Identification of causal effects can be hampered by confounding, selection bias, and other complications. Data fusion is one approach to addressing these difficulties, through the inclusion of auxiliary data on the population of interest. Such data may measure a different set of variables, or be obtained under different experimental or observational conditions than the primary dataset. In particular, selection of experimental units into different datasets may be systematic; similar difficulties are encountered in missing data problems. However, existing methods for combining datasets either do not consider this issue, or assume simple selection mechanisms. In this paper, we propose a general approach, based on graphical causal models, for causal inference from data on the same population that is obtained under different experimental conditions. Our framework allows both arbitrary unobserved confounding, and arbitrary selection processes into different experimental regimes in our data. We describe how systematic selection processes may be organized into a hierarchy similar to censoring processes in missing data: selected completely at random, selected at random, and selected not at random. Finally, we provide a novel general identification algorithm for interventional distributions in this setting.
