Counting process

About: Counting process is a research topic. Over the lifetime, 1052 publications have been published within this topic receiving 36279 citations. The topic is also known as: Zählprozess.

Cox's Regression Model for Counting Processes: A Large Sample Study

Abstract: The Cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life-time distribution of an individual. In this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process. This permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates. Furthermore, this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model. Finally an example of a statistical analysis is included.

The statistical analysis of failure time data

Abstract: Preface.1. Introduction.1.1 Failure Time Data.1.2 Failure Time Distributions.1.3 Time Origins, Censoring, and Truncation.1.4 Estimation of the Survivor Function.1.5 Comparison of Survival Curves.1.6 Generalizations to Accommodate Delayed Entry.1.7 Counting Process Notation.Bibliographic Notes.Exercises and Complements.2. Failure Time Models.2.1 Introduction.2.2 Some Continuous Parametric Failure Time Models.2.3 Regression Models.2.4 Discrete Failure Time Models.Bibliographic Notes.Exercises and Complements.3. Inference in Parametric Models and Related Topics.3.1 Introduction.3.2 Censoring Mechanisms.3.3 Censored Samples from an Exponential Distribution.3.4 Large-Sample Likelihood Theory.3.5 Exponential Regression.3.6 Estimation in Log-Linear Regression Models.3.7 Illustrations in More Complex Data Sets.3.8 Discrimination Among Parametric Models.3.9 Inference with Interval Censoring.3.10 Discussion.Bibliographic Notes.Exercises and Complements.4. Relative Risk (Cox) Regression Models.4.1 Introduction.4.2 Estimation of beta.4.3 Estimation of the Baseline Hazard or Survivor Function.4.4 Inclusion of Strata.4.5 Illustrations.4.6 Counting Process Formulas. 4.7 Related Topics on the Cox Model.4.8 Sampling from Discrete Models.Bibliographic Notes.Exercises and Complements.5. Counting Processes and Asymptotic Theory.5.1 Introduction.5.2 Counting Processes and Intensity Functions.5.3 Martingales.5.4 Vector-Valued Martingales.5.5 Martingale Central Limit Theorem.5.6 Asymptotics Associated with Chapter 1.5.7 Asymptotic Results for the Cox Model.5.8 Asymptotic Results for Parametric Models.5.9 Efficiency of the Cox Model Estimator.5.10 Partial Likelihood Filtration.Bibliographic Notes.Exercises and Complements.6. Likelihood Construction and Further Results.6.1 Introduction.6.2 Likelihood Construction in Parametric Models.6.3 Time-Dependent Covariates and Further Remarks on Likelihood Construction.6.4 Time Dependence in the Relative Risk Model.6.5 Nonnested Conditioning Events.6.6 Residuals and Model Checking for the Cox Model.Bibliographic Notes.Exercises and Complements.7. Rank Regression and the Accelerated Failure Time Model.7.1 Introduction.7.2 Linear Rank Tests.7.3 Development and Properties of Linear Rank Tests.7.4 Estimation in the Accelerated Failure Time Model.7.5 Some Related Regression Models.Bibliographic Notes.Exercises and Complements.8. Competing Risks and Multistate Models.8.1 Introduction.8.2 Competing Risks.8.3 Life-History Processes.Bibliographic Notes.Exercises and Complements.9. Modeling and Analysis of Recurrent Event Data.9.1 Introduction.9.2 Intensity Processes for Recurrent Events.9.3 Overall Intensity Process Modeling and Estimation.9.4 Mean Process Modeling and Estimation.9.5 Conditioning on Aspects of the Counting Process History.Bibliographic Notes.Exercises and Complements.10. Analysis of Correlated Failure Time Data.10.1 Introduction.10.2 Regression Models for Correlated Failure Time Data.10.3 Representation and Estimation of the Bivariate Survivor Function.10.4 Pairwise Dependency Estimation.10.5 Illustration: Australian Twin Data.10.6 Approaches to Nonparametric Estimation of the Bivariate Survivor Function.10.7 Survivor Function Estimation in Higher Dimensions.Bibliographic Notes.Exercises and Complements.11. Additional Failure Time Data Topics.11.1 Introduction.11.2 Stratified Bivariate Failure Time Analysis.11.3 Fixed Study Period Survival Studies.11.4 Cohort Sampling and Case-Control Studies.11.5 Missing Covariate Data.11.6 Mismeasured Covariate Data.11.7 Sequential Testing with Failure Time Endpoints.11.8 Bayesian Analysis of the Proportional Hazards Model.11.9 Some Analyses of a Particular Data Set.Bibliographic Notes.Exercises and Complements.Glossary of Notation.Appendix A: Some Sets of Data.Appendix B: Supporting Technical Material.Bibliography.Author Index.Subject Index.

