Bayesian integration of probability and non-probability samples for logistic regression

Camilla Salvatore, Silvia Biffignandi, Joseph W. Sakshaug, Arkadiusz Wiśniowski, Bella Struminskaya

Research output: Contribution to journalArticlepeer-review


Probability sample 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 non-probability sample 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 probability samples with non-probability samples to improve analytic inference for logistic regression coefficients and potentially reduce survey costs. Specifically, we consider a Bayesian framework, where inference is based on a probability survey with small sample size and supplementary auxiliary information from a less-expensive (but potentially biased) non-probability sample survey fielded in parallel is provided naturally through the prior structure. The performance of several strongly-informative priors constructed from the non-probability sample 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 doesn’t utilize any non-probability information. Potential cost savings (of up to 68%) are evident compared to a probability-only sampling design with the same MSE for different informative priors under different sample size and cost scenarios. The algorithm, detailed results, and interactive cost analysis are provided through a Shiny web app as guidance for survey practitioners.
Original languageEnglish
JournalJournal of Survey Statistics and Methodology
Publication statusAccepted/In press - 3 Oct 2023


  • Bayesian Inference
  • Data Integration
  • Online Access Panel
  • Selection Bias
  • Web Survey


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