Bootstrap test of hypothesis for the multi-state models in survival analysis

Abstract

In the present paper we develop bootstrap tests of hypothesis, based on simulation, for the transition probability matrix arising in the context of a multi-state model. The bootstrap test statistic is based on the paper of Tattar and Vaman (2008 Tattar, P. N., Vaman, H. J. (2008). Testing transition probability matrix of a multi-state model with censored data. Lifetime Data Anal. 14(2):216–230.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which develops a statistic for the testing problems concerning the transition probability matrix of the non homogeneous Markov process.     

Estimation of Survivorship Function Based on Doubly Censored Data with Application to HIV/AIDS

Left censoring concept has been defined in different ways in statistical applications. Turnbull (1974 Turnbull , B. W. ( 1974 ). Nonparametric estimation of a survivorship function with doubly censored data . J. Amer. Statist. Assoc. 69 : 169 – 173 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) defines it in a particular way. Whereas in recent literature, especially in epidemiological studies, it has been defined in another way. This difference between the two approaches is the main reason that despite simplicity, Turnbull method cannot be applicable in all cases of doubly censored data. In this article we present a modified Turnbull method for analysis of doubly censored data adequate with recent definition. Comparison has been done with other statistical methods, including imputation estimator, full likelihood-based and conditional likelihood-based approach using Iranian HIV data.     

A proportional hazards model for the analysis of doubly censored competing risks data

ABSTRACT

Competing risks data are common in medical research in which lifetime of individuals can be classified in terms of causes of failure. In survival or reliability studies, it is common that the patients (objects) are subjected to both left censoring and right censoring, which is refereed as double censoring. The analysis of doubly censored competing risks data in presence of covariates is the objective of this study. We propose a proportional hazards model for the analysis of doubly censored competing risks data, using the hazard rate functions of Gray (1988 Gray, R.J. (1988). A class of k-sample tests for comparing the cumulative incidence of a competing risk. Ann. Statist. 16:1141–1154.[Crossref], [Web of Science ®] , [Google Scholar]), while focusing upon one major cause of failure. We derive estimators for regression parameter vector and cumulative baseline cause specific hazard rate function. Asymptotic properties of the estimators are discussed. A simulation study is conducted to assess the finite sample behavior of the proposed estimators. We illustrate the method using a real life doubly censored competing risks data.     

Modeling bivariate survival data using shared inverse Gaussian frailty model

ABSTRACT

The 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 a random factor (frailty) and the baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and the distribution of frailty. In this paper, we consider inverse Gaussian distribution as frailty distribution and three different baseline distributions, namely the generalized Rayleigh, the weighted exponential, and the extended Weibull distributions. With these three baseline distributions, we propose three different inverse Gaussian shared frailty models. We also compare these models with the models where the above-mentioned distributions are considered without frailty. We develop the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. A search of the literature suggests that currently no work has been done for these three baseline distributions with a shared inverse Gaussian frailty so far. We also apply these three models by using a real-life bivariate survival data set of McGilchrist and Aisbett (1991 McGilchrist, C.A., Aisbett, C.W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to the kidney infection data and a better model is suggested for the data using the Bayesian model selection criteria.     

First- and Second-order Asymptotics for the Tail Distortion Risk Measure of Extreme Risks

The tail distortion risk measure at level p was first introduced in Zhu and Li (2012 Zhu, L., Li, H. (2012). Tail distortion risk and its asymptotic analysis. Insur. Math. Econ. 51(1):115–121.[Crossref], [Web of Science ®] , [Google Scholar]), where the parameter p ∈ (0, 1) indicates the confidence level. They established first-order asymptotics for this risk measure, as p↑1, for the Fréchet case. In this article, we extend their work by establishing both first-order and second-order asymptotics for the Fréchet, Weibull, and Gumbel cases. Numerical studies are also carried out to examine the accuracy of both asymptotics.     

