Thesis (M.S.)--West Virginia University, 2006. / Title from document title page. Document formatted into pages; contains viii, 45 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 44-45).

Thesis (Ph.D.)--University of Toledo, 2005. / Typescript. "A dissertation [submitted] as partial fulfillment of the requirements of the Doctor of Philosophy degree in Mathematics." Bibliography: leaves 100-105.

Thesis (M. Phil.)--University of Hong Kong, 2005. / Title proper from title frame. Also available in printed format.

Judging goodness of fit in multidimensional scaling requires a comprehensive set of diagnostic tools instead of relying on stress rules of thumb. This article elaborates on corresponding strategies and gives practical guidelines for researchers to obtain a clear picture of the goodness of fit of a solution. Special emphasis will be placed on the use of permutation tests. The second part of the article focuses on goodness-of-fit assessment of an important variant of multidimensional scaling called unfolding, which can be applied to a broad range of psychological data settings. Two real-life data sets are presented in order to walk the reader through the entire set of diagnostic measures, tests, and plots. R code is provided as supplementary information that makes the whole goodness-of-fit assessment workflow, as presented in this article, fully reproducible.

Doctor of Philosophy / Department of Statistics / Shie-Shien Yang / Logistic regression model is a branch of the generalized linear models and is widely used in many areas of scientific research. The logit link function and the binary dependent variable of interest make the logistic regression model distinct from linear regression model. The conclusion drawn from a fitted logistic regression model could be incorrect or misleading when the covariates can not explain and /or predict the response variable accurately based on the fitted model- that is, lack-of-fit is present in the fitted logistic regression model. The current goodness-of-fit tests can be roughly categorized into four types. (1) The tests are based on covariate patterns, e.g., Pearson's Chi-square test, Deviance D test, and Osius and Rojek's normal approximation test. (2) Hosmer-Lemeshow's C and Hosmer-Lemeshow's H tests are based on the estimated probabilities. (3) Score tests are based on the comparison of two models, where the assumed logistic regression model is embedded into a more general parametric family of models, e.g., Stukel's Score test and Tsiatis's test. (4) Smoothed residual tests include le Cessie and van Howelingen's test and Hosmer and Lemeshow's test. All of them have advantages and disadvantages. In this dissertation, we proposed a partition logistic regression model which can be viewed as a generalized logistic regression model, since it includes the logistic regression model as a special case. This partition model is used to construct goodness-of- fit test for a logistic regression model which can also identify the nature of lack-of-fit is due to the tail or middle part of the probabilities of success. Several simulation results showed that the proposed test performs as well as or better than many of the known tests.

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The Empirical Likelihood method (ELM) was introduced by A. B. Owen to test hypotheses in the early 1990s. It's a nonparametric method and uses the data directly to do statistical tests and to compute confidence intervals/regions. Because of its distribution free property and generality, it has been studied extensively and employed widely in statistical topics. There are many classical test statistics such as the Cramer-von Mises (CM) test statistic, the Anderson-Darling test statistic, and the Watson test statistic, to name a few. However, none is universally most powerful. This thesis is dedicated to extending the ELM to several interesting statistical topics in hypothesis tests. First of all, we focus on testing the fit of distributions. Based on the CM test, we propose a novel Jackknife Empirical Likelihood test via estimating equations in testing the goodness-of-fit. The proposed new test allows one to add more relevant constraints so as to improve the power. Also, this idea can be generalized to other classical test statistics. Second, when aiming at testing the error distributions generated from a statistical model (e.g., the regression model), we introduce the Jackknife Empirical Likelihood idea to the regression model, and further compute the confidence regions with the merits of distribution free limiting chi-square property. Third, the ELM based on some weighted score equations are proposed for constructing confidence intervals for the coefficient in the simple bilinear model. The effectiveness of all presented methods are demonstrated by some extensive simulation studies.

A test for independence and identical distribution of functional observations is proposed in this thesis. To reduce dimension, curves are projected on the most important functional principal components. Then a test statistic based on lagged cross--covariances of the resulting vectors is constructed. We show that this dimension reduction step introduces asymptotically negligible terms, i.e. the projections behave asymptotically as iid vector--valued observations. A complete asymptotic theory based on correlations of random matrices, functional principal component expansions, and Hilbert space techniques is developed. The test statistic has chi-square asymptotic null distribution. Two inferential tests for error correlation in the functional linear model are put forward. To construct them, finite dimensional residuals are computed in two different ways, and then their autocorrelations are suitably defined. From these autocorrelation matrices, two quadratic forms are constructed whose limiting distributions are chi--squared with known numbers of degrees of freedom (different for the two forms). A test for detecting a change point in the mean of functional observations is developed. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. A comprehensive asymptotic theory for the estimation of a change--point in the mean function of functional observations is developed. The procedures developed in this thesis can be readily computed using the R package fda. All theoretical insights obtained in this thesis are confirmed by simulations and illustrated by real life-data examples.

Exponential Random Graph Models (ERGMs) have been developed for fitting social network data on both static and dynamic levels. However, the lack of large sample asymptotic properties makes it inadequate in assessing the goodness-of-fit of these ERGMs. Simulation-based goodness-of-fit plots were proposed by Hunter et al (2006), comparing the structured statistics of observed network with those of corresponding simulated networks. In this research, we propose an improved approach to assess the goodness of fit of ERGMs. Our method is shown to improve the existing graphical techniques. We also propose a simulation based test statistic with which the model comparison can be easily achieved. / Biostatistics