Baseline Hazard Function Research Articles

ESTROGEN RECEPTOR EXPRESSION ON BREAST CANCER PATIENTS' SURVIVAL UNDER SHAPE RESTRICTED COX REGRESSION MODEL.

For certain subtypes of breast cancer, study findings show that their level of estrogen receptor expression is associated with their risk of cancer death, and also suggests a non-linear effect on the hazard of death. A flexible form of the proportional hazards model, λ(t∣x, z ) = λ(t) exp( z T β )q(x), is desirable to facilitate a rich class of covariate effect on a survival outcome to provide meaningful insight, where the functional form of q(x) is not specified except for its shape. Prior biologic knowledge on the shape of the underlying distribution of the covariate effect in regression models can be used to enhance statistical inference. Despite recent progress, major challenges remain for the semiparametric shape-restricted inference due to lack of practical and efficient computational algorithms to accomplish non-convex optimization. We propose an alternative algorithm to maximize the full log-likelihood with two sets of parameters iteratively under monotone constraints. The first set consists of the non-parametric estimation of the monotone-restricted function q(x), while the second set includes estimating the baseline hazard function and other covariate coefficients. The iterative algorithm in conjunction with the pool-adjacent-violators algorithm makes the computation efficient and practical. The Jackknife resampling effectively reduces the estimator bias, when sample size is small. Simulations show that the proposed method can accurately capture the underlying shape of q(x), and outperforms the estimators when q(x) in the Cox model is mis-specified. We apply the method to model the effect of estrogen receptor on breast cancer patients' survival.

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.

Flexible parametric copula modeling approaches for clustered survival data.

Copula-based survival regression models, which consist of a copula function and marginal distribution (i.e., marginal survival function), have been widely used for analyzing clustered multivariate survival data. Archimedean copula functions are useful for modeling such dependence. For the likelihood inference, one-stage and two-stage estimation methods have been usually used. The two-stage procedure can give inefficient estimation results because of separate estimation of the marginal and copula's dependence parameters. The more efficient one-stage procedure has been mainly developed under a restrictive parametric assumption of marginal distribution due to complexity of the full likelihood with unknown marginal baseline hazard functions. In this paper, we propose a flexible parametric Archimedean copula modeling approach using a one-stage likelihood procedure. In order to reduce the complexity of the full likelihood, the unknown marginal baseline hazards are modeled based on a cubic M-spline basis function that does not require a specific parametric form. Simulation results demonstrate that the proposed one-stage estimation method gives a consistent estimator and also provides more efficient results over existing one- and two-stage methods. The new method is illustrated with three clinical data sets. The Appendix provides an R function so that the proposed method becomes directly accessible to interested readers.

Weighted Lindley frailty model: estimation and application to lung cancer data.

In this paper, we propose a novel frailty model for modeling unobserved heterogeneity present in survival data. Our model is derived by using a weighted Lindley distribution as the frailty distribution. The respective frailty distribution has a simple Laplace transform function which is useful to obtain marginal survival and hazard functions. We assume hazard functions of the Weibull and Gompertz distributions as the baseline hazard functions. A classical inference procedure based on the maximum likelihood method is presented. Extensive simulation studies are further performed to verify the behavior of maximum likelihood estimators under different proportions of right-censoring and to assess the performance of the likelihood ratio test to detect unobserved heterogeneity in different sample sizes. Finally, to demonstrate the applicability of the proposed model, we use it to analyze a medical dataset from a population-based study of incident cases of lung cancer diagnosed in the state of São Paulo, Brazil.

A Comparative Study of Shared Frailty Models for Kidney Infection Data with Generalized Exponential Baseline Distribution

Shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor (frailty) and baseline hazard function which is common to all individuals. There are certain as sumptions about the baseline distribution and distribution of frailty. Mostly assumption of gamma distribution is considered for frailty distribution. To compare the results with gamma frailty model, we introduce three shared frailty models with generalized exponential as baseline distribution. The other three shared frailty models are inverse Gaussian shared frailty model, compound Poisson shared frailty model and compound negative binomial shared frailty model. We fit these models to a real life bivariate survival data set of McGilchrist and Aisbett (1991) related to kidney infection using Markov Chain Monte Carlo (MCMC) technique. Model comparison is made using Bayesian model selection criteria and a better model is suggested for the data.

