Doctor of Philosophy / Department of Statistics / James Neill / The problem of testing for lack of fit in exponential family regression models is considered. Such nonlinear models are the natural extension of Normal nonlinear regression models and generalized linear models. As is usually the case, inadequately specified models have an adverse impact on statistical inference and scientific discovery. Models of interest are curved exponential families determined by a sequence of predictor settings and mean regression function, considered as a sub-manifold of the full exponential family. Constructed general alternative models are based on clusterings in the mean parameter components and allow likelihood ratio testing for lack of fit associated with the mean, equivalently natural parameter, for a proposed null model. A maximin clustering methodology is defined in this context to determine suitable clusterings for assessing lack of fit. In addition, a geometrically motivated goodness of fit test statistic for exponential family regression based on the information metric is introduced. This statistic is applied to the cases of logistic regression and Poisson regression, and in both cases it can be seen to be equal to a form of the Pearson chi[superscript]2 statistic. This same statement is true for multinomial regression. In addition, the problem of testing for equal means in a heteroscedastic Normal model is discussed. In particular, a saturated 3 parameter exponential family model is developed which allows for equal means testing with unequal variances. A simulation study was carried out for the logistic and Poisson regression models to investigate comparative performance of the likelihood ratio test, the deviance test and the goodness of fit test based on the information metric. For logistic regression, the Hosmer-Lemeshow test was also included in the simulations. Notably, the likelihood ratio test had comparable power with that of the Hosmer-Lemeshow test under both m- and n-asymptotics, with superior power for constructed alternatives. A distance function defined between densities and based on the information metric is also given. For logistic models, as the natural parameters go to plus or minus infinity, the densities become more and more deterministic and limits of this distance function are shown to play an important role in the lack of fit analysis. A further simulation study investigated the power of a likelihood ratio test and a geometrically derived test based on the information metric for testing equal means in heteroscedastic Normal models.

Doctor of Philosophy / Department of Statistics / Shie-Shien Yang / When the primary research interest is in the marginal dependence between the response and the covariates, logistic GEE (Generalized Estimating Equation) models are often used to analyze clustered binary data. Relative to ordinary logistic regression, very little work has been done to assess the lack of fit of a logistic GEE model. A new method addressing the LOF of a logistic GEE model was proposed. Simulation results indicate the proposed method performs better than or as well as other currently available LOF methods for logistic GEE models. A SAS macro was developed to implement the proposed method. Nonlinear regression models are widely used in medical science. Before the models can be fit and parameters interpreted, researchers need to decide which design points in a prespecified design space should be included in the experiment. Careful choices at this stage will lead to efficient usage of limited resources. We proposed a cost efficient Bayesian optimal design method for nonlinear combinations of parameters in a nonlinear model with quantitative predictors. An R package was developed to implement the proposed method.

Master of Science / Department of Statistics / Weixing Song / The relationship between a random variable and a random vector is often investigated through the regression modeling. Because of its relative simplicity and ease of interpretation, a particular parametric form is often assumed for the regression function. If the pre-specified function form truly reflects the truth, then the resulting estimators and inference procedures would be reliable and efficient. But if the regression function does not represent the true relationship between the response and the predictors, then the inference results might be very misleading. Therefore, lack-of-fit test should be an indispensable part in regression modeling. This report compares the finite sample performance of several classical lack-of-fit tests in regression models via simulation studies. It has three chapters. The conception of the lack-of-fit test, together with its basic setup, is briefly introduced in Chapter 1; then several classical lack-of-fit test procedures are discussed in Chapter 2; finally, thorough simulation studies are conducted in Chapter 3 to assess the finite sample performance of each procedure introduced in Chapter 2. Some conclusions are also summarized in Chapter 3. A list of MATLAB codes that are used for the simulation studies is given in the appendix.

