#### Sample size for a diagnostic test (2004-07-05)

This page has moved to my new website.

Someone asked me how to determine the sample size for a study involving a diagnostic test.
It seems like a tricky thing, because most studies of diagnostic tests don't have a formal
hypothesis. What you need to do instead is to specify a particular statistic that you are
interested in estimating and then selecting a sample size so that the confidence interval for
this estimate is reasonably precise.

For example, suppose you want to estimate the
sensitivity (Sn) and specificity (Sp) of a
diagnostic test. Your best guess is that sensitivity will be at least 75% and specificity
will be at least 90%. The formula for a confidence interval for Sn or Sp would be

where n_{a} and n_{n} are the number of abnormal (diseased) and normal
(healthy) patients in the study.

A sample of size 50 abnormal and 50 normal patients would give a95% confidence interval of
plus/minus 0.12 for Sn and plus/minus 0.083 for Sp. This seems like a reasonable amount of
precision. A sample of size 75 in each group would provide slightly narrower confidence
intervals (plus/minus 0.098 and plus/minus 0.068 respectively). Your choice of the sample
size depends in large part on the number of patients you can recruit from and also a balance
between maximizing precision and minimizing the amount of time you spend on this project.

Suppose instead that you wanted to estimate the area under the curve (AUC) for a
Received Operating Characteristic Curve (ROC curve). The formula
for a standard error here is a bit messier. The web page

http://www.anaesthetist.com/mnm/stats/roc/

offers a JAVAScript calculator for the standard error of the AUC. Let's suppose that the
AUC is going to be around 0.8. With the same 50 abnormal and normal patients, the standard
error would be 0.044, which is reasonably small. With 75 in each group, the standard error
would be 0.036.

I have a whole section on my web for determining the appropriate
sample size, including a page on sample size for a diagnostic
tests, which I have just updated to include the above example.