Soft calibration for selection bias problems under mixed-effects models

Calibration weighting has been widely used for correcting selection biases in nonprobability sampling, missing data, and causal inference. The main idea is to adjust subject weights that produce covariate balancing between the biased sample and the benchmark. However, hard calibration can produce enormous weights when enforcing the exact balancing of a large set of unnecessary covariates. This is common in situations with mixed effects, e.g., clustered data with many cluster indicators. This article proposes a soft calibration scheme when the outcome and selection indicator follow the mixed-effects models by imposing exact balancing on the fixed effects and approximate balancing on the random effects. We show that soft calibration has intrinsic connections with the best linear unbiased prediction and penalized optimization. Thus, soft calibration can produce a more efficient estimation than hard calibration and exploit the restricted maximum likelihood estimator for selecting the tuning parameter under the mixed-effects model. Furthermore, the asymptotic distribution and a valid variance estimator are derived for soft calibration. We demonstrate the superiority of the proposed estimator over other competitors under a variety of simulation studies and a real-data application.