University of North Carolina at Chapel Hill Graduate School
Doi
Abstract
There is a growing interest in precision medicine, where a potentially censored survival time is often the most important outcome of interest. To discover optimal treatment regimens for such an outcome, we propose a semiparametric proportional hazards model by incorporatingthe interaction between treatment and a single index of covariates through an unknown monotone link function. This model is flexible enough to allow non-linear treatment-covariate interactions and yet provides a clinically interpretable linear rule for treatment decision. We propose a sieve maximum likelihood estimation approach, under which the baseline hazard function is estimated nonparametrically and the unknown link function is estimated via monotone quadratic B-splines. We show that the resulting estimators are consistent and asymptotically normal with a covariance matrix that attains the semiparametric efficiency bound. The optimal treatment rule follows naturally as a linear combination of the maximum likelihood estimators of the model parameters. Through extensive simulation studies and anapplication to an AIDS clinical trial, we demonstrate that the treatment rule derived from the single-index model outperforms the treatment rule under the standard Cox proportionalhazards model. We extend the proposed method to transformation models so that optimal treatment rules can be applied to flexible hazards relationships. The transformation model introduces new challenges to both the estimation procedure and the asymptotic properties of the estimators. We design an estimation procedure with the EM algorithm by recognizing the transformation function as the distribution function of a corresponding missing random variable. We prove that the resulting estimators are consistent and asymptotically normal, with the covariancematrix estimated using the profile likelihood theory. We demonstrate the performance of the transformation single-index model in simulation studies. We show that the proposed treatment rule under the single-index transformation model is more effective than that under the single-index proportional hazards model in delaying the disease relapse of large-bowel carcinoma in a real data analysis. With improvements in technology, researchers are able to collect many clinical and genetic variables; not all the covariates may contribute to the prediction of the optimal treatmentrules. We apply the adaptive Lasso penalty to the log-likelihood of the proposed model and let the data automatically determine the important predictors in the optimal treatment regime.We propose a simple computational approach by quadratic approximation of the original objective function and utilization of the variable selection software package available for theproportional hazards model. We show that the proposed variable selection approach displaysthe oracle property. The performance of the variable selection procedure is demonstrated inextensive simulations and the analysis of a multi-cancer clinical trial.Doctor of Philosoph
