  A common problem in numerous research areas, particularly in clinical trials,
is to test whether the effect of an explanatory variable on an outcome variable
is equivalent across different groups. In practice, these tests are frequently
used to compare the effect between patient groups, e.g. based on gender, age or
treatments. Equivalence is usually assessed by testing whether the difference
between the groups does not exceed a pre-specified equivalence threshold.
Classical approaches are based on testing the equivalence of single quantities,
e.g. the mean, the area under the curve (AUC) or other values of interest.
However, when differences depending on a particular covariate are observed,
these approaches can turn out to be not very accurate. Instead, whole
regression curves over the entire covariate range, describing for instance the
time window or a dose range, are considered and tests are based on a suitable
distance measure of two such curves, as, for example, the maximum absolute
distance between them. In this regard, a key assumption is that the true
underlying regression models are known, which is rarely the case in practice.
However, misspecification can lead to severe problems as inflated type I errors
or, on the other hand, conservative test procedures. In this paper, we propose
a solution to this problem by introducing a flexible extension of such an
equivalence test using model averaging in order to overcome this assumption and
making the test applicable under model uncertainty. Precisely, we introduce
model averaging based on smooth AIC weights and we propose a testing procedure
which makes use of the duality between confidence intervals and hypothesis
testing. We demonstrate the validity of our approach by means of a simulation
study and demonstrate the practical relevance of the approach considering a
time-response case study with toxicological gene expression data.
