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.

causalCmprsk: An R package for nonparametric and Cox-based estimation of average treatment effects in competing risks data

BACKGROUND AND OBJECTIVE

Competing risks data arise in both observational and experimental clinical studies with time-to-event outcomes, when each patient might follow one of the multiple mutually exclusive competing paths. Ignoring competing risks in the analysis can result in biased conclusions. In addition, possible confounding bias of the treatment-outcome relationship has to be addressed, when estimating treatment effects from observational data. In order to provide tools for estimation of average treatment effects on time-to-event outcomes in the presence of competing risks, we developed the R package causalCmprsk. We illustrate the package functionality in the estimation of effects of a right heart catheterization procedure on discharge and in-hospital death from observational data.

METHODS

The causalCmprsk package implements an inverse probability weighting estimation approach, aiming to emulate baseline randomization and alleviate possible treatment selection bias. The package allows for different types of weights, representing different target populations. causalCmprsk builds on existing methods from survival analysis and adapts them to the causal analysis in non-parametric and semi-parametric frameworks.

RESULTS

The causalCmprsk package has two main functions: fit.cox assumes a semiparametric structural Cox proportional hazards model for the counterfactual cause-specific hazards, while fit.nonpar does not impose any structural assumptions. In both frameworks, causalCmprsk implements estimators of (i) absolute risks for each treatment arm, e.g., cumulative hazards or cumulative incidence functions, and (ii) relative treatment effects, e.g., hazard ratios, or restricted mean time differences. The latter treatment effect measure translates the treatment effect from probability into more intuitive time domain and allows the user to quantify, for example, by how many days or months the treatment accelerates the recovery or postpones illness or death.

CONCLUSIONS

The causalCmprsk package provides a convenient and useful tool for causal analysis of competing risks data. It allows the user to distinguish between different causes of the end of follow-up and provides several time-varying measures of treatment effects. The package is accompanied by a vignette that contains more details, examples and code, making the package accessible even for non-expert users.

Segmented correspondence curve regression for quantifying covariate effects on the reproducibility of high-throughput experiments

High-throughput biological experiments are essential tools for identifying biologically interesting candidates in large-scale omics studies. The results of a high-throughput biological experiment rely heavily on the operational factors chosen in its experimental and data-analytic procedures. Understanding how these operational factors influence the reproducibility of the experimental outcome is critical for selecting the optimal parameter settings and designing reliable high-throughput workflows. However, the influence of an operational factor may differ between strong and weak candidates in a high-throughput experiment, complicating the selection of parameter settings. To address this issue, we propose a novel segmented regression model, called segmented correspondence curve regression, to assess the influence of operational factors on the reproducibility of high-throughput experiments. Our model dissects the heterogeneous effects of operational factors on strong and weak candidates, providing a principled way to select operational parameters. Based on this framework, we also develop a sup-likelihood ratio test for the existence of heterogeneity. Simulation studies show that our estimation and testing procedures yield well-calibrated type I errors and are substantially more powerful in detecting and locating the differences in reproducibility across workflows than the existing method. Using this model, we investigated an important design question for ChIP-seq experiments: How many reads should one sequence to obtain reliable results in a cost-effective way? Our results reveal new insights into the impact of sequencing depth on the binding-site identification reproducibility, helping biologists determine the most cost-effective sequencing depth to achieve sufficient reproducibility for their study goals.

IDENTIFICATION OF IMMUNE RESPONSE COMBINATIONS ASSOCIATED WITH HETEROGENEOUS INFECTION RISK IN THE IMMUNE CORRELATES ANALYSIS OF HIV VACCINE STUDIES

In HIV vaccine/prevention research, probing into the vaccine-induced immune responses that can help predict the risk of HIV infection provides valuable information for the development of vaccine regimens. Previous correlate analysis of the Thai vaccine trial aided the discovery of interesting immune correlates related to the risk of developing an HIV infection. The present study aimed to identify the combinations of immune responses associated with the heterogeneous infection risk. We explored a "change-plane" via combination of a subset of immune responses that could help separate vaccine recipients into two heterogeneous subgroups in terms of the association between immune responses and the risk of developing infection. Additionally, we developed a new variable selection algorithm through a penalized likelihood approach to investigate a parsimonious marker combination for the change-plane. The resulting marker combinations can serve as candidate correlates of protection and can be used for predicting the protective effect of the vaccine against HIV infection. The application of the proposed statistical approach to the Thai trial has been presented, wherein the marker combinations were explored among several immune responses and antigens.

