Using Frailty Models to Identify the Longitudinal Biomarkers in Survival Analysis

We introduce a score test to identify longitudinal biomarkers or surrogates for a time to event outcome. This method is an extension of Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 2002 Henderson , R. , Diggle , P. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, a score test is based on a joint likelihood function which combines the likelihood functions of the longitudinal biomarkers and the survival times. Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 2002 Henderson , R. , Diggle , P. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) assumed that the same random effect exists in the longitudinal component and in the Cox model and then they can derive a score test to determine if a longitudinal biomarker is associated with time to an event. We extend this work and our score test is based on a joint likelihood function which allows other random effects to be present in the survival function.

Considering heterogeneous baseline hazards in individuals, we use simulations to explore how the factors can influence the power of a score test to detect the association of a longitudinal biomarker and the survival time. These factors include the functional form of the random effects from the longitudinal biomarkers, in the different number of individuals, and time points per individual. We illustrate our method using a prothrombin index as a predictor of survival in liver cirrhosis patients.     

A Broad Class of Partially Specified Autoregressions on Multi-Casting Data

Extending the bifurcating autoregressive (BAR) process (cf. Cowan and Staudte, 1986 Cowan , R. , Staudte , R. G. ( 1986 ). The bifurcating autoregression model in cell lineage studies . Biometrics 42 : 769 – 783 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to multi-casting (multi-splitting) data, Hwang and Choi (2009 Hwang , S. Y. , Choi , M. S. ( 2009 ). Modeling and large sample estimation for multi-casting autoregression . Statist. Prob. Lett. 79 : 1943 – 1950 .[Crossref], [Web of Science ®] , [Google Scholar]) introduced multi-casting autoregression (MCAR, for short) defined on multi-casting tree structured data. This article is concerned with the case when the MCAR model is partially specified only through conditional mean and variance without directly imposing autoregressive (AR) structure. The resulting class of models will be referred to as P-MCAR (partially specified MCAR). The P-MCAR considerably enlarges the class of multi-casting models including (as special cases) MCAR, random coefficient MCAR, conditionally heteroscedastic multi-casting models and binomial-thinning processes. Moment structures for this broad P-MCAR class are investigated. Least squares (LS) estimation method is discussed and asymptotic relative efficiency (ARE) of the generalized-LS over ordinary-LS is obtained in a closed form. A simulation study is conducted to illustrate results.     

Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

We consider non-parametric estimation of a continuous cdf of a random vector (X ₁, X ₂). With bivariate RC data, it is stated in van der Laan (1996 Van der Laan , M. J. ( 1996 ) Efficient estimation in the bivariate censoring model and repairing NPMLE . Ann. Statist. 24 : 596 – 627 .[Crossref], [Web of Science ®] , [Google Scholar], p. 598¹⁰, Ann. Statist.), Quale et al. (2006 Quale , C. M. , van der Laan , M. J. , Robins , J. R. ( 2006 ). Locally efficient estimation with bivariate right-censored data . JASA. 101 : 1076 – 1084 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar])).” The claim is based on a result in Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], p.1352, Ann. Statist.) that if X ₁ is right censored but not X ₂, then common ways for defining one NPMLE lead to inconsistency. If X ₁ is right censored and X ₂ is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], Ann. Statist.).     

A Comparison of Procedures for Controlling the False Discovery Rate in the Presence of Small Variance Genes: A Simulation Study

The Significance Analysis of Microarrays (SAM; Tusher et al., 2001 Tusher , V. G. , Tibshirani , R. , Chu , G. ( 2001 ). Significance analysis of microarrys applied to the ionizing radiation response . Proceedings of the National Academy of Sciences 98 : 5116 – 5121 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) method is widely used in analyzing gene expression data while controlling the FDR by using resampling-based procedure in the microarray setting. One of the main components of the SAM procedure is the adjustment of the test statistic. The introduction of the fudge factor to the test statistic aims at deflating the large value of test statistics due to the small standard error of gene-expression. Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) pointed out that the fudge factor does not effectively improve the power and the control of the FDR as compared to the SAM procedure without the fudge factor in the presence of small variance genes. Motivated by the simulation results presented in Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), in this article, we extend our study to compare several methods for choosing the fudge factor in the modified t-type test statistics and use simulation studies to investigate the power and the control of the FDR of the considered methods.     

