Health researchers are increasingly adopting nonprobability sampling strategies in survey studies. However, the participation mechanism in such samples is unknown and estimated target parameters and exposure-outcome associations obtained from nonprobability samples can be biased. Current approaches developed to support statistical inference from nonprobability samples are unable to accommodate more than one reference sample. In this paper, we propose a general framework to address participation bias in nonprobability samples using multiple reference surveys. Previously published methods that use one reference survey are special cases within this framework. We focus primarily on the calibration estimators, another important special case in the proposed framework. These estimators have greater flexibility in situations with limited access to survey microdata and are straightforward for practical implementation. We describe two methods for variance estimation that account for all sources of variability of the proposed estimators: (1) the Taylor linearization method, which provides an analytic formula for the variance estimator, and (2) the leave-one-out jackknife method, a replication estimator. We assess the performance of the various methods through an extensive simulation study, which demonstrated satisfactory performance of the raking ratio calibration estimator in situations with highly dispersed participation probabilities in nonprobability samples and markedly smaller variance estimates for continuous outcomes. Finally, we illustrate the application of these methods using data from a real-world study of working adults in Canada.
Landsman et al. (Sun,) studied this question.