Equivalence of Regression Curves

Holger Dette, Kathrin Möllenhoff, Stanislav Volgushev, Frank Bretz

Research output: Contribution to journalArticleAcademicpeer-review

20 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)711-729
Number of pages19
JournalJournal of the American Statistical Association
Issue number522
Publication statusPublished - 3 Apr 2018
Externally publishedYes


  • Constrained parameter estimation
  • Equivalence of curves
  • Keywords and phrases: Dose–response studies
  • Nonlinear regression
  • Parametric bootstrap


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