How do you calculate sensitivity and specificity from a 2×2 table?

Sensitivity = TP / (TP + FN) — the share of truly diseased people the test flags positive. Specificity = TN / (TN + FP) — the share of truly healthy people it clears. TP is true positives, FN false negatives, TN true negatives, FP false positives. Both are properties of the test itself and do not depend on how common the disease is.

Why does PPV drop when disease prevalence is low, even for a good test?

PPV depends on prevalence because a rare disease means most positives come from the large healthy group. At 2% prevalence a test with 85% sensitivity and 98% specificity has a PPV near 46%: the few true positives are swamped by false positives drawn from the many healthy people tested. Raise prevalence and PPV climbs sharply.

Are PPV and precision the same thing?

Numerically, yes: precision in machine learning and positive predictive value in medicine are both TP / (TP + FP). The difference is framing. Precision is usually reported at the dataset's own prevalence, while PPV is routinely re-projected onto the real-world prevalence of the population being tested, which this calculator does via Bayes' theorem.

How do you calculate a positive likelihood ratio (LR+)?

LR+ = sensitivity / (1 − specificity). It says how many times more likely a positive result is in a diseased person than in a healthy one. LR+ above 10 is strong rule-in evidence; around 1 the test is useless. The negative likelihood ratio LR− = (1 − sensitivity) / specificity works the same way for rule-out, where below 0.1 is strong.

What is post-test probability and how does a Fagan nomogram work?

Post-test probability is the chance a patient has the disease after you know their test result. Convert pre-test probability (prevalence) to odds, multiply by the likelihood ratio, then convert back to a probability. A Fagan nomogram is the graphical shortcut: draw a line from pre-test probability through the LR to read off post-test probability. This tool computes it directly.

Why does this calculator use Wilson confidence intervals instead of the simpler Wald ones?

The Wald (normal approximation) interval misbehaves near 0% and 100% and with small samples — exactly the situation in diagnostic validation studies, where sensitivity is often measured on a few dozen diseased cases. The Wilson score interval stays inside 0–100% and gives sensible bounds even at extremes, which is why Altman's Statistics with Confidence recommends it.

What is the diagnostic odds ratio and Youden's J?

The diagnostic odds ratio is (TP·TN) / (FP·FN), equal to LR+ divided by LR−; a higher value means better overall discrimination, and 1 means the test is no better than chance. Youden's J = sensitivity + specificity − 1 runs from 0 (useless) to 1 (perfect) and is a common single-number summary used when choosing a cut-off.

Does this replace a confusion matrix calculator for machine learning?

They overlap but serve different needs. A confusion-matrix tool reports precision, recall, F1 and MCC at the sample's own prevalence. This calculator adds the clinical layer: likelihood ratios, the diagnostic odds ratio, Wilson confidence intervals, and prevalence-adjusted PPV/NPV with post-test probability — the answers a clinician or epidemiologist needs when the deployment prevalence differs from the study.

Health · Epidemiology

Diagnostic Test Statistics Calculator — Sensitivity, Specificity, PPV, NPV & Likelihood Ratios

Enter a 2×2 test table and get sensitivity, specificity, PPV, NPV, positive and negative likelihood ratios, the diagnostic odds ratio and Youden's J — each with its formula and a 95% confidence interval. Then re-project PPV, NPV and post-test probability onto any disease prevalence. Free, no signup, runs in your browser.

By Induwara Ashinsana— Executive Director, Ryzera TechnologiesUpdated Jul 10, 2026

Evaluate a diagnostic test2×2 or rates

Bayes cross-checked

True positives (TP)

Test positive · disease present

False positives (FP)

Test positive · disease absent

False negatives (FN)

Test negative · disease present

True negatives (TN)

Test negative · disease absent

N = 200Diseased = 55Healthy = 145Test+ = 50Test− = 150

Target disease prevalence (optional)

Re-projects PPV, NPV and post-test probability onto a population with this disease prevalence. Leave blank to use the sample's own prevalence (27.5%).

Try a preset

Sensitivity

85.45%

95% CI 73.84%–92.44%

Specificity

97.93%

95% CI 94.09%–99.29%

Accuracy

94.5%

95% CI 90.42%–96.9%

PPV (sample)

94%

95% CI 83.78%–97.94%

NPV (sample)

94.67%

95% CI 89.83%–97.27%

Youden's J

0.834

Likelihood ratios & odds

LR+

41.3

Sens / (1 − Spec)

Strong — a positive result markedly raises the probability of disease.

LR−

0.149

(1 − Sens) / Spec

Moderate — a negative result meaningfully lowers disease probability.

Diagnostic OR

278

(TP·TN) / (FP·FN)

LR+ ÷ LR− — one number for overall discrimination.

Youden's J

0.834

Sens + Spec − 1

Sens + Spec − 1, range 0–1.

