What does a positive COVID

I was thinking about the case numbers reported and how these come as positives from a PCR test. The question one has to ask is what does this mean, and that depends not only on how precise the test is but what is the probability of getting the COVID in a population before you do the test.

To work that out, you need Bayes’ Theorem;

The positive and negative predictive values (PPV and NPV) can be calculated for any prevalence as follows:

sensitivity×prevalence

PPV = —————————————————————sensitivity×prevalence + (1 – specificity)×(1 – prevalence) specificity×(1 – prevalence)

NPV = ————————————————————— (1 – sensitivity)×prevalence + specificity×(1 – prevalence)

If the prevalence of the disease is very low, the positive predictive value will not be close to 1 even if both the sensitivity and specificity are high. Thus in screening the general population it is inevitable that many people with positive test results will be false positives.

Altman, BMJ 1994

What we need is the positive predictive value. So let’s punch some numbers in. As of today, this is the number of cases in NZ; we have had 3645 cases in total. The New Zealand estimated population from the last census is 4,900,800. The ministry of health says our COVID PCR tests are 95% sensitive and 96% specific.

Our COVID test positive rate is 74 per 100 000. This gives a post test probability — positive predictive value — of 4%.

So let’s look at other countries. The UK reports a COVID test positive rate of 355 per 100 000 given that they have similar sensitivity and specificity, this gives a positive predictive value of 4%

Australia has reported 53851 cases with a population of 25 694 393 as of the end of 2020 using the same numbers, you get a positive test rate of 209per 100 000 and a positive predictive value of 4%.

The USA has 39 110 086 cases with a population of 332,785,650. This is a base prevalence of 11%, and a positive predictive value of 14%.

In countries where there has been an epidemic, the tests add little. In screening populations where a zero covid approach has been tried, they are not useful. They are meaningless.

The alternative is to look at the negative predictive values ie the chance that if your test is positive you do not have COVID. In NZ, Australia and the UK that is 95%. In the USA it is still 85%.

It is not cases we should be worried about. It is hospitalizations and deaths.

CountryNo casesPopBase Rateper 100000TestSensSpec Pos Pred. ValNeg. Pred. Val
NZL364549008000.0007437674.3756121PCR0.950.960.040676820.95932318
UK0.00355355PCR0.950.960.04323050.9567695
AUS53851256943930.00209583209.58269PCR0.950.960.04190720.9580928
USA391100863327856500.1175233511752.3355PCR0.950.960.146946250.85305375
The spreadsheet results.