The Fisher Transform of the Pearson Product Moment Correlation Coefficient and Its Square: Cumulants, Moments, and Applications
Bibliographic record
Abstract
This paper extends results on the distribution of the Fisher transform of the correlation coefficient (Fisher, 1921 Fisher , R. A. ( 1921 ). On the “probable error” of a coefficient of correlation deduced from a small sample . Metron 1 : 1 – 32 . [Google Scholar]). Approaches to obtain exact moments of the Fisher transform for both null and non-null correlations are presented. We extend the classic series expansion formulae of Hotelling (1953 Hotelling , H. ( 1953 ). New light on the correlation coefficient and its transforms . Journal of the Royal Statistical Society, Ser. B 15 : 192 – 232 . [Google Scholar]) for the moments of the Fisher transform. These results are considered in the context of quadratic functions of the Fisher transform. Some applications of these results are discussed in the context of correlational hypothesis tests and confidence intervals, and a Monte Carlo experiment is used to demonstrate how application of these results impact the small sample performance of select tests on correlations.
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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 | 0.004 | 0.036 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.007 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".