Bayesian analysis of a 2×2 contingency table with dependent proportions and exact sample size
Bibliographic record
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.018 | 0.088 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".