Statistical Analysis
Youden’s J Index, Likelihood Ratios (LR+ / LR-) & Fagan Post-Test Probability
Evaluate ROC cutoff optimality via Youden’s J = Sens + Spec - 1, compute Positive and Negative Likelihood Ratios, and update pre-test prevalence to post-test probability via Fagan’s odds rule.
Connecting ROC cutoff selection (Youden’s J) to Fagan Bayesian odds updating
When choosing a cutoff on a continuous biomarker or fraud score, maximizing accuracy misleads under rare base rates because labeling everyone negative achieves 99% accuracy at 1% prevalence. W. J. Youden’s (1950) J statistic—J = Sensitivity + Specificity - 1—measures the vertical distance between the ROC curve and the 45-degree chance diagonal, ranging from 0 (useless coin flip) to 1 (perfect separation) independently of disease prevalence.
To translate test results into patient- or transaction-level probabilities, Fagan’s (1975) nomogram multiplies Pre-Test Odds = P_pre / (1 - P_pre) by the Positive Likelihood Ratio LR+ = Sens / (1 - Spec) (or Negative Likelihood Ratio LR- = (1 - Sens) / Spec) to obtain Post-Test Odds, then converts back via P_post = Odds_post / (1 + Odds_post).
| Diagnostic Metric | Formula | Assay (Sens=90%, Spec=85%, Prev=20%) |
|---|---|---|
| Youden’s J Statistic | Sens + Spec - 1 | 0.7500 |
| Positive Likelihood Ratio (LR+) | Sens / (1 - Spec) | 6.0000 |
| Negative Likelihood Ratio (LR-) | (1 - Sens) / Spec | 0.1176 |
| Post-Test Prob Given + / - | Odds_pre*LR / (1 + Odds_pre*LR) | 0.6000 (60.0%) / 0.0286 (2.86%) |
Youden’s J & Positive Likelihood Ratio Formulas
Compute Sens + Spec - 1 for ROC threshold ranking and Sens / (1 - Spec) (guarded when Spec = 1) for Bayesian odds multiplication.
=IF(B3 = 1, "INF_LR", ROUND(B2 / (1 - B3), 4))Fagan Post-Test Probability Formula
Convert pre-test prevalence B4 to odds_pre = B4 / (1 - B4), multiply by LR+ or LR-, and map odds_post / (1 + odds_post) back to [0, 1] probability.
=LET(odds_pre, B4 / (1 - B4), lr_pos, B2 / (1 - B3), odds_post, odds_pre * lr_pos, ROUND(odds_post / (1 + odds_post), 4))Try the example
Use the original values or change them to test your understanding.