In providing an explanation of the answer to Casscells et al question, which is given as follows:
"(In case you’re struggling, one way of thinking about the question is to imagine that 1000 people are given the test. Since the prevalence is one in 1000, one of these people will have the disease. But a false positive rate of 5% means that 50 people will have a positive test result. So the chance of someone with a positive test actually having the disease is one in 50.)"
Mr Martyn ironically exposes himself to the lack of true understanding of how the real answer should be derived.
Based on the the "common sense" reasoning from the actual Casscells et al paper uses the usual equation for the Positive Predictive Value (PPV) being
PPV = TP / (TP + FP)
whereby TP is True Positive and FP is False Positive
Based on this equation:
1 out of any 1000 people has the disease
So 999 people does not have the disease
False Positive Rate of a test for the disease is 5%
Of the 999 people without the disease, about 0.05 X 999 = just under 50 people would have tested positive on the test.
Of 1000 people tested, there can only be a maximum of 1 person who is tested positive with the disease (true positive)
(Just under) 50 people (false positive) plus 1 person (max possible number of people with the disease to have positive test in a group of 1000 ie maximum possible true positive) means there is a maximum possible just under 51 people who would have tested positive out of 1000
So PPV is TP / (TP + FP) = (maximum possible 1) / (just under 51) = not more than 1/51*
* which is the number stated in the 1978 Casscells et al paper and not 1/50 as Mr Martyn suggest. The values are close but nevertheless it is still the wrong working solution.
Rapid Response:
Dear Editors
In providing an explanation of the answer to Casscells et al question, which is given as follows:
"(In case you’re struggling, one way of thinking about the question is to imagine that 1000 people are given the test. Since the prevalence is one in 1000, one of these people will have the disease. But a false positive rate of 5% means that 50 people will have a positive test result. So the chance of someone with a positive test actually having the disease is one in 50.)"
Mr Martyn ironically exposes himself to the lack of true understanding of how the real answer should be derived.
Based on the the "common sense" reasoning from the actual Casscells et al paper uses the usual equation for the Positive Predictive Value (PPV) being
PPV = TP / (TP + FP)
whereby TP is True Positive and FP is False Positive
Based on this equation:
1 out of any 1000 people has the disease
So 999 people does not have the disease
False Positive Rate of a test for the disease is 5%
Of the 999 people without the disease, about 0.05 X 999 = just under 50 people would have tested positive on the test.
Of 1000 people tested, there can only be a maximum of 1 person who is tested positive with the disease (true positive)
(Just under) 50 people (false positive) plus 1 person (max possible number of people with the disease to have positive test in a group of 1000 ie maximum possible true positive) means there is a maximum possible just under 51 people who would have tested positive out of 1000
So PPV is TP / (TP + FP) = (maximum possible 1) / (just under 51) = not more than 1/51*
* which is the number stated in the 1978 Casscells et al paper and not 1/50 as Mr Martyn suggest. The values are close but nevertheless it is still the wrong working solution.
Competing interests: No competing interests