Compound Poisson frailty model with a gamma process prior for the baseline hazard: accounting for a cured fraction

Cox model and traditional frailty models assume that all individuals will eventually experience the event of interest. This assumption is often overlooked, and situations will arise where it is not realistic. We introduce Compound Poisson frailty model for survival analysis to deal with populations in which some of the individuals will not experience the event of interest. This model assumes that the target population is a mixture of individuals with zero frailty and those with positive frailty. In this paper, we consider a compound Poisson frailty model for right-censored event times from a Bayesian perspective and compute the Bayesian estimator using the Markov Chain Monte Carlo method, where a Gamma process prior is adopted for the baseline hazard function. Furthermore, we evaluate the approach using simulation studies and demonstrate the methodology by analyzing the data from achalasia patient cohort.

Regression estimation of the marginal models with general relative risk form for multivariate failure time data

We develop the quadratic inference function method to the parameter estimation of the marginal models with general relative risk form and common baseline hazard function for multivariate failure time data. The usual exponential relative risk form is relaxed to an non-negative twice differentiable form. The proposed estimator, under some regularity conditions, is shown to be consistent and asymptotically normal with a covariance matrix that can be consistently estimated. In addition, a test statistic based on the quadratic inference function for the parameter inference has been provided. The simulation results show that the proposed method taking the correlation between the failure times into the estimation procedure gains more efficiency than the method using the independent structure when the correlation cannot be ignored, and can easily deal with the situation when the cluster size is large. The proposed method is illustrated by analysis of a real data from the Diabetic Retinopathy Study.

A tractable Bayesian joint model for longitudinal and survival data.

We introduce a numerically tractable formulation of Bayesian joint models for longitudinal and survival data. The longitudinal process is modeled using generalized linear mixed models, while the survival process is modeled using a parametric general hazard structure. The two processes are linked by sharing fixed and random effects, separating the effects that play a role at the time scale from those that affect the hazard scale. This strategy allows for the inclusion of nonlinear and time-dependent effects while avoiding the need for numerical integration, which facilitates the implementation of the proposed joint model. We explore the use of flexible parametric distributions for modeling the baseline hazard function which can capture the basic shapes of interest in practice. We discuss prior elicitation based on the interpretation of the parameters. We present an extensive simulation study, where we analyze the inferential properties of the proposed models, and illustrate the trade-off between flexibility, sample size, and censoring. We also apply our proposal to two real data applications in order to demonstrate the adaptability of our formulation both in univariate time-to-event data and in a competing risks framework. The methodology is implemented in rstan.

National immunisation campaigns with oral polio vaccine may reduce all-cause mortality: Analysis of 2004-2019 demographic surveillance data in rural Bangladesh.

BackgroundWest African studies have suggested that national immunisation campaigns with oral polio vaccine (C-OPV) may non-specifically reduce all-cause child mortality rate by 15–25%. We investigated whether C-OPVs had similar non-specific effects in rural Bangladesh from 2004 to 2019.MethodsChakaria, is a health and demographic surveillance system (HDSS) in Southern Bangladesh. From 2004–2011 the HDSS covered a random sample of households; from 2012 to 2019 it covered a random sample of villages. Using Cox proportional hazards models, we calculated hazard ratios (HR) comparing mortality for children under 3 years of age after C-OPV versus before C-OPV to assess the effect of receiving a C-OPV. We allowed for different baseline hazard function in the two periods (2004–2011, 2012–2019), with separate models for each period.FindingsThere were 768 deaths (2.1%) amongst 36,176 children. The HR after C-OPV was 0.69 (95% confidence interval: 0.52–0.90). National campaigns providing vitamin A or measles vaccine did not have similar effects. Each additional dose of C-OPV was associated with a reduction in the mortality rate by 6% (−2 to 13%). The number needed to treat with C-OPV to save one life between 0 and 35 months of age was 88 (81–96).InterpretationThis is the fourth study to show that C-OPV has beneficial non-specific effects on child survival. All studies have shown a beneficial effect of C-OPV on child health. Stopping OPV as planned after polio eradication without any mitigation plan could have detrimental effects for overall child health in low-income countries.FundingThe Chakaria HDSS was funded by international sponsors. No sponsor had any influence on the preparation of the article.

