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The power of two sample tests
Rózsahegyi, Dominik ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
Two-sample tests are one of commonly used statistical tools with which we make decisions if experimentally obtained data from different populations satisfy pre-specified statement. Suitability of using them could be dedicated by the power of test, which states for probability of rejection of invalid statement. The reader gets to know with terms of hypothesis testing which are necessary for introduction of tests. The second chapter introduce tests which could be used when analysed data are complete. If some observations are not exactly known, we call them censored and it is more suitable to use tests listed in the third chapter. We estimate the power for tests in simulations and observe its behavior with different conditions. 1
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Distribution of interpoint distances
Horská, Šárka ; Hlávka, Zdeněk (advisor) ; Komárek, Arnošt (referee)
This thesis investigates basic properties of the interpoint distances be- tween random vectors drawn from multinomial distribution. We also describe a possible application to testing sparse observations, i.e., a setup with small number of observations and large number of categories, where the classical χ2 -test cannot be recommended. As an alternative, utilizing the multinomial interpoint distances, we will present the test statistic proposed by Biswas and Ghosh (2014). 1
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Analysis of fluctuation of labourers
Zeman, Ondřej ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
The main goal of this thesis is to analyse the fluctuation of the employees in a well established Czech manufacturing company. Due to the GDPR regulations, the underlying company is kept anonymised in this thesis. The data were transformed into longitudinal data and the GEE methodology was used for the analysis of the fluctuation. In the first chapter, an introduction to the problem and a short description of the data is provided. The second chapter contains some theoretical description of the GEE methodology and the QIC information criterion. In the third chapter, multiple models for a binary and multinomial response are fitted to the data and their results are described in detail. This allows us to describe the influence of various factors to the fluctuation of the employees in the underlying company. 1
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Joint Models for Longitudinal and Time-to-Event Data
Vorlíčková, Jana ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
Title: Joint Models for Longitudinal and Time-to-Event Data Author: Jana Vorlíčková Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mathematical Statistics Abstract: The joint model of longitudinal data and time-to-event data creates a framework to analyze longitudinal and survival outcomes simultaneously. A commonly used approach is an interconnection of the linear mixed effects model and the Cox model through a latent variable. Two special examples of this model are presented, namely, a joint model with shared random effects and a joint latent class model. In the thesis we focus on the joint latent class model. This model assumes an existence of latent classes in the population that we are not able to observe. Consequently, it is assumed that the longitudinal part and the survival part of the model are independent within one class. The main intention of this work is to transfer the model to the Bayesian framework and to discuss an estimation procedure of parameters using a Bayesian statistic. It consists of a definition of the model in the Bayesian framework, a discussion of prior distributions and the derivation of the full conditional distributions for all parameters of the model. The model's ability to...
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Generalized Wilcoxon Test for Censored Data
Vařejková, Michaela ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
This paper deals with the generalized Wilcoxon test and its use for censored data. The introduction describes standard one-sample and two-samples Wilco- xon tests and their basic properties, censored data and methods of censoring. The main part of the paper is devoted to the introduction of the generalized Wilcoxon test and to its properties. First, a test for singly-censored data is de- scribed; the description of a test for doubly censored data follows. The paper concludes with a simulations part in which statistical properties of the test are demonstrated. The first example compares the generalized test with the stan- dard two-samples Wilcoxon test. The second example shows how the censoring rate affects the power and significance level of the generalized test. 1
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Multiple testing problems
Turzová, Kristína ; Maciak, Matúš (advisor) ; Komárek, Arnošt (referee)
Statistical hypothesis testing is used while analyzing experimental data. This thesis is focused on multiple testing, which means testing many hypotheses simultaneously, and multiple testing problems occurring while running multiple hypotheses tests. These multiple testing problems are described and two errors, FWER (Family-Wise Error Rate) and FDR (False Discovery Rate), are defined. Selected multiple testing corrections are introduced and compared in detail using simulations regarding significance level and power. All of the discussed corrections control for the problem of multiple testing.
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Zero inflated Poisson model
Veselý, Martin ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This paper deals with the zero-inflated Poisson distribution. First the Poisson model is defined and generalized to a zero-inflated model. The basic properties of this generalized model are derived. After- wards the basics of the method of moments and the maximum likelihood method are described. Both of these are used to derive parameter estimates of such distribution. The feasibility of calculating the distribution of moment method estimates is analyzed. Then the asymptotic distribution of maximum likelihood estimates is derived and used to create confidence intervals. In the last chapter a numeric si- mulation of the derived asymptotic properties is performed. Special attention is paid to situations where regularity conditions are not met. 1
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Variance stabilizing transformations
Kuželová, Noemi ; Omelka, Marek (advisor) ; Komárek, Arnošt (referee)
Abstract. We often examine data whose sample mean converges to a normal distribution, but the variance generally depends on an unknown parameter. To get rid of this dependence, we can sometimes use the so-called variance-stabilizing transformation method. Firstly, this thesis explains the method in detail and finds a general procedure to find suitable transformations. Then it will focus on data from Poisson and binomial distributions with unknown parameters. For these data, it finds transformations that stabilize (asymptotic) variance, and compares them with the "improved"transforms from the article Anscombe (1948). Most of the thesis is devoted to the shape of these transformations. Finally, we show in the Poisson distribution simulation that it is really appropriate to use this method and compare the derived transformation with its Anscombe version.
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Sample Quantiles
Hrušková, Iveta ; Komárek, Arnošt (advisor) ; Nagy, Stanislav (referee)
If the distribution of random variable is uknown, we are not able to figure out the value of theoretical quantile. In case there is a random sample from this distribution, it is possible to estimate the value of theoretical quantile. This es- timation is called sample quantile. This work is focused on nine frequently used varieties of sample quantile. They will be compared by means of characteristics that can be examined when speaking about sample quantile. All these varieties will be demonstrated on simple example. Finally, there will be shown that all these versions of sample quantile are consistent estimators of theoretical quan- tile. The construction of confidence interval for theoretical quantile will be the topic of the final part of the work. 1
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Statistical tests in stratified fourfold tables
Vook, Peter ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
This paper deals with statistical tests in stratified fourfold tables. Several tests of conditional indepen- dence are derived in it. A test of homogeneous association is also described. At first, contingency tables with arbitrary dimensions and multinomial distribution are defined. Then we continue with a description of fourfold tables and their binomial representation. In the next section we deal with an odds ratio and its asymptotic distribution. Formal definition of stratification and relevant terms follows afterwards. In the next chapter a derivation of test statistics for conditional independence tests including the well-known Cochran-Mantel-Haenszel test based on a hypergeometric distribution can be found. This chapter also includes a description of Breslow-Day test of homogeneous association. A numerical simulation of chosen tests is performed eventually. 1
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