Estimation for exponential distribution based on competing risk middle censored data

Abstract

For a general censoring scheme called “middle censoring” scheme which was proposed by Jammalamadaka and Mangalam (2003 Jammalamadaka, S.R., Mangalam, V. (2003). Nonparametric estimation for middle censored data. J. Nonparametric Statist. 15:253–265.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in nonparametric set up. In this article, point and interval estimation problems are considered for the exponential distribution when the failure data is middle censored with two independent competing failure risks. Different methods are introduced to estimate the unknown model parameters such as maximum likelihood estimation, midpoint approximation, equivalent quantities estimation. The Bayesian estimation is also considered with gamma priors. Two numerical examples are analyzed to show the performance of the proposed methods.     

Group Variable Selection with Oracle Property by Weight-Fused Adaptive Elastic Net Model for Strongly Correlated Data

This paper is the generalization of weight-fused elastic net (Fu and Xu, 2012 Fu, G., Xu, Q. (2012). Grouping variable selection by weight fused elastic net for multi-collinear data. Communications in Statistics-Simulation and Computation 41(2):205–221.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which performs group variable selection by combining weight-fused LASSO(wfLasso) and elastic net (Zou and Hastie, 2005 Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2):301–320.[Crossref], [Web of Science ®] , [Google Scholar]) penalties. In this study, the elastic net penalty is replaced by adaptive elastic net penalty (AdaEnet) (Zou and Zhang, 2009 Zou, H., Zhang, H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics 37(4):1733–1751.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), and a new group variable selection algorithm with oracle property (Fan and Li, 2001 Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456):1348–1360.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Zou, 2006 Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101(476):1418–1429.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is obtained.     

Bayesian Estimation of Parameters and Comparison of Shared Gamma Frailty Models

In this article, we introduce shared gamma frailty models with three different baseline distributions namely, Weibull, generalized exponential and exponential power distributions. We develop Bayesian estimation procedure using Markov Chain Monte Carlo(MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these three models to a real life bivariate survival dataset of McGilchrist and Aisbett (1991 McGilchrist, C. A. and Aisbett, C. W. 1991. Regression with frailty in survival analysis. Biometrics, 47: 461–466. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to kidney infection data and a better model is suggested for the data.     

Note on posterior inference for the Bingham distribution

The properties of high-dimensional Bingham distributions have been studied by Kume and Walker (2014 Kume, A., and S. G. Walker. 2014. On the Bingham distribution with large dimension. Journal of Multivariate Analysis 124:345–52.[Crossref], [Web of Science ®] , [Google Scholar]). Fallaize and Kypraios (2016 Fallaize, C. J., and T. Kypraios. 2016. Exact Bayesian inference for the Bingham distribution. Statistics and Computing 26:349–60.[Crossref], [Web of Science ®] , [Google Scholar]) propose the Bayesian inference for the Bingham distribution and they use developments in Bayesian computation for distributions with doubly intractable normalizing constants (Møller et al. 2006 Møller, J., A. N. Pettitt, R. Reeves, and K. K. Berthelsen. 2006. An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika 93 (2):451–458.[Crossref], [Web of Science ®] , [Google Scholar]; Murray, Ghahramani, and MacKay 2006 Murray, I., Z. Ghahramani, and D. J. C. MacKay. 2006. MCMC for doubly intractable distributions. In Proceedings of the 22nd annual conference on uncertainty in artificial intelligence (UAI-06), 359–66. AUAI Press. [Google Scholar]). However, they rely heavily on two Metropolis updates that they need to tune. In this article, we propose instead a model selection with the marginal likelihood.     

Exact Inference for the pth-Quantile and the Reliability of the Two-Parameter Exponential Distribution with Singly Type II Censoring: A Standard Approach

Starting from a standard pivot, exact inference for the pth-quantile and for the reliability of the two-parameter exponential distribution in case of singly Type II censored samples is developed in this article. Fernandez (2007 Fernandez , A. J. ( 2007 ). On calculating generalized confidence intervals for the two-parameter exponential reliability function . Statistics 41 : 129 – 135 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first obtained some of the results proposed in this article, but, differently from what are proposed here, and developed his theory starting from a generalized pivot. An illustrative example shows that, with the expressions proposed in this article, it is also possible to overcome some shortcomings raising from the formulas by Fernandez (2007 Fernandez , A. J. ( 2007 ). On calculating generalized confidence intervals for the two-parameter exponential reliability function . Statistics 41 : 129 – 135 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Finally, a new expression for the moments of the pivot is obtained.     

