# Power analysis for inter-rater reliability study (Kappa) with multiple raters

I've spent some time looking through literature about sample size calculation for Cohen's kappa and found several studies stating that increasing the number of raters reduces the number of subjects required to get the same power. I think this is logical when looking at inter-rater reliability by use of kappa statistics. But there is, as far as I can see, no specific calculation or reference for the statement. In this link there is calculation for 2 raters.

• Is anyone familiar with similar calculation for several raters?
• Other factors that would affect the number of subjects required?

I will (probably) have 5 categories of nominal data. There might be combined findings. There will be 3 raters.

Sim, J. and Wright, C. C. (2005) Interpretation, and Sample Size Requirements The Kappa Statistic in Reliability Studies: Use, Interpretation, and Sample Size Requirements, Journal of the American Physical Therapy Association, 85, pp. 257–268.

When seeking to optimize sample size, the investigator needs to choose the appropriate balance between the number of raters examining each subject and the number of subjects. In some instances, it is more practical to increase the number of raters rather than increase the number of subjects. However, according to Shoukri, when seeking to detect a kappa of .40 or greater on a dichotomous variable, it is not advantageous to use more than 3 raters per subject—it can be shown that for a fixed number of observations, increasing the number of raters beyond 3 has little effect on the power of hypothesis tests or the width of confidence intervals. Therefore, increasing the number of subjects is the more effective strategy for maximizing power.

• Power for kappa is a bit weird. We usually aren't looking for statistically significant values of kappa, because the kappa might be highly significant, but still indicate low agreement. Nov 22, 2015 at 22:09
• You are right. But still there are requirements for sample size to ensure validity of results. By a relative error of i.e. 30% and a pa-pe of i.e. 0,6, the table here agreestat.com/blog_irr/sample_size_determination.html suggests 31 subjects. What would this number be for 3 raters?
– Siv
Nov 23, 2015 at 14:23
• I see. They are talking about getting an acceptable standard error. Nov 23, 2015 at 14:41

You can use the R package kappaSize for sample size calculation or power analysis in the cases of 2-6 raters and 3-5 categories.
Three functions in the package are relevant depending on the number of categories: Power3Cats(), Power4Cats() and Power5Cats().