Equivalence of Regression Curves

ABSTRACT

This article investigates the problem whether the difference between two parametric models m₁, m₂ describing the relation between a response variable and several covariates in two different groups is practically irrelevant, such that inference can be performed on the basis of the pooled sample. Statistical methodology is developed to test the hypotheses H₀: d(m₁, m₂) ⩾ ϵ versus H₁: d(m₁, m₂) < ϵ to demonstrate equivalence between the two regression curves m₁, m₂ for a prespecified threshold ϵ, where d denotes a distance measuring the distance between m₁ and m₂. Our approach is based on the asymptotic properties of a suitable estimator $d({\widehat{m}}_1,{\widehat{m}}_2)$ of this distance. To improve the approximation of the nominal level for small sample sizes, a bootstrap test is developed, which addresses the specific form of the interval hypotheses. In particular, data have to be generated under the null hypothesis, which implicitly defines a manifold for the parameter vector. The results are illustrated by means of a simulation study and a data example. It is demonstrated that the new methods substantially improve currently available approaches with respect to power and approximation of the nominal level.