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Competing risk models in survival analysis
Macoun, Jaromír ; Komárek, Arnošt (advisor) ; Kulich, Michal (referee)
We study the extension of methods from classical survival analysis to competing risks. These methods can be used to analyse time-to-event data. Firstly, we establish notation, define fundamental concepts and present basic theorems and properties. The second chap- ter explores semiparametric methods for estimating the cumulative incidence function. We compare two methods of estimation: the first treats competing events as censored, while the second takes competing events into account. At the end of the chapter, we prove the asymptotic distribution of the estimator of the cumulative incidence function. Further, we present semiparametric regression methods for estimating cause-specific and subdistribution hazards. Generalisations of the Cox model are used to estimate regres- sion parameters. We introduce proofs of the martingale property for the subdistribution hazard case with complete data. Lastly, we propose a small simulation study to assess the efficiency of the presented nonparametric estimates. Different scenarios with con- stant cause-specific hazards are simulated and visualized. Additionally, there is one more simulation study for semiparametric estimation methods. Two different Cox models with two covariates for cause-specific hazard are assumed. 1
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Homoscedasticity tests in one-way classification
Gulík, Matyáš ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
The bachelor thesis deals with tests of variance equality in the context of simple classification. It focuses on three tests commonly used in practice. The thesis first provides an overview of concepts and knowledge from probability theory, which are utilized in subsequent chapters. Additionally, a one-way analysis of variance is introduced, which is crucial for the tests of variance equality. In the main part of the thesis, the Levene test is derived, followed by the Brown-Forsythe test, which is its modification. The Bart- lett test is also presented. Finally, simulations were conducted using the R program to determine the ability of the tests to maintain the desired significance level. 1
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Interval estimation of the correlation coefficient
Rusá, Vendula ; Komárek, Arnošt (advisor) ; Kalina, Jan (referee)
Correlation coefficients are a standard measure of the relationship between two ran- dom variables. In this paper, we will present various methods for constructing a (1 − α) level confidence interval for Pearson and Kendall correlation coefficients. We focus on Fisher's z-transformation method and two methods based on empirical likelihood for the Pearson correlation coefficient. For the Kendall correlation coefficient, we will present two methods based on the properties of the influence function for the Kendall correlation coefficient, one of which is also based on empirical likelihood. The added value of the methods based on empirical likelihood is their suitability even for the unknown bivariate distributions. Finally, we conduct a simulation study where we compare the discussed methods in terms of coverage probabilities and average length of confidence intervals for finite ranges. 1
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Confidence regions in nonlinear regression
Marcinko, Tomáš ; Zvára, Karel (advisor) ; Komárek, Arnošt (referee)
The aim of this thesis is a comprehensive description of the properties of a nonlinear least squares estimator for a nonlinear regression model with normally distributed errors and thorough development of various methods for constructing confidence regions and confidence intervals for the parameters of the nonlinear model. Due to the fact that, unlike the case of linear models, there is no easy way to construct an exact confidence region for the parameters, most of these methods are only approximate. A short simulation study comparing observed coverage of various confidence regions and confidence intervals for models with different curvatures and sample sizes is also included. In case of negligible intrinsic curvature the use of likelihood-ratio confidence regions seems the most appropriate.
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Analysis of cross-over clinical trials in the presence of baseline measurements
Helebrand, František ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
This thesis aims to provide a comprehensive overview of methods for estimating treat- ment effects in cross-over designs. It examines approaches that use baseline measurements to estimate the treatment effect, as well as alternative methods that do not use baseline measurements at all. It also proposes new approaches to estimating treatment effects and introduces robust procedures to ensure that biases caused by residual treatment effects from the previous period are within acceptable limits. The theoretical properties of the methods are investigated in a simulation study. Furthermore, a comparison of different methods is performed in cases where a theoretical comparison is not possible. 1
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Estimation of latent distribution for ordinal data
Hržič, Viktor ; Hudecová, Šárka (advisor) ; Komárek, Arnošt (referee)
The main goal of the bachelor thesis is to introduce problematics of ordinal data together with estimations of latent density distribution based on ordinal data. The esti- mations obtained using ordinal data are compared to the ones that are more common and used on daily basis. The reader will be introduced to the maximum likelihood method that is used in the development of each estimate. One chapter is dedicated to alterna- tive approach using Bernstein polynomials. When we take all benefits into account such as easier collecting the data in ordinal form or minimization of possible errors committed by respondent, we obtained very valuable and promising results. 1
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Classification based on mixture models
Janečková, Lucie ; Komárek, Arnošt (advisor) ; Maciak, Matúš (referee)
This thesis deals with classification based on mixture models, mainly on models finite normal. At first, there are introduced basic definitions and characteristics of finite mix- tures. Afterwards there is described the maximum likelihood method and her obstacles in context of finite mixtures, which we are using for unknown parameters estimation. Then there is described EM algorithm, that is used to obtain the maximum likelihood estimator and there are calculated the formulae for one iteration of EM algorithm. In the last part there is shown, how can finite normal mixtures be used for classification. 1
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Dynamic prediction in survival analysis
Mečiarová, Kristína ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
Often the motivation behind building a statistical model is to provide prediction for an outcome of interest. In the context of survival analysis it is important to distingu- ish between two types of time-varying covariates and take into careful consideration the appropriate type of analysis. Joint model for longitudinal and time-to-event data, in con- trast to standard Cox model, enables to account for continuous change of the covariate over time in the survival model. In this thesis two examples of joint models are presen- ted, the shared random-effect model and the joint latent class model. Bayesian estimation of the model parameters and summary of methodology for dynamic prediction of indi- vidual survival probability is provided for the first one of the aforementioned types of models. Application of the theoretical knowledge is illustrated in the analysis of the data on primary biliary cirrhosis. The impact of number of patients, number of longitudinal measurements and per-cent of censoring on the quality of prediction and estimates of the model parameters is examined in the simulation study. 1
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Tolerance limits
Bedřich, Marek ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
This bachelor's thesis deals with tolerance intervals, a statistical tool used to quan- tify the uncertainty of statistical predictions. The introductory part of the text briefly recalls confidence intervals. The thesis then focuses on prediction intervals, which are an intermediate step between confidence intervals and tolerance intervals. Specifically, the prediction interval for normal distribution and nonparametric prediction interval are analyzed. The main part of the thesis then deals with tolerance intervals - the definition, construction of both parametric and nonparametric tolerance intervals, convergence, and actual coverage of the derived intervals. In the final part, an example of the practical application of this tool is presented. 1
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