Escaping the trap: Replacing the trapezoidal rule to better impute censored covariates with their conditional means

Clinical trials to test experimental treatments for Huntington's disease are expensive, so it is prudent to enroll subjects whose symptoms may be most impacted by the treatment during follow-up. However, modeling how symptoms progress to identify such subjects is problematic since time to diagnosis, a key covariate, can be censored. Imputation is an appealing strategy where censored covariates are replaced with their conditional means, the calculation of which requires estimating and integrating over its conditional survival function from the censored value to infinity. To flexibly estimate the survival function, existing approaches use the semiparametric Cox model with Breslow's estimator. Then, for integration, the trapezoidal rule is used, but the trapezoidal rule is not designed for indefinite integrals and leads to bias. We propose a conditional mean calculation that properly handles the indefinite integral with adaptive quadrature. Yet, even with adaptive quadrature, the integrand (the survival function) is undefined beyond the observed data, so we explore methods to extend it. In extensive simulation studies, we show that replacing the trapezoidal rule with adaptive quadrature corrects the bias seen with existing methods. We further illustrate how imputing with corrected conditional means can help prioritize patients for a new Huntington's disease trial.