Fieller's theorem, the likelihood and the delta method

This note discusses the direct likelihood estimation of ratio parameters. The method is based on the use of nonlinear regression models from the exponential family. The variances of the maximum likelihood estimates are identical with values obtained from the corresponding generalized linear models and the delta method (Bishop, Fienberg, and Holland, Discrete Multivariate Analysis: Theory and Practice, Cambridge, Massachusetts: MIT Press, 1975), but are easier to calculate. The approach is illustrated by means of a number of examples, including a rather complex logistic regression model. We also discuss the connection between confidence intervals obtained from Fieller's theorem and large-sample intervals obtained from the information matrix. We conclude from this comparison and from the examples that direct maximum likelihood estimation of ratios offers a useful alternative to traditional methods based on linear models, especially for complex data sets.