Continuous Univariate Distributions Volume 2

In many statistical applications, one encounters populations that form homogenous subgroups regarding one or several characteristics. Across the subgroups, however, heterogeneity may often be found. Mixture distributions are a natural means to model data from such applications. This PhD thesis is based on two projects that focus on such applications. In the first project, spatial nanoscale clusters formed by Ras proteins in the cell membrane are investigated. Such clusters play a crucial role in intracellular communication and are thus of interest in cancer research. In this case, the subgroups are clustered and non-clustered proteins. In the second project, epigenomic data obtained from sequencing experiments are integrated with another genomic or epigenomic input, aiming, e.g., to detect genes that contribute to the development of cancer. Here, the subgroups are defined by a) genes presenting congruent (epi)genomic aberrations in both considered variables, b) genes presenting incongruent aberrations, and c) genes lacking aberrations in at least one of the variables. Employing a Bayesian framework, objects are classified in both projects by fitting finite univariate mixture distributions with a small fixed number of components to values from a score summarizing relevant information about the research question. Such mixture distributions have favorable characteristics in terms of interpretation and present little sensitivity to label switching in Markov Chain Monte Carlo analyses. Mixtures of gamma distributions are considered for Ras proteins, while mixtures of normal and exponential or gamma distributions are a focus for the bioinformatic analysis. In the latter, classification is the primary goal, while in the Ras protein application, estimating key parameters of the spatial clustering is of more interest. The results of both projects are presented in this thesis. For both applications, the methods have been implemented in software and their performance is compared with competing approaches on experimental as well as on simulated data. To warrant an appropriate simulation of Ras protein patterns, a new cluster point process model called the double Matérn cluster process is developed and described in this thesis. Acknowledgements First of all, I wish to thank my advisor Katja Ickstadt for her guidance during my PhD time, in particular for giving me both freedom in many research problems and valuable advice whenever needed, but also for providing an always very comfortable working atmosphere in her research group. Alan E. Gelfand supported my research on spatial Ras protein clusters through helpful discussions on concepts. I therefore would like to thank him for advice and for sharing his large statistical experience. I would like to thank Peter Verveer and members of his former research group at the Max Planck Institute of Molecular Physiology in Dortmund for providing the problem regarding the research on Ras protein clusters, as well as for the fruitful collaboration in general. In particular, I want to thank Yvonne Radon for furnishing me with experimental Ras data and Thomas Klein for making my MATLAB code more efficient. My research on spatial Ras protein clusters formed part of the project ‘Modeling spatial effects of cellular signals’ in the Research Training Group 1032/2 ‘Statistical Modeling’ funded by the Deutsche Forschungsgemeinschaft (DFG). I was granted a PhD scholarship for my research within this Research Training Group and would like to gratefully acknowledge this financial support. I wish to thank Hans-Ulrich Klein, now Harvard Medical School/Brigham and Women’s Hospital, Boston, and formerly University of Münster, Münster, for our fruitful collaboration for years in the analysis of genomic and epigenomic data, in which I feel we have complemented one another very well. I also thank Martin Dugas, his former group leader at the University of Münster, for supporting our work. I thank my colleagues for numerous discussions on statistics. Representing many others, in particular I mention Jakob Wieczorek, with whom I shared an office at the Faculty of Statistics, TU Dortmund University, and Bernd Bischl, with whom I shared many coffees during conversations there in the last years. ii I thank Holger Schwender, Philipp Berger, Julia Schiffner, Sabrina Herrmann, Claudia Köllmann, Lars Koppers and Jenny Peplies for proofreading this thesis. Further thanks go to Eva Brune and Brigitte Kupiec for administrative support. Equally important as work-related aspects, my friends have been providing support in many other ways. Thank you! I am deeply grateful to Jutta and Thomas as I could not imagine better parents . . . and they have made me what I am to a great extent. Thomas also gave me the idea to study statistics by sharing his experience about this exciting profession with me. Jenny has always been a wonderful big sister and friend, and I am glad to have her! Finally, I thank Tamara for patience and emotional support during my PhD time... gracias mi vida... eres maravillosa y te amo!

Finite Bayesian mixture models with applications in spatial cluster analysis and bioinformatics

Finite mixture models assume
that a distribution is a combination of several parametric
distributions. They offer a compromise between the interpretability
of parametric models and the flexibility of nonparametric models.
This thesis considers a Bayesian approach to these models, which
has several advantages. For example, using only weak prior
information, it can solve problems with unbounded likelihood
functions, that can occur in mixture models. The Bayesian approach
also allows an elegant extension of finite to (countable) infinite
mixture models. Depending on the application, the components of
mixture models can either be viewed as just a means to the flexible
modeling of a distribution or as defining subgroups of a population
with different parametric distributions. Regarding the former case
consistency results for Bayesian mixtures are stated. An example
concerning the flexible modeling of a random effects distribution
in a logistic regression is also given. The application considers
the goalkeeper's effect in saving a penalty. In the latter case
mixture models can be used for clustering. Bayesian mixtures then
allow the estimation of the number of clusters at the same time as
the cluster-specific parameters. For cluster analysis the standard
approach for fitting Bayesian mixtures, Markov Chain Monte Carlo
(MCMC), unfortunately leads to inferential difficulties. The labels
associated with the clusters can change during the MCMC run, a
phenomenon called label-switching. The problem gets severe, if the
number of clusters is allowed to vary. Existing methods to deal
with label-switching and a varying number of components are
reviewed and new approaches are proposed for both situations. The
first consists of a variant of the relabeling algorithm of Stephens
(2000). The variant is more general, as it applies to drawn
clusterings and not drawn parameter values. Therefore it does not
depend on the specific form of the component distributions. The
second approach is based on pairwise posterior probabilities and is
an improvement of a commonly used loss function due to Binder
(1978). Minimization of this loss is shown to be equivalent to
maximizing the posterior expected Rand index with the true
clustering. As the adjusted Rand index is preferable to the raw
index, the maximization of the posterior expected adjusted Rand is
proposed. The new approaches are compared to the previous methods
on simulated and real data. The real data used for cluster analysis
are two gene expression data sets and Fisher's iris data.

Arno Fritsch

Papers

Bayesian mixtures for cluster analysis and flexible modeling of distributions