False positive rate visualiser for PCR tests using Bayes Rule.




False Positive Rate

PCR result probabilities

Sensitivity and Specificity performance of a PCR test system are typically combined with a pre-test prediction of the probability (symptoms, exposure etc) that the disease under test is present in the test subject to provide probabilities for True Positive, False Positive, True Negative and False Negative results. This pre-test prediction method is useful for individual tests when there is concern about the problem of False Negatives whereby a person who tested negative may actually be infectious. The British Medical Journal provides a calculator for this diagnostic approach.

However, the diagnostic method described can significantly underestimate of the rate of False Positive results when the prevalence of the disease in the test population is relatively low. While Prevalence is difficult to predict for the evaluation of an individual test result, it is particularly important to consider the condition of low prevalence and its impact on False Positive results when Government policies have this condition as a stated policy objective. The closer a country gets to zero prevalence, the higher the False Positive rate.

Sensitivity measures how often a test correctly generates a positive result for people who have the condition that’s being tested for.

Specificity measures a test’s ability to correctly generate a negative result for people who don’t have the condition that’s being tested for.

To use the tool above, try with some typical PCR specifications of 95% for Sensitivity and 99% for Specificity. Then start sliding the Prevalence control toward 0% to observe the problem. At these values, if 1000 people are tested and 5 people test positive, then the prevelance would be at about 0.5% but at that level of prevelance the false positve rate is a whopping 68%.

It is important to understand that the False Positive Rate is the proportion of postive results, not the overall number of tests, that are likely to be false positives. Also, the Prevelance input is the prevalance of the condition within the set of tests, not the overall population prevelance.

But wait, there's more!

Unfortunately there is another major source of uncertainty with PCR testing, namely the Cycle Threshold of the test. Very briefly, the Cycle Threshold refers to the number of amplification processes the test goes through before a measureable amount of the genetic sequence being tested for is observable.

Misinterpreting results | Cycle threshold explained by Anthony Kuster does a good job of explaining this problem. A more science based text is available at Global Biotech Insights.

Many experts have pointed out that cycle thresholds above 35 are extremely likely to be giving positive results for people who are not infectious and many countries are using thresholds up to 45 cycles. This expalins why many people getting positive tests are asymptomatic but they are still being labelled as 'positive cases'. As such, people are being needlessly quarantined and for countries engaging in contact tracing of positive cases, then these contacts are also being quarantined.

When combined with the problem of false positives, the disparity of cycle thresholds adopted by different countries also make it a pointless exercise to compare 'case' numbers for different countries.


I have implemented the Bayes Theorem equations from Dr. Trefor Bazetts video on estimating False Positives.


I am not a scientist, mathematician, doctor, software engineer, politician or even a musician.

Consequently, the content of this site is to be treated as suspect and further research by the reader is encouraged by the author.