Bayesian Approach to the Analysis of Nonlinear Models: Implementation and Evaluation

Abstract

A traditional approach to parameter estimation and hypothesis testing in nonlinear models is based on least squares procedures. Error analysis depends on large-sample theory; Bayesian analysis is an alternative approach that avoids substantial errors which could result frorn this dependence. This communication is concerned with the implementation of the Bayesian approach as an alternative to least squares nonlinear regression. Special attention is given to the numerical evaluation of multiple integrals and to the behavior of the parameter estimators and their estimated covariances. The Bayesian approach is evaluated in the light of practical as well as theoretical considerations. 1. Inttoduction The traditional approach to the statistical analysis of nonlinear models is first to use some numerical method to minimize the sum-of-squares objective function in order to obtain least squares estimators of the parameters (Draper and Smith, 1966, pp. 267-275; Nelder and Mead, 1965), and then to apply linear regression theory to the linear part of the Taylor series approximation of the model expanded about these estimators in order to obtain the asymptotic covariance matrix (Bard, 1974, pp. 176-179). The distributions of the estimators obtained in this manner are known only in the limit as the sample size approaches infinity (Jennrich, 1969). Hence, analyses based on these statistics may be inappropriate for small-sample problems, such as those arising in pharmacokinetics (Wagner, 1975). An alternative approach to the statistical analysis of nonlinear models is to utilize methods based on Bayes's theorem (Box and Tiao, 1973, pp. 1-73). Parameters are regarded as random variables rather than as unknown constants. If a nonlinear model with known error distribution is assumed, a-correct probability analysis follows and asymptotic theory is not involved. In this communication a Bayesian approach to nonlinear regression is implemented and evaluated. Particular attention is given to numerical integration methods and the calculation of confidence regions using the posterior distributions.