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## The laplace approximation and inference in generalized linear models with two or more random effects

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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