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.     

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.     

Improved Bar-Lev,Bobovitch, and Boukai Randomized Response Models

In this article, two new improved randomized response models have been proposed. The proposed models are found to be more efficient than the recent randomized response model studied by Bar-Lev et al. (2004 Bar-Lev , S. K. , Bobovitch , E. , Boukai , B. ( 2004 ). A note on randomized response models for quantitative data . Metrika 60 : 255 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]). The relative efficiency of the proposed models has been studied with respect to the Bar-Lev et al. (2004 Bar-Lev , S. K. , Bobovitch , E. , Boukai , B. ( 2004 ). A note on randomized response models for quantitative data . Metrika 60 : 255 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]) model under different situations.     

Generalized Scrambling in Quantitative Optional Randomized Response Models

Huang (2010 Huang , K. C. ( 2010 ). Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling . Metrika 71 : 341 – 352 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed an optional randomized response model using a linear combination scrambling which is a generalization of the multiplicative scrambling of Eichhorn and Hayre (1983 Eichhorn , B. H. , Hayre , L. S. ( 1983 ). Scrambled randomized response methods for obtaining sensitive quantitative data . J. Statist. Plann. Infer. 7 : 307 – 316 .[Crossref], [Web of Science ®] , [Google Scholar]) and the additive scrambling of Gupta et al. (2006, 2010). In this article, we discuss two main issues. (1) Can the Huang (2010 Huang , K. C. ( 2010 ). Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling . Metrika 71 : 341 – 352 .[Crossref], [Web of Science ®] , [Google Scholar]) model be improved further by using a two-stage approach?; (2) Does the linear combination scrambling provide any benefit over the additive scrambling of Gupta et al. (2010 Gupta , S. N. , Shabbir , J. , Sehra , S. ( 2010 ). Mean and sensitivity estimation in optional randomized response models . J. Statist. Plann. Infer. 140 : 2870 – 2874 .[Crossref], [Web of Science ®] , [Google Scholar])? We will note that the answer to the first question is “yes” but the answer to the second question is “no.”     

The Performance of Panel Cointegration Methods: Results from a Large Scale Simulation Study

