This dissertation develops influence diagnostics for crossover models. Mixed linear models and generalised mixed linear models are utilised to investigate continuous and count data from crossover studies, respectively. For both types of models, changes in the maximum likelihood estimates of parameters, particularly in the estimated treatment effect, due to minor perturbations of the observed data, are assessed. The novelty of this dissertation lies in the analytical derivation of influence diagnostics using decompositions of the perturbed mixed models. Consequently, the suggested influence diagnostics, referred to as the delta-beta and variance-ratio influences, provide new findings about how the constructed residuals affect the estimation in terms of different parameters of interest. The delta-beta and variance-ratio influence in three different crossover models are studied in Chapters 5-6, respectively. Chapter 5 analyses the influence of subjects in a two-period continuous crossover model. Possible problems with observation-level perturbations in crossover models are discussed. Chapter 6 extends the approach to higher-order crossover models. Furthermore, not only the individual delta-beta and variance-ratio influences of a subject are derived, but also the joint influences of two subjects from different sequences. Chapters 5-6 show that the delta-beta and variance-ratio influences of a particular parameter are decided by the special linear combination of the constructed residuals. In Chapter 7, explicit delta-beta influence on the estimated treatment effect in the two-period count crossover model is derived. The influence is related to the Pearson residuals of the subject. Graphical tools are developed to visualise information of influence concerning crossover models for both continuous and count data. Illustrative examples are provided in each chapter.

We propose a simple adjustment to a Wald confidence interval to estimate a linear function of binomial proportions. This method is an extension to the adjusted Wald confidence intervals for proportions and their differences that have recently been proposed.

No description available.

Arnold and Strauss (1991) 探討2x2列聯表中的3個方格 (cell) 有相同機率θ的問題，他們比較了參數θ的最大概似估計值與最大擬概似估計值，發現參數θ的最大概似估計值與最大擬概似估計值是不相同的。在本論文中，我們將2x2列聯表中的3個方格的參數值 (機率值)，從限制為相同θ，放寬為成某種比例，並證明了在一般情況下參數θ的最大概似估計值與最大擬概似估計值也不相同。我們也提出一些使參數θ的最大概似估計值及最大擬概似估計值相同的特殊條件，諸如三個方格內的觀察值跟機率值成比例或格子內的觀察值有某些特定值。本論文也透過電腦模擬的結果，發現最大概似估計式較最大擬概似估計式來得精確，而且當參數θ在參數空間之中點附近時，最大概似估計值與最大擬概似估計值的差異為最大。 / Arnold and Strauss (1991) study the cases that three of the four cells in the 2x2 contingency table have the same cell probability θ. In particular, Arnold and Strauss (1991) compare the maximum likelihood estimate (MLE) and maximum pseudolikelihood estimate (MPLE) of the parameter θ. They find that MLE and MPLE of the parameter are not the same. In this thesis, we relax the assumptions so that those three cell probabilities may not be the same and each is proportional to a parameter θ. We find that, in general, MLE’s of θ are still not the same as MPLE’s of θ. Some special cases that make MLE the same as MPLE are also given. We also find, through computer simulations, that MLE’s are accurate than MPLE’s and that the difference between MLE and MPLE is getting larger when the parameter θ is closer to the midpoint of its space.

No description available.

The driving risk of novice teenagers is the highest during the initial period after licensure but decreases rapidly. This dissertation develops recurrent-event change-point models to detect the time when driving risk decreases significantly for novice teenager drivers. The dissertation consists of three major parts: the first part applies recurrent-event change-point models with identical change-points for all subjects; the second part proposes models to allow change-points to vary among drivers by a hierarchical Bayesian finite mixture model; the third part develops a non-parametric Bayesian model with a Dirichlet process prior. In the first part, two recurrent-event change-point models to detect the time of change in driving risks are developed. The models are based on a non-homogeneous Poisson process with piecewise constant intensity functions. It is shown that the change-points only occur at the event times and the maximum likelihood estimators are consistent. The proposed models are applied to the Naturalistic Teenage Driving Study, which continuously recorded textit{in situ} driving behaviour of 42 novice teenage drivers for the first 18 months after licensure using sophisticated in-vehicle instrumentation. The results indicate that crash and near-crash rate decreases significantly after 73 hours of independent driving after licensure. The models in part one assume identical change-points for all drivers. However, several studies showed that different patterns of risk change over time might exist among the teenagers, which implies that the change-points might not be identical among drivers. In the second part, change-points are allowed to vary among drivers by a hierarchical Bayesian finite mixture model, considering that clusters exist among the teenagers. The prior for mixture proportions is a Dirichlet distribution and a Markov chain Monte Carlo algorithm is developed to sample from the posterior distributions. DIC is used to determine the best number of clusters. Based on the simulation study, the model gives fine results under different scenarios. For the Naturalist Teenage Driving Study data, three clusters exist among the teenagers: the change-points are 52.30, 108.99 and 150.20 hours of driving after first licensure correspondingly for the three clusters; the intensity rates increase for the first cluster while decrease for other two clusters; the change-point of the first cluster is the earliest and the average intensity rate is the highest. In the second part, model selection is conducted to determine the number of clusters. An alternative is the Bayesian non-parametric approach. In the third part, a Dirichlet process Mixture Model is proposed, where the change-points are assigned a Dirichlet process prior. A Markov chain Monte Carlo algorithm is developed to sample from the posterior distributions. Automatic clustering is expected based on change-points without specifying the number of latent clusters. Based on the Dirichlet process mixture model, three clusters exist among the teenage drivers for the Naturalistic Teenage Driving Study. The change-points of the three clusters are 96.31, 163.83, and 279.19 hours. The results provide critical information for safety education, safety countermeasure development, and Graduated Driver Licensing policy making. / Ph. D.

