Field

Two tests, a threshold, and the rate they are read against

A predictive value is computed from a sensitivity, a specificity and a prevalence, and each of the three is softer than it is quoted as being. Two positives from a 90/95 test on a one-in-a-thousand condition are worth 24.49% if the tests' errors are independent, 10.16% at a correlation of 0.1, and 3.16% at 0.5 — barely more than the 1.77% one positive was worth. The sensitivity and specificity are not two properties of a test but one property read at a threshold somebody chose, and the threshold that minimises harm moves by three standard deviations of the score across the prevalences such a test is used at. The prevalence itself is usually estimated from the same test's positive rate, which at one in a thousand reads 5.09%.

What a second positive is worth, prevalence 0.10%. A 90% sensitive, 95% specific test. One positive gives 1.77%. Two independent positives give 24.49%, which is what multiplying the likelihood ratios says. At a correlation of 0.1 between the tests' errors it is 10.16%, and at 0.5 it is 3.16%.

The second test that is not a second opinion

Two positives from a 90/95 test on a one-in-a-thousand condition give a 24.49% chance of disease if the tests are independent. At a correlation of 0.1 between their errors it is 10.16%, and at 0.5 it is 3.16% — barely more than the 1.77% one positive was worth.

The operating characteristic, and the point that minimises harm at 0.10% prevalence. The published pair — 90% sensitive, 95% specific — is the open mark. With a miss costing 100 times a false alarm and a prevalence of 0.10%, the threshold that minimises expected cost sits at 74.7% sensitivity and 98.81% specificity, with a predictive value of 5.9%.

The test is a point somebody chose

A test reported as 90% sensitive and 95% specific is not two properties of a test. It is one property read at a threshold, and the threshold that minimises harm runs from 3.05 standard deviations of the score at a prevalence of one in ten thousand to −0.12 at one in two — 45% of cases detected at one end and 99.9% at the other.

Estimating a prevalence of 0.10% from 1,000 tests. The positive rate reads 5.09%, which is 50.9 times the truth. The Rogan-Gladen correction averages 0.100% — unbiased — with a standard deviation of 0.820 points against the positive rate's 0.695, and it comes out negative on 48.6% of samples.

The prevalence the test has to estimate

Every predictive value takes a prevalence as given, and the prevalence is usually estimated from the same test's positive rate. At a true prevalence of one in a thousand that rate reads 5.09% — fifty times the truth — and the correction that inverts it is unbiased, 18% more variable, and negative on 48.6% of samples of a thousand.

Thirty surveys of a thousand people, true prevalence 0.1%, test 90% sensitive and 95% specific. Each row is one survey. The thin bar is the corrected estimate ± 1.96 standard errors; the heavy bar is the exact interval for the positive rate carried through the correction; the tick is the one-sided 95% upper limit. 30 of the thirty printed intervals reach below zero and 0 lie entirely below it.

An upper limit is the finding

A survey of a thousand people with a 90%-sensitive, 95%-specific test, estimating a prevalence of one in a thousand, prints a corrected 95% interval that covers the truth 95.17% of the time — and reaches below zero in 97.95% of surveys and lies entirely below zero in 3.35%. Coverage is not the problem. The problem is what the interval can say, and the honest summary of a typical survey is one number: the prevalence is below 1.64%. Bringing that limit down to twice the truth takes 184,521 people at this specificity and 5,083 with a test that never gives a false positive.

A threshold that jumps

When a test's scores are equally spread in the healthy and the diseased, the harm-minimising threshold slides smoothly as the prevalence changes. Make the diseased scores twice as spread — the ordinary shape of a group that mixes mild and severe cases — and keep the published 90/95 pair, and the best threshold flags 81.1% of healthy people at a prevalence just under 25.7% and every healthy person just above it. At three times the spread it jumps from 45.7% to everyone at 13.0%. A threshold that jumps cannot be given as a formula; it has to be given with the second-best point beside it.

Where a thousand validation studies of 100 diseased and 300 healthy put the threshold, at a prevalence of 26%. The best threshold flags 100.0% of healthy people. Reading the study's own operating curve, 7.7% of studies recommend flagging nearly everyone, and the recommended thresholds cost 15.4% more than the best on average; fitting a normal curve to each group's scores first, 47.4% do, at an average excess of 2.5%.

The threshold a validation study can see

When diseased scores are twice as spread as healthy ones, the harm-minimising threshold jumps near a prevalence of 26% from an interior cut to flagging everyone, because the true operating curve keeps climbing in a tail where few diseased people score. A validation study of a hundred diseased and three hundred healthy cannot see that tail: its own curve reaches 100% sensitivity at its lowest diseased score and goes flat. Just past the jump, only 7.7% of such studies recommend flagging everyone, and their recommended thresholds cause 15.4% more harm than the best on average. Fitting a normal curve to each group first sees the tail: 47.4% of studies then recommend flagging everyone and the excess harm falls to 2.5%. More healthy people in the study change nothing; more diseased people do.

A test whose tail is not normal

A validation study that fits a normal curve to each group saw the tail where a threshold jumps — when the diseased scores really were normal. Give the same published test a diseased group with no lower tail at all, and the binormal fit invents one: 89.7% of studies recommend flagging everyone, adding 67.08 false alarms' worth of harm per hundred people to a best threshold that costs 5.61. Give it a lump of mild cases instead, and the binormal fit misses the tail that is there and does worse than the study's own curve. A normality test on the hundred diseased scores sees the missing tail every time and the lump 60.4% of the time, and a study that fits the binormal model only when normality is not rejected is never the worst of the four.

A threshold reported with its interval

Resample a validation study of a hundred diseased and three hundred healthy people, refit the binormal model each time, and read off the threshold each resample recommends. Far from the prevalence at which the best threshold jumps to flagging everyone, the interval is tight — flagging 12% to 24% of healthy people at a prevalence of 5%. Near the jump, the median study's interval runs from flagging under half of the healthy to flagging everyone, and 91% of intervals cross the jump. The two ends differ by fifteen false alarms' worth of harm per hundred people, and no rule for choosing inside the interval beats the study's own choice at every prevalence.

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