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Analysis of change from baseline to post-intervention value
Pacáková, Andrea ; Hlávka, Zdeněk (referee) ; Kulich, Michal (advisor)
The aim of the present work is to compare three di erent estimators of a treatment e ect in clinical randomized studies. The purpose of these studies is to compare the change of a distribution of certain variable between two attendances. Mentioned estimators were developed from the assumption of validity of some model. In this work we gather properties of the estimators when each of all given models is valid. We deal with the consistency of the estimators and with their asymptotic distributions and then we compare the estimators on the basis of their asymptotic variances. In the most of cases is possible to make the comparison in general. In the case when it is not possible, we show a few particular examples. Eventually, we accomplish the simulation study, which certi es theoretical conclusions and extends pieces of knowledge in the cases when it was not possible to make theoretical computation in general.
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Multicollinearity
Dřizgová, Lucie ; Hlávka, Zdeněk (referee) ; Zvára, Karel (advisor)
In our work, we explored multicollinearity problem from a complex point of view - from diagnostic methods to the solving of the problems which are caused by the multicollinearity. We compared the Least Squares method with some alternative methods - Principal Component Regression, Partial Least Squares Regression and Ridge Regression on the theoretical basis. In the last section, we demonstrated all methods on practical example computed in the program R.
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Analysis of Biosensoric Data
Timková, Jana ; Antoch, Jaromír (referee) ; Hlávka, Zdeněk (advisor)
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Testing of composite hypotheses in regression models with small sample size
Simerská, Olga ; Hlávka, Zdeněk (referee) ; Kulich, Michal (advisor)
The thesis, through a simulation study, examines the behaviour of asymptotic tests for testing hypotheses that several coefficients in logistic models are zero. Likelihood ratio, Wald's, and Rao's tests are considered. The necessary theory is formulated to derive the form of the statistics of asymptotic tests for testing composite hypotheses in logistic regression. Based on the numerical treatment of simulated data, the levels of significance of these tests are investigated, with critical values of the chi-squared distribution. The powers of the tests are then compared, modified empirically so that all tests reject the null hypotesis at the 5% level. The main focus is on the dependence of these values on the sample size and parameter settings.
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Isotonic Regression in Sobolev Spaces
Pešta, Michal ; Dostál, Petr (referee) ; Hlávka, Zdeněk (advisor)
We propose a class of nonparametric estimators for the regression models based on least squares over the sets of sufficiently smooth functions. Least squares permit the imposition of additional constraint-isotonia-on nonparametric regression estimation and testing of this constraint. The estimation takes place over the balls of functions which are elements of a suitable Sobolev space-special types of Hilbert spaces that facilitate calculation of the least squares projection. The Hilbertness is allowing us to take projections and hence to decompose spaces into mutually orthogonal complements. Then we transform the problem of searching for the best fitting function in an infinite dimensional space into a finite dimensional optimization problem. We prove that the balls of functions in Sobolev space are bounded and have bounded higher order derivatives. It permits us to estimate over rich set of functions with sufficiently low metric entropy and apply Laws of Large Numbers and Central Limit Theorems.
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Estimation of the Location of Zeros of Regression Functions
Juríček, Jozef ; Zvára, Karel (referee) ; Hlávka, Zdeněk (advisor)
The main interest of this master thesis is the estimation of location of zeros of the regression function and its derivatives by the parametric and nonparametric method. The first section includes either linear and nonlinear regression model of the parametric methods. The estimators are then based on the estimates of parameters. The second part includes nonparametric regression model - kernel estimators of the regression function and its derivatives investigated by Gasser and Müller. Especially, the limit distributions of the estimators of zeros and the choice of smoothing parameter and kernel function are studied. Confidence bands for zeros of regression function and its derivatives are constructed in both sections. Models are studied with independent as well as correlated errors. This master thesis o®ers examples to particular sections that are computed with software R and also sources of some programmed functions.
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