Intracluster Correlation Coefficient
The similarity of observations within a cluster can be quantified by means of the Intracluster Correlation Coefficient (ICC), sometimes also referred to as intraclass correlation coefficient. This is very similar to the well known Pearson’s correlation coefficient; only that we do not simultaneously look at observations of two variables on the same object but we look simultaneously on two values of the same variable, but taken at two different objects. As also the Pearson correlation coefficient, the parametric intra-cluster correlation coefficient is denoted with the Greek \(\bar \rho_{ICC}\) and the sample based estimation by the Latin \(r_{ICC}\). It is calculated as
\[\bar \rho_{ICC}=\frac{cov(y_p,y_q)}{\sqrt{var(y_p)} \sqrt{var(y_q)}}=\frac{cov(y_p y_q)}{var(y)}\]
In the case of the intra-cluster correlation coefficient, we are looking at one and the same variable so that In the case of the intra-cluster correlation coefficient, we are looking at one and the same variable so that \(\sqrt{var(y_p)}\) and \(\sqrt{var(y_q)}\) refer to the same variable and can be combined to \(var(y)\) .
For clusters of equal size (the case dealt with in this chapter) values of the ICC can be in the following range \(- \frac{1}{m-1} \le \bar \rho_{ICC} \le +1\). The upper limit is fixed, but the lower limit depends on the cluster size (the number of elements (sub-plots) that are combined to one cluster plot). The larger the number of sub-plots, the more close to 0 is the lower, negative limit. Through some rearranging of the error variance formula (not presented here), the intra-cluster correlation coefficient can be incorporated. Then, the error variance of the estimated mean is
\[var(\bar y)=\frac{N-n}{N-1} \frac{1}{m} \frac{1}{n} \sigma^2 \left {1+(m-1)\bar \rho_{ICC} \right}\]
