Semiparametric estimation of the nonmixture cure model with auxiliary survival information

With rapidly increasing data sources, statistical methods that make use of external information are gradually becoming popular tools in medical research. In this article, we efficiently synthesize the auxiliary survival information and propose a semiparametric estimation method for the combined empirical likelihood in the framework of the nonmixture cure model, to enhance inference about the associations between exposures and disease outcomes. The auxiliary survival probabilities from external sources are first summarized as unbiased estimation equations, which help produce more efficient estimates of the effects of interest and improve the prediction accuracy for the risk of the event. Then we develop a Bernstein‐based sieve empirical likelihood method to estimate the parametric and nonparametric components simultaneously. Such an estimation procedure allows us to reduce the computation burden while preserving the shape constraint on the baseline distribution function. The resulting estimators for the true associations are strongly consistent and asymptotically normal. Instead of collecting substantial exposure data, the auxiliary survival information at multiple time points is incorporated, which further reduces the mean squared error of the estimators. This contributes to biomarker evaluation and treatment effect analysis within smaller studies. We show how to choose the number of auxiliary survival probabilities appropriately and provide a guideline for practical applications. Simulation studies demonstrate that the estimators enjoy large gains in efficiency. A melanoma dataset is analyzed for illustrating the methodology.