Efficient estimation of Cox model with random change point

In clinical studies, the risk of a disease may dramatically change when some biological indexes of the human body exceed some thresholds. Furthermore, the differences in individual characteristics of patients such as physical and psychological experience may lead to subject-specific thresholds or change points. Although a large literature has been established for regression analysis of failure time data with change points, most of the existing methods assume the same, fixed change point for all study subjects. In this paper, we consider the situation where there exists a subject-specific change point and two Cox type models are presented. The proposed models also offer a framework for subgroup analysis. For inference, a sieve maximum likelihood estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. An extensive simulation study is conducted to assess the empirical performance of the proposed method and indicates that it works well in practical situations. Finally the proposed approach is applied to a set of breast cancer data.

Information criteria for detecting change-points in the Cox proportional hazards model

The Cox proportional hazards model, commonly used in clinical trials, assumes proportional hazards. However, it does not hold when, for example, there is a delayed onset of the treatment effect. In such a situation, an acute change in the hazard ratio function is expected to exist. This paper considers the Cox model with change-points and derives Akaike information criterion (AIC)-type information criteria for detecting those change-points. The change-point model does not allow for conventional statistical asymptotics due to its irregularity, thus a formal AIC that penalizes twice the number of parameters would not be analytically derived, and using it would clearly give overfitting analysis results. Therefore, we will construct specific asymptotics using the partial likelihood estimation method in the Cox model with change-points, and propose information criteria based on the original derivation method for AIC. If the partial likelihood is used in the estimation, information criteria with penalties much larger than twice the number of parameters could be obtained in an explicit form. Numerical experiments confirm that the proposed criteria are clearly superior in terms of the original purpose of AIC, which are to provide an estimate that is close to the true structure. We also apply the proposed criterion to actual clinical trial data to indicate that it will easily lead to different results from the formal AIC.

Survival analysis with a random change-point

Contemporary works in change-point survival models mainly focus on an unknown universal change-point shared by the whole study population. However, in some situations, the change-point is plausibly individual-specific, such as when it corresponds to the telomere length or menopausal age. Also, maximum-likelihood-based inference for the fixed change-point parameter is notoriously complicated. The asymptotic distribution of the maximum-likelihood estimator is non-standard, and computationally intensive bootstrap techniques are commonly used to retrieve its sampling distribution. This article is motivated by a breast cancer study, where the disease-free survival time of the patients is postulated to be regulated by the menopausal age, which is unobserved. As menopausal age varies across patients, a fixed change-point survival model may be inadequate. Therefore, we propose a novel proportional hazards model with a random change-point. We develop a nonparametric maximum-likelihood estimation approach and devise a stable expectation-maximization algorithm to compute the estimators. Because the model is regular, we employ conventional likelihood theory for inference based on the asymptotic normality of the Euclidean parameter estimators, and the variance of the asymptotic distribution can be consistently estimated by a profile-likelihood approach. A simulation study demonstrates the satisfactory finite-sample performance of the proposed methods, which yield small bias and proper coverage probabilities. The methods are applied to the motivating breast cancer study.

Multi-threshold proportional hazards model and subgroup identification

We propose a novel two-stage procedure for change point detection and parameter estimation in a multi-threshold proportional hazards model. In the first stage, we estimate the number of thresholds by formulating the threshold detection problem as a variable selection problem and applying the penalized partial likelihood approach. In the second stage, the change point locations are refined by a grid search and the standard inference for segment regression can then follow. The proposed model and estimation procedure could lend support to subgroup identification and personalized treatment recommendation in medical research. We establish the consistency of the threshold estimators and regression coefficient estimators under technical conditions. The finite sample performance of the method is demonstrated via simulation studies and two cancer data examples.

Nested logistic regression models and ΔAUC applications: Change-point analysis

The area under the receiver operating characteristic curve (AUC) is one of the most popular measures for evaluating the performance of a predictive model. In nested models, the change in AUC (ΔAUC) can be a discriminatory measure of whether the newly added predictors provide significant improvement in terms of predictive accuracy. Recently, several authors have shown rigorously that ΔAUC can be degenerate and its asymptotic distribution is no longer normal when the reduced model is true, but it could be the distribution of a linear combination of some χ12 random variables [1,2]. Hence, the normality assumption and existing variance estimate cannot be applied directly for developing a statistical test under the nested models. In this paper, we first provide a brief review on the use of ΔAUC for comparing nested logistic models and the difficulty of retrieving the reference distribution behind. Then, we present a special case of the nested logistic regression models that the newly added predictor to the reduced model contains a change-point in its effects. A new test statistic based on ΔAUC is proposed in this setting. A simple resampling scheme is proposed to approximate the critical values for the test statistic. The inference of the change-point parameter is done via m-out-of-n bootstrap. Large-scale simulation is conducted to evaluate the finite-sample performance of the ΔAUC test for the change-point model. The proposed method is applied to two real-life datasets for illustration.