Es sei X eine Zufallsvariable mit unbekannter Verteilung P. Zu den Hauptaufgaben der Mathematischen Statistik zählt die Konstruktion von Schätzungen für einen abgeleiteten Parameter theta(P) mit Hilfe einer Beobachtung X=x. Im Fall einer dominierten Verteilungsfamilie ist es möglich, das Maximum-Likelihood-Prinzip (MLP) anzuwenden. Eine Alternative dazu liefert der Bayessche Zugang. Insbesondere erweist sich unter Regularitätsbedingungen, dass die Maximum-Likelihood-Schätzung (MLS) dem Grenzwert einer Folge von Bayesschen Schätzungen (BSen) entspricht. Eine BS kann aber auch im Fall einer nicht dominierten Verteilungsfamilie betrachtet werden, was als Ansatzpunkt zur Erweiterung des MLPs genutzt werden kann. Weiterhin werden zwei Ansätze einer verallgemeinerten MLS (vMLS) von Kiefer und Wolfowitz sowie von Gill vorgestellt. Basierend auf diesen bekannten Ergebnissen definieren wir einen selbstinformativen Grenzwert und einen selbstinformativen a posteriori Träger. Im Spezialfall einer dominierten Verteilungsfamilie geben wir hinreichende Bedingungen an, unter denen die Menge der MLSen einem selbstinformativen a posteriori Träger oder, falls die MLS eindeutig ist, einem selbstinformativen Grenzwert entspricht. Das Ergebnis für den selbstinformativen a posteriori Träger wird dann auf ein allgemeineres Modell ohne dominierte Verteilungsfamilie erweitert. Insbesondere wird gezeigt, dass die Menge der vMLSen nach Kiefer und Wolfowitz ein selbstinformativer a posteriori Träger ist. Weiterhin wird der selbstinformative Grenzwert bzw. a posteriori Träger in einem Modell mit nicht identifizierbarem Parameter bestimmt. Im Mittelpunkt dieser Arbeit steht ein multivariates semiparametrisches lineares Modell. Zunächst weisen wir jedoch nach, dass in einem rein nichtparametrischen Modell unter der a priori Annahme eines Dirichlet Prozesses der selbstinformative Grenzwert existiert und mit der vMLS nach Kiefer und Wolfowitz sowie der nach Gill übereinstimmt. Anschließend untersuchen wir das multivariate semiparametrische lineare Modell und bestimmen die vMLSen nach Kiefer und Wolfowitz bzw. nach Gill sowie den selbstinformativen Grenzwert unter der a priori Annahme eines Dirichlet Prozesses und einer Normal-Wishart-Verteilung. Im Allgemeinen sind die so erhaltenen Schätzungen verschieden. Abschließend gehen wir dann auf den Spezialfall eines semiparametrischen Lokationsmodells ein, in dem die vMLSen nach Kiefer und Wolfowitz bzw. nach Gill und der selbstinformative Grenzwert wieder identisch sind. / We assume to observe a random variable X with unknown probability distribution. One major goal of mathematical statistics is the estimation of a parameter theta(P) based on an observation X=x. Under the assumption that P belongs to a dominated family of probability distributions, we can apply the maximum likelihood principle (MLP). Alternatively, the Bayes approach can be used to estimate the parameter. Under some regularity conditions it turns out that the maximum likelihood estimate (MLE) is the limit of a sequence of Bayes estimates (BE's). Note that BE's can even be defined in situations where no dominating measure exists. This allows us to derive an extension of the MLP using the Bayes approach. Moreover, two versions of a generalised MLE (gMLE) are presented, which have been introduced by Kiefer and Wolfowitz and Gill, respectively. Based on the known results, we define a selfinformative limit and a posterior carrier. In the special case of a model with dominated distribution family, we state sufficient conditions under which the set of MLE's is a selfinformative posterior carrier or, in the case of a unique MLE, a selfinformative limit. The result for the posterior carrier is extended to a more general model without dominated distributions. In particular we show that the set of gMLE's of Kiefer and Wolfowitz is a posterior carrier. Furthermore we calculate the selfinformative limit and posterior carrier, respectively, in the case of a model with possibly nonidentifiable parameters. In this thesis we focus on a multivariate semiparametric linear model. At first we show that, in the case of a nonparametric model, the selfinformative limit coincides with the gMLE of Kiefer and Wolfowitz as well as that of Gill, if a Dirichlet process serves as prior. Then we investigate both versions of gMLE's and the selfinformative limit in the multivariate semiparametric linear model, where the prior for the latter estimator is given by a Dirichlet process and a normal-Wishart distribution. In general the estimators are not identical. However, in the special case of a location model we find again that the three considered estimates coincide.

