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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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Parametric regression models in survival analysis
Otava, Martin ; Komárek, Arnošt (advisor) ; Ševčík, Jaroslav (referee)
Na'/cv prace: Paraniot.ricke regiesiri modely v analv'/e pfezitf Autor: Martin Otava Katedra (li.stav): Katedra pravdepoiiolmosti a niatcmatitke statistiky Vedouci bakalafske prace: Mgr. Amost Komarek, Ph.D. (j-inail vedonciho: komaiek( >karlin.mli.cuni.c /, Abst.rakl: \ pfcdlo/.ene. praci studujcme panunetricke re^resni modely v ana- lyze pre/.ili. Skrze pojem cen/orovani se se/namfme s podst.alou analy/.y pixv.iti a zavudrme si zakladnf pojiiiy ir/ivanr v souvislo^ti s ui. Uka'/niK1 si tvurbu vlicxlnc'lio rc^rcsni'lio inodcln a zpiiwoby odliadti paranictru s diua/,cni na inotixhi inaxiinalni vrrohodnosTi spolrriu"' s itcrarniini nu'l.odaini pro ... vyrt'seni. Vysvrtlnnr si vv'/nani iialiodiu'' chyliy inefriii. Die1 jcjilio roxdrlcni pak vyLvoriiuc nckolik ni/nycli pai'a.iiHit.i1itikycli niodrlu pro odhad li\i.s(oly etiyu do sclhani. Srovnanir inodcly s neparamctrirkym odliadnn, ktdiy nain poimizc ... i t . /da na.s model odpovi'da ri'aliTr. CV-lou [iraci bude provaxcT ihi- ,stra,ce na skiilcrnycli dal(irh slouxicf jako nkii/ka fuii^cjvani metody v pra.xi. KliVova slova: Analv/a prc/iti. iJarainrtrickr inodcly. Title: Accclcralfd iailurc t.inn1 models in survival analysis Ant hor: Mart in Otava Dqiart.nieiit.: Depart.mcnl of Proba.bilit.y a.nd Mat lunnatical St.atisl.ics Supervisor: Mgv. Arnost Komarck,...
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Bayesian and Maximum Likelihood Nonparametric Estimation in Monotone Aalen Model
Timková, Jana ; Volf, Petr (advisor) ; Kraus, David (referee) ; Komárek, Arnošt (referee)
This work is devoted to seeking methods for analysis of survival data with the Aalen model under special circumstances. We supposed, that all regression functions and all covariates of the observed individuals were nonnegative and we named this class of models monotone Aalen models. To find estimators of the unknown regres- sion functions we considered three maximum likelihood based approaches, namely the nonparametric maximum likelihood method, the Bayesian analysis using Beta processes as the priors for the unknown cumulative regression functions and the Bayesian analysis using a correlated prior approach, where the regression functions were supposed to be jump processes with a martingale structure.
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Expectile regression
Ondřej, Josef ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
In this thesis we present an alternative to quantiles, which is known as expectiles. At first we define the notion of expectile of a distribution of ran- dom variable and then we show some of its basic properties such as linearity or monotonic behavior of τ-th expectile eτ in τ. Let (Y, X), Y ∈ R, X ∈ Rp be a ran- dom vector. We define conditional expectile of Y given X = x, which we denote eτ (Y |X = x). We introduce model of expectile regression eτ (Y |X = x) = x⊤ βτ , where βτ ∈ Rp and we examine asymptotic behavior of estimate of the regression coefficients βτ and ways how to calculate it. Further we introduce semiparametric expectile regression, which generalizes the previous case and adds restrictions on the estimate of the regression coefficients which enforce desired properties such as smoothness of fitted curves. We illustrate the use of theoretical results on me- chanographic data, which describe dependence of power and force of a jump on age of children and adolescents aged between 6 and 18. Keywords: expectiles, expectile regression, quantiles, penalized B-splines 1
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EM algorithm
Vacula, Ondřej ; Komárek, Arnošt (advisor) ; Antoch, Jaromír (referee)
This paper discusses the EM algorithm. This algorithm is used, for example, to calculate maximum likelihood estimate of unknown parameter. The algorithm is based on repeated calculations of certain expected value and maximizing specific function. We begin with parameter estimation problem, describe the maximum likelihood method and concept of incomplete data. Then we formulate the EM algorithm and its properties. In the next chapter we apply this knowledge to three selected statistical problems. At first we examine standard mixture model, then the linear mixed model and finally we analyze censored data. Powered by TCPDF (www.tcpdf.org)
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Simultaneous confidence intervals dual to stepwise methods of multiple comparison
Moravec, Jan ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
The central theme of this thesis is the construction of simultaneous confidence regions (SCR) corresponding to stepwise multiple comparison procedures (MCP). The first chapter is devoted to the theory of multiple comparisons, including the class of closed testing procedures which contains every MCP that strongly con- trols the familywise error rate. The second chapter is concerned with the gene- ral principle of construction of SCR corresponding to closed testing procedures. These general results are used in the third and the forth chapter for deriving the SCR corresponding to a subclass of closed testing procedures which are based on weighted Bonferroni tests. The SCR corresponding to the Holm, the Holm(W), the fixed-sequence and the fallback MCP are derived explicitly. The theoretical results are numerically illustrated on a bioequivalence study. In the fifth chapter we briefly discuss the SCR corresponding to the Hommel, the Hochberg and the step-down Dunnett MCP.
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Some problems of exponential smoothing
Čurda, David ; Hanzák, Tomáš (advisor) ; Komárek, Arnošt (referee)
In this work the several exponential smoothing type methods are briefly described, which are often used to smoothing and forecasting in the time series. Selected problems, that occur in described methods, are presented and in some cases there are the suggestions to their solution, which should tend to more suitable smoothing or to the better forecasts. It's shown how the methods are applied on different data and how the forecasts differ from each other. In conclusion the quality of modifications is evaluated.
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