Watch the prevalence of the disease and the specificity of the test. Specificity of a test for a disease is the true negatives divided by the sum of true negatives plus false positives.

Let P = disease prevalence, S = specificity of test, T = likelihood of getting a true positive, N = likelihood of a true negative, and F = likelihood of a false positive. For simplicity, I assume that the test is perfectly sensitive and everyone with the disease gets a positive test result (no false negatives).

All of these numbers here are between 0 and 1. For a perfect test, with no false positives = false alarms, S = 1. No biological test is perfect.

Let's look at the predictive value positive, the likelihood that a positive result on the test indicates actually having the disease.

Define PV = predictive value positive.
PV = T / (T + F)

 P = T, because everyone with the disease gets a positive result.

Every test gives a result, therefore

P + F + N = 1
(which simply means that the prevalence plus the true negatives plus the false positives = 1.)

Solve the above for F:

F = 1 - P - N

The specificity equation, true negatives divided by the sum of true negatives plus false positives:

S = N / (N + F)

The next task is to solve for predictive value in terms of S and P only, not relying on T or F or N.

Substitute F as expressed above:

S = N / (N + 1 - P - N)
S = N / (1 - P)

Now solve for N, true negative.

S x (1 - P) = N

N = S x (1 - P)

The true negative is the specificity multiplied by the fraction of people who do NOT have the disease.

Returning to the equation for F

F = 1 - P - N

Substitute the solution for N

F = 1 - P - S x (1 - P)

F = (1 - S) x (1 - P)

The false positive rate F is the specificity defect, its difference from 1, multiplied by the true frequency of healthy people (1 minus the prevalence rate).

Going back to the predictive value equation

PV = T / (T + F)

Substitute to solve for PV in terms of P the prevalence and S the test specificity

PV = P / [P + (1 - S) x (1 - P)]

This equation is exact and says that the predictive value of a positive test result, the likelihood that having such a result indicates being truly afflicted with the disease, is the disease prevalence corrected by the specificity defect times the disease-free proportion of people.

The greater the denominator above relative to P, the less likely a positive test result indicates truly having the disease. But how big is (1 - S) x (1 - P)?

Right now, the prevalence is apparently less than .01 and we can pretend that 1 - P ≈ 1.

PV ≈ P / [P + (1 - S)]

The above equation is VERY important. For example, if P < (1 - S), then PV is less than ½.

If the specificity defect of a test is less than the prevalence of the assumed-uncommon disease that it tests for, then the predictive value of a positive result of the test is less than ½. Reworded, the finding of a positive test result is more likely a false alarm than proof of having the disease.

Do not trust a coronavirus test unless the test specificity is known.

Copyright 2020 Edward J. Harshman. Circulate this analysis as you wish, but please do not erase my copyright notice or deny me credit.

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