How Worth the Test does its math

Worth the Test · Educational analysis · Updated September 2026

Open this calculator with the numbers it starts on — 1,000 people, a disease that affects 1% of them, and a test that is 90% sensitive and 90% specific — and it reports that a positive result carries only about an 8% chance of real disease. That figure surprises almost everyone, so this page shows exactly how the tool gets there. Every output on the site comes from the formulas below; nothing is a black box, and every number in this guide can be reproduced by typing the same inputs into the calculator.

From a population to a 2×2 table

The model starts with a whole population and splits it by prevalence — the share who actually have the condition, which is also the pre-test probability for a person drawn at random. With N people and prevalence p, the diseased group is N·p and the healthy group is N·(1 − p). For the starting scenario that is 10 people with the disease and 990 without.

Everyone is then screened, and each person lands in one of four cells of a 2×2 table — the tool's confusion matrix view — set by whether they are truly sick and how the test calls them:

• True positives (TP) = diseased × sensitivity — the sick people the test catches. Here 10 × 0.90 = 9.
• False negatives (FN) = diseased × (1 − sensitivity) — sick people the test misses. Here 10 × 0.10 = 1.
• False positives (FP) = healthy × (1 − specificity) — healthy people wrongly flagged. Here 990 × 0.10 = 99.
• True negatives (TN) = healthy × specificity — healthy people correctly cleared. Here 990 × 0.90 = 891.

Those four counts always sum back to N (9 + 1 + 99 + 891 = 1,000), and the two rows sum to the diseased and healthy groups you started with. Sensitivity and specificity are held fixed in this model; real estimates can vary with population and threshold; the split into these four cells is where prevalence quietly takes over.

Reading the table: predictive values

Sensitivity answers "if you are sick, will the test catch it?" A patient holding a positive result wants the reverse: "given this positive, am I actually sick?" That is the positive predictive value, and you read it straight off the table as the share of all positives that are true:

PPV = TP / (TP + FP) = 9 / (9 + 99) = 9 / 108 ≈ 8.3%.

Written in terms of the inputs, that is Bayes' theorem: PPV = (sens·p) / (sens·p + (1 − spec)·(1 − p)). The mirror-image quantity for a negative result is the negative predictive value, the share of negatives that are truly disease-free: NPV = TN / (TN + FN) = 891 / 892 ≈ 99.9%. A negative on this test is strongly reassuring; a positive, at this prevalence, is usually a false alarm — the base-rate fallacy expressed in one calculation.

Likelihood ratios and the odds form of Bayes

Predictive values are useful, but they move with prevalence, so they have to be recomputed for every population. Likelihood ratios strip prevalence out and describe the test alone:

LR+ = sens / (1 − spec) = 0.90 / 0.10 = 9 · LR− = (1 − sens) / spec = 0.10 / 0.90 ≈ 0.11.

An LR+ of 9 means a positive result is nine times as likely to come from a sick person as from a healthy one. To turn that into a probability you apply Bayes' rule in odds form: convert the pre-test probability to odds, multiply by the likelihood ratio, convert back.

• Pre-test odds = p / (1 − p) = 0.01 / 0.99 ≈ 0.0101.
• Post-test odds = 0.0101 × 9 ≈ 0.0909.
• Post-test probability = 0.0909 / (1 + 0.0909) ≈ 0.083.

That 8.3% is the very same PPV the 2×2 table gave. The count path and the odds path are two routes to one number — the equality the tool draws as a live Fagan nomogram, where a straight line from your pre-test probability through the likelihood ratio lands on the post-test probability. Likelihood ratios also chain, so working from pre-test to post-test probability across tests uses the likelihood ratio conditional on preceding results; multiplying marginal LRs requires independence conditional on disease status.

From detection to treatment: who is helped, who is harmed

A test result is not an outcome. The calculator's second half follows the people who test positive into treatment and asks what it actually does for them. The people treated are the positives, optionally scaled by an uptake fraction for those who decline or never follow up:

treated = (TP + FP) × uptake.

Benefit can only reach people who truly have the disease, so it is governed by the number needed to treat: helped = treated true-positives / NNT. Harm can befall anyone who undergoes the treatment or work-up, so it is spread across everyone treated: harmed = treated / NNH. Scaling the starting scenario up to 10,000 people so the tallies stay whole, with NNT = 10 and NNH = 20: 1,080 people are treated, 90 / 10 = 9 are helped, 1,080 / 20 = 54 are harmed, and the minimum-overlap illustration places 1,017 in neither modeled event.

Benefit and harm are marginal expected effects and may overlap. We preserve the NNT and NNH formulas even at extreme settings. The inputs cannot identify the overlap: in a population of T treated people, with B expected benefits and H expected harms, between max(0, B + H − T) and min(B, H) may experience both within the illustration. The bands and squares use the minimum possible overlap, labeled beside the results. For NNT = NNH = 1 in the default 1,000-person scenario, the correct marginals are 9 benefits and 108 harms among 108 treated, with all 9 benefits overlapping harms. We do not shrink those effects to force a disjoint display. NNT and NNH describe population differences, not identifiable individual destinies.

