Four-Verdict Statistical Inference: Extending Royall’s Evidential Paradigm to Contradiction Between Sources

When several sources bear on two hypotheses, Royall’s three evidential outcomes (strong for one, strong for the other, weak) leave no room for a fourth state: some sources are strong evidence for one hypothesis and others for the other. We integrate four verdicts (supported, refuted, undetermined, contradicted) in one likelihood-based report and state which error probability of each verdict is bounded in finite samples, which vanishes only asymptotically, and when “contradicted” stops describing a real conflict between source parameters. Spurious contradiction has probability at most \( m/c \) for \( m \) sources and per-source threshold \( c \), under any dependence, for any correctly specified family. With a truth midway between two simple hypotheses the contradiction probability tends to \( 1-2^{1-m} \); a guarded rule keeps it at most \( m\Phi (-\sqrt{2\ln\,c}) \) for marginally Gaussian sources with any common mean and any dependence, and at most \( m(m-1)\Phi (-2\sqrt{\ln\,c}) \) under pairwise joint normality with non-negative covariances. For composite hypotheses an integrated-over-maximised likelihood ratio gives the bound \( m/c \) for heterogeneous parameters within one hypothesis. A preregistered simulation confirmed the bounds. In the Bacillus Calmette–Guérin (BCG) vaccine trials the verdict for “risk ratio at most 0.5” against “at least 1” is “contradicted” under normal, binomial-profile and conditional likelihoods; because these hypotheses leave an indifference zone, this does not establish that the trials’ true effects conflict.