On Fitting Transformation Model to Survey Data

In this article, we consider fitting a semiparametric linear model to survey data with censored observations. The specific goal of the paper is to extend the methods of Cheng et al. (1995 Cheng, S.C., Wei, L.J., Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika 82(4):835–845.[Crossref], [Web of Science ®] , [Google Scholar]) and Chen et al. (2002 Chen, K., Jin, Z. Ying, Z. (2002). Semiparametric analysis of transformation models with censored data. Biometrika 89:659–668.[Crossref], [Web of Science ®] , [Google Scholar]) to the case when the sample has been drawn from a population using a complex sampling design. Similar to the approach of Lin (2000 Lin, D.Y. (2000). On fitting Cox’s proportional hazards models to survey data. Biometrika 87:37–47.[Crossref], [Web of Science ®] , [Google Scholar]), we regard the survey population as a random sample from an infinite universe and accounts for this randomness in the statistical inference. A simulation study is conducted to investigate the performance of the proposed estimators.     

Maximum Likelihood Estimation in Compound Geometric Failure Model with Changing Parameters From Type-I Two-Stage Progressively Censored and Group Censored Samples

Boardman and Kendell (1970 Boardman , T. J. , Kendell , P. J. ( 1970 ). Estimation in compound failure models . Technometrics 12 : 891 – 908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimation with respect to Type-I censoring when an item is subjected to only one of the two causes of failure assuming exponential model. Patel and Gajjar (1992 Patel , M. N. , Gajjar , A. V. ( 1992 ). Maximum likelihood estimation in compound exponential failure model with changing failure rates from Type-I progressively censored and group censored samples . Commun. Statist. Theor. Meth. 21 ( 10 ): 2899 – 2908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered extension of the Boardman and Kendell's results in case of two-stage progressive censoring. Here we have considered geometric competing risk failure model with two independent causes of failures. Maximum likelihood estimation of the parameters is carried out using Type-I two-stage progressively censored and group censored samples. Asymptotic standard errors of the estimators are obtained for both the cases. Two illustrative examples are cited for ungroup and group competing risk models.     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.     

Skew-Normal ARMA Models with Nonlinear Heteroscedastic Predictors

Multivariate skew-normal (SN) distributions (Azzalini and Dalla Valle, 1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]) enjoy some of the useful properties of normal distributions, have nonlinear heteroscedastic predictors but lack the closure property of normal distributions (the sum of independent SN random variables is not SN). Recently, there has been a proliferation of classes of SN distributions with certain closure properties, one of the most promising being the closed skew-normal (CSN) distributions of González-Farías et al. (2004 González-Farías , G. , Dominguez-Molina , J. A. , Gupta , A. K. ( 2004 ). Additive properties of skew-normal random vectors . J. Statist. Plann. Infer. 126 : 521 – 534 .[Crossref], [Web of Science ®] , [Google Scholar]). We study the construction of stationary SN ARMA models for colored SN noise and show that their finite-dimensional distributions are skew-normal, seldom strictly stationary and their covariance functions differ from their normal ARMA counterparts in that they do not converge to zero for large lags. The situation is better for ARMA models driven by CSN noise, but at the additional cost of considerable computational complexity and a less explicit skewness parameter. In view of these results, the widespread use of such classes of SN distributions in the framework of ARMA models seem doubtful.     

