It's integral: Replacing the trapezoidal rule to remove bias and correctly impute censored covariates with their conditional means

Modeling symptom progression to identify informative subjects for a new Huntington's disease clinical trial is problematic since time to diagnosis, a key covariate, can be heavily censored. Imputation is an appealing strategy where censored covariates are replaced with their conditional means, but existing methods saw over 200% bias under heavy censoring. Calculating these conditional means well requires estimating and then integrating over the survival function of the censored covariate from the censored value to infinity. To flexibly estimate the survival function, existing methods 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 improper integrals and leads to bias. We propose calculating the conditional mean with adaptive quadrature instead, which can handle the improper integral. Yet, even with adaptive quadrature, the integrand (the survival function) is undefined beyond the observed data, so we identify the "Weibull extension" as the best method to extrapolate and then integrate. In simulation studies, we show that replacing the trapezoidal rule with adaptive quadrature and adopting the Weibull extension corrects the bias seen with existing methods. We further show how imputing with corrected conditional means helps to prioritize patients for future clinical trials.