In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures, it operates on data structured as groups rather than data structured as paired observations.
The intraclass correlation is commonly used to quantify the degree to which individuals with a fixed degree of relatedness (e.g. full siblings) resemble each other in terms of a quantitative trait (see heritability). Another prominent application is the assessment of consistency or reproducibility of quantitative measurements made by different observers measuring the same quantity.
Early ICC definition: unbiased but complex formula The earliest work on intraclass correlations focused on the case of paired measurements, and the first intraclass correlation (ICC) statistics to be proposed were modifications of the interclass correlation (Pearson correlation). Consider a data set consisting of N paired data values (xn,1, xn,2), for n = 1, ..., N. The intraclass correlation r originally proposed by Ronald Fisher is
r = 1 N s 2 ∑ n = 1 N ( x n , 1 − x ¯ ) ( x n , 2 − x ¯ ) , {\displaystyle r={\frac {1}{Ns^{2}}}\sum _{n=1}^{N}(x_{n,1}-{\bar {x}})(x_{n,2}-{\bar {x}}),}
where
x ¯ = 1 2 N ∑ n = 1 N ( x n , 1 + x n , 2 ) , {\displaystyle {\bar {x}}={\frac {1}{2N}}\sum _{n=1}^{N}(x_{n,1}+x_{n,2}),}
s 2 = 1 2 N { ∑ n = 1 N ( x n , 1 − x ¯ ) 2 + ∑ n = 1 N ( x n , 2 − x ¯ ) 2 } . {\displaystyle s^{2}={\frac {1}{2N}}\left\{\sum _{n=1}^{N}(x_{n,1}-{\bar {x}})^{2}+\sum _{n=1}^{N}(x_{n,2}-{\bar {x}})^{2}\right\}.}
Later versions of this statistic used the degrees of freedom 2N −1 in the denominator for calculating s2 and N −1 in the denominator for calculating r, so that s2 becomes unbiased, and r becomes unbiased if s is known. The key difference between this ICC and the interclass (Pearson) correlation is that the data are pooled to estimate the mean and variance. The reason for this is that in the setting where an intraclass correlation is desired, the pairs are considered to be unordered. For example, if we are studying the resemblance of twins, there is usually no meaningful way to order the values for the two individuals within a twin pair. Like the interclass correlation, the intraclass correlation for paired data will be confined to the interval [−1, +1]. The intraclass correlation is also defined for data sets with groups having more than 2 values. For groups consisting of three values, it is defined as
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