Soft calibration for selection bias problems under mixed-effects models

TL;DR

This paper introduces a soft calibration method for mixed-effects models that improves estimation efficiency and stability in selection bias correction, outperforming traditional hard calibration techniques.

Contribution

It proposes a novel soft calibration scheme that combines fixed and random effects, linking to best linear unbiased prediction and penalized propensity score estimation.

Findings

Soft calibration yields more efficient estimates than hard calibration.

The method provides valid variance estimation for inference.

Simulation and real data show the method's superiority over competitors.

Abstract

Calibration weighting has been widely used to correct selection biases in non-probability sampling, missing data, and causal inference. The main idea is to calibrate the biased sample to the benchmark by adjusting the subject weights. However, hard calibration can produce enormous weights when an exact calibration is enforced on a large set of extraneous covariates. This article proposes a soft calibration scheme, in which the outcome and the selection indicator follow mixed-effects models. The scheme imposes an exact calibration on the fixed effects and an approximate calibration on the random effects. On the one hand, our soft calibration has an intrinsic connection with best linear unbiased prediction, which results in a more efficient estimation compared to hard calibration. On the other hand, soft calibration weighting estimation can be envisioned as penalized propensity score weight estimation, with the penalty term motivated by the mixed-effects structure. The asymptotic distribution and a valid variance estimator are derived for soft calibration. We demonstrate the superiority of the proposed estimator over other competitors in simulation studies and a real-data application.