This article presents results concerning the performance of both single equation and system panel cointegration tests and estimators. The study considers the tests developed in Pedroni (1999 Pedroni , P. ( 1999 ). Critical values for cointegration tests in heterogeneous panels with multiple regressors . Oxford Bulletin of Economics and Statistics 61 : 653 – 670 .[Crossref], [Web of Science ®] , [Google Scholar], 2004 Pedroni , P. ( 2004 ). Panel cointegration. Asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis . Econometric Theory 20 : 597 – 625 .[Crossref], [Web of Science ®] , [Google Scholar]), Westerlund (2005 Westerlund , J. ( 2005 ). New simple tests for panel cointegration . Econometric Reviews 24 : 297 – 316 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Larsson et al. (2001 Larsson , R. , Lyhagen , J. , Löthgren , M. ( 2001 ). Likelihood-based cointegration tests in heterogeneous panels . Econometrics Journal 4 : 109 – 142 .[Crossref] , [Google Scholar]), and Breitung (2005 Breitung , J. ( 2005 ). A parametric approach to the estimation of cointegration vectors in panel data . Econometric Reviews 24 : 151 – 173 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the estimators developed in Phillips and Moon (1999 Phillips , P. C. B. , Moon , H. R. ( 1999 ). Linear regression limit theory for nonstationary panel data . Econometrica 67 : 1057 – 1111 .[Crossref], [Web of Science ®] , [Google Scholar]), Pedroni (2000 Pedroni , P. ( 2000 ). Fully modified OLS for heterogeneous cointegrated panels . In: Baltagi , B. H. , ed. Nonstationary Panels, Panel Cointegration, and Dynamic Panels . Amsterdam : Elsevier , pp. 93 – 130 .[Crossref] , [Google Scholar]), Kao and Chiang (2000 Kao , C. , Chiang , M.-H. ( 2000 ). On the estimation and inference of a cointegrated regression in panel data . In: Baltagi , B. H. , ed. Nonstationary Panels, Panel Cointegration, and Dynamic Panels . Amsterdam : Elsevier , pp. 179 – 222 .[Crossref] , [Google Scholar]), Mark and Sul (2003 Mark , N. C. , Sul , D. ( 2003 ). Cointegration vector estimation by panel dynamic OLS and long-run money demand . Oxford Bulletin of Economics and Statistics 65 : 655 – 680 .[Crossref], [Web of Science ®] , [Google Scholar]), Pedroni (2001 Pedroni , P. ( 2001 ). Purchasing power parity tests in cointegrated panels . Review of Economics and Statistics 83 : 1371 – 1375 . [Google Scholar]), and Breitung (2005 Breitung , J. ( 2005 ). A parametric approach to the estimation of cointegration vectors in panel data . Econometric Reviews 24 : 151 – 173 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We study the impact of stable autoregressive roots approaching the unit circle, of I(2) components, of short-run cross-sectional correlation and of cross-unit cointegration on the performance of the tests and estimators. The data are simulated from three-dimensional individual specific VAR systems with cointegrating ranks varying from zero to two for fourteen different panel dimensions. The usual specifications of deterministic components are considered.     

Estimators for the Population Mean in the Case of Missing Data

We propose a class of estimators for the population mean when there are missing data in the data set. Obtaining the mean square error equations of the proposed estimators, we show the conditions where the proposed estimators are more efficient than the sample mean, ratio-type estimators, and the estimators in Singh and Horn (2000 Singh , S. , Horn , S. ( 2000 ). Compromised imputation in survey sampling . Metrika 51 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) and Singh and Deo (2003 Singh , S. , Deo , B. (2003). Imputation by power transformation. Statist. Pap. 44:555–579.[Crossref], [Web of Science ®] , [Google Scholar]) in the case of missing data. These conditions are also supported by a numerical example.     

A Note on Importance Resampling for Multi-Dimensional Statistics

Johns (1988 Johns , M. V. (1988). Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association 83:709–714.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Davison (1988 Davison , A. C. ( 1988 ). Discussion of paper by D. V. Hinkley . Journal of the Royal Statistical Society Series B 50 : 356 – 357 . [Google Scholar]), and Do and Hall (1991 Do , K. A. , Hall , P. ( 1991 ). On importance sampling for the bootstrap . Biometrika 78 : 161 – 167 .[Crossref], [Web of Science ®] , [Google Scholar]) used importance sampling for calculating bootstrap distributions of one-dimensional statistics. Realizing that their methods can not be extended easily to multi-dimensional statistics, Fuh and Hu (2004 Fuh , C. D. , Hu , I. ( 2004 ). Efficient importance sampling for events of moderate deviations with applications . Biometrika 91 : 471 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed an exponential tilting formula for statistics of multi-dimension, which is optimal in the sense that the asymptotic variance is minimized for estimating tail probabilities of asymptotically normal statistics. For one-dimensional statistics, Hu and Su (2008 Hu , J. , Su , Z. ( 2008 ). Adaptive resampling algorithms for estimating bootstrap distributions . Journal of Statistical Planning and Inference 138 ( 6 ): 1763 – 1777 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a multi-step variance minimization approach that can be viewed as a generalization of the two-step variance minimization approach proposed by Do and Hall (1991 Do , K. A. , Hall , P. ( 1991 ). On importance sampling for the bootstrap . Biometrika 78 : 161 – 167 .[Crossref], [Web of Science ®] , [Google Scholar]). In this article, we generalize the approach of Hu and Su (2008 Hu , J. , Su , Z. ( 2008 ). Adaptive resampling algorithms for estimating bootstrap distributions . Journal of Statistical Planning and Inference 138 ( 6 ): 1763 – 1777 .[Crossref], [Web of Science ®] , [Google Scholar]) to multi-dimensional statistics, which applies to general statistics and does not resort to asymptotics. Empirical results on a real survival data set show that the proposed algorithm provides significant computational efficiency gains.     