Exploratory analyses of surrogate endpoints in metastatic non-small cell lung cancer

Context: Metastatic lung cancer accounts for 40-50% of diagnosed lung cancers, and has a great lethality, with 4% 5-year net survival. Treatment has been recently modified due to the development of immunotherapy. In order to obtain faster results and to overcome the limitations associated with the use of overall survival (OS), the search for surrogate endpoints (SE) has become an area of research. Using the real-life database ESME-AMLC ( NCT03848052 ), we investigated two candidate surrogate endpoints for OS: progression-free survival (PFS) and time to next treatment (TNT) for patients treated with first-line chemotherapy and/or immunotherapy. This is an exploratory analysis assessing investigating surrogate properties for OS of PFS and TNT in patients diagnosed with metastatic non-small cell lung cancer (NSCLC). Using real world data, we estimated the individual-level association between PFS/TNT and OS as well as the center-level association between treatment effect on PFS/TNT and treatment effect on OS. This observational, retrospective and multi-center study uses data from patients treated since 2015 in French cancer care centers for metastatic NSCLC, with first-line chemotherapy and/or immunotherapy. Baseline was the administration of the first treatment. PFS was defined as the time until first progression of disease or death, TNT as the time until initiation of a second line of treatment or death, and OS as the time until death. Associations between OS and surrogate endpoints were estimated at the individual and center level with a meta-analytical approach. A joint frailty model was used with an individual random effect or with copula functions. Estimated parameters were found using penalized likelihood and baseline hazard functions were approximated using splines functions. Effectiveness of models was assessed with predictions using leave-one-out cross validation. The study population included 5294 patients: 5055 patients treated with chemotherapy alone, and 239 patients treated with immunotherapy alone or in association with chemotherapy. Median OS was 13 months, and median PFS and TNT were 4 months. These estimates were higher for patients treated with immunotherapy, with 16 months versus 12 months for OS, 5 months versus 4 months for PFS, and 8 months versus 4 months for TNT. With the joint frailty model approach using copula functions, estimated individual-level association was 0.57 (0.54–0.61) between OS and PFS, and 0.57 (0.53–0.60) also between OS and TNT. All estimated coefficients for associations at the center-level were close to 1, with a 95%CI equal to (0.91–1.09) between OS and PFS and equal to (1.00–1.00) between OS and TNT. PFS and TNT were moderately correlated with OS at the individual-level. Center-level associations were strong. These results deserve to be confirmed with meta-analyses of clinical trials once published data will become available.

Semiparametric regression with the U-shaped baseline hazard function in the additive hazards model under general censoring mechanisms

When employing natural shape constraint knowledge obtained from prior studies in survival data analysis, both restrictions of parametric models and difficulties of nonparametric approaches, including the selection of bandwidth and tuning parameter, can be avoided. We use this technique to derive a nonparametric estimator of the U-shaped baseline hazard function in the semiparametric additive hazards regression model for partly general interval-censored data. An iterative algorithm is developed for computing estimators of the baseline hazard function and regression coefficients simultaneously. Numerical studies using simulation and real dataset are conducted to compare the performance of the proposed approach incorporating shape information with that of the B-spline and I-spline methods without using shape restriction. They show that the proposed method increases the efficiency of estimator for the baseline hazard function in terms of the mean Hellinger distance and performs better, or at least leads to competitive results for estimating regression coefficients.

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.

Semiparametric analysis of clustered interval-censored survival data using soft Bayesian additive regression trees (SBART).

Popular parametric and semiparametric hazards regression models for clustered survival data are inappropriate and inadequate when the unknown effects of different covariates and clustering are complex. This calls for a flexible modeling framework to yield efficient survival prediction. Moreover, for some survival studies involving time to occurrence of some asymptomatic events, survival times are typically interval censored between consecutive clinical inspections. In this article, we propose a robust semiparametric model for clustered interval-censored survival data under a paradigm of Bayesian ensemble learning, called soft Bayesian additive regression trees or SBART (Linero and Yang, 2018), which combines multiple sparse (soft) decision trees to attain excellent predictive accuracy. We develop a novel semiparametric hazards regression model by modeling the hazard function as a product of a parametric baseline hazard function and a nonparametric component that uses SBART to incorporate clustering, unknown functional forms of the main effects, and interaction effects of various covariates. In addition to being applicable for left-censored, right-censored, and interval-censored survival data, our methodology is implemented using a data augmentation scheme which allows for existing Bayesian backfitting algorithms to be used. We illustrate the practical implementation and advantages of our method via simulation studies and an analysis of a prostate cancer surgery study where dependence on the experience and skill level of the physicians leads to clustering of survival times. We conclude by discussing our method's applicability in studies involving high-dimensional data with complex underlyingassociations.