PPV / NPV vs prevalence

Prevalence	PPV	NPV
1%	29.44%	99.85%
5%	68.49%	99.22%
10%	82.11%	98.38%
25%	93.23%	95.28%
50%	97.64%	87.07%

Formulas: Altman & Bland (BMJ 1994) for sensitivity/specificity/PPV/NPV; Deeks & Altman (BMJ 2004) for likelihood ratios and post-test probability; Wilson score for the 95% confidence intervals. Full citations and worked examples are below the tool.

How it works

A diagnostic test is scored against a reference (“gold”) standard. Cross-tabulating the two gives four counts: true positives (TP), false positives (FP), false negatives (FN) and true negatives (TN). Every statistic below is a ratio of those cells or of the rates derived from them. Definitions follow Altman & Bland (BMJ 1994) and Deeks & Altman (BMJ 2004).

Sensitivity = TP / (TP + FN) and specificity = TN / (TN + FP). These describe the test itself and stay fixed no matter how common the disease is.
PPV = TP / (TP + FP) and NPV = TN / (TN + FN) are the predictive values at the sample's own prevalence. Unlike sensitivity and specificity, they move with prevalence.
Likelihood ratiossummarise a result's weight of evidence: LR+ = Sens / (1 − Spec) and LR− = (1 − Sens) / Spec. The diagnostic odds ratio = (TP·TN) / (FP·FN) = LR+ / LR−, and Youden's J = Sens + Spec − 1.
Prevalence adjustmentuses Bayes' theorem to recompute predictive values for any target prevalence p: PPV(p) = Sens·p / (Sens·p + (1 − Spec)·(1 − p)) and NPV(p) = Spec·(1 − p) / (Spec·(1 − p) + (1 − Sens)·p). The post-test probability after a positive result is exactly PPV(p); equivalently, pre-test odds × LR+ converted back to a probability — the Fagan nomogram in algebra.
Confidence intervals. Sensitivity, specificity, PPV and NPV each carry a 95% Wilson score interval on their own denominator (for example, sensitivity uses n = TP + FN). Wilson is used instead of the Wald approximation because it stays inside 0–100% and behaves at the extremes and small samples typical of validation studies (Altman, Statistics with Confidence).

A built-in cross-check proves the two engines agree: projecting PPV with Bayes at the sample's own prevalence reproduces the direct count formula TP / (TP + FP) to machine precision. All arithmetic uses exact double-precision values; only the display is rounded, so chained steps never accumulate rounding drift.

Worked examples

Rapid dengue NS1 antigen vs PCR (counts mode)

Counts: TP = 47, FP = 3, FN = 8, TN = 142 (N = 200)
Sensitivity = 47 / 55 = 0.8545 → 85.5%
Specificity = 142 / 145 = 0.9793 → 97.9%
PPV (sample) = 47 / 50 = 94.0%; NPV = 142 / 150 = 94.7%
Sample prevalence = 55 / 200 = 27.5%; Accuracy = 189 / 200 = 94.5%
LR+ = 0.8545 / (1 − 0.9793) = 41.3; LR− = 0.1455 / 0.9793 = 0.149
Youden's J = 0.8545 + 0.9793 − 1 = 0.834
Re-projected to community prevalence p = 2%:
PPV(0.02) = (0.8545·0.02) / (0.8545·0.02 + 0.0207·0.98) = 45.7%

Takeaway: A positive rapid test that is 94% reliable in an outbreak sample is right less than half the time in a 2%-prevalence off-season — the reason predictive values must be read against real-world prevalence.

Screening test at very low prevalence (rates mode)

Sensitivity = 99%, Specificity = 95%, target prevalence = 0.1% (1 in 1000)
LR+ = 0.99 / 0.05 = 19.8; LR− = 0.01 / 0.95 = 0.0105
PPV(0.001) = (0.99·0.001) / (0.99·0.001 + 0.05·0.999) = 1.94%
NPV(0.001) = (0.95·0.999) / (0.95·0.999 + 0.01·0.001) ≈ 100.0%

Takeaway: Even a '99% accurate' test yields a positive that is correct only ~1.9% of the time when the disease affects 1 in 1000 — the classic base-rate result behind cautious interpretation of population screening.

Edge case: perfect specificity (counts mode)

Counts: TP = 50, FP = 0, FN = 10, TN = 40 (N = 100)
Sensitivity = 50 / 60 = 83.3%; Specificity = 40 / 40 = 100%
PPV = 50 / 50 = 100%; NPV = 40 / 50 = 80.0%
LR+ = 0.833 / (1 − 1) = division by zero → ∞ (a positive is conclusive)
LR− = (1 − 0.833) / 1 = 0.167; Youden's J = 0.833

Takeaway: With zero false positives, LR+ is infinite, so the tool shows ∞ rather than a misleading number or NaN. Every division-by-zero cell is handled this way.

Frequently asked questions

Sources & references

The formulas on this page are textbook-standard and vendor-neutral. They were last reconciled against the cited references on 2026-07-10. If a result ever disagrees with a source you trust, email me the inputs and I will investigate.

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