Analyzing the dependence structure of microarray data: a copula–based approach

The main aim of this Ph.D. dissertation is the study of clustering dependent data by means of copula functions with particular emphasis on microarray data. Copula functions are a popular multivariate modeling tool in each field where the multivariate dependence is of great interest and their use in clustering has not been still investigated. The first part of this work contains the review of the literature of clustering methods, copula functions and microarray experiments. The attention focuses on the K–means (Hartigan, 1975; Hartigan and Wong, 1979), the hierarchical (Everitt, 1974) and the model–based (Fraley and Raftery, 1998, 1999, 2000, 2007) clustering techniques because their performance is compared. Then, the probabilistic interpretation of the Sklar’s theorem (Sklar’s, 1959), the estimation methods for copulas like the Inference for Margins (Joe and Xu, 1996) and the Archimedean and Elliptical copula families are presented. In the end, applications of clustering methods and copulas to the genetic and microarray experiments are highlighted. The second part contains the original contribution proposed. A simulation study is performed in order to evaluate the performance of the K–means and the hierarchical bottom–up clustering methods in identifying clusters according to the dependence structure of the data generating process. Different simulations are performed by varying different conditions (e.g., the kind of margins (distinct, overlapping and nested) and the value of the dependence parameter ) and the results are evaluated by means of different measures of performance. In light of the simulation results and of the limits of the two investigated clustering methods, a new clustering algorithm based on copula functions (‘CoClust’ in brief) is proposed. The basic idea, the iterative procedure of the CoClust and the description of the written R functions with their output are given. The CoClust algorithm is tested on simulated data (by varying the number of clusters, the copula models, the dependence parameter value and the degree of overlap of margins) and is compared with the performance of model–based clustering by using different measures of performance, like the percentage of well–identified number of clusters and the not rejection percentage of H0 on . It is shown that the CoClust algorithm allows to overcome all observed limits of the other investigated clustering techniques and is able to identify clusters according to the dependence structure of the data independently of the degree of overlap of margins and the strength of the dependence. The CoClust uses a criterion based on the maximized log–likelihood function of the copula and can virtually account for any possible dependence relationship between observations. Many peculiar characteristics are shown for the CoClust, e.g. its capability of identifying the true number of clusters and the fact that it does not require a starting classification. Finally, the CoClust algorithm is applied to the real microarray data of Hedenfalk et al. (2001) both to the gene expressions observed in three different cancer samples and to the columns (tumor samples) of the whole data matrix.

A partial dependence factorial analysis to deal with selection bias in observational studis

This thesis presents a creative and practical approach to dealing with the problem of selection bias. Selection bias may be the most important vexing problem in program evaluation or in any line of research that attempts to assert causality. Some of the greatest minds in economics and statistics have scrutinized the problem of selection bias, with the resulting approaches – Rubin’s Potential Outcome Approach(Rosenbaum and Rubin,1983; Rubin, 1991,2001,2004) or Heckman’s Selection model (Heckman, 1979) – being widely accepted and used as the best fixes. These solutions to the bias that arises in particular from self selection are imperfect, and many researchers, when feasible, reserve their strongest causal inference for data from experimental rather than observational studies. The innovative aspect of this thesis is to propose a data transformation that allows measuring and testing in an automatic and multivariate way the presence of selection bias. The approach involves the construction of a multi-dimensional conditional space of the X matrix in which the bias associated with the treatment assignment has been eliminated. Specifically, we propose the use of a partial dependence analysis of the X-space as a tool for investigating the dependence relationship between a set of observable pre-treatment categorical covariates X and a treatment indicator variable T, in order to obtain a measure of bias according to their dependence structure. The measure of selection bias is then expressed in terms of inertia due to the dependence between X and T that has been eliminated. Given the measure of selection bias, we propose a multivariate test of imbalance in order to check if the detected bias is significant, by using the asymptotical distribution of inertia due to T (Estadella et al. 2005) , and by preserving the multivariate nature of data. Further, we propose the use of a clustering procedure as a tool to find groups of comparable units on which estimate local causal effects, and the use of the multivariate test of imbalance as a stopping rule in choosing the best cluster solution set. The method is non parametric, it does not call for modeling the data, based on some underlying theory or assumption about the selection process, but instead it calls for using the existing variability within the data and letting the data to speak. The idea of proposing this multivariate approach to measure selection bias and test balance comes from the consideration that in applied research all aspects of multivariate balance, not represented in the univariate variable- by-variable summaries, are ignored. The first part contains an introduction to evaluation methods as part of public and private decision process and a review of the literature of evaluation methods. The attention is focused on Rubin Potential Outcome Approach, matching methods, and briefly on Heckman’s Selection Model. The second part focuses on some resulting limitations of conventional methods, with particular attention to the problem of how testing in the correct way balancing. The third part contains the original contribution proposed , a simulation study that allows to check the performance of the method for a given dependence setting and an application to a real data set. Finally, we discuss, conclude and explain our future perspectives.

