On the permutation equivariance principle for causal estimands
Jiaqi Tong, Fan Li
TL;DR
A class of weighted estimands that project unstructured potential outcome means into a vector of permutation equivariant and interpretable estimands capturing all orders of interaction, which is useful in context such as causal mediation and network interference.
Abstract
In many causal inference problems, multiple action variables, such as factors, mediators, or network units, often share a common causal role yet lack a natural ordering. To avoid ambiguity, the scientific interpretation of a vector of estimands should remain invariant under relabeling, an implicit principle we refer to as permutation equivariance. Permutation equivariance can be understood as the property that permuting the variables permutes the estimands in a trackable manner, such that scientific meaning is preserved. We formally characterize this principle and study its combinatorial algebra. We present a class of weighted estimands that project unstructured potential outcome means into a vector of permutation equivariant and interpretable estimands capturing all orders of interaction. To guide practice, we discuss the implications and choices of weights and define residual-free estimands, whose inclusion-exclusion sums capture the maximal effect, which is useful in context such as causal mediation and network interference. We present the application of our general theory to three canonical examples and extend our results to ratio effect measures.