Extending R-Squared to the Generalized Linear Mixed Model for Longitudinal Data

The R2 statistic is a well known measure of association for the linear model that has been extended in various ways to the linear mixed model. Statistical literature suggests multiple measures of R2 must be used to characterize linear mixed models by summarizing goodness-of-fit for both fixed and random effects. Regarding the latter, there are currently no measures of R2 which consistently demonstrate a capacity to avoid over-fitting and under-fitting covariance models. In this dissertation we extend R2 to the generalized linear mixed model and develop new covariance model selection and inference techniques for R2 in the linear mixed model that can also be extended to the generalized linear mixed model. Chapter 2 describes a marginal R2 statistic for the linear mixed model that measures generalized explained variance. Our method utilizes standardized generalized variance to stabilize the estimated denominator degrees of freedom used in the approximate Wald F test. The proposed modification consistently estimates a well-defined population value, exhibits a non-central beta sample distribution, and demonstrates superior performance in a simulation study where R2 statistics are used to assess covariance goodness-of-fit. Chapter 3 introduces a paradigm of conducting statistical tests regarding R2 statistics in the linear mixed model. Simple summary tests of covariance goodness-of-fit for a specific model are explored as well as tests for model selection. This approach is able to compare covariance models that are not hierarchically related (i.e. nested). A simulation study and two applied examples demonstrate the testing procedure’s capacity to fill in the gaps of uncertainty regarding covariance model selection when candidates are non-nested. Chapter 4 discusses a method to extend R2 from the linear mixed model to the generalized linear mixed model. The approach utilizes penalized quasi-likelihood estimation and is the first to enable computation of semi-partial R2 statistics for fixed effects in the generalized linear mixed model. A simulation study assesses the performance of the proposed method. Extensions based on the linear mixed model results from Chapters 2 and 3 are explored.Doctor of Philosoph

Goodness-of-Fit Test Issues in Generalized Linear Mixed Models

Linear mixed models and generalized linear mixed models are random-effects models widely applied to analyze clustered or hierarchical data. Generally, random effects are often assumed to be normally distributed in the context of mixed models. However, in the mixed-effects logistic model, the violation of the assumption of normally distributed random effects may result in inconsistency for estimates of some fixed effects and the variance component of random effects when the variance of the random-effects distribution is large. On the other hand, summary statistics used for assessing goodness of fit in the ordinary logistic regression models may not be directly applicable to the mixed-effects logistic models. In this dissertation, we present our investigations of two independent studies related to goodness-of-fit tests in generalized linear mixed models. First, we consider a semi-nonparametric density representation for the random effects distribution and provide a formal statistical test for testing normality of the random-effects distribution in the mixed-effects logistic models. We obtain estimates of parameters by using a non-likelihood-based estimation procedure. Additionally, we not only evaluate the type I error rate of the proposed test statistic through asymptotic results, but also carry out a bootstrap hypothesis testing procedure to control the inflation of the type I error rate and to study the power performance of the proposed test statistic. Further, the methodology is illustrated by revisiting a case study in mental health. Second, to improve assessment of the model fit in the mixed-effects logistic models, we apply the nonparametric local polynomial smoothed residuals over within-cluster continuous covariates to the unweighted sum of squares statistic for assessing the goodness-of-fit of the logistic multilevel models. We perform a simulation study to evaluate the type I error rate and the power performance for detecting a missing quadratic or interaction term of fixed effects using the kernel smoothed unweighted sum of squares statistic based on the local polynomial smoothed residuals over x-space. We also use a real data set in clinical trials to illustrate this application

A goodness-of-fit test for parametric and semi-parametric models in multiresponse regression

We propose an empirical likelihood test that is able to test the goodness of fit of a class of parametric and semi-parametric multiresponse regression models. The class includes as special cases fully parametric models; semi-parametric models, like the multiindex and the partially linear models; and models with shape constraints. Another feature of the test is that it allows both the response variable and the covariate be multivariate, which means that multiple regression curves can be tested simultaneously. The test also allows the presence of infinite-dimensional nuisance functions in the model to be tested. It is shown that the empirical likelihood test statistic is asymptotically normally distributed under certain mild conditions and permits a wild bootstrap calibration. Despite the large size of the class of models to be considered, the empirical likelihood test enjoys good power properties against departures from a hypothesized model within the class.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ208 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Are benefits from oil - stocks diversification gone? New evidence from a dynamic copula and high frequency data

Oil is perceived as a good diversification tool for stock markets. To fully understand this potential, we propose a new empirical methodology that combines generalized autoregressive score copula functions with high frequency data and allows us to capture and forecast the conditional time-varying joint distribution of the oil -- stocks pair accurately. Our realized GARCH with time-varying copula yields statistically better forecasts of the dependence and quantiles of the distribution relative to competing models. Employing a recently proposed conditional diversification benefits measure that considers higher-order moments and nonlinear dependence from tail events, we document decreasing benefits from diversification over the past ten years. The diversification benefits implied by our empirical model are, moreover, strongly varied over time. These findings have important implications for asset allocation, as the benefits of including oil in stock portfolios may not be as large as perceived

Non-linear dependency of the subjective perceived intensity of steering wheel rotational vibration

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2009 ElsevierThe present study has established equal sensation curves for steering wheel hand-arm rotational vibration. Psychophysical response tests of 20 participants were performed in a steering wheel rotational vibration simulator using the category-ratio Borg CR10 scale procedure for direct estimation of perceived vibration intensity. The test stimuli used were sinusoidal vibrations at 22 third octave band centre frequencies in the range from 3 to 400 Hz, with acceleration amplitudes in the range from 0.06 to 30 m/s(2) r.m.s. A multivariate regression analysis was performed on the mean perceived intensity Borg CR10 values as a function of the two independent parameters of the vibration frequency and amplitude. The results suggested a non-linear dependency of the subjective perceived intensity on both the steering wheel rotational vibration frequency and amplitude. The equal sensation curves were found to be characterised by a decreased sensitivity to hand-arm vibration with increasing frequency from 10 to 400 Hz, but by an increased sensitivity with increasing frequency from 4 to 10 Hz. A 6th order polynomial model has been proposed as a best fit regression model from which the equal sensation curves for steering wheel rotational vibration are derived.Relevance to industry: For the manufactures of automobiles, steering systems and other automobile components this study provides a mathematical model from which one or more new frequency weightings for the use in evaluating the perceived intensity of steering wheel rotational vibration are derived. (C) 2008 Elsevier B.V. All rights reserved

Spectral goodness of fit for network models

We introduce a new statistic, 'spectral goodness of fit' (SGOF) to measure how well a network model explains the structure of an observed network. SGOF provides an absolute measure of fit, analogous to the standard R-squared in linear regression. Additionally, as it takes advantage of the properties of the spectrum of the graph Laplacian, it is suitable for comparing network models of diverse functional forms, including both fitted statistical models and algorithmic generative models of networks. After introducing, defining, and providing guidance for interpreting SGOF, we illustrate the properties of the statistic with a number of examples and comparisons to existing techniques. We show that such a spectral approach to assessing model fit fills gaps left by earlier methods and can be widely applied