Testing nonlinear regression parameters under heteroscedastic, normally distributed errors

Likelihood ratio tests for parameters estimated assuming normally distributed errors are examined under a variety of homoscedastic and heteroscedastic variance assumptions. It is assumed that gamma ij, the jth observation from the ith population, is distributed as N[mu(chi ij, beta i), (sigma i mu(chi ij, beta i)theta i)2]. By varying sigma 2i and theta i, this model contains the most obvious examples of homoscedastic and heteroscedastic variability. Under this model, the null hypothesis of equal beta i's can easily be tested using the likelihood ratio criterion. Also considered is the two-sample chi-square statistic, chi 2t = U'S-1 U, where U is the difference vector of nonlinear least squares parameter estimates and S is an estimate of the covariance matrix of U. Monte Carlo simulation using the von Bertalanffy growth curve as an example is used to evaluate several test statistics for Type I error rates under different sampling assumptions.