Baseline Cumulative Hazard Function Research Articles

Nested case–control sampling without replacement

Nested case–control design (NCC) is a cost-effective outcome-dependent design in epidemiology that collects all cases and a fixed number of controls at the time of case diagnosis from a large cohort. Due to inefficiency relative to full cohort studies, previous research developed various estimation methodologies but changing designs in the formulation of risk sets was considered only in view of potential bias in the partial likelihood estimation. In this paper, we study a modified design that excludes previously selected controls from risk sets in view of efficiency improvement as well as bias. To this end, we extend the inverse probability weighting method of Samuelsen which was shown to outperform the partial likelihood estimator in the standard setting. We develop its asymptotic theory and a variance estimation of both regression coefficients and the cumulative baseline hazard function that takes account of the complex feature of the modified sampling design. In addition to good finite sample performance of variance estimation, simulation studies show that the modified design with the proposed estimator is more efficient than the standard design. Examples are provided using data from NIH-AARP Diet and Health Cohort Study.

Deep partially linear cox model for current status data.

Deep learning has continuously attained huge success in diverse fields, while its application to survival data analysis remains limited and deserves further exploration. For the analysis of current status data, a deep partially linear Cox model is proposed to circumvent the curse of dimensionality. Modeling flexibility is attained by using deep neural networks (DNNs) to accommodate nonlinear covariate effects and monotone splines to approximate the baseline cumulative hazard function. We establish the convergence rate of the proposed maximum likelihood estimators. Moreover, we derive that the finite-dimensional estimator for treatment covariate effects is $\sqrt{n}$-consistent, asymptotically normal, and attains semiparametric efficiency. Finally, we demonstrate the performance of our procedures through extensive simulation studies and application to real-world data on news popularity.

Partly linear single-index cure models with a nonparametric incidence link function.

In cancer studies, it is commonplace that a fraction of patients participating in the study are cured, such that not all of them will experience a recurrence, or death due to cancer. Also, it is plausible that some covariates, such as the treatment assigned to the patients or demographic characteristics, could affect both the patients' survival rates and cure/incidence rates. A common approach to accommodate these features in survival analysis is to consider a mixture cure survival model with the incidence rate modeled by a logistic regression model and latency part modeled by the Cox proportional hazards model. These modeling assumptions, though typical, restrict the structure of covariate effects on both the incidence and latency components. As a plausible recourse to attain flexibility, we study a class of semiparametric mixture cure models in this article, which incorporates two single-index functions for modeling the two regression components. A hybrid nonparametric maximum likelihood estimation method is proposed, where the cumulative baseline hazard function for uncured subjects is estimated nonparametrically, and the two single-index functions are estimated via Bernstein polynomials. Parameter estimation is carried out via a curated expectation-maximization algorithm. We also conducted a large-scale simulation study to assess the finite-sample performance of the estimator. The proposed methodology is illustrated via application to two cancer datasets.

Regression analysis of group-tested current status data

Summary Group testing is an effective way to reduce the time and cost associated with conducting large-scale screening for infectious diseases. Benefits are realized through testing pools formed by combining specimens, such as blood or urine, from different individuals. In some studies, individuals are assessed only once and a time-to-event endpoint is recorded, for example, the time until infection. Combining group testing with this type of endpoint results in group-tested current status data (Petito & Jewell, 2016). To analyse these complex data, we propose methods that estimate a proportional hazard regression model based on test outcomes from measuring the pools. A sieve maximum likelihood estimation approach is developed that approximates the cumulative baseline hazard function with a piecewise constant function. To identify the sieve estimator, a computationally efficient expectation-maximization algorithm is derived by using data augmentation. Asymptotic properties of both the parametric and nonparametric components of the sieve estimator are then established by applying modern empirical process theory. Numerical results from simulation studies show that our proposed method performs nominally and has advantages over the corresponding estimation method based on individual testing results. We illustrate our work by analysing a chlamydia dataset collected by the State Hygienic Laboratory at the University of Iowa.

A flexible model based on piecewise linear approximation for the analysis of left truncated right censored data with covariates, and applications to Worcester Heart Attack Study data and Channing House data.

Left truncated right censored (LTRC) data arise quite commonly from survival studies. In this article, a model based on piecewise linear approximation is proposed for the analysis of LTRC data with covariates. Specifically, the model involves a piecewise linear approximation for the cumulative baseline hazard function of the proportional hazards model. The principal advantage of the proposed model is that it does not depend on restrictive parametric assumptions while being flexible and data-driven. Likelihood inference for the model is developed. Through detailed simulation studies, the robustness property of the model is studied by fitting it to LTRC data generated from different processes covering a wide range of lifetime distributions. A sensitivity analysis is also carried out by fitting the model to LTRC data generated from a process with a piecewise constant baseline hazard. It is observed that the performance of the model is quite satisfactory in all those cases. Analyses of two real LTRC datasets by using the model are provided as illustrative examples. Applications of the model in some practical prediction issues are discussed. In summary, the proposed model provides a comprehensive and flexible approach to model a general structure for LTRC lifetime data.

