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.

翻译：校准权重被广泛用于纠正非概率抽样、缺失的数据和因果推断中的选择偏差,主要想法是调整在偏差抽样和基准之间产生共差平衡的主体权重,然而,硬校准在对大量不必要的共差进行精确平衡时可产生巨大的权重,这在具有混合效应的情况下是常见的,例如,集成数据与许多组群指标相结合。本条提议在结果和选择指标遵循混合效应模型时采用软校准方案,对固定效应进行精确平衡,对随机效应进行近似平衡。我们表明软校准与最佳线性不偏差预测和惩罚性优化有内在联系。因此,软校准可产生比硬校准更有效率的估算,并利用有限的最大可能性估算器来选择混合效应模型下的调幅参数。此外,软校准还得得出无症状分布和有效差异估测仪。我们展示了在一系列模拟研究和实际数据应用中拟议的估测算器优于其他竞争者。