A quantile cure model with partially functional covariate effects.

Quantile regressions offer several attractive features, including the ability to allow covariate effects to vary at different quantile levels and to effectively handle heteroscedasticity in data, which makes it a viable alternative for analyzing data with continuous outcomes in recent years. It has been used in modeling survival data with and without a cured fraction. In this work, we propose novel estimating equation approaches to estimate a mixture cure model where the latency survival time is modeled using a quantile regression. Our proposed estimation methods provide double robustness, meaning that a misspecification in one part of the mixture cure model will not affect the estimation in the other part. The methods relax the restrictive global log-linear assumption that is typically found in existing quantile regressions, and they allow for both quantile-varying and quantile-invariant effects in the regression when the log-linear assumption holds within a certain range of quantile levels. We established the asymptotic properties of the proposed estimators, and our simulation studies demonstrated their double robustness and efficiency gains. An application of the proposed model and methods to data from a lung cancer study revealed that uncured patients with adenocarcinoma have significantly longer quantiles in the survival time than uncured patients with squamous cell carcinoma, which had not been reported in previous analyses of the data due to the limitations of the existing methods.