Nonparametric Bayesian modeling for hazard functions in spatial survival data

Survival analysis is a collection of statistical procedures for data analysis where the outcome variable of interest is time until an event occurs. The Cox proportional hazard is the most widely used in survival analysis as it takes advantage that it accounts for the proportionate risk of covariates without estimating baseline hazards. However, it is necessary to obtain the baseline hazard function as well as regression parameters to describe the exact hazards of a study population. When data is collected by clinical sites or geographical regions, we should reflect the spatial structures into the model. In this paper we adopted a nonparametric Bayesian hierarchical model to estimate the hazard function in spatial survival data. We assume the cumulative baseline hazard function as a monotone step function, considering flexible priors called stick-breaking priors. We provide modeling for the hazard function that can describe spatial structure, and apply it to survival data which is observed in 24 administrative districts in the northwest of England.

A flexible parametric approach for analyzing arbitrarily censored data that are potentially subject to left truncation under the proportional hazards model.

The proportional hazards (PH) model is, arguably, the most popular model for the analysis of lifetime data arising from epidemiological studies, among many others. In such applications, analysts may be faced with censored outcomes and/or studies which institute enrollment criterion leading to left truncation. Censored outcomes arise when the event of interest is not observed but rather is known relevant to an observation time(s). Left truncated data occur in studies that exclude participants who have experienced the event prior to being enrolled in the study. If not accounted for, both of these features can lead to inaccurate inferences about the population under study. Thus, to overcome this challenge, herein we propose a novel unified PH model that can be used to accommodate both of these features. In particular, our approach can seamlessly analyze exactly observed failure times along with interval-censored observations, while aptly accounting for left truncation. To facilitate model fitting, an expectation-maximization algorithm is developed through the introduction of carefully structured latent random variables. To provide modeling flexibility, a monotone spline representation is used to approximate the cumulative baseline hazard function. The performance of our methodology is evaluated through a simulation study and is further illustrated through the analysis of two motivating data sets; one that involves child mortality in Nigeria and the other prostate cancer.

Nonparametric Bayesian modeling for baseline hazard functions

Survival analysis is primarily used to identify the time-to-event for events of interest. The Cox proportional hazards model takes advantage that it accounts for the proportionate risks of covariates without estimating exact baseline hazards. However, estimation of exact hazard distribution is always accompanied by estimating baseline hazard function as well as regression parameters. In this study, we adopted nonparametric Bayesian hierarchical model with flexible priors in estimating cumulative baseline hazard function. We assume a monotone step function for the cumulative baseline hazard function, where the number, size, and location of jumps are random. By Estimating the step function through stick-breaking construction, we can obtain a totally data-driven step function.

A new approach to estimation of the proportional hazards model based on interval-censored data with missing covariates

This paper discusses the fitting of the proportional hazards model to interval-censored failure time data with missing covariates. Many authors have discussed the problem when complete covariate information is available or the missing is completely at random. In contrast to this, we will focus on the situation where the missing is at random. For the problem, a sieve maximum likelihood estimation approach is proposed with the use of I-spline functions to approximate the unknown cumulative baseline hazard function in the model. For the implementation of the proposed method, we develop an EM algorithm based on a two-stage data augmentation. Furthermore, we show that the proposed estimators of regression parameters are consistent and asymptotically normal. The proposed approach is then applied to a set of the data concerning Alzheimer Disease that motivated this study.

Causal inference with a mediated proportional hazards regression model

The natural direct and indirect effects in causal mediation analysis with survival data having one mediator is addressed by VanderWeele. He derived an approach for (1) an accelerated failure time regression model in general cases and (2) a proportional hazards regression model when the time-to-event outcome is rare. If the outcome is not rare, then VanderWeele did not derive a simple closed-form expression for the log-natural direct and log-natural indirect effects for the proportional hazards regression model because the baseline cumulative hazard function does not approach zero. We develop two approaches to extend VanderWeele’s approach, in which the assumption of a rare outcome is not required. We obtain the natural direct and indirect effects for specific time points through numerical integration after we calculate the cumulative baseline hazard by (1) applying the Breslow method in the Cox proportional hazards regression model to estimate the unspecified cumulative baseline hazard; (2) assuming a piecewise constant baseline hazard model, yielding a parametric model, to estimate the baseline hazard and cumulative baseline hazard. We conduct simulation studies to compare our two approaches with other methods and illustrate our two approaches by applying them to data from the ASsessment, Serial Evaluation, and Subsequent Sequelae in Acute Kidney Injury (ASSESS-AKI) Consortium.

