Penalized estimation for proportional hazards models with current status data

Sample size determination for a study with variable follow-up time

For a study to detect the outcome change at the follow-up visit from baseline, the pre-test and post-test design is commonly used to assess the treatment-control difference. Several existing methods were developed for sample size calculation including the subtraction method, analysis of covariance (ANCOVA), and linear mixed model. The first two methods can be used when the follow-up time is the same as scheduled. Although the linear mixed model can analyze the repeated measures by including the actual visit time to account for the variability of the follow-up time, it often assumes a constant treatment-control difference at any follow-up time which may not be correct in practice. We propose to develop a new statistical model to compare the treatment-control difference at the planned follow-up time while controlling for the follow-up time variation. The spline functions are used to estimate the trajectories of the treatment arm and the control arm. We compared the performance of these methods with regards to type I error rate, statistical power, and sample size under various conditions. These four methods all control for the type I error rate. The new method and the ANCOVA method are often more powerful than the other two methods, and they have similar statistical power when a linear disease progression is satisfied. For a study with non-linear disease progression, the new method can be more powerful than the ANCOVA method. We used data from a completed Alzheimer's disease trial to illustrate the application of the proposed method.

An extended proportional hazards model for interval-censored data subject to instantaneous failures

The proportional hazards (PH) model is arguably one of the most popular models used to analyze time to event data arising from clinical trials and longitudinal studies. In many such studies, the event time is not directly observed but is known relative to periodic examination times; i.e., practitioners observe either current status or interval-censored data. The analysis of data of this structure is often fraught with many difficulties since the event time of interest is unobserved. Further exacerbating this issue, in some such studies the observed data also consists of instantaneous failures; i.e., the event times for several study units coincide exactly with the time at which the study begins. In light of these difficulties, this work focuses on developing a mixture model, under the PH assumptions, which can be used to analyze interval-censored data subject to instantaneous failures. To allow for modeling flexibility, two methods of estimating the unknown cumulative baseline hazard function are proposed; a fully parametric and a monotone spline representation are considered. Through a novel data augmentation procedure involving latent Poisson random variables, an expectation-maximization (EM) algorithm is developed to complete model fitting. The resulting EM algorithm is easy to implement and is computationally efficient. Moreover, through extensive simulation studies the proposed approach is shown to provide both reliable estimation and inference. The motivation for this work arises from a randomized clinical trial aimed at assessing the effectiveness of a new peanut allergen treatment in attaining sustained unresponsiveness in children.

Efficient estimation of a linear transformation model for current status data via penalized splines

We propose a flexible and computationally efficient penalized estimation method for a semi-parametric linear transformation model with current status data. To facilitate model fitting, the unknown monotone function is approximated by monotone B-splines, and a computationally efficient hybrid algorithm involving the Fisher scoring algorithm and the isotonic regression is developed. A goodness-of-fit test and model diagnostics are also considered. The asymptotic properties of the penalized estimators are established, including the optimal rate of convergence for the function estimator and the semi-parametric efficiency for the regression parameter estimators. An extensive numerical experiment is conducted to evaluate the finite-sample properties of the penalized estimators, and the methodology is further illustrated with two real studies.

A partially linear proportional hazards model for current status data

For analyzing current status data, a flexible partially linear proportional hazards model is proposed. Modeling flexibility is attained through using monotone splines to approximate the baseline cumulative hazard function, as well as B-splines to accommodate nonlinear covariate effects. To facilitate model fitting, a computationally efficient and easy to implement expectation-maximization algorithm is developed through a two-stage data augmentation process involving carefully structured latent Poisson random variables. Asymptotic normality and the efficiency of the spline estimator of the regression coefficients are established, and the spline estimators of the nonparametric components are shown to possess the optimal rate of convergence under suitable regularity conditions. The finite-sample performance of the proposed approach is evaluated through Monte Carlo simulation and it is further illustrated using uterine fibroid data arising from a prospective cohort study on early pregnancy.