Comparing causal parameters with many treatments and positivity violations

TL;DR

This paper examines how to compare treatment effects reliably when many treatments and positivity violations occur, proposing new parameters and estimators that are robust and identifiable under weaker assumptions.

Contribution

It introduces a comparability criterion for causal parameters, identifies parameters satisfying it under mild positivity, and develops doubly robust estimators for these parameters.

Findings

Proposed a criterion for meaningful treatment comparison.

Identified parameters like trimmed means satisfying the criterion.

Developed doubly robust estimators with parametric convergence rates.

Abstract

Comparing outcomes across treatments is essential in medicine and public policy. To do so, researchers typically estimate a set of parameters, possibly counterfactual, with each targeting a different treatment. Treatment-specific means are commonly used, but their identification requires a positivity assumption, that every subject has a non-zero probability of receiving each treatment. This is often implausible, especially when treatment can take many values. Causal parameters based on dynamic stochastic interventions offer robustness to positivity violations. However, comparing these parameters may fail to reflect the effects of the underlying target treatments because the parameters can depend on outcomes under non-target treatments. To clarify when two parameters targeting different treatments yield a useful comparison of treatment efficacy, we propose a comparability criterion: if the conditional treatment-specific mean for one treatment is greater than that for another, then the corresponding causal parameter should also be greater. Many standard parameters fail to satisfy this criterion, but we show that only a mild positivity assumption is needed to identify parameters that yield useful comparisons. We then provide two simple examples that satisfy this criterion and are identifiable under the milder positivity assumption: trimmed and smooth trimmed treatment-specific means with multi-valued treatments. For smooth trimmed treatment-specific means, we develop doubly robust-style estimators that attain parametric convergence rates under nonparametric conditions. We illustrate our methods with an analysis of dialysis providers in New York State.