  This paper 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 $H_0 : d(m_1,m_2)\geq \epsilon$
versus $H_1 : d(m_1,m_2) < \epsilon$ to demonstrate equivalence between the two
regression curves $m_1,m_2$ for a pre-specified threshold $\epsilon$, 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 $d(\hat{m}_1;
\hat{m}_2)$ of this distance. In order 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 has
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.
