by Lam Man Kin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Bibliography: leaves 69-71. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Generalized Linear Models --- p.7 / Chapter §2.1 --- The Model --- p.7 / Chapter §2.2 --- Estimation of the Parameters --- p.10 / Chapter Chapter 3 --- Tests for the Dispersion Parameters --- p.15 / Chapter S3.1 --- Three Asymptotically Equivalent Tests based on MLE --- p.15 / Chapter §3.2 --- Application of the Tests for the Dispersion Parameters --- p.22 / Chapter Chapter 4 --- Finite Sample Study of the Three Tests based on Simulation --- p.29 / Chapter §4.1 --- Introduction --- p.29 / Chapter §4.2 --- The Simulation --- p.31 / Chapter §4.3 --- The Results --- p.34 / Chapter Chapter 5 --- An Example --- p.38 / Chapter Chapter 6 --- Conclusions and Discussions --- p.42 / Tables / References

This thesis proposes an approximate maximum likelihood estimator and likelihood ratio test for parameters in a generalized linear model when two or more random effects are present. Substantial progress in parameter estimation for such models has been made with methods involving generalized least squares based on the approximate marginal mean and covariance matrix. However, tests and confidence intervals based on this approach have been limited to what is provided through asymptotic normality of estimates. The proposed solution is based on maximizing a Laplace approximation to the log-likelihood function. This approximation is remarkably accurate and has previously been demonstrated to work well for obtaining likelihood based estimates and inferences in generalized linear models with a single random effect. This thesis concentrates on extensions to the case of several random effects and the comparison of the likelihood ratio inference from this approximate likelihood analysis to the Wald-like inferences for existing estimators. The shapes of the Laplace approximate and true log-likelihood functions are practically identical, implying that maximum likelihood estimates and likelihood ratio inferences are obtained from the Laplace approximation to the log-likelihood. Use of the Laplace approximation circumvents the need for numerical integration, which can be practically impossible to compute when there are two random effects. However, both the Laplace and exact (via numerical integration) methods require numerical optimization, a sometimes slow process, for obtaining estimates and inferences. The proposed Laplace method for estimation and inference is demonstrated for three real (and some simulated) data sets, along with results from alternative methods which involve use of marginal means and covariances. The Laplace approximate method and another denoted as Restricted Maximum Likelihood (REML) performed rather similarly for estimation and hypothesis testing. The REML approach produced faster analyses and was much easier to implement while the Laplace implementation provided likelihood ratio based inferences rather than those relying on asymptotic normality. / Graduation date: 1995

Thesis (Ph. D.)--University of Queensland, 2001. / Includes bibliographical references.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 155-159). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 155-159). Also available on the Internet.

When dealing with exponential family distributions, a constant dispersion is often assumed since it simplifies both model formulation and estimation. In contrast, heteroscedasticity is a common feature of almost every empirical data set. In this dissertation, the dispersion parameter is no longer considered as constant throughout the entire sample, but defined as the expected deviance of the individual response yi and its expected value _i such that it will be expressed as a linear combination of some covariates and their coefficients. At the same time, the dispersion regression is an essential part of a double Generalized Linear Model in which mean and dispersion are modelled in two interlinked and pseudo-simultaneously estimated submodels. In other words, the deviance is a function of the response mean which on the other hand depends on the dispersion. Due to the mutual dependency, the estimation algorithm will be iterated as long as the improvement of the one parameter leads to significant changes of the other until it is not the case. If appropriate covariates are chosen, the model’s goodness of fit should be improved by the property that the dispersion is estimated by external information instead of being a constant. In the following, the advantage of dispersion modelling will be shown by its application on three different types of data: a) zero-inflated data, b) non-linear time series data, and c) clinical trials data. All these data follow distributions of the exponential family for which the application of the Generalized Linear Model is justified, but require certain extensions of modelling methodologies. In this dissertation, The enhanced goodness of fit given that the constant dispersion assumption is dropped will be shown in the above listed examples. In fact, by formulating and carrying out score and Wald tests on testing for the possible occurrence of varying dispersion, evidence of heterogeneous dispersion could be found to be present in the data sets considered. Furthermore, although model formulation, asymptotic properties and computational effort are more extensive when dealing with the double models, the benefits and advantages in terms of improved fitting results and more efficient parameter estimates appear to justify the additional effort not only for the types of data introduced, but also generally for empirical data analysis, on different types of data as well. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

