Bayesian inference for nonprobability samples with nonignorable missingness.

Nonprobability samples, especially web survey data, have been available in many different fields. However, nonprobability samples suffer from selection bias, which will yield biased estimates. Moreover, missingness, especially nonignorable missingness, may also be encountered in nonprobability samples. Thus, it is a challenging task to make inference from nonprobability samples with nonignorable missingness. In this article, we propose a Bayesian approach to infer the population based on nonprobability samples with nonignorable missingness. In our method, different Logistic regression models are employed to estimate the selection probabilities and the response probabilities; the superpopulation model is used to explain the relationship between the study variable and covariates. Further, Bayesian and approximate Bayesian methods are proposed to estimate the response model parameters and the superpopulation model parameters, respectively. Specifically, the estimating functions for the response model parameters and superpopulation model parameters are utilized to derive the approximate posterior distribution in superpopulation model estimation. Simulation studies are conducted to investigate the finite sample performance of the proposed method. The data from the Pew Research Center and the Behavioral Risk Factor Surveillance System are used to show better performance of our proposed method over the other approaches. [ABSTRACT FROM AUTHOR]