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# Estimating the confidence interval for true prevalence

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 Gladen-estimator. Hence:

#### Algebraically speaking -

 True prevalence (ptrue) = 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 (ptrue) = p q × 1  n J2
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/J2 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(ptrue) = p ± 1.96 √ p q n J2
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).  Except where otherwise specified, all text and images on this page are copyright InfluentialPoints, all rights reserved. Images not copyright InfluentialPoints credit their source on web-pages attached via hypertext links from those images.