Decomposing Pearson's χ<sup>2</sup> test: A linear regression and its departure from linearity
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Bibliographic record
Abstract
Abstract In case–control genetic association studies, a standard practice is to perform the Cochran‐Armitage (CA) trend test under the assumption of the additive model because of its robustness. We could even identify situations in which it outperformed the analysis model consistent with the underlying inheritance mode. In this article, we analytically reveal the statistical basis that leads to the phenomenon. By elucidating the origin of the CA trend test as a linear regression model, we decompose Pearson's χ 2 ‐test statistic into two components—one is the CA trend test statistic that measures the goodness of fit of the linear regression model, and the other measures the discrepancy between data and the linear regression model. Under this framework, we show that the additive coding scheme, as well as the multiplicative coding scheme, increases the coefficient of determination of the regression model by increasing the spread of data points. We also obtain the conditions under which the CA trend test statistic equals the MAX statistic and Pearson's χ 2 ‐test statistic.
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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 | 0.000 | 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) | 0.000 | 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