<p>Network meta-analysis (NMA) is a process by which several treatments can be simultaneously compared for relative effectiveness. When conducted in a Bayesian framework, the probability that each treatment is ranked 1st, 2nd and so on can be calculated. A square matrix of these probabilities, referred to as the rank probability matrix, can be structured with rows representing treatments and columns representing ranks. In this thesis, a simulation study was conducted to explore properties of five proposed rank probability matrix summary measures: determinant, Frobenius norm, trace, diagonal maximum and diagonal minimum. Each measure is standardized to approach 1 for absolute certainty. The goal of this simulation is to identify strengths and weaknesses of these measures for varying networks. The measures are applied to previously published NMA data for further investigation. The simulation study and real data analysis revealed pros and cons of each summary measure; the Frobenius norm was found most effective. All summary measures yielded higher values with increases in symmetry, relative effect size and number of studies in the network. If the rank probability matrix is used as the primary output of a network meta-analysis (as is often the case), a simple measure of the overall confidence in the rankings is beneficial. Future research will require exploration into the distributions of these measures.</p> / Master of Science (MSc)

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This senior thesis project explores and generalizes some fundamental machine learning algorithms from the Euclidean space to the statistical manifold, an abstract space in which each point is a probability distribution. In this thesis, we adapt the optimal separating hyperplane, the k-means clustering method, and the hierarchical clustering method for classifying and clustering probability distributions. In these modifications, we use the statistical distances as a measure of the dissimilarity between objects. We describe a situation where the clustering of probability distributions is needed and useful. We present many interesting and promising empirical clustering results, which demonstrate the statistical-distance-based clustering algorithms often outperform the same algorithms with the Euclidean distance in many complex scenarios. In particular, we apply our statistical-distance-based hierarchical and k-means clustering algorithms to the univariate normal distributions with k = 2 and k = 3 clusters, the bivariate normal distributions with diagonal covariance matrix and k = 3 clusters, and the discrete Poisson distributions with k = 3 clusters. Finally, we prove the k-means clustering algorithm applied on the discrete distributions with the Hellinger distance converges not only to the partial optimal solution but also to the local minimum.

The current generation of structural equation modeling (SEM) is loosely split in two divergent groups - covariance-based and variance-based structural equation modeling. The relative newness of variance-based SEM has limited the development of techniques that extend its applicability to non-metric data. This study focuses upon the extension of general linear model techniques within the variance-based platform of partial least squares structural equation modeling (PLS-SEM). This modeling procedure receives it name through the iterative PLS‑SEM algorithm's estimates of the coefficients for the partial ordinary least squares regression models in both the measurement model and the overall structural model. This research addresses the following research questions: (1) What are the appropriate measures for data segmentation within PLS‑SEM? (2) What are the appropriate steps for the analysis of rank-ordered path coefficients within PLS‑SEM? and (3) What is an appropriate model selection index for PLS‑SEM? The limited type of data to which PLS-SEM is applicable suggests an opportunity to extend the method for use with different data and as a result a broader number of applications. This study develops and tests several methodologies that are prevalent in the general linear model (GLM). The proposed data segmentation approaches posited and tested through post hoc analysis of structural model. Monte Carlo simulation allows demonstrating the improvement of the proposed model fit indices in comparison to the established indices found within the SEM literature. These posited PLS methods, that are logical transfers of GLM methods, are tested using examples. These tests enable demonstrating the methods and recommending reporting requirements.

<p>Statistical models often rely on several assumptions including distributional assumptions on outcome variables and relational assumptions where we model the relationship between outcomes and independent variables. Further assumptions are also made depending on the complexity of the data and the model being used. Model diagnostics is, therefore, a crucial component of any model fitting problem. Residuals play important roles in model diagnostics. Residuals are not only used to check adequacy of model fit, but they also are excellent tools to validate model assumptions as well as identify outliers and influential observations. Residuals in univariate models are studied extensively and are routinely used for model diagnostics. In multivariate models residuals are not commonly used to assess model fit, although a few approaches have been proposed to check multivariate normality. However, in the analysis of longitudinal data, the resulting residuals are correlated and are not normally distributed. It is, therefore, not clear as to how ordinary residuals can be used for model diagnostics. Under sufficiently large sample size, a transformation of ordinary residuals are proposed to check the normality assumption. The transformation is based solely on removing correlation among the residuals. However, we show that these transformed residuals fail in the presence of model mis-specification. In this thesis, we investigate residuals in the analysis of longitudinal data. We consider ordinary residuals, Fitzmaurice’s transformed (uncorrelated) residuals as well as von Rosen’s decomposed residuals. Using simulation studies, we show how the residuals behave under multivariate normality and when this assumption is violated. We also investigate their properties under correct fitting as well as wrongly fitted models. Finally, we propose new residuals by transforming von Rosen’s decomposed residuals. We show that these residuals perform better than Fitzmourice’s transformed residuals in the presence of model mis-specification. We illustrate our approach using two real data sets.</p> / Master of Science (MSc)