Change point detection in Cox proportional hazards mixture cure model

The mixture cure model has been widely applied to survival data in which a fraction of the observations never experience the event of interest, despite long-term follow-up. In this paper, we study the Cox proportional hazards mixture cure model where the covariate effects on the distribution of uncured subjects' failure time may jump when a covariate exceeds a change point. The nonparametric maximum likelihood estimation is used to obtain the semiparametric estimates. We employ a two-step computational procedure involving the Expectation-Maximization algorithm to implement the estimation. The consistency, convergence rate and asymptotic distributions of the estimators are carefully established under technical conditions and we show that the change point estimator is n consistency. The m out of n bootstrap and the Louis algorithm are used to obtain the standard errors of the estimated change point and other regression parameter estimates, respectively. We also contribute a test procedure to check the existence of the change point. The finite sample performance of the proposed method is demonstrated via simulation studies and real data examples.

Survival analysis with change-points in covariate effects

We apply a maximal likelihood ratio test for the presence of multiple change-points in the covariate effects based on the Cox regression model. The covariate effect is assumed to change smoothly at one or more unknown change-points. The number of change-points is inferred by a sequential approach. Confidence intervals for the regression and change-point parameters are constructed by a bootstrap method based on Bernstein polynomials conditionally on the number of change-points. The methods are assessed by simulations and are applied to two datasets.

Estimation in the Cox survival regression model with covariate measurement error and a changepoint

The Cox regression model is a popular model for analyzing the relationship between a covariate vector and a survival endpoint. The standard Cox model assumes a constant covariate effect across the entire covariate domain. However, in many epidemiological and other applications, the covariate of main interest is subject to a threshold effect: a change in the slope at a certain point within the covariate domain. Often, the covariate of interest is subject to some degree of measurement error. In this paper, we study measurement error correction in the case where the threshold is known. Several bias correction methods are examined: two versions of regression calibration (RC1 and RC2, the latter of which is new), two methods based on the induced relative risk under a rare event assumption (RR1 and RR2, the latter of which is new), a maximum pseudo-partial likelihood estimator (MPPLE), and simulation-extrapolation (SIMEX). We develop the theory, present simulations comparing the methods, and illustrate their use on data concerning the relationship between chronic air pollution exposure to particulate matter PM₁₀ and fatal myocardial infarction (Nurses Health Study (NHS)), and on data concerning the effect of a subject's long-term underlying systolic blood pressure level on the risk of cardiovascular disease death (Framingham Heart Study (FHS)). The simulations indicate that the best methods are RR2 and MPPLE.

Bayesian approach for cure models with a change-point based on covariate threshold: application to breast cancer data

In this study, a Bayesian approach was suggested to estimate a change-point according to a covariate threshold when some patients never experienced the event of interest. Gibbs sampler algorithm with latent binary cure indicators was used to simplify the implementation of Markov chain Monte Carlo method. Then, the accuracy of new model was demonstrated by simulation studies to compute the point and interval estimates of parameters. Finally, an effective threshold was suggested in age at surgery time to experience the metastasis when the model was applied for a data set of breast cancer patients.

A Continuous Threshold Expectile Model

Expectile regression is a useful tool for exploring the relation between the response and the explanatory variables beyond the conditional mean. A continuous threshold expectile regression is developed for modeling data in which the effect of a covariate on the response variable is linear but varies below and above an unknown threshold in a continuous way. The estimators for the threshold and the regression coefficients are obtained using a grid search approach. The asymptotic properties for all the estimators are derived, and the estimator for the threshold is shown to achieve root-n consistency. A weighted CUSUM type test statistic is proposed for the existence of a threshold at a given expectile, and its asymptotic properties are derived under both the null and the local alternative models. This test only requires fitting the model under the null hypothesis in the absence of a threshold, thus it is computationally more efficient than the likelihood-ratio type tests. Simulation studies show that the proposed estimators and test have desirable finite sample performance in both homoscedastic and heteroscedastic cases. The application of the proposed method on a Dutch growth data and a baseball pitcher salary data reveals interesting insights. The proposed method is implemented in the R package cthreshER.

Robust bent line regression

We introduce a rank-based bent linear regression with an unknown change point. Using a linear reparameterization technique, we propose a rank-based estimate that can make simultaneous inference on all model parameters, including the location of the change point, in a computationally efficient manner. We also develop a score-like test for the existence of a change point, based on a weighted CUSUM process. This test only requires fitting the model under the null hypothesis in absence of a change point, thus it is computationally more efficient than likelihood-ratio type tests. The asymptotic properties of the test are derived under both the null and the local alternative models. Simulation studies and two real data examples show that the proposed methods are robust against outliers and heavy-tailed errors in both parameter estimation and hypothesis testing.

