it's likely that ~5% of the original positive results (if the studies were properly conducted) would turn out to be unsubstantiated after all.
Not quite. Assume P of the tests are truly positive, and N are truly negative.
We expect to see approximately 0.05 x N + A x P positives in the entire sample, where A is the probability of a false negative. So the fraction of true positives not replicated is likely to be 0.05 x N / (0.05 x N + A x P).
Comments
it's likely that ~5% of the original positive results (if the studies were properly conducted) would turn out to be unsubstantiated after all.
Not quite. Assume P of the tests are truly positive, and N are truly negative.
We expect to see approximately 0.05 x N + A x P positives in the entire sample, where A is the probability of a false negative. So the fraction of true positives not replicated is likely to be 0.05 x N / (0.05 x N + A x P).