New Approach of Directional Dependence in Exchange Markets Using Generalized FGM Copula Function

This article presents an application of copula methodology in exchange markets. In this article, we consider the concept of directional dependence given by Sungur (2005 Sungur , E. A. ( 2005 ). Some observations on copula regression functions . Communications in Statistics—Theory and Methods 34 : 1967 – 1978 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We also consider and study directional dependence for generalized Farlie–Gumbel–Morgenstern (FGM) distributions, which are a member of the Rodríguez-Lallena and Úbeda-Flores (2004 Rodríguez-Lallena , J. A. , Úbeda-Flores , M. ( 2004 ). A new class of bivariate copulas . Statistical Probability Letters 66 : 315 – 325 .[Crossref], [Web of Science ®] , [Google Scholar]) family, C(u, v) = uv + f(u)g(v). Examples of the generalized FGM distributions are provided with exchange market data of the Euro, Canadian dollar, Korean Won, Japanese Yen, and Hong Kong dollar against the U.S. dollar.     

Testing factor–covariate interaction in rank repeated-measures analysis of covariance models

This article derives the asymptotic properties of rank-based tests for the covariate effects in rank repeated-measures analysis of covariance (ANCOVA) models (Fan and Zhang 2017 Fan, C., and D. Zhang. 2017. Rank repeated measures analysis of covariance. Communications in Statistics - Theory and Methods 46:1158–83.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) employing generalized estimating equation (GEE) techniques. One interested application of the proposed tests is to check the validity of the assumption of homogeneous covariate effects in different levels of the factors. Performance of the proposed tests has been confirmed by simulation studies and illustrated using the famous seizure count data. While the article mainly focuses on interaction tests, the scope of the proposed tests includes testing any contrast of the covariate effect such as the null of no overall covariate effect.     

Inference in log-alpha-power and log-skew-normal multivariate models

ABSTRACT

Random vectors with positive components are common in many applied fields, for example, in meteorology, when daily precipitation is measured through a region Marchenko and Genton (2010 Marchenko, Y., Genton, M. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21:318–340.[Crossref], [Web of Science ®] , [Google Scholar]). Frequently, the log-normal multivariate distribution is used for modeling this type of data. This modeling approach is not appropriate for data with high asymmetry or kurtosis. Consequently, more flexible multivariate distributions than the log-normal multivariate are required. As an alternative to this distribution, we propose the log-alpha-power multivariate and log-skew-normal multivariate models. The first model is an extension for positive data of the fractional order statistics model Durrans (1992 Durrans, S. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]). The second one is an extension of the log-skew-normal model studied by Mateu-Figueras and Pawlowsky-Glahn (2007 Mateu-Figueras, G., Pawlowsky-Glahn, V. (2007). The skew-normal distribution on the simplex. Commun. Stat.-Theory Methods 36:1787–1802.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We study parameter estimation for these models by means of pseudo-likelihood and maximum likelihood methods. We illustrate the proposal analyzing a real dataset.     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.     

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99: 23122327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010 Lin, G.D., Huang, J.S. (2010). A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals. J. Multivariate Anal. 101: 22272233.[Crossref], [Web of Science ®] , [Google Scholar]) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013 Bairamov, I., Bayramoglu, K. (2013). From Huang-Kotz distribution to Baker’s distribution. J. Multivariate Anal. 113: 106115.[Crossref], [Web of Science ®] , [Google Scholar]) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.     

Closeness of k-records to Progressive Type-II Censored Order Statistics for Location-scale Families

In this article, the Pitman closeness of upper and lower k-records to progressive Type-II censored order statistics for location-scale families is investigated. In each case, the special properties of the probability of Pitman closeness are obtained and the corresponding monotonicity properties are discussed. Moreover, the closest k-record to a specific progressive Type-II censored data is obtained. Finally, for the standard exponential and standard uniform distributions, explicit expressions for the probability of Pitman closeness are derived. For various censoring schemes, the results of the numerical computations are displayed in tables. Most of the results in Ahmadi and Balakrishnan (2013) Ahmadi, J., Balakrishnan, N. (2013). On the nearness of record values to order statistics from Pitman measure of closeness. Metrika 76:521–541.[Crossref], [Web of Science ®] , [Google Scholar] can be achieved as special cases.