Likelihood Contour Method for the Calculation of Asymptotic Upper Confidence Limits on the Risk Function for Quantitative Responses

This article develops a computationally and analytically convenient form of the profile likelihood method for obtaining one-sided confidence limits on scalar-valued functions phi = phi(psi) of the parameters psi in a multiparameter statistical model. We refer to this formulation as the likelihood contour method (LCM). In general, the LCM procedure requires iterative solution of a system of nonlinear equations, and good starting values are critical because the equations have at least two solutions corresponding to the upper and lower confidence limits. We replace the LCM equations by the lowest order terms in their asymptotic expansions. The resulting equations can be solved explicitly and have exactly two solutions that are used as starting values for obtaining the respective confidence limits from the LCM equations. This article also addresses the problem of obtaining upper confidence limits for the risk function in a dose-response model in which responses are normally distributed. Because of normality, considerable analytic simplification is possible and solution of the LCM equations reduces to an easy one-dimensional root-finding problem. Simulation is used to study the small-sample coverage of the resulting confidence limits.