Regression models describing the joint distribution of multivariate responses conditional on covariate information have become an important aspect of contemporary regression analysis. However, a limitation of such models are the rather simplistic assumptions often made, for example, a constant dependence structure not varying with covariates or the restriction to linear dependence between the responses. 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, it scales well-beyond bivariate response situations, which were the main focus of most earlier investigations. We illustrate the benefits in a trivariate analysis of childhood undernutrition and demonstrate empirically that complex truly multivariate data-generating processes can be inferred from observations.

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. We propose a general framework for multivariate conditional transformation models that overcomes such limitations and describes the full joint distribution in simple, interpretable terms. Among the particular merits of the framework are that it can be embedded into likelihood-based inference and allows the dependence structure to vary with the covariates. In addition, the framework scales 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 even complex multivariate data-generating processes can be inferred from observations.

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By extending single-species distribution models, multi-species distribution models and joint species distribution models are able to describe the relationship between environmental variables and a community of species. It is also possible to model either the marginal distribution of each species (multi-species models) in the community or their joint distribution (joint species models) under certain assumptions, but a model describing both entities simultaneously has not been available. We propose a novel model that allows description of both the joint distribution of multiple species and models for the marginal single-species distributions within the framework of multivariate transformation models. Model parameters can be estimated from abundance data by two approximate maximum-likelihood procedures.Using a model community of three fish-eating birds, we demonstrate that interspecific food competition over the course of a year can be modeled using count transformation models equipped with three time-dependent Spearman's rank correlation parameters. We use the same data set to compare the performance of our model to that of a competitor model from the literature on species distribution modeling. Multispecies count transformation models provide an alternative to multi-and joint species distribution models. In addition to marginal transformation models capturing singlespecies distributions, the interaction between species can be expressed by Spearman's rank correlations in an overarching model formulation that allows simultaneous inferences for all model parameters. A software implementation is available in the cotram add-on package to the R system for statistical computing.

Summary Clustered observations are ubiquitous in controlled and observational studies and arise naturally in multicenter trials or longitudinal surveys. We present a novel model for the analysis of clustered observations where the marginal distributions are described by a linear transformation model and the correlations by a joint multivariate normal distribution. The joint model provides an analytic formula for the marginal distribution. Owing to the richness of transformation models, the techniques are applicable to any type of response variable, including bounded, skewed, binary, ordinal, or survival responses. We demonstrate how the common normal assumption for reaction times can be relaxed in the sleep deprivation benchmark data set and report marginal odds ratios for the notoriously difficult toe nail data. We furthermore discuss the analysis of two clinical trials aiming at the estimation of marginal treatment effects. In the first trial, pain was repeatedly assessed on a bounded visual analog scale and marginal proportional-odds models are presented. The second trial reported disease-free survival in rectal cancer patients, where the marginal hazard ratio from Weibull and Cox models is of special interest. An empirical evaluation compares the performance of the novel approach to general estimation equations for binary responses and to conditional mixed-effects models for continuous responses. An implementation is available in the tram add-on package to the $\texttt{R}$ system and was benchmarked against established models in the literature.