A Comprehensive and Systematic Analysis of Minimal Residual Disease (MRD) Monitoring in Follicular Lymphoma: Results from the Fondazione Italiana Linfomi (FIL) FOLL12 Trial

A Comprehensive and Systematic Analysis of Minimal Residual Disease (MRD) Monitoring in Follicular Lymphoma: Results from the Fondazione Italiana Linfomi (FIL) FOLL12 Trial

Exposure assessment for Cox proportional hazards cure models with interval-censored survival data.

Mixture cure models have been developed as an effective tool to analyze failure time data with a cure fraction. Used in conjunction with the logistic regression model, this model allows covariate-adjusted inference of an exposure effect on the cured probability and the hazard of failure for the uncured subjects. However, the covariate-adjusted inference for the overall exposure effect is not directly provided. In this paper, we describe a Cox proportional hazards cure model to analyze interval-censored survival data in the presence of a cured fraction and then apply a post-estimation approach by using model-predicted estimates difference to assess the overall exposure effect on the restricted mean survival time scale. For baseline hazard/survival function estimation, simple parametric models as fractional polynomials or restricted cubic splines are utilized to approximate the baseline logarithm cumulative hazard function, or, alternatively, the full likelihood is specified through a piecewise linear approximation for the cumulative baseline hazard function. Simulation studies were conducted to demonstrate the unbiasedness of both estimation methods for the overall exposure effect estimates over various baseline hazard distribution shapes. The methods are applied to analyze the interval-censored relapse time data from a smoking cessation study.

New methods for the additive hazards model with the informatively interval-censored failure time data.

The additive hazards model is one of the most commonly used models for regression analysis of failure time data and many inference procedures have been developed for it under various situations. In particular, Wang etal. (2018a, Computational Statistics and Data Analysis, 125, 1-9) discussed the situation where one observes informatively interval-censored data and proposed a likelihood estimation approach. However , it involves estimation of the unknown baseline cumulative hazard function and thus may be time-consuming . Corresponding to this, we propose two new procedures, an estimating equation-based one and an empirical likelihood-based one, and both do not need estimation of the cumulative hazard function and can be easily implemented. The asymptotic properties of the proposed methods are established and an extensive simulation study suggests that they work well in practical situations. An application is alsoprovided.

Semiparametric analysis of zero-inflated recurrent events with a terminal event.

Recurrent event data frequently arise in longitudinal studies and observations on recurrent events could be terminated by a major failure event such as death. In many situations, there exist a large fraction of subjects without any recurrent events of interest. Among these subjects, some are unsusceptible to recurrent events, while others are susceptible but have no recurrent events being observed due to censoring. In this article, we propose a zero-inflated generalized joint frailty model and a sieve maximum likelihood approach to analyze zero-inflated recurrent events with a terminal event. The model provides a considerable flexibility in formulating the effects of covariates on both recurrent events and the terminal event by specifying various transformation functions. In addition, Bernstein polynomials are employed to approximate the unknown cumulative baseline hazard (intensity) function. The estimation procedure can be easily implemented and is computationally fast. Extensive simulation studies are conducted and demonstrate that our proposed method works well for practical situations. Finally, we apply the method to analyze myocardial infarction recurrences in the presence of death in a clinical trial with cardiovascular outcomes.

Generalized accelerated hazards mixture cure models with interval-censored data

Existing semiparametric mixture cure models with interval-censored data often assume a survival model, such as the Cox proportional hazards model, proportional odds model, accelerated failure time model, or their transformations for the susceptible subjects. There are cases in practice that such conventional assumptions may be inappropriate for modeling survival outcomes of susceptible subjects. We propose a more flexible class of generalized accelerated hazards mixture cure models for analysis of interval-censored failure times in the presence of a cure fraction. We develop a sieve maximum likelihood estimation in which the unknown cumulative baseline hazard function is approximated by means of B-splines and bundled with regression parameters. The proposed estimator possesses the properties of consistency and asymptotic normality, and can achieve the optimal global convergence rate under some conditions. Simulation results demonstrate that the proposed estimator performs satisfactorily in finite samples. The application of the proposed method is illustrated by the analysis of smoking cessation data from a lung health study.

