Improvement of the Liu Estimator in Weighted Mixed Regression

ABSTRACT

The maximum likelihood approach to the proportional hazards model is considered. The purpose is to find a general approach to the analysis of the proportional hazards model, whether the baseline distribution is absolutely continuous, discrete, or a mixture. The advantage is that ties are treated without pain, while the performance for continuous data is almost the same as Cox's partial likelihood. The potential disadvantage with many nuisance parameters is taken care of by profiling them out for risk sets containing only one failure.     

Bayesian partial likelihood approach for tied observations

The Bayesian analysis based on the partial likelihood for Cox's proportional hazards model is frequently used because of its simplicity. The Bayesian partial likelihood approach is often justified by showing that it approximates the full Bayesian posterior of the regression coefficients with a diffuse prior on the baseline hazard function. This, however, may not be appropriate when ties exist among uncensored observations. In that case, the full Bayesian and Bayesian partial likelihood posteriors can be much different. In this paper, we propose a new Bayesian partial likelihood approach for many tied observations and justify its use.     

LIKELIHOOD APPROXIMATIONS AND DISCRETE MODELS FOR TIED SURVIVAL DATA

ABSTRACT

Ties among event times are often recorded in survival studies. For example, in a two week laboratory study where event times are measured in days, ties are very likely to occur. The proportional hazards model might be used in this setting using an approximated partial likelihood function. This approximation works well when the number of ties is small. On the other hand, discrete regression models are suggested when the data are heavily tied. However, in many situations it is not clear which approach should be used in practice. In this work, empirical guidelines based on Monte Carlo simulations are provided. These recommendations are based on a measure of the amount of tied data present and the mean square error. An example illustrates the proposed criterion.     

A real survival analysis application via variable selection methods for Cox's proportional hazards model

Variable selection is fundamental to high-dimensional statistical modeling in diverse fields of sciences. In our health study, different statistical methods are applied to analyze trauma annual data, collected by 30 General Hospitals in Greece. The dataset consists of 6334 observations and 111 factors that include demographic, transport, and clinical data. The statistical methods employed in this work are the nonconcave penalized likelihood methods, Smoothly Clipped Absolute Deviation, Least Absolute Shrinkage and Selection Operator, and Hard, the maximum partial likelihood estimation method, and the best subset variable selection, adjusted to Cox's proportional hazards model and used to detect possible risk factors, which affect the length of stay in a hospital. A variety of different statistical models are considered, with respect to the combinations of factors while censored observations are present. A comparative survey reveals several differences between results and execution times of each method. Finally, we provide useful biological justification of our results.     

The glm log-rank test:general linear modeling of log-rank scores as a method of analysis for survival data

The use of general linear modeling (GLM) procedures based on log-rank scores is proposed for the analysis of survival data and compared to standard survival analysis procedures. For the comparison of two groups, this approach performed similarly to the traditional log-rank test. In the case of more complicated designs - without ties in the survival times - the approach was only marginally less powerful than tests from proportional hazards models, and clearly less powerful than a likelihood ratio test for a fully parametric model; however, with ties in the survival time, the approach proved more powerful than tests from Cox's semi-parametric proportional hazards procedure. The method appears to provide a reasonably powerful alternative for the analysis of survival data, is easily used in complicated study designs, avoids (semi-)parametric assumptions, and is quite computationally easy and inexpensive to employ.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Testing for Covariate Effect in the Cox Proportional Hazards Regression Model

This article presents methods for testing covariate effect in the Cox proportional hazards model based on Kullback–Leibler divergence and Renyi's information measure. Renyi's measure is referred to as the information divergence of order γ (γ ≠ 1) between two distributions. In the limiting case γ → 1, Renyi's measure becomes Kullback–Leibler divergence. In our case, the distributions correspond to the baseline and one possibly due to a covariate effect. Our proposed statistics are simple transformations of the parameter vector in the Cox proportional hazards model, and are compared with the Wald, likelihood ratio and score tests that are widely used in practice. Finally, the methods are illustrated using two real-life data sets.     

A Cautionary Note on the Analysis of Extreme Data with Cox Regression

This article analyzes a small censored data set to demonstrate the potential dangers of using statistical computing packages without understanding the details of statistical methods. The data, consisting of censored response times with heavy ties in one time point, were analyzed with a Cox regression model utilizing SAS PHREG and BMDP2L procedures. The p values, reported from both SAS PHREG and BMDP2L procedures, for testing the equality of two treatments vary considerably. This article illustrates that (1) the Breslow likelihood used in both BMDP2L and SAS PHREG procedures is too conservative and can have a critical effect on an extreme data set, (2) Wald's test in the SAS PHREG procedure may yield absurd results from most likelihood models, and (3) BMDP2L needs to include more than just the Breslow likelihood in future development.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Global Partial Likelihood for Nonparametric Proportional Hazards Models

As an alternative to the local partial likelihood method of Tibshirani and Hastie and Fan, Gijbels, and King, a global partial likelihood method is proposed to estimate the covariate effect in a nonparametric proportional hazards model, λ(t|x) = exp{ψ(x)}λ(0)(t). The estimator, ψ?(x), reduces to the Cox partial likelihood estimator if the covariate is discrete. The estimator is shown to be consistent and semiparametrically efficient for linear functionals of ψ(x). Moreover, Breslow-type estimation of the cumulative baseline hazard function, using the proposed estimator ψ?(x), is proved to be efficient. The asymptotic bias and variance are derived under regularity conditions. Computation of the estimator involves an iterative but simple algorithm. Extensive simulation studies provide evidence supporting the theory. The method is illustrated with the Stanford heart transplant data set. The proposed global approach is also extended to a partially linear proportional hazards model and found to provide efficient estimation of the slope parameter. This article has the supplementary materials online.     