Bayesian Capture-Recapture Analysis: An Application in Modeling Semelparity of a Neotropical Didelphid Marsupial

We develop a series of Bayesian statistical models for estimating survival of a neotropic didelphid marsupial, the Brazilian gracile mouse opossum (Gracilinanus microtarsus). These models are based on the Cormack–Jolly–Seber model (Cormack, 1964 Cormack , R. M. ( 1964 ). Estimates of survival from the sighting of marked animals . Biometrika 51 : 429 – 438 .[Crossref], [Web of Science ®] , [Google Scholar]; Jolly 1965 Jolly , G. M. ( 1965 ). Explicit estimates from capture-recapture data with both death and immigration stochastic model . Biometrika 52 : 225 – 247 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Seber 1965 Seber , G. A. F. ( 1965 ). A note on the multiple recapture census . Biometrika 52 : 249 – 259 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) with both survival and recapture rates expressed as a function of covariates using a logit link. The proposed models allow taking into account heterogeneity in capture probability caused by the existence of different groups of individuals in the population. The models were applied to two cohorts (Cohort, 2000, 2001) with the first one including 14 and the second one 15 sampling occasions. The best models for each of the cohorts indicate that G. microtarsus is best described as partially semelparous, a condition in which mortality after the first mating is high but graded over time, with a fraction of males surviving for a second breeding season (Boonstra, 2005 Boonstra , R. ( 2005 ). Equipped for life: the adaptive role of the stress axis in male mammals . Journal of Mammalogy 86 : 236 – 247 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Additive Mixed Effect Model for Clustered Doubly Censored Data

Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) proposed an additive mixed effect model to analyze clustered right-censored data. In this article, we demonstrate that the approach of Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) can be extended to clustered doubly censored data. Furthermore, when both left- and right-censoring variables are always observed, we propose alternative estimators using the approach of Cai and Cheng (2004 Cai, T. and Cheng, S. C. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] , [Google Scholar]). A simulation study is conducted to investigate the performance of the proposed estimators.     

Aspects of Dependence in Generalized Farlie-Gumbel-Morgenstern Distributions

In this article, we obtained a dependence measure for generalized Farlie-Gumbel-Morgenstern (FGM) family in view of Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar]) and then compared this measure with Spearman's rho and Kendall's tau in FGM family. Moreover, we evaluated the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar], 1990 Kochar , S. G. , Gupta , R. P. ( 1990 ). Distribution-free tests based on sub-sample extrema for testing against positive dependence . Australian Journal of Statistics 32 : 45 – 51 .[Crossref] , [Google Scholar]) based on exact distribution of a U-statistics. This is derived via a simulation study for sample of sizes n = 6, 8, 10, 12, 16, and 20. Also, we compared our simulation results with those achieved by Amini et al. (2010 Amini , M. , Jabbari , H. , Mohtashami Borzadaran , G. R. , Azadbakhsh , M. ( 2010 ). Power comparison of independence test for the Farlie-Gumbel-Moregenstern family . Communications of the Korean Statistical Society 17 ( 4 ): 493 – 505 .[Crossref] , [Google Scholar]) and Güven and Kotz (2008 Güven , B. , Kotz , S. ( 2008 ). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions . Journal of Computational and Applied Mathematics 212 : 102 – 111 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Improved strategy to collect sensitive data by using geometric distribution as a randomization device

