  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.
