Dalzell's theorem and the analysis of proportions: A methodological note
Bibliographic record
Abstract
The golden section is a well-known proportion that occurs when something (e.g. a line) is divided into two unequal parts such that the smaller (m) is to the larger (M) as the larger is to the sum of the two (i.e. m/M = M/(M + m) = .618). Dalzell's theorem holds that the absolute value of the difference between M/(M + m) and .618 will tend to be smaller than the corresponding difference between m/M and .618. This means that the use of M/(M + m) ratios leads to results that are more supportive of the golden section hypothesis than does the use of m/M ratios. Notice that M/(M + m) corresponds to the proportion of Ms that will occur; while m/M corresponds to the odds that m will occur. While these are mathematically equivalent, in practice they may lead to different interpretations of the same data. Although originally envisaged as applying to the golden section, Dalzell's theorem may have implications for any study that uses either a proportion or the odds as a dependent measure. The use of proportions may produce results that are closer to a predicted value than will the use of the odds as a dependent measure.
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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 | 0.065 | 0.176 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.004 |
| Bibliometrics | 0.007 | 0.009 |
| Science and technology studies | 0.004 | 0.044 |
| Scholarly communication | 0.012 | 0.018 |
| Open science | 0.005 | 0.009 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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".