Sensitivity analysis for incremental effects, with application to a study of victimization & offending

TL;DR

This work addresses unmeasured confounding in causal inference with incremental interventions by developing bounds under Rosenbaum's sensitivity framework for single-time-point data and extending to longitudinal settings via a marginal sensitivity model. It introduces a doubly robust, cross-fitted estimator for the incremental bound, with asymptotic normality under mild nuisance-rate conditions, and characterizes the behavior of bound lengths relative to the sensitivity parameters $\Gamma$ and $\delta$. The paper also provides an empirical application in Add Health to study victimization and subsequent offending, demonstrating robustness of conclusions to unmeasured confounding and illustrating how incremental interventions yield nuanced, range-bounded causal estimates. For longitudinal data, it identifies sharp bounds under the marginal model but acknowledges the lack of a practical estimator, pointing to directions for future methodological development and applied work with policy relevance.

Abstract

Sensitivity analysis for unmeasured confounding under incremental propensity score interventions remains relatively underdeveloped. Incremental interventions define stochastic treatment regimes by multiplying the odds of treatment, offering a flexible framework for causal effect estimation. To study incremental effects when there are unobserved confounders, we adopt Rosenbaum's sensitivity model in single time point settings, and propose a doubly robust estimator for the resulting effect bounds. The bound estimators are asymptotically normal under mild conditions on nuisance function estimation. We show that incremental effect bounds can be narrower or wider than those for mean potential outcomes, and that the bounds must lie between the expected minimum and maximum of the conditional bounds on E(Y^0|X) and E(Y^1|X). For time-varying treatments, we consider the marginal sensitivity model. Although sharp bounds for incremental effects are identifiable from longitudinal data under this model, practical estimators have not yet been established; we discuss this challenge and provide partial results toward implementation. Finally, we apply our methods to study the effect of victimization on subsequent offending using data from the National Longitudinal Study of Adolescent to Adult Health (Add Health), illustrating the robustness of our findings in an empirical setting.

Figures (8)

• Figure 1: Plot of bounds as $\Gamma$ varies. The solid lines correspond to Gaussian noise $\epsilon\sim N(0, 0.5^2)$ in $Y$, and the dashed lines correspond to noise that is uniformly distribution with the same variance, $\text{Unif}(-0.5\sqrt{3}, 0.5\sqrt{3})$.
• Figure 2: The absolute bias of different estimators as the estimation rate $\alpha$ varies.
• Figure 3: Estimated chance of offending if victimization odds were multiplied by $\delta$, assuming no unmeasured confounding.
• Figure 4: Plot of effect bounds as $\delta$ varies. The dashed line shows a possible effect value that remains the same under different $\delta$, for $\Gamma=3$.
• Figure 5: Plot of effect bounds as the confounding level $\Gamma$ increases. The black point with $\Gamma=3$ shows a possible value in all effect bounds as $\delta$ varies. The other point with $\Gamma=2.1$ is in all 95% confidence intervals.

Theorems & Definitions (14)

• Definition 1
• Proposition 1
• Lemma 1
• Definition 2
• Theorem 1
• Remark 1
• Proposition 2
• Remark 2
• Proposition 3
• Proposition 4