Youden index instrument
| Definition | True positive rate minus false positive rate at a threshold, whose maximum over thresholds is the pointwise ceiling that prunes the formula search. Chapter 5 section 5.4. Also Youden. |
|---|---|
| Example | True positive rate 0.75 and false positive rate 0.25 give Youden index 0.5. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id youden-index, kind instrument. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 5.4. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/YoudenF1.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 5.4.
\[\begin{gathered} F_1^{\max}\ \le\ \sup_{t\in[J,\,1]}\ \frac{2t\pi}{t\pi+\pi+(t-J)(1-\pi)},\qquad J=\max_{\tau}\big(\mathrm{TPR}-\mathrm{FPR}\big),\qquad \pi=\text{prevalence}, \\ J\le 2A-1\ \text{when the ROC curve is concave, and not in general.} \end{gathered}\]
Conditions
- True positive rate minus false positive rate at a threshold, whose maximum over thresholds bounds the best F1 the score can reach at any threshold. The bound is exact for concave ROC curves and fails for a curve at AUROC 0.75.
- The pointwise ceiling on a node of the formula search is proved, and using it to prune the node’s descendants is a heuristic that missed the optimum on two of five datasets in the rerun.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Youden, index for rating diagnostic tests, 1950, as chapter 5 section
5.4 of Data Mining as Observation reads it, with the pointwise
ceiling in theory-radar/paper/astar_paper.tex:94-110.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/YoudenF1.lean,
theorems f1_eq, f1_le_of_youden,
auroc_eq, youden_eq,
youden_exceeds_auroc_form, at observation-data-mining
f3914f0; what the check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation chapters 5.
Related
ROC curve; AUROC; F1; safe pruning.
See also
Book equations stated beside the entry’s terms, not defining it: 0.28, 0.38.
Sources-table rows that share a record with the entry without naming it: chapter 5 section 5.4, chapter 6 section 6.3.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.