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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 under Two-phase Stratified and Cluster Sampling
Šedová, Michaela ; Kulich, Michal (advisor) ; Picek, Jan (referee) ; Omelka, Marek (referee)
Title: Parameter Estimation under Two-phase Stratified and Cluster Sampling Author: Mgr. Michaela Šedová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Mgr. Michal Kulich, Ph.D. Abstract: In this thesis we present methods of parameter estimation under two-phase stratified and cluster sampling. In contrast to classical sampling theory, we do not deal with finite population parameters, but focus on model parameter inference, where the ob- servations in a population are considered to be realisations of a random variable. However, we consider the sampling schemes used, and thus we incorporate much of survey sampling theory. Therefore, the presented methods of the parameter estimation can be understood as a combination of the two approaches. For both sampling schemes, we deal with the concept where the population is considered to be the first-phase sample, from which a sub- sample is drawn in the second phase. The target variable is then observed only for the subsampled subjects. We present the mean value estimation, including the statistical prop- erties of the estimator, and show how this estimation can be improved if some auxiliary information, correlated with the target variable, is observed for the whole population. We extend the method to the regression problem....
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Modeling progression of HIV disease
Žohová, Ivana ; Kulich, Michal (advisor) ; Zvára, Karel (referee)
In the present work we study modeling of HIV disease progression via multistate Markov model. The difficulty in this approach is how to define HIV disease states. These are usually defined in terms of CD4+ T lymphocyte counts, but this marker is a subject to biological fluctuation and, in real life, measurement errors as well. Estimating the model on such a data will lead to intensity estimates depending on frequency of observations. That is why we usually smooth the data before fitting the Markov model. In this work we studied two different approaches - linear mixed-effects model and local polynomial kernel estimator. All modeling is performed on real data and also an illustrative simulation example is included. Another issue considered in this work is determination of sero-conversion time. The sero-conversion distribution is derived based on time of last negative observation, first positive observation and last performed measurement.
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