Reliability Analysis with Proportional Hazard Model in Aeronautics

Reliability analysis is a specific field of statistics that studies failure time and its probability on a system. This branch of statistics takes an important place in many domains, especially in aeronautics. One of the most commonly used methods of reliability analysis is the hazard function which is defined for a specific instance in time, given the information up to that moment. Commonly, the most used parametric proportional hazard model is Weibull proportional hazard model. Weibull distribution makes restrictive assumptions of the baseline hazard function like monotonicity and gives average values, which can cover a large type of failures due to aircraft age and fatigue in life operation. The present work deals with an unusual phenomena, especially in the aircraft entrance in a turbulence area, see Zbrozek (Weather R Meteorol Soc 13(7):215–227, 1958). This type of situation can lead to the need of increasing–decreasing proportional hazard models, taking into account harsh conditions during the flight. The functions which could model these non-monotone phenomena are scarce in literature but remain very crucial due to their impact on flight safety. In this paper, we propose a new baseline hazard function using extreme values theory in proportional hazard model and we adjusted the flexible Weibull increasing–decreasing baseline hazard function proposed by Park* and Park (Commun Stat Theory Methods 47(4):767–778, 2018) to the proposed baseline hazard function. This function could be implemented in control system of navigation particularly the take-off and landing phase and used as a support tool for pilot or aviation controller running turbulence area. The newly suggested function is non-monotone and will be named generalized extreme values baseline hazard function and we prove that this function satisfies hazard properties and could be used as a tool in the prediction of system operations. An application on aircraft flights is done and the effects of environmental factors are studied in aircraft performance modeling with the proposed model.

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.

Mediation analysis for mixture Cox proportional hazards cure models.

Mediation analysis aims to decompose a total effect into specific pathways and investigate the underlying causal mechanism. Although existing methods have been developed to conduct mediation analysis in the context of survival models, none of these methods accommodates the existence of a substantial proportion of subjects who never experience the event of interest, even if the follow-up is sufficiently long. In this study, we consider mediation analysis for the mixture of Cox proportional hazards cure models that cope with the cure fraction problem. Path-specific effects on restricted mean survival time and survival probability are assessed by introducing a partially latent group indicator and applying the mediation formula approach in a three-stage mediation framework. A Bayesian approach with P-splines for approximating the baseline hazard function is developed to conduct analysis. The satisfactory performance of the proposed method is verified through simulation studies. An application of the Alzheimer's disease (AD) neuroimaging initiative dataset investigates the causal effects of APOE- allele on AD progression.

Online Updating of Survival Analysis

When large amounts of survival data arrive in streams, conventional estimation methods become computationally infeasible since they require access to all observations at each accumulation point. We develop online updating methods for carrying out survival analysis under the Cox proportional hazards model in an online-update framework. Our methods are also applicable with time-dependent covariates. Specifically, we propose online-updating estimators as well as their standard errors for both the regression coefficients and the baseline hazard function. Extensive simulation studies are conducted to investigate the empirical performance of the proposed estimators. A large colon cancer dataset from the Surveillance, Epidemiology, and End Results program and a large venture capital dataset with time-dependent covariates are analyzed to demonstrate the utility of the proposed methodologies. Supplemental files for this article are available online.

Regularisation in discrete survival models : a comparison of lasso and gradient boosting

We present the results of a simulation study performed to compare the accuracy of a lassotype penalization method and gradient boosting in estimating the baseline hazard function and covariate parameters in discrete survival models. The mean square error results reveal that the lasso-type algorithm performs better in recovering the baseline hazard and covariate parameters. In particular, gradient boosting underestimates the sizes of the parameters and also has a high false positive rate. Similar results are obtained in an application to real-life data.

Cox Piecewise Constant Hazard Model with Bayesian Method

Cox PH model is one of the survival models that is widely used for analyzing time-to-event data. Cox PH model consists of two main components, the baseline hazard consisting of time-dependent component; and the exponential function accomodating explanatory variables. The baseline hazard is not estimated in the Cox PH model, thus not accommodating the need for hazard rate estimation. Therefore, in this paper we discuss the estimation of baseline hazard through piecewise constant hazard using Bayesian method. Gamma distribution is assumed for the piecewise constant baseline hazard, and normal distribution is assumed for the regression coefficient. Sampling from the posterior is conducted using Markov chain Monte Carlo through Gibbs sampling. Echocardiogram data containing 106 observations and 6 explanatory variables were used in analysis. The result showed that the baseline hazard functions were estimated and each of parameters in the model is converged as shown by the trace plot and posterior density plot.