It is important to be able to compare variances efficiently and accurately regardless of the parent populations. This study proposes a new right tailed test for the ratio of two variances using the Edgeworth’s expansion. To study the Type I error rate and Power performance, simulation was performed on the new test with various combinations of symmetric and skewed distributions. It is found to have more controlled Type I error rates than the existing tests. Additionally, it also has sufficient power. Therefore, the newly derived test provides a good robust alternative to the already existing methods.

One of the most important decisions to make when performing an exploratory factor or principal component analysis regards the number of factors to retain. Parallel analysis is considered to be the best course of action in these circumstances as it consistently outperforms other factor extraction methods (Zwick & Velicer, 1986). Even so, parallel analysis could benefit from further research and refinement to improve its accuracy. Characteristics such as factor loadings, correlations between factors, and number of variables per factor all have been shown to adversely impact the effectiveness of parallel analysis as a means of identifying the number of factors (Pearson, Mundfrom, & Piccone, 2013). Critically, even the choice of criteria on which to evaluate factors (such as the eigenvalue at the 50th or 95th percentile) can have deleterious effects on the number of factors extracted (Peres-Neto, Jackson, & Somers, 2004). One area of parallel analysis yet to be researched is the magnitude of the difference between the actual eigenvalue and the random data-based eigenvalue. Currently, even if the margin between the actual eigenvalue and the random data-based eigenvalue is nominal, the factor is considered to be meaningful. As such, it may behoove researchers to enforce a higher standard, such as a greater margin between the two eigenvalues than just a simple difference. Accordingly, the purpose of this study was to evaluate the efficacy of a 10% margin criterion as compared to an absolute margin. These margins were evaluated in conjunction with the 50th, 90th, 95th, and 99th percentile eigenvalue criteria on a population correlation matrix designed to engender underextraction. Previous research (Matsumoto & Brown, 2017) explored the same conditions on a population correlation matrix designed to elicit overextraction. They found that the most stringent standard (99th percentile eigenvalue plus a 10% margin) was the most accurate. For the present study however, it was hypothesized that the most accurate results would be obtained from a standard less stringent than the 99th percentile eigenvalue plus a 10% margin. The results suggest that when a correlation matrix has properties which may illicit underextraction, the use of less stringent criteria may lead to greater accuracy in identifying the number of factors and that the incorporation of an additional margin criterion may not improve the accuracy of the analysis.

The purpose of this thesis is to propose some test statistics for testing the skewness and kurtosis parameters of a distribution, not limited to a normal distribution. Since a theoretical comparison is not possible, a simulation study has been conducted to compare the performance of the test statistics. We have compared both parametric methods (classical method with normality assumption) and non-parametric methods (bootstrap in Bias Corrected Standard Method, Efron’s Percentile Method, Hall’s Percentile Method and Bias Corrected Percentile Method). Our simulation results for testing the skewness parameter indicate that the power of the tests differs significantly across sample sizes, the choice of alternative hypotheses and methods we chose. For testing the kurtosis parameter, the simulation results suggested that the classical method performs well when the data are from both normal and beta distributions and bootstrap methods are useful for uniform distribution especially when the sample size is large.

Syncope is a common Emergency Department (ED) presentation. An important proportion of syncope patients are at risk of developing serious adverse events (SAEs), such as deaths or arrhythmias following ED disposition. Currently, no clinically-useful decision tool exists to reliably identify high-risk patients. This study derived a clinical decision tool to identify syncope patients at risk of developing SAEs after ED disposition. This study also examined key methodological considerations involved in deriving decision tools by comparing two different methodological approaches: a traditional and modern approach. The traditional approach led to an eight-variable decision tool that allowed simple clinical interpretation and use. The modern approach, which aims to avoid data-driven methodology and statistical overfitting, was used to derive a ten-variable decision tool. Both decision tools displayed acceptable and comparable performance in internal validation studies (c-statistic 0.87, 95% confidence interval 0.84-0.89). A future external validation study is required to comprehensively compare the methods.

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