Bayesian survival analysis using gamma processes with adaptive time partition

In Bayesian semi-parametric analyses of time-to-event data, non-parametric process priors are adopted for the baseline hazard function or the cumulative baseline hazard function for a given finite partition of the time axis. However, it would be controversial to suggest a general guideline to construct an optimal time partition. While a great deal of research has been done to relax the assumption of the fixed split times for other non-parametric processes, to our knowledge, no methods have been developed for a gamma process prior, which is one of the most widely used in Bayesian survival analysis. In this paper, we propose a new Bayesian framework for proportional hazards models where the cumulative baseline hazard function is modelled a priori by a gamma process. A key feature of the proposed framework is that the number and position of interval cutpoints are treated as random and estimated based on their posterior distributions.

HIV incidence declines in a rural South African population: a G-imputation approach for inference

BackgroundAd hoc assumptions about the unobserved infection event, which is known only to occur between the latest-negative and earliest-positive test dates, can lead to biased HIV incidence rate estimates. Using a G-imputation approach, we infer the infection dates from covariate data to estimate the HIV incidence rate in a hyper-endemic South African setting.MethodsA large demographic surveillance system has annually tested a cohort of HIV-uninfected participants living in the KwaZulu-Natal province. Using this data, we estimated a cumulative baseline hazard function and the effects of time-dependent covariates on the interval censored infection dates. For each HIV-positive participant in the cohort, we derived a cumulative distribution function and sampled multiple infection dates conditional on the unique covariate values. We right censored the data at the imputed dates, calculated the annual HIV incidence rate per 100 person-years, and used Rubin’s rules to obtain the 95% confidence intervals.ResultsA total of 20,011 uninfected individuals with a repeat HIV test participated in the incidence cohort between 2005 and 2018. We observed 2,603 infections per 58,769 person-years of follow-up among women and 845 infections per 41,178 person-years of follow-up among men. Conditional on age and circumcision status (men only), the female HIV incidence rate declined by 25%, from 5.0 to 3.7 infections per 100 person-years between 2014 and 2018. During this period, the HIV incidence rate among men declined from 2.1 to 1.1 infections per 100 person-years—a reduction of 49%. We observed similar reductions in male and female HIV incidence conditional on condom-use, marital status, urban residential status, migration history, and the HIV prevalence in the surrounding community.ConclusionWe have followed participants in one of the world’s largest and longest running HIV cohorts to estimate long-term trends in the population-wide incidence of infection. Using a G-imputation approach, we present further evidence for HIV incidence rate declines in this hyper-endemic South African setting.

Generalized method of moment for case-cohort under additive hazards model

In rare disease study, the failure rate is low in a large cohort of subjects due to constraint by funding and research time. Case-cohort design is a widely used sampling scheme for cost-effective study. The weighted pseudo-score (PS) estimators are proposed in the additive hazards model with data generated under case-cohort design. In this article, we present two more effective estimators: the generalized moment estimations with two different weights. Combining the subgroup information with the PS estimator, the generalized methods of moment are applied for the regression parameters. We establish the asymptotic properties of the proposed estimators and conclude that the estimators are more efficient than the weighted PS estimator. We also show the asymptotic properties for the estimated cumulative baseline hazard function. Simulation and real data studies demonstrate the finite-sample performances of the proposed method and the corresponding theories.

Estimation of the cumulative baseline hazard function for dependently right-censored failure time data

In this paper, we study the properties of a special class of frailty models when the frailty is common to several failure times. The models are closely linked to Archimedean copula models. We establish a useful formula for cumulative baseline hazard functions and develop a new estimator for cumulative baseline hazard functions in bivariate frailty regression models. Based on our proposed estimator, we present a graphical model checking procedure. We fit a leukemia data set using our model and end our paper with some discussions.

Joint analysis of longitudinal and interval-censored failure time data

Methods for joint analysis of longitudinal measures and right-censored survival outcomes have received much attention in the literature. However, in clinic and epidemiology research, the event of interest is only examined at the inspection time, resulting in the interval-censored failure time data. In this article, we construct a joint model for the longitudinal outcomes and the failure time, which is subject to interval censoring. We account for the dependence between the longitudinal measures and failure time through the incorporation of random effects. The semiparametric regression model with random effects and the Cox frailty model is used for the longitudinal outcome and the failure time, respectively. Spline functions are applied to approximate the infinite-dimensional baseline mean function and baseline cumulative hazard function in the joint model. The EM algorithm is used to find the maximum likelihood estimators and the estimated standard deviations are computed from the log observed-data likelihood. Extensive simulation studies are conducted to assess the performance of the proposed estimators. Simulation results show that the proposed method performs quite well. Finally, we illustrate the proposed approach by analyzing the chronic kidney disease.