I am looking at a two-stage process of diagnosis.
Stage 1 is a screen that identifies people with the diagnosis with some error, Stage 2 is a detailed examination (a gold standard)
In Stage 1, N people were screened, and S were screened positively.
When the S people screened positively were examined in detail it was discovered that G actually had the disease
A 10% random sample of the those screened negative were also examined in detail, and it was found that J actually had the disease (the false negatives from the first stage)
The prevalence of the disease is the sum of the true positives and the false negatives from the first stage, or (G/S)(S/N) + (J/(N-S))((N-S)/S)) = (G+J)/N
My question is, what is the standard error of this prevalence estimate?