Tests of Independence for a $2 \times 2$ Contingency Table with Random Margins
Bibliographic record
Abstract
Fisher's exact test is commonly used for testing the hypothesis of independence between the row and column variables in a $r \times c$ contingency table. It is a ``small-sample'' test since it is used when the sample size is not large enough for the Pearsonian chi-square test to be valid. Fisher's exact test conditions on both margins of a $2 \times 2$ table leading to a hypergeometric distribution of the cell counts under independence. Moreover, it is conservative in the sense that its actual significance level falls short of the nominal level. In this paper, we modify Fisher's exact test by lifting the restriction of fixed margins and allow the margins to be random. In doing so, we propose two new tests - a likelihood ratio test in a frequentist framework and a Bayes factor test in a Bayesian framework, both of which are based on a new multinomial distributional framework. We apply the three tests on data from the Worcester Heart Attack study and compare their power functions in assessing gender difference in the therapeutic management of patients with acute myocardial infarction (AMI).
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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 | 0.045 | 0.257 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.015 | 0.001 |
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".