Models for mediation are widely used in psychology, behavior science and education because they help researchers understand how a causal effect happens through one or several mediating variables. And more complex mediation models that incorporate multiple mediators are increasingly being assessed. This report uses a generated dataset to provide an overview of the assessment of direct effects and indirect effects in multiple mediator models. Use of a multiple comparison-based procedure for testing a set of hypotheses simultaneously while controlling the experiment-wise type I error rate is used to calculate a confidence interval for each pairwise contrast of mediated effects. Three approaches will be used to test hypotheses concerning the contrast between pairs of mediator effects. These approaches include 1) an assumption of zero covariance between parameters from different models, 2) assumption of a non-zero covariance between parameters from different models and 3) use of bootstrapping. Results are provided and discussed. / text

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In this thesis, pairwise multiple comparisons and multiple comparisons with a control are studied when the observations have short-tailed symmetric distributions. Under non-normality, the testing procedure is given and Huber estimators, trimmed mean with winsorized standard deviation, modified maximum likelihood estimators and ordinary sample mean and sample variance used in this procedure are reviewed. Finally, robustness properties of the stated estimators are compared with each other and it is shown that the test based on the modified maximum likelihood estimators has better robustness properties under short-tailed symmetric distribution.

The problem of this study was to determine the differences in experimentwise and comparisonwise Type I error rate among six multiple comparison procedures when applied to twenty-eight combinations of normally distributed data. These were the Least Significant Difference, the Fisher-protected Least Significant Difference, the Student Newman-Keuls Test, the Duncan Multiple Range Test, the Tukey Honestly Significant Difference, and the Scheffe Significant Difference. The Spjøtvoll-Stoline and Tukey—Kramer HSD modifications were used for unequal n conditions. A Monte Carlo simulation was used for twenty-eight combinations of k and n. The scores were normally distributed (µ=100; σ=10). Specified multiple comparison procedures were applied under two conditions: (a) all experiments and (b) experiments in which the F-ratio was significant (0.05). Error counts were maintained over 1000 repetitions. The FLSD held experimentwise Type I error rate to nominal alpha for the complete null hypothesis. The FLSD was more sensitive to sample mean differences than the HSD while protecting against experimentwise error. The unprotected LSD was the only procedure to yield comparisonwise Type I error rate at nominal alpha. The SNK and MRT error rates fell between the FLSD and HSD rates. The SSD error rate was the most conservative. Use of the harmonic mean of the two unequal sample n's (HSD-TK) yielded uniformly better results than use of the minimum n (HSD-SS). Bernhardson's formulas controlled the experimentwise Type I error rate of the LSD and MRT to nominal alpha, but pushed the HSD below the 0.95 confidence interval. Use of the unprotected HSD produced fewer significant departures from nominal alpha. The formulas had no effect on the SSD.

Master of Science / Department of Statistics / Haiyan Wang / The selection of relevant genes for classification of phenotypes for diseases with gene expression data have been extensively studied. Previously, most relevant gene selection was conducted on individual gene with limited sample size. Modern technology makes it possible to obtain microarray data with higher resolution of the chromosomes. Considering gene sets on an entire block of a chromosome rather than individual gene could help to reveal important connection of relevant genes with the disease phenotypes. In this report, we consider feature selection and classification while taking into account of the spatial location of probe sets in classification of Duke’s stages B and C using DNA copy number data or gene expression data from colorectal cancers. A novel method was presented for feature selection in this report. A chromosome was first partitioned into blocks after the probe sets were aligned along their chromosome locations. Then a test of interaction between Duke’s stage and probe sets was conducted on each block of probe sets to select significant blocks. For each significant block, a new multiple comparison procedure was carried out to identify truly relevant probe sets while preserving the neighborhood location information of the probe sets. Support Vector Machine (SVM) and K-Nearest Neighbor (KNN) classification using the selected final probe sets was conducted for all samples. Leave-One-Out Cross Validation (LOOCV) estimate of accuracy is reported as an evaluation of selected features. We applied the method on two large data sets, each containing more than 50,000 features. Excellent classification accuracy was achieved by the proposed procedure along with SVM or KNN for both data sets even though classification of prognosis stages (Duke’s stages B and C) is much more difficult than that for the normal or tumor types.

