Causal Transportability of Experiments on Controllable Subsets of Variables: z-Transportability

We introduce z-transportability, the problem of estimating the causal effect of a set of vari- ables X on another set of variables Y in a target domain from experiments on any sub- set of controllable variables Z where Z is an arbitrary subset of observable variables V in a source domain. z-Transportability general- izes z-identifiability, the problem of estimat- ing in a given domain the causal effect of X on Y from surrogate experiments on a set of variables Z such that Z is disjoint from X. z- Transportability also generalizes transporta- bility which requires that the causal effect of X on Y in the target domain be estimable from experiments on any subset of all ob- servable variables in the source domain. We first generalize z-identifiability to allow cases where Z is not necessarily disjoint from X. Then, we establish a necessary and sufficient condition for z-transportability in terms of generalized z-identifiability and transporta- bility. We provide a sound and complete al- gorithm that determines whether a causal ef- fect is z-transportable; and if it is, produces a transport formula, that is, a recipe for es- timating the causal effect of X on Y in the target domain using information elicited from the results of experimental manipulations of Z in the source domain and observational data from the target domain. Our results also show that do-calculus is complete for z- transportability.