Mathematical Modeling to Investigate Antiarrhythmic Drug Side Effects: Rate-Dependence Role in Ionic Currents and Action Potentials Shape in the O’Hara Model

Sudden cardiac death due to ventricular arrhythmia is one of the leading causes of mortality in the world. In the last decades, it has proven that anti-arrhythmic drugs, which prolong the refractory period by means of prolongation of the cardiac action potential duration (APD), play a good role in preventing of relevant human arrhythmias. However, it has long been observed that the “class III antiarrhythmic effect” diminish at faster heart rates and that this phenomenon represent a big weakness, since it is the precise situation when arrhythmias are most prone to occur. It is well known that mathematical modeling is a useful tool for investigating cardiac cell behavior. In the last 60 years, a multitude of cardiac models has been created; from the pioneering work of Hodgkin and Huxley (1952), who first described the ionic currents of the squid giant axon quantitatively, mathematical modeling has made great strides. The O’Hara model, that I employed in this research work, is one of the modern computational models of ventricular myocyte, a new generation began in 1991 with ventricular cell model by Noble et al. Successful of these models is that you can generate novel predictions, suggest experiments and provide a quantitative understanding of underlying mechanism. Obviously, the drawback is that they remain simple models, they don’t represent the real system. The overall goal of this research is to give an additional tool, through mathematical modeling, to understand the behavior of the main ionic currents involved during the action potential (AP), especially underlining the differences between slower and faster heart rates. In particular to evaluate the rate-dependence role on the action potential duration, to implement a new method for interpreting ionic currents behavior after a perturbation effect and to verify the validity of the work proposed by Antonio Zaza using an injected current as a perturbing effect.

Measurement of the helicity dependence of deuteron photodisintegration for photon energies below 450 MeV

In 1998 a pilot experiment was carried out to study the helicity dependence of photoreaction cross sections using circularly polarized real photons on longitudinally polarized deuterons in a deuterated butanol target. The knowledge of these cross sections is required to test the validity of the Gerasimov-Drell-Hearn sum rule on the deuteron and the neutron. The focus of this thesis is on the results for the differential and total cross sections for the photodisintegration reaction for various photon energies in the range from 200 to 450 MeV using data taken with the detector system DAPHNE. The current understanding of the NN interaction as represented by the calculations by M. Schwamb could be confirmed within the given uncertainties. In addition, the detector DAPHNE has been prepared for the main experiment in 2003. The according work is presented together with results of the quality-test measurements of the renewed detector components.

Study of the helicity dependence of single pion photoproduction on the deuteron

Der erste experimentelle Test der GDH-Summenregel für das Proton wurde 1998 am MAMI-Beschleuniger der Universität Mainz durchgeführt. Ferner wurde ein Pilot-Experiment mit einem polarisierten Deuteron-Target vorgenommen. Dieselbe Kollaboration führte 2003 ein auf das Deuteron ausgerichtetes Experiment durch mit der Absicht, die GDH-Summenregel für das Neutron zu untersuchen. Der in beiden Experimenten verwendete Aufbau erlaubt nicht nur die Messung des totalen Wirkungsquerschnitts, sondern auch gleichzeitig einzelne Teilreaktionen zu studieren. In dieser Arbeit werden die Daten des Deuteron-Pilot-Experiments von 1998 analysiert. Desweiteren wird eine Studie der Helizitätsabhängigkeit der differenziellen Wirkungsquerschnitte für drei Pion-Photoproduktionskanäle des Deuterons in der oberen Hälfte der Delta-Resonanz präsentiert. Diese Ergebnisse werden mit einem theoretischen Modell verglichen. Dabei wurde eine hinreichend gute Übereinstimmung für die unpolarisierten Reaktionen gefunden, während für die polarisierten Kanäle kleinere Diskrepanzen beobachtet wurden. Der Targetpolarisationsgrad ist einer der relevanten Parameter, der für eine absolute Normalisierung der Wirkungsquerschnitte notwendig ist. Die Analyse dieses Parameters für die 2003er Daten wird in der vorliegenden Arbeit vorgestellt. Zur Zeit ist in Mainz ein Frozen-Spin-Target im Bau. Es wird als Target für polarisierte Protonen oder polarisierte Deuteronen für zukünftige Experimente mit dem Crystal Ball zur Verfügung stehen. Die Vorbereitungen der verschiedenen Subsysteme dieses Aufbaus stellten einen wichtigen Teil dieser Arbeit dar. Die fundamentalen Grundlagen der Methode und deren technische Umsetzung, sowie der momentane Status der Aufbauarbeiten am Target werden im Detail präsentiert.

