<p><P>The nonlinear mixed model is an important tool for analyzingpharmacokinetic and other repeated-measures data.In particular, these models are used when the measured response for anindividual,,has a nonlinear relationship with unknown, random, individual-specificparameters,.Ideally, the method of maximum likelihood is used to find estimates forthe parameters ofthe model after integrating out the random effects in the conditionallikelihood. However, closed form solutions tothe integral are generally not available. As a result, methods have beenpreviously developed to find approximatemaximum likelihood estimates for the parameters in the nonlinear mixedmodel. These approximate methods include FirstOrder linearization, Laplace's approximation, importance sampling, andGaussian quadrature. The methods are availabletoday in several software packages for models of limited sophistication;constant conditional error variance is requiredfor proper utilization of most software. In addition, distributionalassumptions are needed. This work investigates howrobust two of these methods, First Order linearization and Laplace'sapproximation, are to these assumptions. The findingis that Laplace's approximation performs well, resulting in betterestimation than first order linearization when bothmodels converge to a solution. </P><P>A method must provide good estimates of the likelihood at points inthe parameter space near the solution. This workcompares this ability among the numerical integration techniques,Gaussian quadrature, importance sampling, and Laplace'sapproximation. A new "scaled" and "centered" version of Gaussianquadrature is found to be the most accurate technique.In addition, the technique requires evaluation of the integrand at onlya few abscissas. Laplace's method also performswell; it is more accurate than importance sampling with even 100importance samples over two dimensions. Even so,Laplace's method still does not perform as well as Gaussian quadrature.Overall, Laplace's approximation performs betterthan expected, and is shown to be a reliable method while stillcomputationally less demanding. </P><P>This work also introduces a new method to maximize the likelihood.This method can be sharpened to any desired levelof accuracy. Stochastic approximation is incorporated to continuesampling until enough information is gathered to resultin accurate estimation. This new method is shown to work well for linearmixed models, but is not yet successful for thenonlinear mixed model.</P><P>
| Identifer | oai:union.ndltd.org:NCSU/oai:NCSU:etd-20000719-081254 |
| Date | 19 July 2000 |
| Creators | Hartford, Alan Hughes |
| Contributors | Marie Davidian, John Monahan, Pierre Gremaud, Sastry Pantula, Carla Savage |
| Publisher | NCSU |
| Source Sets | North Carolina State University |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://www.lib.ncsu.edu/theses/available/etd-20000719-081254 |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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