Optimal Probability Weights for Inference With Constrained Precision

Probability weights are used in many areas of research including complex survey designs, missing data analysis, and adjustment for confounding factors. They are useful analytic tools but can lead to statistical inefficiencies when they contain outlying values. This issue is frequently tackled by replacing large weights with smaller ones or by normalizing them through smoothing functions. While these approaches are practical, they are also prone to yield biased inferences. This article introduces a method for obtaining optimal weights, defined as those with smallest Euclidean distance from target weights among all sets of weights that satisfy a constraint on the variance of the resulting weighted estimator. The optimal weights yield minimum-bias estimators among all estimators with specified precision. The method is based on solving a constrained nonlinear optimization problem whose Lagrange multipliers and objective function can help assess the trade-off between bias and precision of the resulting weighted estimator. The finite-sample performance of the optimally weighted estimator is assessed in a simulation study, and its applicability is illustrated through an analysis of heterogeneity over age of the effect of the timing of treatment-initiation on long-term treatment efficacy in patient infected by human immunodeficiency virus in Sweden.