  Discovering causal relationships from observational data is a fundamental yet
challenging task. Invariant causal prediction (ICP, Peters et al., 2016) is a
method for causal feature selection which requires data from heterogeneous
settings and exploits that causal models are invariant. ICP has been extended
to general additive noise models and to nonparametric settings using
conditional independence tests. However, the latter often suffer from low power
(or poor type I error control) and additive noise models are not suitable for
applications in which the response is not measured on a continuous scale, but
reflects categories or counts. Here, we develop transformation-model (TRAM)
based ICP, allowing for continuous, categorical, count-type, and
uninformatively censored responses (these model classes, generally, do not
allow for identifiability when there is no exogenous heterogeneity). As an
invariance test, we propose TRAM-GCM based on the expected conditional
covariance between environments and score residuals with uniform asymptotic
level guarantees. For the special case of linear shift TRAMs, we also consider
TRAM-Wald, which tests invariance based on the Wald statistic. We provide an
open-source R package 'tramicp' and evaluate our approach on simulated data and
in a case study investigating causal features of survival in critically ill
patients.