Doctor of Philosophy / Department of Statistics / Weixing Song / The regression model has been given a considerable amount of attention and played a significant role in data analysis. The usual assumption in regression analysis is that the variances of the error terms are constant across the data. Occasionally, this assumption of homoscedasticity on the variance is violated; and the data generated from real world applications exhibit heteroscedasticity. The practical importance of detecting heteroscedasticity in regression analysis is widely recognized in many applications because efficient inference for the regression function requires unequal variance to be taken into account. The goal of this thesis is to propose new testing procedures to assess the adequacy of fitting parametric variance function in heteroscedastic regression models. The proposed tests are established in Chapter 2 using certain minimized L[subscript]2 distance between a nonparametric and a parametric variance function estimators. The asymptotic distribution of the test statistics corresponding to the minimum distance estimator under the fixed model and that of the corresponding minimum distance estimators are shown to be normal. These estimators turn out to be [sqrt]n consistent. The asymptotic power of the proposed test against some local nonparametric alternatives is also investigated. Numerical simulation studies are employed to evaluate the nite sample performance of the test in one dimensional and two dimensional cases. The minimum distance method in Chapter 2 requires the calculation of the integrals in the test statistics. These integrals usually do not have a tractable form. Therefore, some numerical integration methods are needed to approximate the integrations. Chapter 3 discusses a nonparametric empirical smoothing lack-of-fit test for the functional form of the variance in regression models that do not involve evaluation of integrals. empirical smoothing lack-of-fit test can be treated as a nontrivial modification of Zheng (1996)'s nonparametric smoothing test and Koul and Ni (2004)'s minimum distance test for the mean function in the classic regression models. The asymptotic normality of the proposed test under the null hypothesis is established. Consistency at some fixed alternatives and asymptotic power under some local alternatives are also discussed. Simulation studies are conducted to assess the nite sample performance of the test. The simulation studies show that the proposed empirical smoothing test is more powerful and computationally more efficient than the minimum distance test and Wang and Zhou (2006)'s test.

Svensk arbetsmarknad är snedfördeladmellan män och kvinnor. Tidigare forskning har visat att könsstereotypiskt språk (agentic/communal) kan bidra till att upprätthålla denna snedfördelning, skapa statusskillnader mellan kvinnor och män samt leda till diskriminering av kvinnor redan i en rekryteringsprocess. Med ett flerfaktoriellt experiment (N=194) av dubbel-blind-design med randomiserade deltagare undersöktes, huruvida bedömningen av en arbetssökande påverkas av könsstereotypiskt språk, samt vilken attityd deltagarna hade till pronomenHen. Deltagare betingades med agentic eller communal platsannons samt en av fyra personbeskrivningar.Sedan skattades den sökandes agentic och communal egenskaper. Deltagarna svarade på frågor gällande sexism samt uttryckte sin åsikt om Hen. Två av studiens fyra hypoteser kunde styrkas, de övriga två förkastades. Studien visar att könsstereotypiskt språk har betydelse för hur en sökande bedöms, att Den sökande uppfattas som en man och att sexism predicerar negativa åsikter om Hen.

In this study we develop a new method for testing the null hypothesis that the predictor function in a canonical link regression model has a prescribed linear form. The class of models, which we will refer to as canonical link regression models, constitutes arguably the most important subclass of generalized linear models and includes several of the most popular generalized linear models. In addition to the primary contribution of this study, we will revisit several other tests in the existing literature. The common feature among the proposed test, as well as the existing tests, is that they are all based on orthogonal series estimators and used to detect departures from a null model. Our proposal for a new lack-of-fit test is inspired by the recent contribution of Hart and is based on a Laplace approximation to the posterior probability of the null hypothesis. Despite having a Bayesian construction, the resulting statistic is implemented in a frequentist fashion. The formulation of the statistic is based on characterizing departures from the predictor function in terms of Fourier coefficients, and subsequent testing that all of these coefficients are 0. The resulting test statistic can be characterized as a weighted sum of exponentiated squared Fourier coefficient estimators, whereas the weights depend on user-specified prior probabilities. The prior probabilities provide the investigator the flexibility to examine specific departures from the prescribed model. Alternatively, the use of noninformative priors produces a new omnibus lack-of-fit statistic. We present a thorough numerical study of the proposed test and the various existing orthogonal series-based tests in the context of the logistic regression model. Simulation studies demonstrate that the test statistics under consideration possess desirable power properties against alternatives that have been identified in the existing literature as being important.

Master of Science / Department of Statistics / James W. Neill / Cluster based lack of fit tests for linear regression models are generally effective in detecting model inadequacy due to between- or within-cluster lack of fit. Typically, lack of fit exists as a combination of these two pure types, and can be extremely difficult to detect depending on the nature of the mixture. Su and Yang (2006) and Miller and Neill (2007) have proposed lack of fit tests which address this problem. Based on a simulation comparison of the two testing procedures, it is concluded that the Su and Yang test can be expected to be effective when the true model is locally well approximated within each specified cluster and the lack of fit is not due to an unspecified predictor variable. The Miller and Neill test accommodates a broader alternative, which can thus result in comparatively lower but effective power. However, the latter test demonstrated the ability to detect model inadequacy when the lack of fit was a function of an unspecified predictor variable and does not require a specified clustering for implementation.

