Second-Order Inference for the Mean of a Variable Missing at Random

TL;DR

This paper introduces a second-order estimator for the mean of a variable with missing data, improving accuracy and robustness over existing methods by leveraging second-order expansions within the TMLE framework.

Contribution

It develops a novel second-order TMLE estimator that requires weaker assumptions and demonstrates improved coverage probability in simulations and real data applications.

Findings

Second-order TMLE improves confidence interval coverage by up to 85%.

A first-order estimator inspired by second-order expansion enhances finite sample performance.

The methods are illustrated with a real dataset on anticoagulant effects, with available R code.

Abstract

We present a second-order estimator of the mean of a variable subject to missingness, under the missing at random assumption. The estimator improves upon existing methods by using an approximate second-order expansion of the parameter functional, in addition to the first-order expansion employed by standard doubly robust methods. This results in weaker assumptions about the convergence rates necessary to establish consistency, local efficiency, and asymptotic linearity. The general estimation strategy is developed under the targeted minimum loss-based estimation (TMLE) framework. We present a simulation comparing the sensitivity of the first and second order estimators to the convergence rate of the initial estimators of the outcome regression and missingness score. In our simulation, the second-order TMLE improved the coverage probability of a confidence interval by up to 85%. In addition, we present a first-order estimator inspired by a second-order expansion of the parameter functional. This estimator only requires one-dimensional smoothing, whereas implementation of the second-order TMLE generally requires kernel smoothing on the covariate space. The first-order estimator proposed is expected to have improved finite sample performance compared to existing first-order estimators. In our simulations, the proposed first-order estimator improved the coverage probability by up to 90%. We provide an illustration of our methods using a publicly available dataset to determine the effect of an anticoagulant on health outcomes of patients undergoing percutaneous coronary intervention. We provide R code implementing the proposed estimator.