Abstract

In this article, we improvise Singh and Grewal (2013 Singh, S., and I. S. Grewal. 2013. Geometric distribution as a randomization device implemented in the Kuk’s model. International Journal of Contemporary Mathematical Sciences 8:243–8.[Crossref] , [Google Scholar]) and Hussain et al. (2016 Hussain, Z., J. Shabbir, Z. Pervez, S. F. Shah, and M. Khan. 2016. Generalized geometric distribution of order k: A flexible choice to randomize the response. Communications in Statistics: Simulation and Computation 46:4708–21.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) techniques by introducing a new two-stage randomization response process. Using the proposed new technique, we achieve better efficiency and increasing protection of privacy of respondents than the Kuk (1990 Kuk, A. Y. C. 1990. Asking sensitive questions indirectly. Biometrika 77 (2):436–8.[Crossref], [Web of Science ®] , [Google Scholar]), Singh and Grewal (2013 Singh, S., and I. S. Grewal. 2013. Geometric distribution as a randomization device implemented in the Kuk’s model. International Journal of Contemporary Mathematical Sciences 8:243–8.[Crossref] , [Google Scholar]) and Hussain et al. (2016 Hussain, Z., J. Shabbir, Z. Pervez, S. F. Shah, and M. Khan. 2016. Generalized geometric distribution of order k: A flexible choice to randomize the response. Communications in Statistics: Simulation and Computation 46:4708–21.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) models. The relative efficiency and protection of the respondents of the proposed two-stage randomization device have been investigated through simulation study, and the situations are reported where the proposed estimator performs better than its competitors. The SAS code used to investigate the performance of the proposed strategy are also provided.     

A side sensitive modified group runs control chart to detect shifts in the process mean

Gadre and Rattihalli [5 Gadre, M. P. and Rattihalli, R. N. 2006. Modified group runs control charts to detect increases in fraction non-conforming and shifts in the process mean. Commun. Stat. Simul. Comput., 35: 225–240. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] have introduced the Modified Group Runs (MGR) control chart to identify the increases in fraction non-conforming and to detect shifts in the process mean. The MGR chart reduces the out-of-control average time-to-signal (ATS), as compared with most of the well-known control charts. In this article, we develop the Side Sensitive Modified Group Runs (SSMGR) chart to detect shifts in the process mean. With the help of numerical examples, it is illustrated that the SSMGR chart performs better than the Shewhart's X¯ chart, the synthetic chart [12 Wu, Z. and Spedding, T. A. 2000. A synthetic control chart for detecting small shifts in the process mean. J. Qual. Technol., 32: 32–38. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]], the Group Runs chart [4 Gadre, M. P. and Rattihalli, R. N. 2004. A group runs control chart for detecting shifts in the process mean. Econ. Qual. Control, 19: 29–43. [Crossref] [Google Scholar]], the Side Sensitive Group Runs chart [6 Gadre, M. P. and Rattihalli, R. N. 2007. A side sensitive group runs control chart for detecting shifts in the process mean. Stat. Methods Appl., 16: 27–37. [Crossref], [Web of Science ®] [Google Scholar]], as well as the MGR chart [5 Gadre, M. P. and Rattihalli, R. N. 2006. Modified group runs control charts to detect increases in fraction non-conforming and shifts in the process mean. Commun. Stat. Simul. Comput., 35: 225–240. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]]. In some situations it is also superior to the Cumulative Sum chart p9 Page, E. S. 1954. Continuous inspection schemes. Biometrika, 41: 100–114. [Crossref], [Web of Science ®] [Google Scholar]] and the exponentially weighed moving average chart [10 Roberts, S. W. 1959. Control chart tests based on geometric moving averages. Technometrics, 1: 239–250. [Taylor & Francis Online] [Google Scholar]]. In the steady state also, its performance is better than the above charts.     

