Primary Biliary Cirrhosis Data Set Research Articles

Copula modeling for the estimation of measures of marker classification and predictiveness performance with survival outcomes.

The discriminative and predictive power of a continuous-valued marker for survival outcomes can be summarized using the receiver operating characteristic and predictiveness curves, respectively. In this paper, fully parametric and semi-parametric copula-based constructions of the joint model of the marker and the survival time are developed for characterizing, plotting, and analyzing both curves along with other underlying performance measures. The formulations require a copula function, a parametric specification for the margin of the marker, and either a parametric distribution or a non-parametric estimator for the margin of the time to event, to respectively characterize the fully parametric and semi-parametric joint models. Estimation is carried out using maximum likelihood and a two-stage procedure for the parametric and semi-parametric models, respectively. Resampling-based methods are used for computing standard errors and confidence bounds for the various parameters, curves, and associated measures. Graphical inspection of residuals from each conditional distribution is employed as a guide for choosing a copula from a set of candidates. The performance of the estimators of various classification and predictiveness measures is assessed in simulation studies, assuming different copula and censoring scenarios. The methods are illustrated with the analysis of two markers using the familiar primary biliary cirrhosis data set.

Gsem: A Stata command for parametric joint modelling of longitudinal and accelerated failure time models

BackgroundThe number of studies using joint modelling of longitudinal and survival data have increased in the past two decades, but analytical techniques and software shortcomings have remained. A joint model is often used for analysis of a combination of longitudinal sub-model and survival sub-model using shared random effects. Cox regression commonly referring to the survival sub-model, should not be used when proportional hazards assumptions are not satisfied. In such cases, the parametric survival model is preferable. MethodsWe describe different parametric survival models for survival sub-model of joint modelling. We demonstrate how these models can be fit using gsem command (used for generalized structural equation model) in Stata that allows the model to be jointly continuous longitudinal and parametric survival data. With this code, linear mixed effect model is used for the longitudinal sub-model of the joint model, allowing random and fixed effects of the time. In gsem command for survival sub-models, there are five different choices: exponential, Weibull, log-normal, log-logistic and gamma accelerated failure time models. ResultsIn this paper, we have described properties of gsem command for parametric joint modelling and have shown an application for parametric joint models on the 312 patients with primary biliary cirrhosis, which is a major health problem in the western world. ConclusionsWe showed how parametric joint models can be used with the gsem command which has been the only Stata code in the literature to fit the parametric joint models, for the generalized structural equation model, and we used the primary biliary cirrhosis dataset for the detailed application of the command. Thus, the gsem command becomes more useful for fitting parametric joint models.

Joint Modeling of a Longitudinal Measurement and Parametric Survival Data with Application to Primary Biliary Cirrhosis Study

Although longitudinal and survival data are collected in the same study, they are usually analyzed separately. Measurement errors and missing data problems arise because of separate analysis of these two data. Therefore, joint model should be used instead of separate analysis. The standard joint model frequently used in the literature is obtained by combining the linear mixed effect model of longitudinal data and Cox regression model with survival data. Nevertheless, to use the Cox regression model for survival data, the assumption of proportional hazards must be provided. Parametric survival sub-models should be used instead of the Cox regression model for the survival sub-model of the JM where the assumption is not provided. In this article, parametric joint modeling of longitudinal data and survival data that do not provide the assumption of proportional hazards are investigated. For the survival data, the model with Exponential, Weibull, Log-normal, Log-logistic, and Gamma accelerated failure time models and the linear mixed effect model are combined with random effects and the models were applied in primary biliary cirrhosis data set obtained from Mayo Clinic. After determining the best parametric joint model according to Akaike and Bayesian information criterions, the best available model was compared with standard joint model and of separate analysis of survival data and longitudinal data. As a results, in the studies where longitudinal and survival data are obtained together, it is seen that the parametric joint model gives more better results than the standard joint model when the proportional hazard assumption is not provided.

Flexible and Interpretable Models for Survival Data

As data sets continue to increase in size, there is growing interest in methods for prediction that are both flexible and interpretable. A flurry of recent work on this topic has focused on additive modeling in the regression setting, and in particular, on the use of data-adaptive non-linear functions that can be used to flexibly model each covariate’s effect, conditional on the other features in the model. In this paper, we extend this recent line of work to the survival setting. We develop an additive Cox proportional hazards model, in which each additive function is obtained by trend filtering, so that the fitted functions are piece-wise polynomial with adaptively-chosen knots. An efficient proximal gradient descent algorithm is used to fit the model. We demonstrate its performance in simulations and in application to a primary biliary cirrhosis data set, as well as a data set consisting of time to publication for clinical trials in the biomedical literature.