The general linear model, the weighted linear model, and the generalized linear model are presented in detail. Diagnostic tools for the linear models are considered. In general the standard analysis for linear models does not account for measurement error. / Thesis (M.Sc.) - University of KwaZulu-Natal, Pietermaritzburg, 2007.

No description available.

It is well known that when the input variables of the linear regression model are subject to noise contamination, the model parameters can not be estimated uniquely. This, in the statistical literature, is referred to as the identifiability problem of the errors-in-variables models. Further, in linear regression there is an explicit assumption of the existence of a single linear relationship. The statistical properties of the errors-in-variables models under the assumption that the noise variances are either known or that they can be estimated are well documented. In many situations, however, such information is neither available nor obtainable. Although under such circumstances one can not obtain a unique vector of parameters, the space, Ω, of the feasible solutions can be computed. Additionally, assumption of existence of a single linear relationship may be presumptuous as well. A multi-equation model similar to the simultaneous-equations models of econometrics may be more appropriate. The goals of this dissertation are the following: (1) To present analytical techniques or algorithms to reduce the solution space, Ω, when any type of prior information, exact or relative, is available; (2) The data covariance matrix, Σ, can be examined to determine whether or not Ω is bounded. If Ω is not bounded a multi-equation model is more appropriate. The methodology for identifying the subsets of variables within which linear relations can feasibly exist is presented; (3) Ridge regression technique is commonly employed in order to reduce the ills caused by collinearity. This is achieved by perturbing the diagonal elements of Σ. In certain situations, applying ridge regression causes some of the coefficients to change signs. An analytical technique is presented to measure the amount of perturbation required to render such variables ineffective. This information can assist the analyst in variable selection as well as deciding on the appropriate model; (4) For the situations when Ω is bounded, a new weighted regression technique based on the computed upper bounds on the noise variances is presented. This technique will result in identification of a unique estimate of the model parameters.

This dissertation considers two approaches for testing hypotheses in unbalanced mixed linear models. The first approach is to construct a design with some type of structure or "partial" balance, so that some of the optimal properties of a completely balanced design hold. It is shown that for a particular type of partially balanced design certain hypothesis tests are optimal. The second approach is to study how the unbalancedness of a design affects a hypothesis test in terms of level and power. Measures of imbalance are introduced and simulation results are presented that demonstrate the relationship of the level and power of a test and the measures. The first part of this thesis focuses on error orthogonal designs which are a type of partially balanced design. It is shown that with an error orthogonal design and under certain additional conditions, ANOVA F-tests about certain linear combinations of the variance components and certain linear combinations of the fixed effects are uniformly most powerful (UMP) similar and UMP unbiased. The ANOVA F-tests for the variance components are also invariant, so that the tests are also UMP invariant similar and UMP invariant unbiased. For certain simultaneous hypotheses about linear combinations of the fixed effects, the ANOVA F-tests are UMP invariant unbiased. The second part of this thesis considers a mixed model with a random nested effect, and studies the effects of an unbalanced design on the level and power of a hypothesis test of the nested variance component being equal to zero. Measures of imbalance are introduced for each of the four conditions necessary to obtain an exact test. Simulations are done for two different models to determine if there is a relationship between any of the measures and the level and power for both a naive test and a test using Satterthwaite's approximation. It is found that a measure based on the coefficients of the expected mean squares is indicative of how a test is performing. This measure is also simple to compute, so that it can easily be employed to determine the validity of the expected level and power. / Graduation date: 1999