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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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Confidence intervals for two-parameter exponential distribution
Špinka, Karel ; Kulich, Michal (advisor) ; Antoch, Jaromír (referee)
Abstract. In this work, we examine both point and interval estimators of two-parameter exponential distribution. We determine whether point estimates are unbiased, consistent, or both, and derive exact distributions from which confidence intervals can be construc- ted. We also demonstrate another method of creating confidence sets whose volume is, given certain conditions, the smallest possible on a specific confidence level. 1
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Modeling transition intensities of a non-homogenous Markov chain via the Cox model
Jandl, Vojtěch ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
We study the extension of methods from the classical two-state survival analysis to the multi-state setting. Such models are applicable in a variety of fields in situations, for which the classical case does not suffice due to the fact that omission of some states is not possible. First of all, we explore one sample methods, in particular the extensions of the well known Nelson-Aalen and Kaplan-Meier estimators. Then, we deal with regression models for transition intensities, which include a generalisation of the Cox model, and the Lin-Ying additive model, for which we derive the asymptotic properties of the estimator of regression parameters. Lastly, to illustrate the practicality of presented methods, we propose an experiment that could help a website understand the behaviour of its members. A small simulation study is also a part of the last chapter in order to demonstrate the asymptotic properties of the underlying model empirically. 1
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Semiparametric Analysis of Nested Case-Control Design
Strachoňová, Karla ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
Studying rare diseases often deals with small percentage of cases requiring a large amount of subjects in the medical study. The common analysis by the Cox proportional hazards model may be very time-consuming and financially inefficient. Nested case- control design presents a sampling method offering fewer data needed for the analysis while keeping the estimator of the Cox model consistent and asymptotic normal. In this thesis, we introduce nested case-control design, we describe in detail the method for sampling controls for cases, we present the partial likelihood and the maximum partial likelihood estimator of the regression parameter and we prove the consistency and the asymptotic normality of the estimator. Then, we introduce the counter-matching design as an extension of the nested case-control design and the pseudolikelihood approach under nested case-control design. In the last chapter, we perform a simulation study comparing the four designs. The contribution of this thesis is the detailed introduction to nested case- control design and its alternatives, more detailed proofs of the asymptotic properties of the maximum partial likelihood of the regression parameter of nested case-control design and the comparison of the four approaches through the simulation study. 1
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Stochastic models for genetic analysis
Selementová, Martina ; Kulich, Michal (advisor) ; Zichová, Jitka (referee)
Nazev prace: Slochasticke niodcly pro geneliekou analyzu Autor: MartinaSelementova Katedra: Katedra pravdepodobnosti a matematicke statistiky Vedouci bakalafske prace: Mgr. Michal Kulich, PhD, e-mail vedouciho: kuliclX^karlin.mff.cuni.cz Abstrakt: V pfedlozene praci studujerne elementarni stochasticke melody pro modelovani genetickych populaci. Nejprve uvadime zakladni pojmy obecne gcnetiky pro snadnejsi oricntaci v textu. V naslcdujicich kapitolach se zabyvame dynamikou vyvoje populace a zakladnimi problemy vyberu v populaci. lllavnim tcmatem je nahodne kfizeni v populaci, ktcre dale aplikujeme na pohlavne va/ane geny, letraploidy. autosterilizacni geny a na pfi'pad dvou lokusu. Ve druhe casti se xabyvame problemy vybcru: zamC'fili jsme se pf-edevsim na vyber xalozeny na gcnolypu aplikovany na pohlavi a pohlavne vazane geny a dale na lamiliarni vyber a konkretnf pn'klad Rh faktoru. Klicova slova: nahodne kfizeni, vyber Title: Stochastic Models for Genetic Analysis Author: Martina Selementova Department: Department oi'Probability and Mathematical Statistics Supervisor: Mgr. Michal Kulich, PhD. Supervisor's e-mail adress: kulich(«)karlin.mff.cuni.cz Abstract: In the presented work we study elementary stochastic methods for modeling genetic populations. First we show some basic notion of general genetics for...
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Introduction to Bayesian Data Analysis
Štádlerová, Kateřina ; Kulich, Michal (advisor) ; Anděl, Jiří (referee)
of the bachelor's thesis Title: Introduction to Bayesian Data Analysis Author: Kateřina Štádlerová Department: Department of Probability and Mathematical Statistics Supervisor: doc. Mgr. Michal Kulich, Ph.D., Department of Probability and Mathematical Statistics Abstract: The paper deals with basic principles of Bayesian methods. These me- thods have very broad range of use in statistical problems concerning estimation and hypothesis testing. However, their use is much wider; these methods are used in anti-spam filters of electronic mail or in the game theory. Definitions, theo- rems, proofs and examples are included in the paper for this purpose to enable easier understanding of particular topics. The paper is helpful mainly because of the fact that as yet there are not many books in Czech language dealing with Bayesian methods. 1
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Statistical analysis of censored data
Jakubcová, Monika ; Klášterecký, Petr (advisor) ; Kulich, Michal (referee)
Nazov prace: Statist icka analy/n cenzorovauych dat Antor: Monika .Ja.kubcova Katedra : Katedra pravdo])odol)iiosli a matematickej statisliky Vediici bakalarskej prace: Mgr. Petr Kla.sterecky e-itin.il veduccho: kla.stor.(i ka.rlin.niff.cuni.cz Abstrakt: Predlozena praca. podava prehlad o niektorych xakladnycli nieto- dach stat.isl.ickeho spracovania cenzorovaiiych dat. V iivodiiej kapitole uve- dieme akyni sposubom ccnzoruviUK'1 po/orovanio vxnika a pupi.seme zakladne typy cen/orova.nia,. Dalej sa oboznaminio s t'liukeijiini, ktcjre eeiixorova.ne data, popisujii a, icli v/ajomnymi v/,ta,hini. Nasleihu,1 nvedieine ne])arame- trlcke inet6(ly odluuln tyeht.o hinkcit. Uvedene odliady neskor ])ouzijeine pri koiistrukcii inter\alov S])olahli\"os(.i a ]>ri testovani hy]»)to/,. V xaverecnej kapitoh1 pripajame priklad, na ktorom deiuonst rnjeinc niektore z popi'saiiyoh inetod a spoyob praee s ( eii/orovaiiyiiii dat.a.ini v pro^,ra,ine R.. Kfueove slova: renzorovanie, eonzoroiiaiie data, funkria. pre/itia, Kaplanov- Meierov otlliad. log-rank test Title: Statistical analysis of censored data Author: Monika. Jakubcova Department.: Department of Probability and Mathematical Statistics Supervisor: Mgr. Petr KhlstenK'ky Suj)ervisor's e-mail address: klasterTikarlin.infr.cmii.cz Abstract: The presented work gives an...
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Tests of normality of time series
Stibůrek, David ; Anděl, Jiří (advisor) ; Kulich, Michal (referee)
This work considers testing normality of time series in AR and ARMA processes. Firstly we investigate properties of common normality tests, which assume independency. The main goal is to examine levels and powers of tests in dependence on distances of the roots of the characteristic polynom from unit circle. After this we study the tests, which don't assume independency. In the case of AR processes, we get good results by testing normality of residuals. More complex tests can also give good results, but these tests need many observations and are difficult from the numerical point of view.
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