Covid Test

My girlfriend and I recently went for a Covid test, given that the pair of us were either both positive or both negative (as we live together) and the test is 50% accurate, how accurate is the overall test?

@campaigner said
My girlfriend and I recently went for a Covid test, given that the pair of us were either both positive or both negative (as we live together) and the test is 50% accurate, how accurate is the overall test?

50% accurate...that's a junk test!

25% for the group.

P( C+/-) = probability you have/don't have COVID
P( T+/-) = probability you test positive/negative

You either have it or you don't have it: P(C+) = P(C-) = 1/2

P(T+| C+ ) = 1/2 = P(T-| C- )


P(C+) P(T+| C+ )^2 + P(C-) P(T-| C- )^2 = 1/2 * ( 1/4 + 1/4 ) = 1/4

This ignores Bayes Theorem and disease prevalence.

@campaigner said
My girlfriend and I recently went for a Covid test, given that the pair of us were either both positive or both negative (as we live together) and the test is 50% accurate, how accurate is the overall test?

There are 4 possible results
pp nn pn np = 1/4 = 25%

@venda

Yep, that simplifies it, Venda

Each test is 1/2 chance of being accurate, so chance of both accurate is (1/2)^2 = 1/4

@Campaigner
On the assumption that if one of you has the disease both do, and that if the test is positive then it is correct (sensitivity) then the combined test produces a 75% chance of disease detection.

50% accurate does not specify which direction the test fails. There is specificity (true negatives) and sensitivity (true positives) - so what do you mean by "accurate"?

@deepthought said
@Campaigner
On the assumption that if one of you has the disease both do, and that if the test is positive then it is correct (sensitivity) then the combined test produces a 75% chance of disease detection.

50% accurate does not specify which direction the test fails. There is specificity (true negatives) and sensitivity (true positives) - so what do you mean by "accurate"?

That's a good point. If they are both equal ( specificity and sensitivity ) at 50% then its not a test, its a coin flip.