In this dissertation, several essays in the field of bio-security are presented. The estimation of the probability of an FMD outbreak by type and location of premises is important for decision making. In Essay I, we estimate and predict the probability/risk of an FMD outbreak spreading to the various premises in the study area. We first used a Poisson regression model with adjustment dispersion associated with random simulation results from the AusSpead model to estimate the parameters of the model. Our estimation and prediction show that large cattle loss could be concentrated in three counties-Deaf Smith, Parmer, and Castro. These results are based on approximately 70% of the feedlots with over 10,000 cattle located in the three counties previously mentioned. In Essay II, our objective is to determine the best mitigation strategies in minimizing animal loss based on AusSpead simulation model. We tested 15 mitigation strategies by using multiple comparison. The results show that the best mitigation strategies for all four scenarios are regular surveillance, slaughter of the infected animals, and early detection. We then used the Mixed Integer Programming to estimate costs of disposing of animal carcasses and transportation. Results show that the unit disposal cost will vary with carcass scale and the unit transportation cost also varies with the distribution of the infected premises and disposal locations. FMD seems to have varying impacts on equity markets. In Essay III, we studied returns at three different levels of the stock market. We determined results in a structural break, and then estimated the impact of the announcement of confirmed cases of FMD disease on the volatility of stock market returns by using a GARCH-Mean model. Our results show that the structure break occurs on the day with the largest number of confirmed cases for meat product firms rather than the day of the first confirmed case. We found that the conditional volatilities over the FMD period are higher than those over the sample period. The announcement of confirmed cases had the largest marginal impact on meat products. Investors may always consider maintaining a portfolio consisting of index funds or hedge funds.

The central theme of this thesis is the construction of simultaneous confidence regions (SCR) corresponding to stepwise multiple comparison procedures (MCP). The first chapter is devoted to the theory of multiple comparisons, including the class of closed testing procedures which contains every MCP that strongly con- trols the familywise error rate. The second chapter is concerned with the gene- ral principle of construction of SCR corresponding to closed testing procedures. These general results are used in the third and the forth chapter for deriving the SCR corresponding to a subclass of closed testing procedures which are based on weighted Bonferroni tests. The SCR corresponding to the Holm, the Holm(W), the fixed-sequence and the fallback MCP are derived explicitly. The theoretical results are numerically illustrated on a bioequivalence study. In the fifth chapter we briefly discuss the SCR corresponding to the Hommel, the Hochberg and the step-down Dunnett MCP.

O presente trabalho teve por objetivo fazer uma reflexão acerca do uso de testes de comparações múltiplas e da análise de regressão no estudo de experimentos unifatoriais cujos tratamentos são níveis de um fator quantitativo, para comparar os resultados e informações que são trazidas por cada uma dessas análises, verificando suas eventuais vantagens e limitações. De acordo com os objetivos propostos pelo presente trabalho, foi feita, depois de realizada a revisão bibliográfica sobre a análise de regressão e alguns dos testes de comparação de médias, um levantamento acerca de artigos cujo objetivo principal era o de fazer uma investigação de trabalhos publicados em jornais, revistas ou periódicos nos quais se utilizou algum procedimento de comparação de médias verificando assim a adequação desses testes às análises estatísticas realizadas. Essa revisão demonstrou que um número significativo de pesquisadores utiliza de procedimentos de comparações múltiplas em análises estatísticas de experimentos unifatoriais nos quais os tratamentos envolvidos são níveis de um fator quantitativo, o que é considerado por alguns como um procedimento inadequado. Assim sendo, foram analisados também dados de experimentos unifatoriais com tratamentos dessa ordem, que foram submetidos a uma análise de regressão e também a um procedimento de comparação múltipla das médias, com o objetivo de verificar quais as vantagens e limitações de cada um desses procedimentos na análise do experimento em questão. Nessa comparação ficou claro que o uso de procedimentos de comparações múltiplas na análise de experimentos unifatoriais envolvendo tratamentos quantitativos pode resultar na redução de informações e também da eficiência dos resultados, quando procedimentos mais apropriados, nesse caso, a análise de regressão, estão disponíveis para analisar dados dessa natureza. / The present work had like purpose to make a reflection about the use of multiple comparison tests and of the regression analysis on learning of unifactorial experiments whose treatments are levels of a quantitative factor, to compare the results and information are brought for each one of the analysis, verifying the eventual advantages and limitations of them. According to the purposes of the present work, was realized, later the bibliographical revision about regression analysis and some of the mean comparison tests was done, a survey about articles whose principal aim was to make a raising of works published at newspapers, magazines or periodicals where was used some mean comparison procedure verifying the adaptation of these tests to the statistical analysis realized. This revision demonstrated that a revealing number of searchers use multiple comparison procedures at analysis of unifactorial experiments whose treatments involved are levels of a quantitative factor, what is considered for some searchers like an inadequate procedure. Of this way, the data of unifactorial experiments, whose treatments were levels of a quantitative factor, were analyzed too, that were submitted to a regression analysis and to a multiple comparison procedure, with the aim of verifying the advantages and limitations of each one of these procedures at the analysis of the experiment. At this comparison, was clear that the use of multiple comparison procedures at analysis of experiments involving quantitative experiments can result in loss of information and reduced efficiency of the results, when more appropriate procedures, in this case, the regression analysis, are available to analyze this kind of data.

