Equivalence of Regression Curves

This article investigates the problem whether the difference between two parametric models m ₁, m ₂ 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 H ₀: d(m ₁, m ₂) ⩾ ϵ versus H ₁: d(m ₁, m ₂) < ϵ to demonstrate equivalence between the two regression curves m ₁, m ₂ for a prespecified threshold ϵ, where d denotes a distance measuring the distance between m ₁ and m ₂. Our approach is based on the asymptotic properties of a suitable estimator (Formula presented.) 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.