scholarly journals Estimating Proportional Hazards Model Using Frequentist and Bayesian Approaches

2014 ◽  
Vol 8 (2) ◽  
Author(s):  
Noraslinda Mohamed Ismail ◽  
Zarina Mohd Khalid ◽  
Norhaiza Ahmad
Keyword(s):  
Bayesian Approach ◽  
Survival Data ◽  
Hazard Function ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Risk Function ◽  
Model Parameters ◽  
Right Censored Data ◽  
Hazards Model ◽  
Leukemia Data

In statistics, the proportional hazards model (PHM) is one of a class of survival models. This model estimates the effects of different covariates influencing the time-to-event data in which the hazard function has been assumed to be the product of the baseline hazard function and a non-negative function of covariates. In this study, we investigate the hazard function, also known as the risk function or intensity function, which is employed in modelling the survival data and waiting times. The model parameters can be estimated via frequentist or Bayesian approach. However, the Bayesian approach is well known to have the advantages over frequentist methods when the data are small in size and involve censored individuals. In this paper, the PHM for right-censored data from Bayesian perspective will be discussed and the Markov Chain Monte Carlo (MCMC) method will be used to estimate the posterior distributions of the model parameters using Leukemia data.

Multiplicative Piecewise Gamma in Survival Data Analysis

2014 ◽  
Vol 70 (1) ◽  
Author(s):  
Noraslinda Mohamed Ismail ◽  
Zarina Mohd Khalid ◽  
Norhaiza Ahmad
Keyword(s):  
Survival Data ◽  
Hazard Function ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Bayesian Estimator ◽  
Proportional Hazard ◽  
Proportional Hazard Model ◽  
Hazards Model ◽  
Proposed Model ◽  
Leukemia Data

The proportional hazard model is the most general of the regression models since it is not based on any assumptions concerning the nature or shape of the underlying survival distribution. The model assumes that the underlying hazard rate is a function of the covariates (independent variables) and there are no assumptions about the nature or shape of the hazard function. Proportional hazards model in survival analysis is used to estimate the effects of different covariates which was influenced by the survival data. This paper proposes the new multiplicative piecewise gamma in the hazard function using OpenBugs Statistical Packages. The proposed model is a flexible survival model for any types of non-informative censored data in estimating the parameters using Bayesian approach and also an alternative model to the existing model. We used the Markov Chain Monte Carlo method in computing the Bayesian estimator on leukemia data. The results obtained show that the proposed model can be an alternative to the existing multiplicative model since it can estimate the parameters using any types of survival data compared to the existing model that can only be used for leukemia data.  

scholarly journals Use of the concordance index for predictors of censored survival data

2016 ◽  
Vol 27 (8) ◽  
pp. 2359-2373 ◽  
Author(s):  
Adam R Brentnall ◽  
Jack Cuzick
Keyword(s):  
Survival Data ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Prognostic Biomarkers ◽  
Concordance Index ◽  
Right Censored Data ◽  
Censored Survival Data ◽  
Hazards Model ◽  
Pareto Model ◽  
Substantial Bias

The concordance index is often used to measure how well a biomarker predicts the time to an event. Estimators of the concordance index for predictors of right-censored data are reviewed, including those based on censored pairs, inverse probability weighting and a proportional-hazards model. Predictive and prognostic biomarkers often lose strength with time, and in this case the aforementioned statistics depend on the length of follow up. A semi-parametric estimator of the concordance index is developed that accommodates converging hazards through a single parameter in a Pareto model. Concordance index estimators are assessed through simulations, which demonstrate substantial bias of classical censored-pairs and proportional-hazards model estimators. Prognostic biomarkers in a cohort of women diagnosed with breast cancer are evaluated using new and classical estimators of the concordance index.

