  Valid statistical inference is challenging when the sample is subject to
unknown selection bias. Data integration can be used to correct for selection
bias when we have a parallel probability sample from the same population with
some common measurements. How to model and estimate the selection probability
or the propensity score (PS) of a non-probability sample using an independent
probability sample is the challenging part of the data integration. We approach
this difficult problem by employing multiple candidate models for PS combined
with empirical likelihood. By incorporating multiple propensity score models
into the internal bias calibration constraint in the empirical likelihood
setup, the selection bias can be eliminated so long as the multiple candidate
models contain a true PS model. The bias calibration constraint under the
multiple PS models is called multiple bias calibration. Multiple PS models can
include both missing-at-random and missing-not-at-random models. Asymptotic
properties are discussed, and some limited simulation studies are presented to
compare the proposed method with some existing competitors. Plasmode simulation
studies using the Culture \& Community in a Time of Crisis dataset demonstrate
the practical usage and advantages of the proposed method.
