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Group sequential tests in clinical trials
Jílek, Josef ; Kulich, Michal (advisor) ; Komárek, Arnošt (referee)
Group sequential tests are an important statistical method. The analysis of data are performed continuously, which allows us to terminate the test before all observations are collected. For example these tests are used in medicine. When testing new drugs or procedures, this method brings about financial savings as well as ethical advantages. There are many ways of conducting group sequential tests with different qualities. Based on the perused literature, both basic and more complex types of group sequential tests are introduced in this paper. It discribes their principle and respective examples are provided. With this information it is possible to design and conduct a particular test. It's merits and demerits are compared for every method in real situations. The result is a tabular scale of different tests, from which it is possible to select a particular test for a given situation.
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Matrix Algebra in Statistics
Navrátil, František ; Kulich, Michal (advisor) ; Antoch, Jaromír (referee)
of the bachelor thesis Title: Matrix Algebra in Statistics Author: František Navrátil Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Mgr. Michal Kulich Ph.D. Abstract: The thesis deals with the theory of matrix algrebra, which is applicable in probability and statistics. The aim of the thesis is to summarize it in a clear and understandable way, so that the student familiar with the basics of matrix theory can expand his knowledge and use it in further studies. Therefore, the thesis contains many definitions and proved theorems, and examples to help understanding the theory. Applications are mentioned. It also provides references for further reading. The thesis begins with a brief summary of basic definitions and results in matrix algebra, which are covered in the usual courses on linear algebra. Subsequent chapters are specific, inter alia, for probability and statistics - in particular, they focus on special types of matrices and their properties, important matrix decompositions, functions of matrices and matrix difierentiation. Keywords: matrix algebra, statistics, idempotent matrix, spectral decomposition, Kronecker product
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Variable selection based on penalized likelihood
Chlubnová, Tereza ; Kulich, Michal (advisor) ; Maciak, Matúš (referee)
Selection of variables and estimation of regression coefficients in datasets with the number of variables exceeding the number of observations consti- tutes an often discussed topic in modern statistics. Today the maximum penalized likelihood method with an appropriately selected function of the parameter as the penalty is used for solving this problem. The penalty should evaluate the benefit of the variable and possibly mitigate or nullify the re- spective regression coefficient. The SCAD and LASSO penalty functions are popular for their ability to choose appropriate regressors and at the same time estimate the parameters in a model. This thesis presents an overview of up to date results in the area of characteristics of estimates obtained by using these two methods for both small number of regressors and multidimensional datasets in a normal linear model. Due to the fact that the amount of pe- nalty and therefore also the choice of the model is heavily influenced by the tuning parameter, this thesis further discusses its selection. The behavior of the LASSO and SCAD penalty functions for different values and possibili- ties for selection of the tuning parameter is tested with various numbers of regressors on simulated datasets.
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Odhad momentů při intervalovém cenzorování typu I
Ďurčík, Matej ; Komárek, Arnošt (advisor) ; Kulich, Michal (referee)
Title: Moments Estimation under Type I Interval Censoring Author: Matej Ďurčík Department: Faculty of Probability and Mathematical Statistics Supervisor: RNDr. Arnošt Komárek Ph.D. Abstract: In this thesis we apply the uniform deconvolution model to the interval censoring problem. We restrict ourselves only on interval censoring case 1. We show how to apply uniform deconvolution model in estimating the probability distribution characteristics in the interval censoring case 1. Moreover we derive limit distributions of the estimators of mean and variance. Then we compare these estimators to the asymptotically efficient estimators based on the nonparametric maximum likelihood estimation by simulation studies under some certain distributions of the random variables. 1
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Computational Methods for Maximum Likelihood Estimation in Generalized Linear Mixed Models
Otava, Martin ; Komárek, Arnošt (advisor) ; Kulich, Michal (referee)
of the diploma thesis Title: Computational Methods for Maximum Likelihood Estimation in Generalized Linear Mixed Models Author: Bc. Martin Otava Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mathematical Statistics Abstract: Using maximum likelihood method for generalized linear mixed models, the analytically unsolvable problem of maximization can occur. As solution, iterative and ap- proximate methods are used. The latter ones are core of the thesis. Detailed and general introducing of the widely used methods is emphasized with algorithms useful in practical cases. Also the case of non-gaussian random effects is discussed. The approximate methods are demonstrated using the real data sets. Conclusions about bias and consistency are supported by the simulation study. Keywords: generalized linear mixed model, penalized quasi-likelihood, adaptive Gauss- Hermite quadrature 1
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Parameter estimation in case-cohort studies
Klášterecký, Petr ; Kulich, Michal (advisor) ; Volf, Petr (referee) ; Komárek, Arnošt (referee)
The concern of this thesis is parameter estimation in regression models in survival analysis, particularly in case-cohort studies. In case-cohort studies, observations are sampled to form a subcohort which is followed and analysed. As a result, the cost of performing such studies is reduced but standard procedures for parameter estimation need to be modified. This is usually done by incorporating weights into the estimating equations so that individual sampling probabilities are accounted for. In this thesis we show that this method can lead to biased estimators when the subcohort sampling probability is low and suggest an alternative estimator based on logistic regression.
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Effect of measurement error on the shape of the regression function in nonlinear models
Drábková, Alena ; Kulich, Michal (advisor) ; Zvára, Karel (referee)
In this thesis we study the effect of regressors measured with an error on an estimated coefficients in a generalized linear model. We infer the true shape of the mean and of the variance function in the given model. We show that assumptions of a generalized linear model are not fulfilled universally if we use variables measured with an error. Despite this, the error-in-variable model can still be useful for testing dependence of original correct regressor. Further on in the thesis, the asymptotic values of coefficients are approximated, assuming g(E(Yi|Wi)) is a quadratic function. Examples for all results are provided through simulations.
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Classification based on longitudinal observations
Bandas, Lukáš ; Komárek, Arnošt (advisor) ; Kulich, Michal (referee)
The concern of this thesis is to discuss classification of different objects based on longitudinal observations. In the first instance the reader is introduced to a linear mixed-effects model which is useful for longitudinal data modeling. Description of discriminant analysis methods follows. These methods ares usually used for classification based on longitudinal observations. Individual methods are introduced in the theoretic aspect. Random effects approach is generalized to continuous time. Subsequently the methods and features of the linear mixed-effects model are applied to real data. Finally features of the methods are studied with help of simulations.
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Applications of EM-algorithm
Komora, Antonín ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
EM algorithm is a very valuable tool in solving statistical problems, where the data presented is incomplete. It is an iterative algorithm, which in its first step estimates the missing data based on the parameter estimate from the last iteration and the given data and it does so by using the conditional expectation. In the second step it uses the maximum likelihood estimation to find the value that maximizes the logarithmic likelihood function and passes it along to the next iteration. This is repeated until the point, where the value increment of the logarithmic likelihood function is small enough to stop the algorithm without significant errors. A very important characteristic of this algorithm is its monotone convergence and that it does so under fairly general conditions. However the convergence itself is not very fast, and therefore at times requires a great number of iterations.
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