The Identification of Potential Longitudinal Biomarkers and Measurements of Effectiveness for Biomarkers as Surrogates in Multivariate Survival Data

A proposed method based on frailty models is used to identify longitudinal biomarkers or surrogates for a multivariate survival. This method is an extention of earlier models by Wulfsohn and Tsiatis (1997 Wulfsohn , M. S. , Tsiatis , A. A. ( 1997 ). A joint model for survival and longitudinal data measured with error . Biometrics 53 ( 1 ): 330 – 339 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Song et al. (2002 Song , X. , Davidian , M. , Tsiatis , A. A. ( 2002 ). A Semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data . Biometrics 58 ( 4 ): 742 – 753 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, similar to Henderson et al. (2002 Henderson , R. , Diggle , P. J. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), a joint likelihood function combines the likelihood functions of the longitudinal biomarkers and the multivariate survival times. We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudianl biomarkers influence the power to detect the association of a longitudinal biomarker and the multivariate survival time. The proposed method is illustrate by using the gastric cancer data.     

Semiparametric analysis of 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. In this note, using Martingale arguments of Chen et al. [3 Chen, K., Jin, Z. and Ying, Z. 2002. Semiparametric analysis of transformation models with censored data. Biometrika, 89: 659–668. [Crossref], [Web of Science ®] [Google Scholar]], we propose an estimator (denoted by ?β) for estimating regression coefficients of transformation model when L is always observed. Under Cox proportional hazards model, the proposed estimator is equivalent to the partial likelihood estimator for left-truncated and right-censored data if the left-censoring variables L were regarded as left-truncated variables. In this case, the estimator ?β can be obtained by the standard software. A simulation study is conducted to investigate the performance of ?β. For the purpose of comparison, the simulation study also includes the estimator proposed by Cai and Cheng [2 Cai, T. and Cheng, S. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] [Google Scholar]] for the case when L and U are always observed.     

Computing the Exact Fisher Information Matrix of Periodic State-Space Models

The purpose of this article is to develop algorithms for computing the exact Fisher information matrix of periodic time-varying state-space models. We first present a relatively simple recursive algorithm which computes the elements of the exact information matrix without involving numerical differentiation, since all required derivatives are analytically evaluated. The proposed algorithm extends the procedure due to Cavanaugh and Shumway (1996 Cavanaugh , J. E. , Shumway , R. H. ( 1996 ). On computing the expected Fisher information matrix for state-space model parameters . Statist. Probab. Lett. 26 : 347 – 355 .[Crossref], [Web of Science ®] , [Google Scholar]) to the periodic state-space framework. Exploiting the approach used in Klein et al. (2000 Klein , A. , Mélard , G. , Zahaf , T. ( 2000 ). Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules . Linear Alg. Applic. 321 : 209 – 232 .[Crossref], [Web of Science ®] , [Google Scholar]), a second algorithm is proposed in order to obtain the exact information matrix as a whole instead of element by element. The algorithms are first developed in a general framework and then specialized to the case of a periodic Gaussian vector autoregressive moving-average (PVARMA) model.     

The performance of the truncated mixed control chart

Abstract

When the mixed chart proposed by Aslam et al. (2015 Aslam, M., M. Azam, N. Khan, and C.-H. Jun. 2015. A mixed control chart to monitor the process. International Journal of Production Research 53 (15):4684–93. doi:10.1080/00207543.2015.1031354.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is in use, the sample items are classified as defective or not defective and, depending on the number of defectives, the quality characteristic X of the sample items are also measured. In this case, an Xbar chart decides the state of the process. The previous conforming/non-conforming classification truncates the X distribution and, because of that, the mathematical development to obtain the ARLs is complex. Aslam et al. (2015 Aslam, M., M. Azam, N. Khan, and C.-H. Jun. 2015. A mixed control chart to monitor the process. International Journal of Production Research 53 (15):4684–93. doi:10.1080/00207543.2015.1031354.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) didn’t pay attention to the fact that the X distribution is truncated and, due to that, they obtained incorrect ARLs.