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Random Processes in Reliability Analysis
Chovanec, Kamil ; Volf, Petr (advisor) ; Prokešová, Michaela (referee)
Title: Random Processes in Reliability Analysis Author: Kamil Chovanec Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Supervisor's e-mail address: volf@utia.cas.cz Abstract: The thesis is aimed at the reliability analysis with special em- phasis at the Aalen additive model. The result of hypothesis testing in the reliability analysis is often a process that converges to a Gaussian martingale under the null hypothesis. We can estimate the variance of the martingale using a uniformly consistent estimator. The result of this estimation is a new hypothesis about the process resulting from the original hypothesis. There are several ways to test for this hypothesis. The thesis presents some of these tests and compares their power for various models and sample sizes using Monte Carlo simulations. In a special case we derive a point that maximizes the asymptotic power of two of the tests. Keywords: Martingale, Aalen's additive model, hazard function 1
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Survival analysis with STATISTICA
Kaderjáková, Zuzana ; Hudecová, Šárka (advisor) ; Hurt, Jan (referee)
Survival analysis is a separate statistical area. This paper discusses the~interpretation of basic concepts, principles and methods used and implemented in the software STATISTICA. First, we introduce censoring and ways of characterizing a distribution of survival time. We present Kaplan-Meier estimate of a survival function and also a method of mortality tables. Later, we discuss basic methods of comparison of the survival time distribution in two groups and their suitability for different situations. The paper also deals with application of the survival analysis methods in the financial sector, where we introduce Cox proportional hazards model. Finally, we apply theoretical knowledge to a real data set.
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Kernel estimates of hazard function
Selingerová, Iveta ; Horová, Ivanka (advisor) ; Prášková, Zuzana (referee)
Kernel estimates of hazard function Abstract This doctoral dissertation is devoted to methods for analysis of censored data in survival analysis. The main attention is focused on the hazard function that reflects the instantaneous probability of the event occurrence within the next time instant. The thesis introduces two approaches for a kernel esti- mation of this function. In practice, the hazard function can be affected by other variables. The most frequently used model suggested by D. R. Cox is presented and moreover two types of kernel estimates to estimate a condi- tional hazard function are proposed. For kernel estimates, there is derived some statistical properties and proposed methods of bandwidths selection. The part of the thesis is extensive simulation study where theoretical results are verified and the proposed methods are compared. The last chapter of the thesis is devoted to an analysis of real data sets obtained from different fields.
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Kernel estimates of hazard function
Selingerová, Iveta ; Horová, Ivanka (advisor) ; Prášková, Zuzana (referee)
Kernel estimates of hazard function Abstract This doctoral dissertation is devoted to methods for analysis of censored data in survival analysis. The main attention is focused on the hazard function that reflects the instantaneous probability of the event occurrence within the next time instant. The thesis introduces two approaches for a kernel esti- mation of this function. In practice, the hazard function can be affected by other variables. The most frequently used model suggested by D. R. Cox is presented and moreover two types of kernel estimates to estimate a condi- tional hazard function are proposed. For kernel estimates, there is derived some statistical properties and proposed methods of bandwidths selection. The part of the thesis is extensive simulation study where theoretical results are verified and the proposed methods are compared. The last chapter of the thesis is devoted to an analysis of real data sets obtained from different fields.
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Survival analysis with STATISTICA
Kaderjáková, Zuzana ; Hudecová, Šárka (advisor) ; Hurt, Jan (referee)
Survival analysis is a separate statistical area. This paper discusses the~interpretation of basic concepts, principles and methods used and implemented in the software STATISTICA. First, we introduce censoring and ways of characterizing a distribution of survival time. We present Kaplan-Meier estimate of a survival function and also a method of mortality tables. Later, we discuss basic methods of comparison of the survival time distribution in two groups and their suitability for different situations. The paper also deals with application of the survival analysis methods in the financial sector, where we introduce Cox proportional hazards model. Finally, we apply theoretical knowledge to a real data set.
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Random Processes in Reliability Analysis
Chovanec, Kamil ; Volf, Petr (advisor) ; Prokešová, Michaela (referee)
Title: Random Processes in Reliability Analysis Author: Kamil Chovanec Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Supervisor's e-mail address: volf@utia.cas.cz Abstract: The thesis is aimed at the reliability analysis with special em- phasis at the Aalen additive model. The result of hypothesis testing in the reliability analysis is often a process that converges to a Gaussian martingale under the null hypothesis. We can estimate the variance of the martingale using a uniformly consistent estimator. The result of this estimation is a new hypothesis about the process resulting from the original hypothesis. There are several ways to test for this hypothesis. The thesis presents some of these tests and compares their power for various models and sample sizes using Monte Carlo simulations. In a special case we derive a point that maximizes the asymptotic power of two of the tests. Keywords: Martingale, Aalen's additive model, hazard function 1
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Introduction to Survival Analysis
Valenta, Zdeněk
Survival analysis is concerned with analyzing time-to-event data where the event of interest usually represents some type of “failure”. In clinical medicine, the event of interest may be e.g. death of a patient from well specified causes, autoimmune rejection of the graft by the transplant recipient or other type of graft failure in transplant studies. In certain situations, however, the true survival outcomes may not be observable, because we have observed a so called “censoring event” which prevented the event of interest from occurring. Such censoring event may represent, for instance, loss of a particular subject from follow-up, occurrence of administrative censoring, which typically takes place in clinical trials, or we may indeed observe other type of “failure”, e.g. death from fatal injuries rather than from cardiovascular causes which were of primary interest in a particular clinical trial. In this article we will stress the importance of a key assumption relating censoring process to survival outcomes and review principle univariate survival analysis methods for uncorrelated data. We will review popular models for analyzing univariate survival data, many of which enable us quantifying effect the prognostic variables independently exert on survival outcomes. Model examples will cover the classes of non-parametric, parametric and semi-parametric methods. We will also review underlying assumptions of individual models and stress the importance of using appropriate models in analyzing univariate time-to-event data.
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Nonparametric estimations in survival analysis
Svoboda, Martin ; Malá, Ivana (advisor) ; Tomášek, Ladislav (referee)
This work introduces nonparametric models which are used in time to event data analysis. It is focused on applying these methods in medicine where it is called survival analysis. The basic techniques and problems, which can appear in survival analysis, are presented and explained here. The Kaplan -- Meier estimator of survival function is discussed in the main part. This is the most frequented method used for estimating the survival function in patients who have undergone a specific treatment. The Kaplan -- Meier estimator is also a common device in the statistical packets. In addition to estimation of survival function, the estimation of hazard function and cumulative hazard function is presented. The hazard function shows the intensity of an individual experiencing the particular event in a short time period. Special problems occur when analyzing time to event data. A distinctive feature, often present in such data, is known as censoring. That is the situation when the individual does not experience the event of interest at the time of study. The thesis covers also an empiric part, where the results of an analysis of patients with the larynx carcinoma diagnosis are shown. These patients were treated in a hospital located in České Budějovice. This analysis is based on a theory presented in the previous chapters.
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