient vector of X in the conditional model of T given $(X^{ \mathrm{\scriptscriptstyle \top } }, Z^{ \mathrm{\scriptscriptstyle \top } })^{ \mathrm{\scriptscriptstyle \top } }$ is equal to zero. Traditional tests such as directly comparing likelihood ratios to chi-squared distributions may produce erroneous type-I error rates under model misspecification. As an alternative, we propose a resampling-based method to approximate the distribution of the likelihood ratios. A key advantage of the proposed test is its double robustness: it achieves approximately correct type-I error rates when either the conditional outcome model or the working model of ${\rm pr} (X|Z)$ is correctly specified. Additionally, machine learning techniques can be incorporated to improve test performance. Simulation studies and the application to the Alzheimer's Disease Neuroimaging Initiative (ADNI) data demonstrate the finite-sample performance of the proposed tests.