Acute Myocardial Infarction (AMI), commonly known as heart attack, is a leading cause of death for adult men and women in the world. Studying mortality after AMI is therefore an important problem in epidemiology. This thesis develops statistical methodology for examining geographic patterns in mortality following AMI. Specifically, we develop parametric Accelerated Failure Time (AFT) models for censored survival data, where space-varying regression is used to investigate spatial patterns of mortality after AMI. In addition to important covariates such as age and gender, the regression models proposed here also incorporate spatial random e ects that describe the residual heterogeneity associated with di erent local health geographical units. We conduct model inference under a hierarchical Bayesian modeling framework using Markov Chain Monte Carlo algorithms for implementation. We compare an array of models and address the goodness-of- t of the parametric AFT model through simulation studies and an application to a longitudinal AMI study in Quebec. The application of our AFT model to the Quebec AMI data yields interesting ndings concerning aspects of AMI, including spatial variability. This example serves as a strong case for considering the parametric AFT model developed here as a useful tool for the analysis of spatially correlated survival data.

This thesis provides several new calibration methods for the empirical log-likelihood ratio. The commonly used Chi-square calibration is based on the limiting distribu¬tion of this ratio but it constantly suffers from the undercoverage problem. The finite sample distribution of the empirical log-likelihood ratio is recognized to have a mix¬ture structure with a continuous component on [0, +∞) and a probability mass at +∞. Consequently, new calibration methods are developed to take advantage of this mixture structure; we propose new calibration methods based on the mixture distrib¬utions, such as the mixture Chi-square and the mixture Fisher's F distribution. The E distribution introduced in Tsao (2004a) has a natural mixture structure and the calibration method based on this distribution is considered in great details. We also discuss methods of estimating the E distributions.

The process of particle growth in a warm cloud caused by coalescence is studied. The purely probabilistic model introduced by Gillespie [J. Atmos. Sci. 29 (1972) 1496-1510j is used and solved exactly by the aid of the Monte Carlo algorithm developed by Gillespie [J. Atmos. Sci. 32 (1975) 1977-1989]. Another approach uses the kinetic coalescence equation which is solved numerically using finite difference methods. It is known that the stochastic completeness of the kinetic coalescence equation depends on the extent of correlations between particles. Our objective is to compare these two models and analyze the suitability of the kinetic coalescence equation to simulate the coalescence process using a Brownian diffusion collision kernel. The stochastic coalescence model introduced by Gillespie is discussed in detail. A description of Gillespie's Monte Carlo simulation procedure and the numerical code that implements this algorithm in Fortran are provided. This algorithm is applied to the coalescence kernel for Brownian diffusion and initial Poisson and uniform droplet size distributions. Numerical methods which can he applied to the continuous and the discrete forms of the kinetic equation are described. The discrete form of this equation is solved by using Euler's and the fourth order Runge-Kutta methods. Solutions from the two models at early and later times are examined and the effect of the number of droplets used in simulations is investigated. It is shown that solutions agree well for early and later times using large and relatively small number of droplets initially. The problem of the growth of a large particle as it settles through a monodisperse suspension of small elemental particles is considered. It is demonstrated that the solution to the stochastic equation predicts about twice the growth rate of a large particle than the kinetic model. To validate solutions obtained by the stochastic algorithm, the convergence of the solution to Poisson distribution as time increases is studied. It is shown that the normalized average concentration obtained from the initial uniform and Pois¬son distributions in the stochastic coalescence model can be approximated by the Marshall-Palmer distribution function well known in the cloud physics community. The results of numerical simulations of the coalescence process using Brownian diffusion suggest that the kinetic equation in general produces an average size spec-trum that well matches the stochastic average spectrum. However, in the case of poorly mixed suspensions when correlations between particles are more important, these two models predict different size distributions, which is expected.

This study investigates and compares the effects of student characteristics-socioeconomic status, sex, family structure and immigration background- on 15-year-old mathematics achievement in Canada and Hong Kong through HLM. Using PISA data in 2003, the results showed that 20% and 49% of the variance in mathematics achievement was accounted for by schools in Canada and Hong Kong. respectively. All student-level variables were significant in Canada model except family structure whereas only sex and immigration background were significant in Hong Kong model. At school level, the significant school aggregate variables had much larger effects on school average mathematics achievement in Hong Kong than those in Canada. The findings suggest that school composition has an effect on mathematics achievement over and above that of individual characteristics.

