Normal Distribution
nothing is called 95% until it has been counted
Concept

Sensitivity and specificity — where it appears

The two rates that describe a yes-or-no classifier: the share of true cases it flags and the share of non-cases it clears. A forecaster that only ever says 0 or 1 is such a classifier, and the area under its ROC curve is exactly the average of the two.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

screening · Baserate What a positive test means, sensitivity 90%, specificity 95%. At a prevalence of one in a thousand, 98 of every hundred positives are false. At one in 10, 33 are. The test has not changed.

What a positive test is worth

A test that is 90% sensitive and 95% specific sounds accurate. For a condition affecting one person in a thousand, 98% of its positive results are wrong, and a worse test on a commoner condition beats a better test on a rare one.

paradox · Baserate 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.

screening · Baserate 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.

screening · Baserate One curve, and two forecasters that are each a single point. The ROC curve of an honest forecaster whose signal has correlation 1 with the latent state — every threshold on its probability, from the bivariate normal — and two forecasters that only ever say 0 or 1. The one an absolute-error score pays for says 1 wherever the honest probability exceeds a half: it has a true-positive rate of 0.6742 and a false-positive rate of 0.1375, and its "curve" is the two straight segments through that point, with area (TPR + TNR)/2 = 0.7684. The honest curve's area is 0.8683. Thresholding at the base rate of 0.3744 instead of at a half gives the largest area any two-valued forecaster can have here, 0.7818, and it is still 0.0866 short.

The liar with two answers

The forecaster an absolute-error score pays for says only 0 or 1, and on the ROC square it is a single point: its area is (TPR + TNR)/2 = 0.7684, against the honest forecaster's 0.8683, and it falls below the honest one on 200 of 200 counted records. No relabelling of its two answers returns what it threw away — the best recovers a Brier score short of the honest one by exactly the 0.022154 of resolution lost — and below a signal correlation of 0.7332 the same score prefers saying no every time to an honest forecast.

calibrate · Calibration

Named alongside it

The objects these essays reach for when they reach for this one.

Base ratePredictive valueScreeningROC curveBase rate fallacyBrier scoreClosed formConditional independenceDecision thresholdDiscriminationEstimator biasExpected loss

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