Exhaustive adjustment-set selection and causal effects for distribution-valued outcomes, with inference under latent Gaussian spatio-temporal dependence

Abstract

Selecting adjustment covariates for a distribution-valued outcome must preserve dependence that scalar summaries can discard, and surveillance outcomes are also dependent across space and time. Under unconfoundedness and a relative-overlap condition we identify average treatment effects, stochastic and incremental intervention effects, and natural direct and indirect effects for such outcomes, with doubly and triply robust scores. For independent data, adjustment sets are selected exhaustively by distance-covariance sufficient dimension reduction: we give a root-n projector expansion, a basis-invariant criterion tensor, and a simultaneous outer confidence collection calibrated by fractional sampling with a finite-population correction. For dependence, units are functions of a latent Gaussian field, exponential in space and Ornstein-Uhlenbeck in time, plus independent noise. Such data are $ρ$-mixing at an exponential rate, and the covariance between any two square-integrable scores is a power series in the latent correlation with positive semidefinite coefficients. Consequences include a central limit theorem for all the causal estimators, the exact undercoverage of independence-based intervals, a consistent Mehler covariance estimator under finite chaos order, and an asymptotically conservative convex-chord estimator; block fractional sampling extends the collection under an assumed linearization. In simulations, fractional sampling raises weak-signal collection coverage from .69-.78 to at least .98 while retaining nearly all candidate sets, and the dependence-aware estimators substantially reduce undercoverage, although none is exact when few units fall within a correlation range. A descriptive analysis of U.S. syphilis surveillance and Medicaid expansion illustrates the estimands and compares dependence-aware with conventional inference.