Nonparametric Maximum Likelihood Estimators of Time-Dependent Accuracy Measures for Survival Outcome Under Two-Stage Sampling Designs

Logistic regression analysis of two-phase studies using generalized method of moments

Two-phase designs can reduce the cost of epidemiological studies by limiting the ascertainment of expensive covariates or/and exposures to an efficiently selected subset (phase-II) of a larger (phase-I) study. Efficient analysis of the resulting data set combining disparate information from phase-I and phase-II, however, can be complex. Most of the existing methods, including semiparametric maximum-likelihood estimator, require the information in phase-I to be summarized into a fixed number of strata. In this paper, we describe a novel method for the analysis of two-phase studies where information from phase-I is summarized by parameters associated with a reduced logistic regression model of the disease outcome on available covariates. We then setup estimating equations for parameters associated with the desired extended logistic regression model, based on information on the reduced model parameters from phase-I and complete data available at phase-II after accounting for nonrandom sampling design. We use generalized method of moments to solve overly identified estimating equations and develop the resulting asymptotic theory for the proposed estimator. Simulation studies show that the use of reduced parametric models, as opposed to summarizing data into strata, can lead to more efficient utilization of phase-I data. An application of the proposed method is illustrated using the data from the U.S. National Wilms Tumor Study.

Evaluating the time-dependent predictive accuracy for event-to-time outcome with a cure fraction

In medical studies, it is often observed that a portion of subjects will never experience the event of interest and thus can be treated as cured or long-term survivors. Many populations of early-stage cancer patients contain both uncured and cured individuals that should be modeled using cure models. In prognostic studies, the cure status (uncure or cure) is an issue of interest for medical practitioners, and the disease status (death or alive) of an individual is not a fixed characteristic and it varies along the time. These statuses are usually predicted by a prognostic risk score. The time-dependent receiver operating characteristic (ROC) curve is a powerful tool to evaluate these predicting performances dynamically. In the context with a cure fraction, quantifying and estimating the predictive performances of the risk score is a challenge since the disease status and cure status are both unknown among individuals who are censored. In this paper, to assess the predictive accuracy for the survival outcome with a cure fraction, we propose a time-dependent ROC curve semi-parametric estimator based on the sieve maximum likelihood (ML) estimation under the mixture cure model. We also apply a Bernstein-based smoothing method in the estimation procedure, and this estimator can lead to substantial gain in efficiency. In addition, we derive the time-dependent area under the ROC curve (AUC) to summarize the discriminatory capacity of the risk score globally. Finally, we evaluate the finite sample performance of the proposed methods by extensive simulations and illustrate the estimation using two real data sets, one from a melanoma study and the other from stomach cancer.