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Introduction to Linear Mixed Models
Šaroch, Vojtěch ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
of the bachelor thesis Title: Introduction to Linear Mixed Models Author: Vojtěch Šaroch Department: Department of Probability and Mathematical Statistics, MFF UK Supervisor: doc. Mgr. Michal Kulich Ph.D. Abstract: The thesis describes general procedures of estimation and hypothesis testing for linear statistical models. The models compare groups of observation due to dependent variable. Analysis of variance and linar mixed models are commonly used in the major science like pharmacology, biochemistry, economy and others. The thesis is appropriate for general public, because no advanced knowledge of probability and statistics are required. Particular methods are introduced gently and contain some practical examples for easier understanding of theory. Keywords: Analysis of variance (ANOVA), fixed and random effect, linear mixed model 1
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Introduction to Order Statistics Theory
Hanuš, Antonín ; Kulich, Michal (advisor) ; Klebanov, Lev (referee)
This thesis deals with the theory of order statistics. Its aim is to summarize the basic knowledge concerning the distribution of the order statistics of random variables that are absolutely continuous with respect to the Lebesgue Measure and afterwards use those order statistics for some specific distributions. The first chapter describes the derivation of the density and distribution function of order statistics in several ways as well as dealing with some functions of order statistics and their conditional distribution. The second chapter is devoted to the moments of order statistics and formulae for their calculation and to the relations between them. In the conclusion the previous theoretical findings are applied to the uniform, exponential and normal distributions. 1
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Methods for Analyzing Change From Baseline to Final Assessment
Pekařová, Lucie ; Kulich, Michal (advisor) ; Hušková, Marie (referee)
In this thesis, we analyze treatment effect estimate in randomized clinical studies. Treatment effect estimates are constructed on the basis of three models. The first part of this thesis is about the behaviour of these estimates when the treatment effects vary with patients. We find out that all types of estimates are consistent and we derived their asymptotic distribution. The estimates are compared by their asymptotic variances. The theoretical conclusions are confirmed by a simulation study. The second part describes the case where measurements of baseline and final values contain an error. Two estimates are analyzed. We find out that both estimates are consistent. We derive their asymptotic distribution and compare their variances.
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Simpson's paradox
Balhar, Jan ; Komárek, Arnošt (advisor) ; Kulich, Michal (referee)
Title: Simpson's paradox Author: Jan Balhar Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Arnošt Komárek, Ph.D. Supervisor's e-mail address: arnost.komarek@mff.cuni.cz Abstract: In this work we deal with Simpson's paradox and its more general version, called association reversal. We present definitions of these terms and necessary and sufficient conditions for their occurrence. Due to this, we get to issue of measuring relationship between two characters in 2x2 contigency table, we specifically mention advantages of odds ratio. We also try to answer, what relationship between two characters is, in case of Simpson's paradox, the right one. When looking for answer, we find, that ordinary statistical methods are not sufficient. It is necessary to identify causal relationships between characters. Therefore we get to issue of causality definition. Finally, we present some examples of Simpson's paradox in practice. Keywords: Simpson's paradox, association reversal, confounding, causality.
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Properties of two-phase testing procedures
Krausová, Eliška ; Kulich, Michal (advisor) ; Omelka, Marek (referee)
In the present work we study properties of two-phase testing procedures which formally verify assumptions by performing some test ( first phase) and subsequently calculate a test statistic selected according to the results of the previous test (second phase). In the beginning we describe two-phase testing procedures and mention some literature, in which they are recommended. We try to derive formula for the combined level and power of the whole two-phase testing procedure. After that we illustrate their properties through simulation studies.
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Basic Multivariate Distributions
Sýkorová, Sabina ; Kulich, Michal (advisor) ; Hurt, Jan (referee)
The thesis deals with the basic discrete and continuous multivariate distributions, which play an important role in statistical analyses of models in applied fields. It focuses mainly on the derivation of these distributions using various techniques by which univariate distributions are generalized to higher dimensions. At the beginning of the thesis the multivariate normal distribution is defined, than it deals with distributions that are derived by direct generalization of univariate distributions. These are multivariate log-normal, multivariate Student's, multivariate Pareto, Dirichlet, and multinomial distributions. Furthermore it describes a common components method by which a multivariate Poisson distribution and a multivariate gamma distribution are derived. In the last chapter we introduce a multivariate exponential distribution derived by a stochastic generalization technique.
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Confidence Intervals for Quantiles
Horejšová, Markéta ; Kulich, Michal (advisor) ; Hlávka, Zdeněk (referee)
In this thesis, various construction methods for simultaneous confidence intervals for quantiles are explained. Among nonparametric approaches, a special emphasis is dedicated to a recent method based on a multinomial distribution for calculating the overall confidence level of confidence intervals for all quantiles of interest using an efficient recursive algorithm, which is also described. Furthermore, a method based on Kolmogorov-Smirnov statistic or an asymptotic method using empirical distribution function and order statistics for quantile estimate are presented. A special parametric method for several quantiles of a normally distributed population is introduced along with a few of its modifications. Subsequently, a simulation is run to test the real coverage of the described theoretical methods. Powered by TCPDF (www.tcpdf.org)
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