Multi-State-Markov (MSM) models can be used to characterize the behaviour of categorical outcomes measured repeatedly over time. Kalbfleisch and Lawless (1985) and Gentleman et al. (1994) examine the MSM model under the assumption of time-homogeneous transition intensities. In the context of non-homogeneous intensities, current methods use piecewise constant approximations which are less than ideal. We propose a local likelihood method, based on Tibshirani and Hastie (1987) and Loader (1996), to estimate the transition intensities as continuous functions of time. In particular the local EM algorithm suggested by Betensky et al. (1999) is employed to estimate the in-homogeneous intensities in the presence of missing data. A simulation comparing the piecewise constant method with the local EM method is examined using two different sets of underlying intensities. In addition, model assessment tools such as bandwidth selection, grid size selection, and bootstrapped percentile intervals are examined. Lastly, the method is applied to an HIV data set to examine the intensities with regard to depression scores. Although computationally intensive, it appears that this method is viable for estimating non-homogeneous intensities and outperforms existing methods.

Multi-State-Markov (MSM) models can be used to characterize the behaviour of categorical outcomes measured repeatedly over time. Kalbfleisch and Lawless (1985) and Gentleman et al. (1994) examine the MSM model under the assumption of time-homogeneous transition intensities. In the context of non-homogeneous intensities, current methods use piecewise constant approximations which are less than ideal. We propose a local likelihood method, based on Tibshirani and Hastie (1987) and Loader (1996), to estimate the transition intensities as continuous functions of time. In particular the local EM algorithm suggested by Betensky et al. (1999) is employed to estimate the in-homogeneous intensities in the presence of missing data. A simulation comparing the piecewise constant method with the local EM method is examined using two different sets of underlying intensities. In addition, model assessment tools such as bandwidth selection, grid size selection, and bootstrapped percentile intervals are examined. Lastly, the method is applied to an HIV data set to examine the intensities with regard to depression scores. Although computationally intensive, it appears that this method is viable for estimating non-homogeneous intensities and outperforms existing methods.

Biclustering is a technique for clustering rows and columns of a data matrix simultaneously. Over the past few years, we have seen its applications in biology-related fields, as well as in many data mining projects. As opposed to classical clustering methods, biclustering groups objects that are similar only on a subset of variables. Many biclustering algorithms on continuous data have emerged over the last decade. In this dissertation, we will focus on two Bayesian biclustering algorithms we developed for discrete data, more specifically categorical data and ordinal data. / Statistics

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Data from rating scale assessments have rank-invariant properties only, which means that the data represent an ordering, but lack of standardized magnitude, inter-categorical distances, and linearity. Even though the judgments often are coded by natural numbers they are not really metric. The aim of this thesis is to further develop the nonparametric rank-based Svensson methods for paired ordinal data that are based on the rank-invariant properties only. The thesis consists of five papers. In Paper I the asymptotic properties of the measure of systematic disagreement in paired ordinal data, the Relative Position (RP), and the difference in RP between groups were studied. Based on the findings of asymptotic normality, two tests for analyses of change within group and between groups were proposed. In Paper II the asymptotic properties of rank-based measures, e.g. the Svensson’s measures of systematic disagreement and of additional individual variability were discussed, and a numerical method for approximation was suggested. In Paper III the asymptotic properties of the measures for paired ordinal data, discussed in Paper II, were verified by simulations. Furthermore, the Spearman rank-order correlation coefficient (rs) and the Svensson’s augmented rank-order agreement coefficient (ra) were compared. By demonstrating how they differ and why they differ, it is emphasized that they measure different things. In Paper IV the proposed test in Paper I for comparing two groups of systematic changes in paired ordinal data was compared with other nonparametric tests for group changes, both regarding different approaches of categorising changes. The simulation reveals that the proposed test works better for small and unbalanced samples. Paper V demonstrates that rank invariant approaches can also be used in analysis of ordinal data from multi-item scales, which is an appealing and appropriate alternative to calculating sum scores.

This thesis which consists of four papers is concerned with estimation methods in factor analysis and structural equation models. New estimation methods are proposed and investigated. In paper I an approximation of the penalized maximum likelihood (ML) is introduced to fit an exploratory factor analysis model. Approximated penalized ML continuously and efficiently shrinks the factor loadings towards zero. It naturally factorizes a covariance matrix or a correlation matrix. It is also applicable to an orthogonal or an oblique structure. Paper II, a simulation study, investigates the properties of approximated penalized ML with an orthogonal factor model. Different combinations of penalty terms and tuning parameter selection methods are examined. Differences in factorizing a covariance matrix and factorizing a correlation matrix are also explored. It is shown that the approximated penalized ML frequently improves the traditional estimation-rotation procedure. In Paper III we focus on pseudo ML for multi-group data. Data from different groups are pooled and normal theory is used to fit the model. It is shown that pseudo ML produces consistent estimators of factor loadings and that it is numerically easier than multi-group ML. In addition, normal theory is not applicable to estimate standard errors. A sandwich-type estimator of standard errors is derived. Paper IV examines properties of the recently proposed polychoric instrumental variable (PIV) estimators for ordinal data through a simulation study. PIV is compared with conventional estimation methods (unweighted least squares and diagonally weighted least squares). PIV produces accurate estimates of factor loadings and factor covariances in the correctly specified confirmatory factor analysis model and accurate estimates of loadings and coefficient matrices in the correctly specified structure equation model. If the model is misspecified, robustness of PIV depends on model complexity, underlying distribution, and instrumental variables.