Statistical Models Based on Counting Processes

Abstract: Modern survival analysis and more general event history analysis may be effectively handled in the mathematical framework of counting processes, stochastic integration, martingale central limit theory and product integration. This book presents this theory, which has been the subject of an intense research activity during the past one-and-a-half decades. The exposition of the theory is integrated with the careful presentation of many practical examples, based almost exlusively on the authors' experience, with detailed numerical and graphical illustrations. "Statistical Models Based on Counting Processes" may be viewed as a research monograph for mathematical statisticians and biostatisticians, although almost all methods are given in sufficient detail to be used in practice by other mathematically oriented researchers studying event histories (demographers, econometricians, epidemiologists, actuariala mathematicians, reliability engineers, biologists). Much of the material has so far only been available in the journal literature (if at all), and a wide variety of researchers will find this an invlauable survey of the subject.

Counting Processes and Survival Analysis

Abstract: Preface. 0. The Applied Setting. 1. The Counting Process and Martingale Framework. 2. Local Square Integrable Martingales. 3. Finite Sample Moments and Large Sample Consistency of Tests and Estimators. 4. Censored Data Regression Models and Their Application. 5. Martingale Central Limit Theorem. 6. Large Sample results of the Kaplan-Meier Estimator. 7. Weighted Logrank Statistics. 8. Distribution Theory for Proportional Hazards Regression. Appendix A: Some Results from stieltjes Integration and Probability Theory. Appendix B: An Introduction to Weak convergence. Appendix C: The Martingale Central Limit Theorem: Some Preliminaries. Appendix D: Data. Appendix E: Exercises. Bibliography. Notation. Author Index. Subject Index.

Martingale-based residuals for survival models

Abstract: SUMMARY Graphical methods based on the analysis of residuals are considered for the setting of the highly-used Cox (1972) regression model and for the Andersen-Gill (1982) generalization of that model. We start with a class of martingale-based residuals as proposed by Barlow & Prentice (1988). These residuals and/or their transforms are useful for investigating the functional form of a covariate, the proportional hazards assumption, the leverage of each subject upon the estimates of 13, and the lack of model fit to a given subject. 1 1. Model Consider a set of n independent subjects such that the counting process Ni {Ni(t), t } O} for the ith subject in the set indicates the number of observed events experienced over time t. The sample paths of the Ni are step functions with jumps of size +1 and with Ni(0) =0. We assume that the intensity function for Ni(t) is given by Yi(t)dA{t, Zi(t)} = Yi(t) eP'Z(t) dAO(t), (1) where Yi(t) is a 0-1 process indicating whether the ith subject is a risk at time t, 13 is a vector of regression coefficients, Zi(t) is a p dimensional vector of covariate processes, and Ao is the baseline cumulative hazard function. Several familar survival models fit into this framework. The Andersen & Gill (1982) generalization of the Cox (1972) model arises when AO(t) is completely unspecified. The further restriction that Yi(t) = 1 until the first event or censoring, and is 0 thereafter yields the Cox model. The parametric form AO(t) = t yields a Poisson model, or an exponential if restricted to a single event per subject, and AO(t) = tP a Weibull model. Our attention will focus primarily on the Andersen-Gill and Cox models; however, the