Empirical Bayes learning from selectively reported confidence intervals

We develop a statistical framework for empirical Bayes learning from selectively reported confidence intervals, applied here to provide context for interpreting results published in MEDLINE abstracts. A collection of 326,060 z-scores from MEDLINE abstracts (2000-2018) provides context for interpreting individual studies; we formalize this as an empirical Bayes task complicated by selection bias. We address selection bias through a selective tilting approach that extends empirical Bayes confidence intervals to truncated sampling mechanisms. Sign information is unreliable (a positive z-score need not indicate benefit, and investigators may choose contrast directions post hoc), so we work with absolute z-scores and identify only the distribution of absolute signal-to-noise ratios (SNRs). Our framework provides coverage guarantees for functionals including posterior estimands describing idealized replications and the symmetrized posterior mean, which we justify decision-theoretically as optimal among sign-equivariant (odd) estimators and minimax among priors inducing the same absolute SNR distribution.