Proportional hazards model with a change point for clustered event data

In many epidemiology studies, family data with survival endpoints are collected to investigate the association between risk factors and disease incidence. Sometimes the risk of the disease may change when a certain risk factor exceeds a certain threshold. Finding this threshold value could be important for disease risk prediction and diseases prevention. In this work, we propose a change-point proportional hazards model for clustered event data. The model incorporates the unknown threshold of a continuous variable as a change point in the regression. The marginal pseudo-partial likelihood functions are maximized for estimating the regression coefficients and the unknown change point. We develop a supremum test based on robust score statistics to test the existence of the change point. The inference for the change point is based on the m out of n bootstrap. We establish the consistency and asymptotic distributions of the proposed estimators. The finite-sample performance of the proposed method is demonstrated via extensive simulation studies. Finally, the Strong Heart Family Study dataset is analyzed to illustrate the methods.

ASYMPTOTICS FOR CHANGE-POINT MODELS UNDER VARYING DEGREES OF MIS-SPECIFICATION

Change-point models are widely used by statisticians to model drastic changes in the pattern of observed data. Least squares/maximum likelihood based estimation of change-points leads to curious asymptotic phenomena. When the change-point model is correctly specified, such estimates generally converge at a fast rate (n) and are asymptotically described by minimizers of a jump process. Under complete mis-specification by a smooth curve, i.e. when a change-point model is fitted to data described by a smooth curve, the rate of convergence slows down to n^1/3 and the limit distribution changes to that of the minimizer of a continuous Gaussian process. In this paper we provide a bridge between these two extreme scenarios by studying the limit behavior of change-point estimates under varying degrees of model mis-specification by smooth curves, which can be viewed as local alternatives. We find that the limiting regime depends on how quickly the alternatives approach a change-point model. We unravel a family of 'intermediate' limits that can transition, at least qualitatively, to the limits in the two extreme scenarios. The theoretical results are illustrated via a set of carefully designed simulations. We also demonstrate how inference for the change-point parameter can be performed in absence of knowledge of the underlying scenario by resorting to subsampling techniques that involve estimation of the convergence rate.

Assessing potentially time-dependent treatment effect from clinical trials and observational studies for survival data, with applications to the Women's Health Initiative combined hormone therapy trial

For risk and benefit assessment in clinical trials and observational studies with time-to-event data, the Cox model has usually been the model of choice. When the hazards are possibly non-proportional, a piece-wise Cox model over a partition of the time axis may be considered. Here, we propose to analyze clinical trials or observational studies with time-to-event data using a certain semiparametric model. The model allows for a time-dependent treatment effect. It includes the important proportional hazards model as a sub-model and can accommodate various patterns of time-dependence of the hazard ratio. After estimation of the model parameters using a pseudo-likelihood approach, simultaneous confidence intervals for the hazard ratio function are established using a Monte Carlo method to assess the time-varying pattern of the treatment effect. To assess the overall treatment effect, estimated average hazard ratio and its confidence intervals are also obtained. The proposed methods are applied to data from the Women's Health Initiative. To compare the Women's Health Initiative clinical trial and observational study, we use the propensity score in building the regression model. Compared with the piece-wise Cox model, the proposed model yields a better model fit and does not require partitioning of the time axis.

Bootstrapping a change-point Cox model for survival data

This paper investigates the (in)-consistency of various bootstrap methods for making inference on a change-point in time in the Cox model with right censored survival data. A criterion is established for the consistency of any bootstrap method. It is shown that the usual nonparametric bootstrap is inconsistent for the maximum partial likelihood estimation of the change-point. A new model-based bootstrap approach is proposed and its consistency established. Simulation studies are carried out to assess the performance of various bootstrap schemes.

Bayesian random threshold estimation in a Cox proportional hazards cure model

In this paper, we develop a Bayesian approach to estimate a Cox proportional hazards model that allows a threshold in the regression coefficient, when some fraction of subjects are not susceptible to the event of interest. A data augmentation scheme with latent binary cure indicators is adopted to simplify the Markov chain Monte Carlo implementation. Given the binary cure indicators, the Cox cure model reduces to a standard Cox model and a logistic regression model. Furthermore, the threshold detection problem reverts to a threshold problem in a regular Cox model. The baseline cumulative hazard for the Cox model is formulated non-parametrically using counting processes with a gamma process prior. Simulation studies demonstrate that the method provides accurate point and interval estimates. Application to a data set of oropharynx cancer patients suggests a significant threshold in age at diagnosis such that the effect of gender on disease-specific survival changes after the threshold.

