Bayesian Integration of Probability and Nonprobability Samples for Logistic Regression

Abstract

Abstract Probability sample (PS) surveys are considered the gold standard for population-based inference but face many challenges due to decreasing response rates, relatively small sample sizes, and increasing costs. In contrast, the use of nonprobability sample (NPS) surveys has increased significantly due to their convenience, large sample sizes, and relatively low costs, but they are susceptible to large selection biases and unknown selection mechanisms. Integrating both sample types in a way that exploits their strengths and overcomes their weaknesses is an ongoing area of methodological research. We build on previous work by proposing a method of supplementing PSs with NPSs to improve analytic inference for logistic regression coefficients and potentially reduce survey costs. Specifically, we use a Bayesian framework for inference. Inference relies on a probability survey with a small sample size, and through the prior structure we incorporate supplementary auxiliary information from a less-expensive (but potentially biased) NPS survey fielded in parallel. The performance of several strongly informative priors constructed from the NPS information is evaluated through a simulation study and real-data application. Overall, the proposed priors reduce the mean-squared error (MSE) of regression coefficients or, in the worst case, perform similarly to a weakly informative (baseline) prior that does not utilize any nonprobability information. Potential cost savings (of up to 68 percent) are evident compared to a probability-only sampling design with the same MSE for different informative priors under different sample sizes and cost scenarios. The algorithm, detailed results, and interactive cost analysis are provided through a Shiny web app as guidance for survey practitioners.