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
| Identifer | oai:union.ndltd.org:ORGSU/oai:ir.library.oregonstate.edu:1957/35078 |
| Date | 29 November 1994 |
| Creators | Pratt, James L. |
| Contributors | Schafer, Daniel |
| Source Sets | Oregon State University |
| Language | en_US |
| Detected Language | English |
| Type | Thesis/Dissertation |
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