Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory.

In recent years there has been increased research activity in the area of Func- tional Data Analysis. Methodology from finite dimensional multivariate analysis has been extended to the functional data setting giving birth to Functional ANOVA, Functional Principal Components Analysis, etc. In particular, some studies have pro- posed inferential techniques for various functional models that have connections to well known areas such as mixed-effects models or spline smoothing. The methodol- ogy used in these cases is computationally intensive since it involves the estimation of coefficients in linear models, adaptive selection of smoothing parameters, estimation of variances components, etc. This dissertation proposes a wide-ranging modeling framework that includes many functional linear models as special cases. Three widely used tools are con- sidered: mixed-effects models, penalized least squares, and Bayesian prediction. We show that, in certain important cases, the same numerical answer is obtained for these seemingly different techniques. In addition, under certain assumptions, an applica- tion of a Kalman filter algorithm is shown to improve the order of computations, by two orders of magnitude, for point and interval estimates (with n being the sample size). A functional data analysis setting is used to exemplify our results.

Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory.

Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory. / Science, Faculty of / Statistics, Department of / Graduate

In a generalized linear mixed model (GLMM), the random effects are typically uncorrelated and assumed to follow a normal distribution. However, findings from recent studies on how the misspecification of the random effects distribution affects the estimated model parameters are inconclusive. In the thesis, we extend the randomization approach for deriving linear models to the GLMM framework. Based on this approach, we develop an algorithm for estimating the model parameters of the randomization-based GLMM (RBGLMM) for the completely randomized design (CRD) which does not require normally distributed random effects. Instead, the discrete uniform distribution on the symmetric group of permutations is used for the random effects. Our simulation results suggest that the randomization-based algorithm may be an alternative when the assumption of normality is violated. In the second part of the thesis, we consider an RB-GLMM for the randomized complete block design (RCBD) with random block effects. We investigate the effect of misspecification of the correlation structure and of the random effects distribution via simulation studies. In the simulation, we use the variance covariance matrices derived from the randomization approach. The misspecified model with uncorrelated random effects is fitted to data generated from the model with correlated random effects. We also fit the model with normally distributed random effects to data simulated from models with different random effects distributions. The simulation results show that misspecification of both the correlation structure and of the random effects distribution has hardly any effect on the estimates of the fixed effects parameters. However, the estimated variance components are frequently severely biased and standard errors of these estimates are substantially higher.

Alzheimer’s disease (AD) progression and hypotheses of the cognitive reserve theory were investigated by testing for a relation between educational attainment and rate of decline in patients with Mild Cognitive Impairment, possible AD, probable AD, and other progressive neurodegenerative dementias. Patient data (n = 726) were acquired from a clinical database at the Minneapolis VAMC GRECC Memory Loss Clinic. Analyses using mixed effect regression models found education was significantly related to an accelerated rate of decline in global cognition (MMSE: -0.022, SE = 0.007, p = .003) and a steeper linear rate of decline in functional ability (Cognitive Performance Test: -0.034, SE = 0.011, p = .005). Cox proportional hazard models found little evidence to support an association between educational attainment and relative mortality risk. These results are consistent with previous findings and predictions of the cognitive reserve theory.

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Walmart and other “big box” stores seek to expand in rural markets, possibly due to cheap land and lack of zoning laws. In August 2000, Walmart opened a store in Ephraim, a small rural town in central Utah. It is of interest to understand how Walmart's entrance into the local market changes the sales tax revenue base for Ephraim and for the surrounding municipalities. It is thought that small “Mom and Pop” stores go out of business because they cannot compete with Walmart's prices, leading to a decrease in variety, selection, convenience, and most importantly, sales tax revenue base in areas surrounding Ephraim. This shift in sales tax base is assessed using mixed models. It is found that the entrance of Walmart in Sanpete County has a significant change on sales tax revenue, specifically in the retail industry. A method of calculating the loss for each city is discussed and a sensitivity analysis is performed. This project also documents what has been done to assemble the data set. In addition to discussing the assumptions made to clean the data, explanations of area and industry definition exploration are explained and defended.

Microarray experiments are used to perform gene expression profiling on a large scale. E- and A-optimality of mixed designs was established for experiments with up to 26 different varieties and with the restriction that the number of arrays available is equal to the number of varieties. Because the IBD setting only allows for a single blocking factor (arrays), the search for optimal designs was extended to the Row-Column Design (RCD) setting with blocking factors dye (row) and array (column). Relative efficiencies of these designs were further compared under analysis of variance (ANOVA) models. We also compared the performance of classification analysis for the interwoven loop and the replicated reference designs under four scenarios. The replicated reference design was favored when gene-specific sample variation was large, but the interwoven loop design was preferred for large variation among biological replicates. We applied mixed model methodology to detection and estimation of gene differential expression. For identification of differential gene expression, we favor contrasts which include both variety main effects and variety by gene interactions. In terms of t-statistics for these contrasts, we examined the equivalence between the one- and two-step analyses under both fixed and mixed effects models. We analytically established conditions for equivalence under fixed and mixed models. We investigated the difference of approximation with the two-step analysis in situations where equivalence does not hold. The significant difference between the one- and two-step mixed effects model was further illustrated through Monte Carlo simulation and three case studies. We implemented the one-step analysis for mixed models with the ASREML software. / Ph. D.