We consider the semigroup of principal integral ideals, P. in a number field and study its associated Toeplitz representation. From this specific representation, a certain covariance relation is obtained and subsequently arbitrary isometric representations of P which satisfy this relation are analyzed. This leads to the study of the universal C*-algebra C*(P) satisfying these relations and to the following results. We first express C*(P) as a crossed product of an abelian C*-algebra by endomorphisms associated to P. We then give an explicit characterization of faithful representations of this crossed product, from which it follows as an immediate corollary that the Toeplitz C*-algebra is in fact isomorphic to the universal C*-algebra.

We study aspects of noncommutative geometry on hyperbolic dynamical systems known as Smale spaces. In particular, there are two C*-algebras, defined on the stable and unstable groupoids arising from the hyperbolic dynamics. These give rise to two additional crossed product C*-algebras known as the stable and unstable Ruelle algebras. We show that the Ruelle algebras exhibit noncommutative Poincaré duality. As a consequence we obtain isomorphisms between the K-theory and K-homology groups of the stable and unstable Ruelle algebras. A second result defines spectral triples on these C*-algebras and we show that the spectral dimension recovers the topological entropy of the Smale space itself. Finally we define a natural Fredholm module on the Ruelle algebras in the special case that the Smale space is a shift of finite type. Using unitary operators arising from the Pimsner-Voiculescu sequence we compute the index pairing with our Fredholm module for specific examples.

We construct geometric models for K-homology with coefficients based on the theory of Z/k-manifolds. To do so, we generalize the operations and relations Baum and Douglas put on spinc-manifolds to spinc Z/kZ-manifolds. We then de fine a model for K-homology with coefficients in Z/k using cycles of the form ((Q,P), (E,F), f) where (Q, P) is a spinc Z/k-manifold, (E, F) is a Z/k-vector bundle over (Q, P) and f is a continuous map from (Q, P) into the space whose K-homology we are modelling. Using results of Rosenberg and Schochet, we then construct an analytic model for K-homology with coefficients in Z/k and a natural map from our geometric model to this analytic model. We show that this map is an isomorphism in the case of finite CW-complexes. Finally, using direct limits, we produced geometric models for K-homology with coefficients in any countable abelian group.

Ergodic optimization is the study of which ergodic measures maximize the integral of a particular function. For sufficiently regular functions, e.g. Lipschitz/Holder continuous functions, it is conjectured that the set of functions optimized by measures supported on a periodic orbit is dense. Yuan and Hunt made great progress towards showing this for Lipschitz functions. This thesis presents clear proofs of Yuan and Hunt’s theorems in the case of the Shift as well as introducing a subset of Lipschitz functions, the super-continuous functions, where the set of functions optimized by measures supported on a periodic orbit is open and dense.

This research carries out a survival analysis for patients with breast cancer. The influence of clinical and pathologic features, as well as molecular markers on survival time are investigated. Special attention focuses on whether the molecular markers can provide additional information in helping predict clinical outcome and guide therapies for breast cancer patients. Three outcomes, breast cancer specific survival (BCSS), local relapse survival (LRS) and distant relapse survival (DRS), are examined using two datasets, the large dataset with missing values in markers (n=1575) and the small (complete) dataset consisting of patient records without any missing values (n=910). Results show that some molecular markers, such as YB1, could join ER, PR and HER2 to be integrated into cancer clinical practices. Further clinical research work is needed to identify the importance of CK56. The 10 year survival probability at the mean of all the covariates (clinical variables and markers) for BCSS, LRS, and DRS is 77%, 91%, and 72% respectively. Due to the presence of a large portion of missing values in the dataset, a sophisticated multiple imputation method is needed to estimate the missing values so that an unbiased and more reliable analysis can be achieved. In this study, three multiple imputation (MI) methods, data augmentation (DA), multivariate imputations by chained equations (MICE) and AREG, are employed and compared. Results shows that AREG is the preferred MI approach. The reliability of MI results are demonstrated using various techniques. This work will hopefully shed light on the determination of appropriate MI methods for other similar research situations.

The second-order least-squares estimator (SLSE), which was proposed by Wang (2003), is asymptotically more efficient than the least-squares estimator (LSE) if the third moment of the error distribution is nonzero. However, it is not robust against outliers. In this paper. we propose two robust second-order least-squares estimators (RSLSE) for linear regression models. RSLSE-I and RSLSE-II, where RSLSE-I is robust against X-outliers and RSLSE-II is robust. against X-outliers and Y-outliers. The basic idea is to choose proper weight matrices, which give a zero weight to an outlier. The RSLSEs are asymptotically normally distributed and are highly efficient with high breakdown point.. Moreover, we compare the RSLSEs with the LSE, the SLSE and the robust MM-estimator through simulation studies and real data examples. The results show that they perform very well and are competitive to other robust regression estimators.