Sigmoid curves have found broad applicability in biological sciences and biopharmaceutical research during the last decades. A well planned experiment design is essential to accurately estimate the parameters of the model. In contrast to a large literature and extensive results on optimal designs for linear models, research on the design for nonlinear, including sigmoid curve, models has not kept pace. Furthermore, most of the work in the optimal design literature for nonlinear models concerns the characterization of minimally supported designs. These minimal, optimal designs are frequently criticized for their inability to check goodness of fit, as there are no additional degrees of freedom for the testing. This design issue can be a serious problem, since checking the model adequacy is of particular importance when the model is selected without complete certainty. To assess for lack of fit, we must add at least one extra distinct design point to the experiment. The goal of this dissertation is to identify optimal or highly efficient designs capable of checking the fit for sigmoid curve models. In this dissertation, we consider some commonly used sigmoid curves, including logistic, probit and Gompertz models with two, three, or four parameters. We use D-optimality as our design criterion. We first consider adding one extra point to the design, and consider five alternative designs and discuss their suitability to test for lack of fit. Then we extend the results to include one more additional point to better understand the compromise among the need of detecting lack of fit, maintaining high efficiency and the practical convenience for the practitioners. We then focus on the two-parameter Gompertz model, which is widely used in fitting growth curves yet less studied in literature, and explore three-point designs for testing lack of fit under various error variance structures. One reason that nonlinear design problems are so challenging is that, with nonlinear models, information matrices and optimal designs depend on the unknown model parameters. We propose a strategy to bypass the obstacle of parameter dependence for the theoretical derivation. This dissertation also successfully characterizes many commonly studied sigmoid curves in a generalized way by imposing unified parameterization conditions, which can be generalized and applied in the studies of other sigmoid curves. We also discuss Gompertz model with different error structures in finding an extra point for testing lack of fit. / Statistics

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide desirable response characteristics in finished products. D-Optimal minimal designs have been considered for a variety of mixture models, including Scheffe's linear, quadratic, and cubic models. Usually, these D-Optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the Lack of Fit tests. Also, the majority of the design points in D-Optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. In this dissertation, extensions of the D-Optimal minimal designs are developed to allow additional interior points in the design space to enable prediction of the entire response surface. First, the extensions of the D-Optimal minimal designs for two commonly used second-degree mixture models are considered. Second, the methodology for adding interior points to general mixture models is generalized. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. When compared with the standard mixture designs, the proposed extended D-Optimal minimal design provides higher power for the Lack of Fit tests with comparable D-efficiency. / Statistics

The effects of support in steel bridges can present significant challenges during the construction. The tendency of girders to twist or layovers during the construction can present a particularly challenging problem regarding detailing cross-frames that provide bracing to steel girders. Methods of detailing cross-frames have been investigated in the past to identify some of the issues related to the behavior of straight and skewed steel bridges. However, the absence of a complete and simplified design approach has led to disputes between stakeholders, costly repairs and delays in the construction. The main objective of this research is to develop a complete and simplified design approach considering construction, fabrication and detailing of skewed bridges. This objective is achieved by comparing different detailing methods, understanding the mechanism by which skew effects develop in steel bridges, recommending simplified methods of analysis to evaluate them, and developing a complete and simplified design procedure for skew bridges. Girder layovers, flange lateral bending stress, cross-frame forces, component of vertical deflections, component of vertical reactions and lateral reactions or lateral displacements are affected by detailing methods and are referred as lack-of-fit effects. The main conclusion of this research is that lack-of-fit effects for the Final Fit detailing method at the steel dead load stage are equal and opposite to the lack-of-fit effects for the Erected Fit detailing method at the total dead load stage. This conclusion has helped using 2D grid analyses for estimating these lack-of-fit effects for different detailing methods. 3D erection simulations are developed for estimating fit-up forces required to attach the cross-frames to girders. The maximum fit-up force estimated from the 2D grid analysis shows a reasonable agreement with the one obtained from the erection simulations. The erection sequence that reduces the maximum fit-up force is also found by erection simulations. The line girder analysis is recommended for calculating cambers for the Final Fit detailing method. A combination of line girder analysis and 2D grid analysis is recommended for calculating cambers for the Erected Fit detailing method. Finally, flowcharts are developed that facilitate the selection of a detailing method and show the necessary design checks.