Bayesian analysis of a 2×2 contingency table with dependent proportions and exact sample size
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Bibliographic record
Abstract
Abstract In the analysis of a 2×2 contingency table with dependent proportions, several measures used are based on the two conditional probabilities, π1| 1 and π1| 2, and the marginal probabilities, π1+ and π+1, such as the relative risk , the marginal difference π d =π1+−π+1, the marginal ratio θ=π1+/π+1, and the odds ratio ψ=(π1 | 1/π2 | 1)/(π1 | 2/π2 | 2). In this article, we first establish the exact expressions of the distributions of π d , θ, ρ, and ψ, expressed either as multiple integrals or as closed form formulas, in a Bayesian estimation context, with a Dirichlet prior. Using these expressions, we then compute the exact sample sizes required so that the average lengths of the highest posterior density intervals of these measures, or of their maxima, are less than preset quantities. Other criteria commonly used in Bayesian statistics and Bayesian decision theory are also be considered. Keywords: Bayesian approachDifference of proportionsRisk ratioOdds ratioHighest posterior densitySample size2×2 tableNormal approximation Acknowledgements Research was partially supported by NSERC grant A9249 (Canada). The authors wish to thank J. Martin for providing very effective computation support. The authors have also benefited from various comments made by two referees on a previous version of this article.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it