Penalized empirical likelihood for the sparse Cox regression model

The current penalized regression methods for selecting predictor variables and estimating the associated regression coefficients in the sparse Cox model are mainly based on partial likelihood. In this paper, a bias-corrected empirical likelihood method is proposed for the sparse Cox model in conjunction with appropriate penalty functions when the dimensionality of data is high. Theoretical properties of the resulting estimator for the large sample are proved. Simulation studies suggest that penalized empirical likelihood works better than partial likelihood in terms of selecting correct predictors without introducing more model errors. The well-known primary biliary cirrhosis data set is used to illustrate the proposed penalized empirical likelihood method.

Unsupervised learning of mixture regression models for longitudinal data

This paper is concerned with learning of mixture regression models for individuals that are measured repeatedly. The adjective “unsupervised” implies that the number of mixing components is unknown and has to be determined, ideally by data driven tools. For this purpose, a novel penalized method is proposed to simultaneously select the number of mixing components and to estimate the mixture proportions and unknown parameters in the models. The proposed method is capable of handling both continuous and discrete responses by only requiring the first two moment conditions of the model distribution. It is shown to be consistent in both selecting the number of components and estimating the mixture proportions and unknown regression parameters. Further, a modified EM algorithm is developed to seamlessly integrate model selection and estimation. Simulation studies are conducted to evaluate the finite sample performance of the proposed procedure. And it is further illustrated via an analysis of a primary biliary cirrhosis data set.

Frequentist model averaging estimation for the censored partial linear quantile regression model

In this article, we propose a focused information criterion (FIC) and develop a frequentist model averaging estimation procedure for a partial linear regression model when the response is randomly right-censored. The proposed procedure is based on the quantile regression and can depict the comprehensive character of the distribution of the response by means of modeling different quantiles. The large sample properties of the proposed estimators are established, and their finite sample properties are examined through simulation studies. An application to a primary biliary cirrhosis data set is provided.

A KLD-LSSVM based computational method applied for feature ranking and classification of primary biliary cirrhosis stages

Medical datasets having variety of features can increase the complexity of decision process. Prioritising these features with respect to medical sickness is a prime task for effective assessment of patients' health. In this study, a two-step computational method based on Kullback-Leibler divergence (KLD) method and least squares support vector machine (LSSVM) is presented as an integrated model and a LSSVM-based approach is projected as an individual model. KLD was employed to rank the features and radial basis kernel function-based LSSVM approach was deployed to classify primary biliary cirrhosis (PBC) stages. Performance of several machine-learning algorithms, individually as well as in integration with KLD, was evaluated on a real-life biomedical PBC dataset. Simulation results indicated that proposed LSSVM and KLD-LSSVM-based frameworks had shown robustness to the noisy data and had outperformed other individual and integrated methods, respectively. It is concluded that the proposed methodologies can be productively applied to real-life health examination datasets containing a variety of features and multiple decision classes.

Beran-based approach for single-index models under censoring

In this paper we propose a new method for estimating parameters in a single-index model under censoring based on the Beran estimator for the conditional distribution function. This, likelihood-based method is also a useful and simple tool used for bandwidth selection. Additionally, we perform an extensive simulation study comparing this new Beran-based approach with other existing method based on Kaplan---Meier integrals. Finally, we apply both methods to a primary biliary cirrhosis data set and propose a bootstrap test for the parameters.

Empirical Likelihood Inference on Survival Functions under Proportional Hazards Model

Under the framework of the Cox model, it is often of interest to assess a subject’s survival prospect through the individualized predicted survival function, and the corresponding pointwise or simultaneous confidence bands as well. The standard approach to the confidence bands relies on the weak convergence of the estimated survival function to a Gaussian process. Such normal approximation based confidence band may have poor small sample coverage accuracy and generally requires an appropriate transformation to improve its performance. In this paper, we propose an empirical likelihood ratio based pointwise and simultaneous confidence bands that are transformation preserving and therefore eliminate the need of any transformations. The effectiveness of the proposed method is illustrated by a simulation study and an application to the Mayo Clinical primary biliary cirrhosis dataset.

The Mizon–Richard Encompassing Test for the Cox and Aalen Additive Hazards Models

The Cox hazards model (Cox, 1972, Journal of the Royal Statistical Society, Series B34, 187-220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model (Aalen, 1980, in Lecture Notes in Statistics-2, 1-25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon-Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model (Martinussen and Scheike, 2002, Biometrika 89, 283-298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.