To Örebro University

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio c = p/n -> infinity. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix F = S2-1S1-the product of a sample covariance matrix S-1 (from the independent variable array (be(Z(i))(1xn)) with the inverse of another covariance matrix S-2 (from the independent variable array (Z(j not equal i))(pxn))-are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components S-1 and S-2 of the F-matrix is not required.

Under a model having a Kronecker product covariance structure with compound symmetry, hypothesis testing for a correlation is investigated. Several tests are suggested and practical recommendations are made based on their type I error probabilities and powers.

Centers and vertices principal component analyses are common methods to explain variations within multivariate interval data. We introduce multivariate equicorrelated structures to vertices' covariance. Assuming the structure, we show equivalence between centers and vertices methods by proving their eigensystems proportional.

OBJECTIVES: The aim was to psychometrically test and evaluate the Swedish functional health literacy scale and the Swedish communicative and critical health literacy scale in patients undergoing bariatric surgery.

DESIGN: A prospective cross-sectional psychometric study.

SETTING: Patients from three bariatric centres in Sweden were consecutively included in this study.

PARTICIPANTS: A total of 704 patients undergoing bariatric surgery filled in the questionnaires preoperatively. Inclusion criteria were scheduled for primary bariatric surgery (Roux-en-Y gastric bypass or sleeve gastrectomy) and greater than 17 years, proficiency in Swedish.

PRIMARY AND SECONDARY MEASURES: Psychometric outcomes of the Swedish Functional Health Literacy scale and the Swedish Communicative and Critical Health Literacy scale.

RESULTS: There was a higher proportion of females (74.4%, n=523) to males (25.6%, n=180). The mean age was 42 years (SD 11.5). Limited functional health literacy and limited communicative and critical health literacy (including both inadequate and problematic health literacy) was reported in 55% (n=390) and 40% (n=285), respectively. Cronbach alpha for the Swedish Functional Health Literacy scale was α=0.86 and for the Swedish Communicative and Critical Health Literacy scale, α=0.87. Construct validity showed weak to negative correlations between the Swedish Functional Health Literacy scale and income, education and SF-36/RAND36 summary scores. Confirmatory factor analysis showed a one-factor solution for the Swedish Functional Health Literacy scale and a two-factor solution for the Swedish Communicative and Critical Health Literacy scale.

CONCLUSIONS: The Swedish Functional Health Literacy scale and the Swedish Communicative and Critical Health Literacy scale are valid and reliable to use for patients undergoing bariatric surgery in a Swedish context. Measuring dimensions of health literacy can be used as a guide for the development of health literacy friendly patient information in patients undergoing bariatric surgery.

This thesis concerns the study of multilevel models with specific patterned covariance structures and addresses the issues of maximum likelihoodestimation. In particular, circular symmetric hierarchical datastructures are considered.

In the first paper (Paper I), we consider so-called dihedral block symmetrymodels and extend them to models which covariance structures reflects both circularity and exchangeability present in the data. The main contribution of Paper I are two derived patterns of the covariancematrices which characterizes models under consideration. The relationship between these two patterned covariance matrices was investigatedand it has been verified they are similar matrices. New expressions for the eigenvalues of block circular symmetric matrices are obtained which take into account the block structure. Paper II deals with estimation ofbalanced multilevel models with block circular symmetric covariance matrices. The spectral properties of such patterned covariance matrices are established. Maximum likelihood estimation is considered through the spectral decomposition of the patterned covariance matrix. The main results of Paper II concern the spectrum of the covariance matrix inthe model of interest and the existence of explicit maximum likelihood estimators for the covariance parameters.

This thesis concerns inference problems in balanced random effects models with a so-called block circular Toeplitz covariance structure. This class of covariance structures describes the dependency of some specific multivariate two-level data when both compound symmetry and circular symmetry appear simultaneously.

We derive two covariance structures under two different invariance restrictions. The obtained covariance structures reflect both circularity and exchangeability present in the data. In particular, estimation in the balanced random effects with block circular covariance matrices is considered. The spectral properties of such patterned covariance matrices are provided. Maximum likelihood estimation is performed through the spectral decomposition of the patterned covariance matrices. Existence of the explicit maximum likelihood estimators is discussed and sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and the corresponding maximum likelihood estimators are presented.

This thesis also deals with hypothesis testing of block covariance structures, especially block circular Toeplitz covariance matrices. We consider both so-called external tests and internal tests. In the external tests, various hypotheses about testing block covariance structures, as well as mean structures, are considered, and the internal tests are concerned with testing specific covariance parameters given the block circular Toeplitz structure. Likelihood ratio tests are constructed, and the null distributions of the corresponding test statistics are derived.

In this article, we address the problem of simultaneous testing hypothesis about mean and covariance matrix for repeated measures data when both the mean vector and covariance matrix are patterned. In particular, tests about the mean vector under block circular and doubly exchangeable covariance structures have been considered. The null distributions are established for the corresponding likelihood ratio test statistics and expressions for the exact or near-exact probability density and cumulative distribution functions are obtained. The application of the results is illustrated by both a simulation study and a real-life data example.

Hierarchical linear models with a block circular covariance structure are considered. Sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and maximum likelihood estimators are presented. The theory is illustrated through covariance matrices of small sizes and a real-life example.

Models that capture symmetries present in the data have been widely used in different applications, with early examples from psychometric and medical research. The aim of this article is to study a random effects model focusing on the covariance structure that is block circular symmetric. Useful results are obtained for the spectra of these structured matrices.

In this article, the results concerning hypothesis testing in multivariate normal models with block circular covariance structures are obtained. Hypotheses about a general block structure of the covariance matrix and specific covariance parameters have been of main interest. In addition, the tests about patterned mean vectors have been considered.The corresponding likelihood ratio statistics are derived and their null distributions are studied.

In this article we consider a multilevel model with block circular symmetric covariance structure. Maximum likelihood estimation of the parameters of this model is discussed. We show that explicit maximum likelihood estimators of variance components exist under certain restrictions on the parameter space.

Models that describe symmetries present in the error structure of observations have been widely used in dierent applications, with early examples from psychometric and medical research. The aim of this article is to study a multilevel model with a covariance structure that is block circular symmetric. Useful results are obtained for the spectra of these structured matrices.

In this article we consider a multilevel model with block circular symmetric covariance structure. Maximum likelihood estimation of the parameters of this model is discussed. We show that explicit maximum likelihood estimators of variance components exist under certain restrictions on the parameter space.