"Publication bias" is a relatively new statistical phenomenon that only arises when one attempts through a meta-analysis to review all studies, significant or insignificant, in order to provide a total perspective on a particular issue. This has recently received some notoriety as an issue in the evaluation of the relative risk of lung cancer associated with passive smoking, following legal challenges to a 1992 Environmental Protection Agency analysis which concluded that such exposure is associated with significant excess risk of lung cancer.
We introduce a Bayesian approach which estimates and adjusts for publication bias. Estimation is based on a data-augmentation principle within a hierarchical model, and the number and outcomes of unobserved studies are simulated using Gibbs sampling methods. This technique yields a quantitative adjustment for the passive smoking meta-analysis. We estimate that there may be both negative and positive but insignificant studies omitted, and that failing to allow for these would mean that the estimated excess risk may be overstated by around 30%, both in U.S. studies and in the global collection of studies.
Geof H. Givens. D. D. Smith. R. L. Tweedie. "Publication bias in meta-analysis: a Bayesian data-augmentation approach to account for issues exemplified in the passive smoking debate." Statist. Sci. 12 (4) 221 - 250, November 1997. https://doi.org/10.1214/ss/1030037958