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Equivalence of Regression Curves

Author

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  • Holger Dette
  • Kathrin Möllenhoff
  • Stanislav Volgushev
  • Frank Bretz

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(m^1,m^2)$d(\hat{m}_1, \hat{m}_2)$ 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.

Suggested Citation

  • Holger Dette & Kathrin Möllenhoff & Stanislav Volgushev & Frank Bretz, 2018. "Equivalence of Regression Curves," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 711-729, April.
  • Handle: RePEc:taf:jnlasa:v:113:y:2018:i:522:p:711-729
    DOI: 10.1080/01621459.2017.1281813
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    Citations

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    Cited by:

    1. Bastian, P. & Dette, H. & Koletzko, L., 2024. "Testing equivalence of multinomial distributions — A constrained bootstrap approach," Statistics & Probability Letters, Elsevier, vol. 206(C).
    2. Julie K. Furberg & Christian B. Pipper & Thomas Scheike, 2021. "Testing equivalence of survival before but not after end of follow-up," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 27(2), pages 216-243, April.
    3. Kathrin Möllenhoff & Frank Bretz & Holger Dette, 2020. "Equivalence of regression curves sharing common parameters," Biometrics, The International Biometric Society, vol. 76(2), pages 518-529, June.
    4. Eustasio del Barrio & Hristo Inouzhe & Carlos Matrán, 2020. "Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 65-87, October.
    5. Kathrin Möllenhoff & Kirsten Schorning & Franziska Kappenberg, 2023. "Identifying alert concentrations using a model‐based bootstrap approach," Biometrics, The International Biometric Society, vol. 79(3), pages 2076-2088, September.

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