The “test said negative, but the patient is crashing” moment
You have a new biomarker: procalcitonin for bacterial sepsis. The paper reports sensitivity 85%, specificity 70%. Your patient is septic — but the test came back negative. Do you trust it? Should you treat anyway?
The answer depends on pretest probability. If sepsis is likely (pretest 70%), a negative test only slightly lowers the probability; if it’s unlikely (pretest 10%), a negative test is reassuring. This is the core of Bayesian reasoning — and it’s where most clinicians stumble.
The 2×2 table for diagnostics
Disease (+) | Disease (−) | Total | |
Test (+) | TP (true positive) | FP (false positive) | TP + FP |
Test (−) | FN (false negative) | TN (true negative) | FN + TN |
Total | TP + FN | FP + TN | N |
From this table, you calculate:
- Sensitivity = TP / (TP + FN) — “How many sick patients test positive?”
- Specificity = TN / (FP + TN) — “How many healthy patients test negative?”
- Positive Predictive Value (PPV) = TP / (TP + FP) — depends on prevalence.
- Negative Predictive Value (NPV) = TN / (FN + TN) — depends on prevalence.
ROC curves: sensitivity vs specificity trade‑off
The ROC (Receiver Operating Characteristic) curve plots sensitivity (y) vs 1 − specificity (x) at every possible cutoff of the test.
- AUC = 0.5: test is useless (same as coin flip).
- AUC = 0.7–0.8: acceptable.
- AUC = 0.8–0.9: excellent.
- AUC > 0.9: near perfect.
In the ICU, biomarker AUC values of 0.75–0.85 are common. AUC is a summary of discrimination, but it does not tell you which cutoff is best for clinical use. You must choose a cutoff based on the clinical context: in sepsis screening, high sensitivity (low false negatives) is preferred; in confirming infection before antibiotics, high specificity (low false positives) is preferred.
Likelihood ratios: the Bayesian shortcut
LR is the most clinically useful metric because it can be applied to any pretest probability.
- Positive LR (LR+) = sensitivity / (1 − specificity)
- Negative LR (LR−) = (1 − sensitivity) / specificity
How to use LR (quick mental math)
- Estimate pretest probability (e.g., 30%).
- Convert to pretest odds = p / (1 − p) = 0.30 / 0.70 = 0.43.
- Multiply by LR: posttest odds = pretest odds × LR.
- Convert back to probability = odds / (1 + odds).
Example: Pretest probability of sepsis = 30%. Procalcitonin LR+ = 2.5.
- Pretest odds = 0.43.
- Posttest odds = 0.43 × 2.5 = 1.075.
- Posttest probability = 1.075 / 2.075 ≈ 52%.
A positive test raises probability from 30% to 52% — modest. If pretest was 70%, posttest = 85%. The same test behaves differently in different clinical contexts.
Appraisal checklist for diagnostic studies:
- Was the study designed as a diagnostic accuracy study? (Not an interventional trial disguised as diagnostic.)
- Was the reference standard clearly defined and applied independently? (Blinded to index test result.)
- Was the index test interpreted independently from the reference standard? (To avoid incorporation bias.)
- Was the study population representative of your clinical setting? (Spectrum bias if only very sick or very well patients enrolled.)
- Were confidence intervals reported for sensitivity, specificity, and AUC?
- Was the cutoff pre‑specified, or was it chosen after seeing the data? (Data‑driven cutoff = overfitting.)
Go deeper
- StatPearls – “Sensitivity and Specificity” (NBK557653): free, clear definitions, calculations, and clinical examples. https://www.ncbi.nlm.nih.gov/books/NBK557653/
- BMJ Statistics Notes – “Diagnostic tests” (Altman & Bland): concise on LR, PPV/NPV, and ROC interpretation. https://www.bmj.com/content/bmj_stats_notes
- OpenIntro Statistics – Chapter 11 “Categorical data and diagnostic testing” (free PDF): worked examples with R code for sensitivity, specificity, and ROC curves. https://www.openintro.org/stat/textbook.php
- PMC6613434 – “Likelihood ratios in diagnostic testing” (free full text): practical clinical guide to using LR in practice. https://doi.org/10.1186/s13613-019-0554-3 (PMC free)
Next: Chapter 15 — Pragmatic trials