We saw in Unit 2 that if the sensitivity and specificity of a test is known, we can get an approximately unbiased estimate of the true prevalence using the Rogan Gladenestimator. Hence:
Algebraically speaking 
True prevalence (p_{true}) 
= 
p + Sp − 1 

Se + Sp − 1 
where
 p is the apparent prevalence
 Sp is the specificity of the test
 Se is the sensitivity of the test

The question then is how do we attach a confidence interval to this estimate.
The variance of the true prevalence is given by:
Algebraically speaking 
Variance (p_{true}) 
= 
p q 
× 
1 


n 
J^{2} 
where
 p is the apparent prevalence
 n is the sample size
 J is Youden's index which is Se + Sp − 1.

In other words, the variance of the true prevalence is 1/J^{2} times the variance of the apparent prevalence. For a large sample (n >100) and a moderate prevalence (0.3 < p < 0.7) you
can then use the simple normal approximation to obtain an approximate confidence interval:
Algebraically speaking 
95% CI(p_{true}) 
= 
p ± 1.96 
√ 

p q 

n J^{2}

where:
 p, q, n and J are as above.

We stress that this interval is approximate because it assumes that the values of the sensitivity and specificity are known rather than estimated. If they are only estimates (as is usually the case), then strictly speaking a more complex formulation is required. In addition the sample sizes used to estimate sensitivity and specificity must also be known. Further details can be found in Greiner & Gardner (2000).