# Sample size for a diagnostic study

## 1999/09/03

**[StATS]: Sample size for a diagnostic study (September 3

• 1999)**

*Dear Professor Mean

• How big should a study of a diagnostic test be? I want to estimate a sample size for the sensitivity and specifity of a test. I guess confidence intervals would address this
• but is there a calculation analogous to a power analysis that would apply to figure out the size of the groups beforehand? – Jovial John*

Dear Jovial,

Sample size is not important. Just ask for enough in the research grant so that you can buy your consulting statistician a new computer. Just kidding. You are actually pretty close to having the right answer.

Power calculations are appropriate only when you have a research hypothesis. The emphasis in a study of a diagnostic test is estimation. You want accurate estimates of sensitivity

• specificity and/or likelihood ratios.

**When the focus is on estimation

• you determine the sample size through the width of the confidence interval**. You specify how precise (how narrow) you want your confidence intervals to be. This determines your sample size.

**For sensitivity and specificity

• use the standard formulas for a binomial proportion**. The formulas can be found in any introductory statistics book. For a likelihood ratio
• the formulas are a bit more complex
• but the same principle applies.

Example

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. You assess abnormal and normal under the gold standard

• of course.

A sample of size 50 abnormal and 50 normal patients would give a 95% 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

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.

Summary

Jovial John wants to know how many subjects to include in a research study of a new diagnostic test. Professor Mean explains that you should select a sample size that will make the confidence interval for sensitivity and/or specificity sufficiently narrow.

This page was written by Steve Simon while working at Children’s Mercy Hospital. Although I do not hold the copyright for this material

justification](../category/SampleSizeJustification.html). testing](../category/DiagnosticTesting.html)

• or Category: Sample size intervals
• [Category: Diagnostic for pages similar to this one at [Category: Confidence with general help resources. You can also browse Children’s Mercy Hospital website. Need more information? I have a page reproducing it here as a service
• as it is no longer available on the Hospital. Although I do not hold the copyright for this material
• I am This page was written by Steve Simon while working at Children’s Mercy