Change point-cure models with application to estimating the change-point effect of age of diagnosis among prostate cancer patients

Previous research on prostate cancer survival trends in the United States National Cancer Institute's Surveillance Epidemiology and End Results database has indicated a potential change-point in the age of diagnosis of prostate cancer around age 50. Identifying a change-point value in prostate cancer survival and cure could have important policy and health care management implications. Statistical analysis of this data has to address two complicating features: (1) change-point models are not smooth functions and so present computational and theoretical difficulties; and (2) models for prostate cancer survival need to account for the fact that many men diagnosed with prostate cancer can be effectively cured of their disease with early treatment. We develop a cure survival model that allows for change-point effects in covariates to investigate a potential change-point in the age of diagnosis of prostate cancer. Our results do not indicate that age under 50 is associated with increased hazard of death from prostate cancer.

Inverse regression estimation for censored data

An inverse regression methodology for assessing predictor performance in the censored data setup is developed along with inference procedures and a computational algorithm. The technique developed here allows for conditioning on the unobserved failure time along with a weighting mechanism that accounts for the censoring. The implementation is nonparametric and computationally fast. This provides an efficient methodological tool that can be used especially in cases where the usual modeling assumptions are not applicable to the data under consideration. It can also be a good diagnostic tool that can be used in the model selection process. We have provided theoretical justification of consistency and asymptotic normality of the methodology. Simulation studies and two data analyses are provided to illustrate the practical utility of the procedure.

On Asymptotically Optimal Tests Under Loss of Identifiability in Semiparametric Models

We consider tests of hypotheses when the parameters are not identifiable under the null in semiparametric models, where regularity conditions for profile likelihood theory fail. Exponential average tests based on integrated profile likelihood are constructed and shown to be asymptotically optimal under a weighted average power criterion with respect to a prior on the nonidentifiable aspect of the model. These results extend existing results for parametric models, which involve more restrictive assumptions on the form of the alternative than do our results. Moreover, the proposed tests accommodate models with infinite dimensional nuisance parameters which either may not be identifiable or may not be estimable at the usual parametric rate. Examples include tests of the presence of a change-point in the Cox model with current status data and tests of regression parameters in odds-rate models with right censored data. Optimal tests have not previously been studied for these scenarios. We study the asymptotic distribution of the proposed tests under the null, fixed contiguous alternatives and random contiguous alternatives. We also propose a weighted bootstrap procedure for computing the critical values of the test statistics. The optimal tests perform well in simulation studies, where they may exhibit improved power over alternative tests.

A Class of Semiparametric Mixture Cure Survival Models with Dependent Censoring

Modern cancer treatments have substantially improved cure rates and have generated a great interest in and need for proper statistical tools to analyze survival data with non-negligible cure fractions. Data with cure fractions are often complicated by dependent censoring, and the analysis of this type of data typically involves untestable parametric assumptions on the dependence of the censoring mechanism and the true survival times. Motivated by the analysis of prostate cancer survival trends, we propose a class of semiparametric transformation cure models that allows for dependent censoring without making parametric assumptions on the dependence relationship. The proposed class of models encompasses a number of common models for the latency survival function, including the proportional hazards model and the proportional odds model, and also allows for time-dependent covariates. An inverse censoring probability reweighting scheme is used to derive unbiased estimating equations. We validate small sample properties with simulations and demonstrate the method with a data application.

A Cox-type regression model with change-points in the covariates

We consider a Cox-type regression model with change-points in the covariates. A change-point specifies the unknown threshold at which the influence of a covariate shifts smoothly, i.e., the regression parameter may change over the range of a covariate and the underlying regression function is continuous but not differentiable. The model can be used to describe change-points in different covariates but also to model more than one change-point in a single covariate. Estimates of the change-points and of the regression parameters are derived and their properties are investigated. It is shown that not only the estimates of the regression parameters are square root n-consistent but also the estimates of the change-points in contrast to the conjecture of other authors. Asymptotic normality is shown by using results developed for M-estimators. At the end of this paper we apply our model to an actuarial dataset, the PBC dataset of Fleming and Harrington (Counting processes and survival analysis, 1991) and to a dataset of electric motors.