O presente trabalho teve por objetivo fazer uma reflexão acerca do uso de testes de comparações múltiplas e da análise de regressão no estudo de experimentos unifatoriais cujos tratamentos são níveis de um fator quantitativo, para comparar os resultados e informações que são trazidas por cada uma dessas análises, verificando suas eventuais vantagens e limitações. De acordo com os objetivos propostos pelo presente trabalho, foi feita, depois de realizada a revisão bibliográfica sobre a análise de regressão e alguns dos testes de comparação de médias, um levantamento acerca de artigos cujo objetivo principal era o de fazer uma investigação de trabalhos publicados em jornais, revistas ou periódicos nos quais se utilizou algum procedimento de comparação de médias verificando assim a adequação desses testes às análises estatísticas realizadas. Essa revisão demonstrou que um número significativo de pesquisadores utiliza de procedimentos de comparações múltiplas em análises estatísticas de experimentos unifatoriais nos quais os tratamentos envolvidos são níveis de um fator quantitativo, o que é considerado por alguns como um procedimento inadequado. Assim sendo, foram analisados também dados de experimentos unifatoriais com tratamentos dessa ordem, que foram submetidos a uma análise de regressão e também a um procedimento de comparação múltipla das médias, com o objetivo de verificar quais as vantagens e limitações de cada um desses procedimentos na análise do experimento em questão. Nessa comparação ficou claro que o uso de procedimentos de comparações múltiplas na análise de experimentos unifatoriais envolvendo tratamentos quantitativos pode resultar na redução de informações e também da eficiência dos resultados, quando procedimentos mais apropriados, nesse caso, a análise de regressão, estão disponíveis para analisar dados dessa natureza. / The present work had like purpose to make a reflection about the use of multiple comparison tests and of the regression analysis on learning of unifactorial experiments whose treatments are levels of a quantitative factor, to compare the results and information are brought for each one of the analysis, verifying the eventual advantages and limitations of them. According to the purposes of the present work, was realized, later the bibliographical revision about regression analysis and some of the mean comparison tests was done, a survey about articles whose principal aim was to make a raising of works published at newspapers, magazines or periodicals where was used some mean comparison procedure verifying the adaptation of these tests to the statistical analysis realized. This revision demonstrated that a revealing number of searchers use multiple comparison procedures at analysis of unifactorial experiments whose treatments involved are levels of a quantitative factor, what is considered for some searchers like an inadequate procedure. Of this way, the data of unifactorial experiments, whose treatments were levels of a quantitative factor, were analyzed too, that were submitted to a regression analysis and to a multiple comparison procedure, with the aim of verifying the advantages and limitations of each one of these procedures at the analysis of the experiment. At this comparison, was clear that the use of multiple comparison procedures at analysis of experiments involving quantitative experiments can result in loss of information and reduced efficiency of the results, when more appropriate procedures, in this case, the regression analysis, are available to analyze this kind of data.

同時比較多個樣本間的差異，可用ANOVA來檢定，但ANOVA只能得到樣本間有差異的訊息，無法明確指出是哪些樣本間有差異，需要使用多重比較找出樣本間的差異。本文主要探討相關的離散型資料的多重比較，以型I誤差與檢定力兩指標找出最適的多重比較法。本文依序探討獨立的連續型資料、相關的連續型資料、獨立的離散型資料、相關的離散型資料，並針對相關型的資料提出修正法。綜合型I誤差與檢定力兩指標來看，在樣本間的平均差異小時，Shaffer’s first procedure Test (1986)、Procedure 4 by Bergmann and Hommel (1988)為兩兩比較下較佳的修正法，Hochberg Test (1988)為多對ㄧ比較下較佳的修正法；樣本間平均差異大時，Bonferroni 為兩兩比較下較佳的修正法，Hochberg (1988)、Simes (1986)為多對ㄧ比較下較佳的修正法。 / Analysis of variance (ANOVA) is usually applied to check whether there are differences among more than two treatments. However, even there are differences, multiple comparison procedures are still needed to determine which pair(s) of treatments are different. In this study, we use simulation to compare the frequently used multiple comparison procedures, including many-to-one and pair-wise, and type-I error and power are used to measure the performance of procedures. Two types of data were considered, independently and correlated distributed data. If the differences among treatments are small, Shaffer’s first procedure test (1986) and Procedure 4 by Bergmann and Hommel (1988) are the best in pair-wise case, and Hochberg test (1988) is the best in many-to-one case. If the differences among treatments are large, the Bonferroni procedure is the best in pair-wise case, and the procedures by Hochberg (1988) and Simes (1986) are the best in many-to-one case.