Sensitivity analysis for Box-Cox power transformation model: Contrast parameters

Abstract

SUMMARY Sensitivity analysis is an important tool for the evaluation of scientific models. We consider the Box-Cox power transformation model and study the sensitivity for the estimated slope vector when the transformation parameter is perturbed. This is a key issue in the debate among Bickel & Doksum (1981), Box & Cox (1982) and Hinkley & Runger (1984) on the application of the power transformation model. Hinkley & Runger (1984) conjectured that Box & Cox's (1964) z transformation would eliminate asymptotically the sensitivity for the estimated slope vector. We establish this conjecture under appropriate symmetry conditions on the joint distribution for the regressors, x. When the true transformation is logarithmic, the conjecture holds if the distribution for x is axially symmetric. For other power transformations, the conjecture holds if x is multivariate normal. We also consider a weaker version of Hinkley & Runger's conjecture, namely, the z transformation would achieve the best possible reduction in the sensitivity. We establish this weaker conjecture under less restrictive symmetry conditions which only involve the marginal distribution for xfl. Both conjectures might fail when those symmetry conditions are not satisfied. Two examples are given to illustrate the limitations of the conjectures. Sensitivity analysis is an important tool for the evaluation of scientific models. Many scientific models include some nuisance parameters which are not known precisely. It is usually desirable for the model output to be stable with respect to those parameters. The performance of the model can be questionable if the model output is highly sensitive to those parameters. It is therefore useful to carry out a sensitivity analysis, in which those parameters are perturbed, the model output is derived under the perturbed parameters and compared with the model output prior to the perturbation. We will study the sensitivity analysis for the Box-Cox power transformation model with respect to the specific transformation taken. Let {(yi, xi), i = 1, . . . , n} be the observed data, where yi denotes a positive scalar outcome, and xi denotes a p-dimensional row vector of regressors. Let the transformed outcome be