The derived model screens per benefit is N / helped. In the 10,000-person worked example this is 10,000 / 9 = 1,111.11…; rounded upward as a whole-person number, 1,112. This is not an independently estimated trial NNS. A screening-program NNS or number needed to diagnose must not be entered as a treatment NNT.

Honest numbers: absolute versus relative

The same benefit can be dressed up or down depending on which number you quote. A treatment with an NNT of 20 delivers an absolute risk reduction of 1/20 = 5% — five fewer bad outcomes per hundred treated. If the untreated risk were 10%, that identical five-point drop is a relative risk reduction of 5/10 = 50%, which sounds far more impressive while describing the very same effect. Relative figures alone routinely mislead, so the tool holds every claim to two rules: always present the absolute risk reduction, the NNT, and the baseline risk together, and pair every benefit count with the matching harm count on the same footing. Whether a given trade of helped against harmed is worth making is a value judgment this site deliberately leaves to you and your clinician; the harms and biases of screening are their own subject.

Repeat testing: why false positives pile up

Screening is rarely a one-time event, and each round is a fresh chance at a false alarm. If rounds were statistically independent, the probability of at least one false positive after n rounds is:

P(≥1 false positive) = 1 − specificity^n.

For the 90%-specific default test over ten rounds that is 1 − 0.90^10 ≈ 65.1% — a healthy person is more likely than not to be flagged at least once. The tool draws this as a rising curve in its repeat-testing view.

The independence formula is not a general upper bound. For n ≥ 1 rounds with the same false-positive probability q, but unspecified dependence, cumulative risk lies between q and min(1, nq). At 90% specificity and ten rounds, the possible range is 10%–100%, with 65.1% under independence. Repeated errors can depend on each other, and the person must remain disease-free for this interpretation. The chart also shows historical reference estimates from Elmore and Croswell; these come from different studies and are not points fitted to the user’s specificity.

Guardrails and honest rounding

A live calculator has to stay well-behaved at the edges. Probabilities are clamped to the [0, 1] range; NNT and NNH are floored at 1; and every division is guarded, so an empty group shows an em dash rather than a meaningless "NaN". Perfect specificity is a special case worth naming: when specificity is exactly 1 there are no false positives, the 1 − spec denominator vanishes, and LR+ becomes infinite — a possible positive result has PPV 100% within the exact model. If no positives are possible, PPV is undefined, and if sensitivity is also zero, LR+ is undefined.

The icon array and every displayed count are whole people, but the raw arithmetic produces fractions. Rounding each cell on its own would let the parts drift away from the whole — three groups of 33.3 rounding to 99, not 100. The tool instead uses the largest-remainder method: it rounds every cell down, then hands the leftover units to the cells with the largest fractional parts until the counts sum exactly to their parent total. Whatever the inputs, the four cells of the 2×2 add back to N and the outcome groups add back to the number treated.

How we know the math is right

All of this lives in a small set of pure functions — no interface, no styling, just inputs and outputs — in the project's src/lib/stats/ module, which carries its own written specification. Keeping the math separate from the display makes it testable, and it is tested: at the time of writing, the engine is covered by automated checks of the engine, examples, URL state, and published content. Some are worked examples pinned to exact expected values, including the 8.3% PPV and the 65.1% serial figure used above. Others are property-based "fuzz" tests that generate thousands of random scenarios — 6,000 full population cascades in one block alone — and assert that the invariants never break: people are conserved at every stage, the integer counts always reconcile to their totals, no probability escapes the [0, 1] range, and the odds path and the 2×2 path agree to within rounding. That last check is what lets this page promise the two routes to PPV give the same answer — it is verified automatically, not merely asserted here. If you want to inspect the machinery, the calculator's own worked example and references open the same numbers in the live tool.

Calculation audit and corrections — September 8, 2026

This revision corrects four substantive interpretation errors: screening NNS/NND values were being used as treatment NNTs; the mammography detection rate omitted missed cancers when used as prevalence; the repeated-testing independence curve was called an upper bound; and extreme treatment settings rescaled the requested benefit and harm. The affected examples and guides now distinguish their denominators and assumptions.

Numerical fixes cover impossible-result Bayesian updates, beta-distribution endpoints and credible intervals, fractional outcome readouts, and consistent whole-person counts across diagrams. Scenario URLs retain numerical precision. The comparison panel reports equal-denominator changes, so a change in cohort size cannot masquerade as a change in risk.

The automated checks establish arithmetic and software behavior. They do not certify medical recommendations or validate every historical estimate for a current patient population. Medical examples are educational and linked to their source; the site does not claim independent medical peer review.