Táto práca sa zaoberá odstraňovaním šumu, ktorý vzniká z tzv. multilateračných meraní leteckých cieľov. Na tento účel bude využitá najmä teória Bayesovských odhadov. Odvodí sa aposteriórna hustota skutočnej (presnej) polohy lietadla. Spolu s polohou (alebo aj rýchlosťou) lietadla bude odhadovaná tiež geometria trajektórie lietadla, ktorú lietadlo v aktuálnom čase sleduje a tzv. procesný šum, ktorý charakterizuje ako moc sa skutočná trajektória môže od tejto líšiť. Odhad spomínaného procesného šumu je najdôležitejšou časťou tejto práce. Je odvodený prístup maximálnej vierohodnosti a Bayesovský prístup a ďalšie rôzne vylepšenia a úpravy týchto prístupov. Tie zlepšujú odhad pri napr. zmene manévru cieľa alebo riešia problém počiatočnej nepresnosti odhadu maximálnej vierohodnosti. Na záver je ukázaná možnosť kombinácie prístupov, t.j. odhad spolu aj geometrie aj procesného šumu.

Probabilistic models of decision making under various forms of uncertainty have been applied in recent years to numerous behavioral and model-based fMRI studies. These studies were highly successful in enabling a better understanding of behavior and delineating the functional properties of brain areas involved in decision making under uncertainty. However, as different studies considered different models of decision making under uncertainty, it is unclear which of these computational models provides the best account of the observed behavioral and neuroimaging data. This is an important issue, as not performing model comparison may tempt researchers to over-interpret results based on a single model. Here we describe how in practice one can compare different behavioral models and test the accuracy of model comparison and parameter estimation of Bayesian and maximum-likelihood based methods. We focus our analysis on two well-established hierarchical probabilistic models that aim at capturing the evolution of beliefs in changing environments: Hierarchical Gaussian Filters and Change Point Models. To our knowledge, these two, well-established models have never been compared on the same data. We demonstrate, using simulated behavioral experiments, that one can accurately disambiguate between these two models, and accurately infer free model parameters and hidden belief trajectories (e.g., posterior expectations, posterior uncertainties, and prediction errors) even when using noisy and highly correlated behavioral measurements. Importantly, we found several advantages of Bayesian inference and Bayesian model comparison compared to often-used Maximum-Likelihood schemes combined with the Bayesian Information Criterion. These results stress the relevance of Bayesian data analysis for model-based neuroimaging studies that investigate human decision making under uncertainty.

Probabilistic models of decision making under various forms of uncertainty have been applied in recent years to numerous behavioral and model-based fMRI studies. These studies were highly successful in enabling a better understanding of behavior and delineating the functional properties of brain areas involved in decision making under uncertainty. However, as different studies considered different models of decision making under uncertainty, it is unclear which of these computational models provides the best account of the observed behavioral and neuroimaging data. This is an important issue, as not performing model comparison may tempt researchers to over-interpret results based on a single model. Here we describe how in practice one can compare different behavioral models and test the accuracy of model comparison and parameter estimation of Bayesian and maximum-likelihood based methods. We focus our analysis on two well-established hierarchical probabilistic models that aim at capturing the evolution of beliefs in changing environments: Hierarchical Gaussian Filters and Change Point Models. To our knowledge, these two, well-established models have never been compared on the same data. We demonstrate, using simulated behavioral experiments, that one can accurately disambiguate between these two models, and accurately infer free model parameters and hidden belief trajectories (e.g., posterior expectations, posterior uncertainties, and prediction errors) even when using noisy and highly correlated behavioral measurements. Importantly, we found several advantages of Bayesian inference and Bayesian model comparison compared to often-used Maximum-Likelihood schemes combined with the Bayesian Information Criterion. These results stress the relevance of Bayesian data analysis for model-based neuroimaging studies that investigate human decision making under uncertainty.