  Regression models describing the joint distribution of multivariate response
variables conditional on covariate information have become an important aspect
of contemporary regression analysis. However, a limitation of such models is
that they often rely on rather simplistic assumptions, e.g. a constant
dependency structure that is not allowed to vary with the covariates or the
restriction to linear dependence between the responses only. We propose a
general framework for multivariate conditional transformation models that
overcomes these limitations and describes the entire distribution in a
tractable and interpretable yet flexible way conditional on nonlinear effects
of covariates. The framework can be embedded into likelihood-based inference,
including results on asymptotic normality, and allows the dependence structure
to vary with covariates. In addition, the framework scales well beyond
bivariate response situations, which were the main focus of most earlier
investigations. We illustrate the application of multivariate conditional
transformation models in a trivariate analysis of childhood undernutrition and
demonstrate empirically that our approach can be beneficial compared to
existing benchmarks such that complex truly multivariate data-generating
processes can be inferred from observations.
