Dalzell's theorem and the analysis of proportions: A methodological note
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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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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.006 | 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