abstract: Mixture experiments are useful when the interest is in determining how changes in the proportion of an experimental component affects the response. This research focuses on the modeling and design of mixture experiments when the response is categorical namely, binary and ordinal. Data from mixture experiments is characterized by the perfect collinearity of the experimental components, resulting in model matrices that are singular and inestimable under likelihood estimation procedures. To alleviate problems with estimation, this research proposes the reparameterization of two nonlinear models for ordinal data -- the proportional-odds model with a logistic link and the stereotype model. A study involving subjective ordinal responses from a mixture experiment demonstrates that the stereotype model reveals useful information about the relationship between mixture components and the ordinality of the response, which the proportional-odds fails to detect. The second half of this research deals with the construction of exact D-optimal designs for binary and ordinal responses. For both types, the base models fall under the class of Generalized Linear Models (GLMs) with a logistic link. First, the properties of the exact D-optimal mixture designs for binary responses are investigated. It will be shown that standard mixture designs and designs proposed for normal-theory responses are poor surrogates for the true D-optimal designs. In contrast with the D-optimal designs for normal-theory responses which locate support points at the boundaries of the mixture region, exact D-optimal designs for GLMs tend to locate support points at regions of uncertainties. Alternate D-optimal designs for binary responses with high D-efficiencies are proposed by utilizing information about these regions. The Mixture Exchange Algorithm (MEA), a search heuristic tailored to the construction of efficient mixture designs with GLM-type responses, is proposed. MEA introduces a new and efficient updating formula that lessens the computational expense of calculating the D-criterion for multi-categorical response systems, such as ordinal response models. MEA computationally outperforms comparable search heuristics by several orders of magnitude. Further, its computational expense increases at a slower rate of growth with increasing problem size. Finally, local and robust D-optimal designs for ordinal-response mixture systems are constructed using MEA, investigated, and shown to have high D-efficiency performance. / Dissertation/Thesis / Doctoral Dissertation Industrial Engineering 2016

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In social science research, data is often collected using questionnaires with Likert scales, resulting in ordinal data. Confirmatory factor analysis (CFA) is the most common type of analysis, which assumes continuous data and multivariate normality, the assumptions violated for ordinal data. Simulation studies have shown that Robust Maximum Likelihood (RML) works well when the normality assumption is violated. Diagonally Weighted Least Squares (DWLS) estimation is especially recommended for categorical data. Bayesian estimation (BE) methods are also potentially effective for ordinal data. The current study employs a CFA model and Monte Carlo simulation to evaluate the performance of three estimation methods with ordinal data under various conditions in terms of the levels of asymmetry, sample sizes, and number of categories. The results indicate that, for ordinal data, DWLS outperforms RML and BE. RML is effective for ordinal data when the category numbers are sufficiently large. Bayesian methods do not demonstrate a significant advantage with different values of factor loadings, and category distributions had minimal impact on the estimation results.

One of the most frequently encountered types of data is where the response variables are measured on an ordinal scale. Although there have been substantial developments in the statistical techniques for the analysis of ordinal data, methods appropriate for repeatedly assessed ordinal data collected from field experiments are limited. A series of biennial field screening trials for evaluating cultivar resistance of potato to the disease, late blight, caused by the fungus Phytophthora infestans (Mont.) de Bary has been conducted by the New Zealand Institute of Crop and Food Research since 1983. In each trial, the progression of late blight was visually assessed several times during the planting season using a nine-point ordinal scale based on the percentage of necrotic tissues. As for many other agricultural field experiments, spatial differences between the experimental units is one of the major concerns in the analysis of data from the potato late blight trial. The aim of this thesis is to construct a statistical model which can be used to analyse the data collected from the series of potato late blight trials. We review existing methodologies for analysing ordinal data with mixed effects particularly those methods in the Bayesian framework. Using data collected from the potato late blight trials we develop a Bayesian hierarchical model for the analyses of repeatedly assessed ordinal scores with spatial effects, in particular the time dependence of the scores assessed on the same experimental units was modelled by a sigmoid logistic curve. Data collected from the potato late blight trials demonstrated the importance of spatial effects in agricultural field trials. These effects cannot be neglected when analysing such data. Although statistical methods can be refined to account for the complexity of the data, appropriate trial design still plays a central role in field experiments. / Accompanying dataset is at http://hdl.handle.net/2292/5240