Maximum likelihood estimation in the proportional hazards model of random censorship

The maximum likelihood estimator (MLE) for the survival function S_Tunder the proportional hazards model of censorship is derived and shown to differ from the Abdushukurov-Cheng-Lin estimator when the class of allowable distributions includes all continuous and discrete distributions. The estimators are compared via an example. The MLE is calculated using a Newton-Raphson iterative procedure and implemented via a FORTRAN algorithm.     

Extended Hazard Regression Model for Reliability and Survival Analysis

We propose an extended hazard regression model which allows the spread parameter to be dependent on covariates. This allows a broad class of models which includes the most common hazard models, such as the proportional hazards model, the accelerated failure time model and a proportional hazards/accelerated failure time hybrid model with constant spread parameter. Simulations based on sub-classes of this model suggest that maximum likelihood performs well even when only small or moderate-size data sets are available and the censoring pattern is heavy. The methodology provides a broad framework for analysis of reliability and survival data. Two numerical examples illustrate the results.     

Consistency of The MMGLE under the piecewise proportional hazards models with interval-censored data

Wong et al. [(2018), ‘Piece-wise Proportional Hazards Models with Interval-censored Data’, Journal of Statistical Computation and Simulation, 88, 140–155] studied the piecewise proportional hazards (PWPH) model with interval-censored (IC) data under the distribution-free set-up. It is well known that the partial likelihood approach is not applicable for IC data, and Wong et al. (2018) showed that the standard generalised likelihood approach does not work either. They proposed the maximum modified generalised likelihood estimator (MMGLE) and the simulation results suggest that the MMGLE is consistent. We establish the consistency and asymptotically normality of the MMGLE.     

An extended semiparametric model for short-term and long-term hazard ratios

In survival analysis, one way to deal with non-proportional hazards is to model short-term and long-term hazard ratios. The existing model of this nature has no control over how fast the hazard ratio is changing over time. We add a parameter to the existing model to allow the hazard ratio to change over time at different speed. A nonparametric maximum likelihood approach is used to estimate the model parameters. The existing model is a special case of the extended model when the speed parameter is 0, which leads naturally to a way of testing the adequacy of the existing model. Simulation results show that there can be substantial bias in the estimation of the short-term and long-term hazard ratio if the speed parameter is fixed incorrectly at 0 rather than estimated. The extended model is fitted to three real data sets to shed new insights, including the observation that converging hazards does not necessarily imply the odds are proportional.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions

This paper deals with the Bayesian estimation of generalized exponential distribution in the proportional hazards model of random censorship under asymmetric loss functions. It is well known for the two-parameter lifetime distributions that the continuous conjugate priors for parameters do not exist; we assume independent gamma priors for the scale and the shape parameters. It is observed that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Tierney–Kadane's approximation and Gibbs sampling to approximate the Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed methods and one real data analysis is performed for illustration. Bayesian methods are compared with maximum likelihood and it is observed that the Bayes estimators perform better than the maximum-likelihood estimators in some cases.     

Proportional cause-specific reversed hazards model

The proportional reversed hazards model explains the multiplicative effect of covariates on the baseline reversed hazard rate function of lifetimes. In the present study, we introduce a proportional cause-specific reversed hazards model. The proposed regression model facilitates the analysis of failure time data with multiple causes of failure under left censoring. We estimate the regression parameters using a partial likelihood approach. We provide Breslow's type estimators for the cumulative cause-specific reversed hazard rate functions. Asymptotic properties of the estimators are discussed. Simulation studies are conducted to assess their performance. We illustrate the applicability of the proposed model using a real data set.     

Piecewise proportional hazards models with interval-censored data

We consider the piecewise proportional hazards (PWPH) model with interval-censored (IC) relapse times under the distribution-free set-up. The partial likelihood approach is not applicable for IC data, and the generalized likelihood approach has not been studied in the literature. It turns out that under the PWPH model with IC data, the semi-parametric MLE (SMLE) of the covariate effect under the standard generalized likelihood may not be unique and may not be consistent. In fact, the parameter under the PWPH model with IC data is not identifiable unless the identifiability assumption is imposed. We propose a modification to the likelihood function so that its SMLE is unique. Under the identifiability assumption, our simulation study suggests that the SMLE is consistent. We apply the method to our cancer relapse time data and conclude that the bone marrow micrometastasis does not have a significant prognostic factor.     

Simplex Mixed‐Effects Models for Longitudinal Proportional Data

Abstract. Continuous proportional outcomes are collected from many practical studies, where responses are confined within the unit interval (0,1). Utilizing Barndorff‐Nielsen and Jørgensen's simplex distribution, we propose a new type of generalized linear mixed‐effects model for longitudinal proportional data, where the expected value of proportion is directly modelled through a logit function of fixed and random effects. We establish statistical inference along the lines of Breslow and Clayton's penalized quasi‐likelihood (PQL) and restricted maximum likelihood (REML) in the proposed model. We derive the PQL/REML using the high‐order multivariate Laplace approximation, which gives satisfactory estimation of the model parameters. The proposed model and inference are illustrated by simulation studies and a data example. The simulation studies conclude that the fourth order approximate PQL/REML performs satisfactorily. The data example shows that Aitchison's technique of the normal linear mixed model for logit‐transformed proportional outcomes is not robust against outliers.