Modelling heterogeneity in survival data

1991 ◽  
Vol 28 (03) ◽  
pp. 695-701 ◽  
Author(s):  
Philip Hougaard
Keyword(s):  
Survival Data ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Mixture Distribution ◽  
Survival Models ◽  
Risk Of Death ◽  
Hazards Model ◽  
Occupational Mortality ◽  
Heterogeneity Distribution ◽  
Biased Estimates

Ordinary survival models implicitly assume that all individuals in a group have the same risk of death. It may, however, be relevant to consider the group as heterogeneous, i.e. a mixture of individuals with different risks. For example, after an operation each individual may have constant hazard of death. If risk factors are not included, the group shows decreasing hazard. This offers two fundamentally different interpretations of the same data. For instance, Weibull distributions with shape parameter less than 1 can be generated as mixtures of constant individual hazards. In a proportional hazards model, neglect of a subset of the important covariates leads to biased estimates of the other regression coefficients. Different choices of distributions for the unobserved covariates are discussed, including binary, gamma, inverse Gaussian and positive stable distributions, which show both qualitative and quantitative differences. For instance, the heterogeneity distribution can be either identifiable or unidentifiable. Both mathematical and interpretational consequences of the choice of distribution are considered. Heterogeneity can be evaluated by the variance of the logarithm of the mixture distribution. Examples include occupational mortality, myocardial infarction and diabetes.

A Proportional Hazards Neural Network for Performing Reliability Estimates and Risk Prognostics for Mobile Systems Subject to Stochastic Covariates

2005 ◽  
Author(s):  
George M. Lloyd ◽  
Timothy Hasselman ◽  
Thomas Paez
Keyword(s):  
Survival Data ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Mobile Systems ◽  
Linear Network ◽  
Model Free ◽  
Hazards Model ◽  
Reliability Estimates ◽  
Non Linear ◽  
Stochastic Covariates

We present a proportional hazards model (PHM) that establishes a framework suitable for performing reliability estimates and risk prognostics on complex multi-component systems which are transferred at arbitrary times among a discrete set of non-stationary stochastic environments. Such a scenario is not at all uncommon for portable and mobile systems. It is assumed that survival data, possibly interval censored, is available at several “typical” environments. This collection of empirical survival data forms the foundation upon which the basic effects of selected covariates are incorporated via the proportional hazards model. Proportional hazards models are well known in medical statistics, and can provide a variety of data-driven risk models which effectively capture the effects of the covariates. The paper describes three modifications we have found most suitable for this class of systems: development of suitable survival estimators that function well under realistic censoring scenarios, our modifications to the PHM which accommodate time-varying stochastic covariates, and implementation of said model in a non-linear network context which is itself model-free. Our baseline hazard is a parameterized reliability model developed from the empirical reliability estimates. Development of the risk score for arbitrary covariates arising from movement among different random environments is through interaction of the non-linear network and training data obtained from a Markov chain simulation based on stochastic environmental responses generated from Karhunen-Loe`ve models.

Subdistribution hazard models for competing risks in discrete time

Biostatistics ◽  
2018 ◽  
Vol 21 (3) ◽  
pp. 449-466 ◽  
Author(s):  
Moritz Berger ◽  
Matthias Schmid ◽  
Thomas Welchowski ◽  
Steffen Schmitz-Valckenberg ◽  
Jan Beyersmann
Keyword(s):  
Competing Risks ◽  
Discrete Time ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Model Parameters ◽  
Subdistribution Hazard ◽  
Ml Estimation ◽  
Modeling Approach ◽  
Hazards Model ◽  
Event Times

Summary A popular modeling approach for competing risks analysis in longitudinal studies is the proportional subdistribution hazards model by Fine and Gray (1999. A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association94, 496–509). This model is widely used for the analysis of continuous event times in clinical and epidemiological studies. However, it does not apply when event times are measured on a discrete time scale, which is a likely scenario when events occur between pairs of consecutive points in time (e.g., between two follow-up visits of an epidemiological study) and when the exact lengths of the continuous time spans are not known. To adapt the Fine and Gray approach to this situation, we propose a technique for modeling subdistribution hazards in discrete time. Our method, which results in consistent and asymptotically normal estimators of the model parameters, is based on a weighted ML estimation scheme for binary regression. We illustrate the modeling approach by an analysis of nosocomial pneumonia in patients treated in hospitals.

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