Causal inference for multiple treatments via sufficiency and ratios of generalized propensity scores

Abstract

The coarsest balancing score for multiple treatments T is the minimal sufficient statistic s for the covariates' distributions, DT ; of populations {Pt; t ∈ T } receiving each treatment t ∈ T : A unit in Pr with covariates x is a good match with respect to T for a unit in Pt with covariates y; when x and y provide similar information for DT ; i.e. when s(x) ≈ s(y): For finite, countably finite and often continuous treatments, s(x) is shown to be equivalent to ~e; a vector of propensities' ratios. The units in {Pt; t ∈ T } can be divided into subpopulations where causal comparisons are simultaneously valid. Satisfactory s-matchings are obtained for simulated covariates in R3: The use of ~e's estimate rather than s's estimate allows to avoid the x-curse of dimensionality, but the available data's size in each case is critical for the final choice.