  A common problem in clinical trials is to test whether the effect of an
explanatory variable on a response of interest is similar between two groups,
e.g. patient or treatment groups. In this regard, similarity is defined as
equivalence up to a pre-specified threshold that denotes an acceptable
deviation between the two groups. This issue is typically tackled by assessing
if the explanatory variable's effect on the response is similar. This
assessment is based on, for example, confidence intervals of differences or a
suitable distance between two parametric regression models. Typically, these
approaches build on the assumption of a univariate continuous or binary outcome
variable. However, multivariate outcomes, especially beyond the case of
bivariate binary response, remain underexplored. This paper introduces an
approach based on a generalised joint regression framework exploiting the
Gaussian copula. Compared to existing methods, our approach accommodates
various outcome variable scales, such as continuous, binary, categorical, and
ordinal, including mixed outcomes in multi-dimensional spaces. We demonstrate
the validity of this approach through a simulation study and an
efficacy-toxicity case study, hence highlighting its practical relevance.
