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 multipa rameter statistical model. We refer to this formulation as the likelihood c ontour method (LCM). In general, the LCM procedure requires iterative solut ion of a system of nonlinear equations, and good starting values are critic al because the equations have at least two solutions corresponding to the u pper and lower confidence limits. We replace the LCM equations by the lowes t order terms in their asymptotic expansions. The resulting equations can b e solved explicitly and have exactly two solutions that are used as startin g values for obtaining the respective confidence limits from the LCM equati ons. This article also addresses the problem of obtaining upper confidence limit s for the risk function in a dose-response model in which responses are nor mally distributed. Because of normality, considerable analytic simplificati on is possible and solution of the LCM equations y reduces to an easy one-d imensional root-finding problem. Simulation is used to study the small-samp le coverage of the resulting confidence limits.