Uncertainty, Hypothesis Testing, and Model Selection

Abstract

Nonlinear regression methods often rely on the assumption that the nonlinear mean function can be approximated by a linear function locally. This linear approximation may or may not be used in the estimation algorithm (e.g., the Gauss-Newton algorithm), but more importantly it is often used to obtain standard errors, confidence intervals, and t-tests, all of which will depend on how good the approximation is.

The quality of the linear approximation can be summarised by means of two features of the model referred to as intrinsic curvature and parameter effects curvature. We will not provide rigorous definitions here. More details can be found in Bates and Watts (1988, Chapter 7). The two curvature measures reflect different aspects of the linear approximation: (1) The planar assumption ensures that it is possible to approximate the nonlinear mean function at a given point using a tangent plane. (2) The uniform coordinate assumption means that a linear coordinate system is imposed on the approximating tangent plane. Intrinsic curvature is related to the planar assumption, and it depends on the dataset considered and the mean function but not on the parameterisation used in the mean function. Parameter effects curvature is related to the uniform coordinate assumption, and it depends on all aspects of the model, including the parameterisation. Large values of these two curvature measures indicate a poor linear approximation. The function rms.curv() in the package MASS can be used to calculate the two measures for a given nls() model fit (Venables and Ripley, 2002b).