Elektromagnetische Pionproduktion in manifest Lorentz-invarianter baryonischer chiraler Störungstheorie

In dieser Arbeit wurde die elektromagnetische Pionproduktion unter der Annahme der Isospinsymmetrie der starken Wechselwirkung im Rahmen der manifest Lorentz-invarianten chiralen Störungstheorie in einer Einschleifenrechnung bis zur Ordnung vier untersucht. Dazu wurden auf der Grundlage des Mathematica-Pakets FeynCalc Algorithmen zur Berechnung der Pionproduktionsamplitude entwickelt. Bis einschließlich der Ordnung vier tragen insgesamt 105 Feynmandiagramme bei, die sich in 20 Baumdiagramme und 85 Schleifendiagramme unterteilen lassen. Von den 20 Baumdiagrammen wiederum sind 16 als Polterme und vier als Kontaktgraphen zu klassifizieren; bei den Schleifendiagrammen tragen 50 Diagramme ab der dritten Ordnung und 35 Diagramme ab der vierten Ordnung bei. In der Einphotonaustauschnäherung lässt sich die Pionproduktionsamplitude als ein Produkt des Polarisationsvektors des (virtuellen) Photons und des Übergangsstrommatrixelements parametrisieren, wobei letzteres alle Abhängigkeiten der starken Wechselwirkung beinhaltet und wo somit die chirale Störungstheorie ihren Eingang findet. Der Polarisationsvektor hingegen hängt von dem leptonischen Vertex und dem Photonpropagator ab und ist aus der QED bekannt. Weiterhin lässt sich das Übergangsstrommatrixelement in sechs eichinvariante Amplituden zerlegen, die sich im Rahmen der Isospinsymmetrie jeweils wiederum in drei Isospinamplituden zerlegen lassen. Linearkombinationen dieser Isospinamplituden erlauben letztlich die Beschreibung der physikalischen Amplituden. Die in dieser Rechnung auftretenden Einschleifenintegrale wurden numerisch mittels des Programms LoopTools berechnet. Im Fall tensorieller Integrale erfolgte zunächst eine Zerlegung gemäß der Methode von Passarino und Veltman. Da die somit erhaltenen Ergebnisse jedoch i.a. noch nicht das chirale Zählschema erfüllen, wurde die entsprechende Renormierung mittels der reformulierten Infrarotregularisierung vorgenommen. Zu diesem Zweck wurde ein Verfahren entwickelt, welches die Abzugsterme automatisiert bestimmt. Die schließlich erhaltenen Isospinamplituden wurden in das Programm MAID eingebaut. In diesem Programm wurden als Test (Ergebnisse bis Ordnung drei) die s-Wellenmultipole E_{0+} und L_{0+} in der Schwellenregion berechnet. Die Ergebnisse wurden sowohl mit Messdaten als auch mit den Resultaten des "klassischen" MAID verglichen, wobei sich i. a. gute Übereinstimmungen im Rahmen der Fehler ergaben.

Study of the helicity dependence of pi 0 photoproduction on proton at MAMI

The excitation spectrum is one of the fundamental properties of every spatially extended system. The excitations of the building blocks of normal matter, i.e., protons and neutrons (nucleons), play an important role in our understanding of the low energy regime of the strong interaction. Due to the large coupling, perturbative solutions of quantum chromodynamics (QCD) are not appropriate to calculate long-range phenomena of hadrons. For many years, constituent quark models were used to understand the excitation spectra. Recently, calculations in lattice QCD make first connections between excited nucleons and the fundamental field quanta (quarks and gluons). Due to their short lifetime and large decay width, excited nucleons appear as resonances in scattering processes like pion nucleon scattering or meson photoproduction. In order to disentangle individual resonances with definite spin and parity in experimental data, partial wave analyses are necessary. Unique solutions in these analyses can only be expected if sufficient empirical information about spin degrees of freedom is available. The measurement of spin observables in pion photoproduction is the focus of this thesis. The polarized electron beam of the Mainz Microtron (MAMI) was used to produce high-intensity, polarized photon beams with tagged energies up to 1.47 GeV. A "frozen-spin" Butanol target in combination with an almost 4π detector setup consisting of the Crystal Ball and the TAPS calorimeters allowed the precise determination of the helicity dependence of the γp → π0p reaction. In this thesis, as an improvement of the target setup, an internal polarizing solenoid has been constructed and tested. A magnetic field of 2.32 T and homogeneity of 1.22×10−3 in the target volume have been achieved. The helicity asymmetry E, i.e., the difference of events with total helicity 1/2 and 3/2 divided by the sum, was determined from data taken in the years 2013-14. The subtraction of background events arising from nucleons bound in Carbon and Oxygen was an important part of the analysis. The results for the asymmetry E are compared to existing data and predictions from various models. The results show a reasonable agreement to the models in the energy region of the ∆(1232)-resonance but large discrepancies are observed for energy above 600 MeV. The expansion of the present data in terms of Legendre polynomials, shows the sensitivity of the data to partial wave amplitudes up to F-waves. Additionally, a first, preliminary multipole analysis of the present data together with other results from the Crystal Ball experiment has been as been performed.

Strategies for evaluating the economic value of drugs in alcohol dependence treatment

To assess existing health economic strategies, which are used to evaluate the economic value of drugs to treat alcohol dependence (AD) such as acamprosate, naltrexone and any other pharmaceuticals.