Interclass correlation

In statistics, the intraclass correlation (or the intraclass correlation coefficient ) is a measure of correlation, consistency or conformity for a data set when it has multiple groups.

The intra-class correlation is used to estimate the correlation of one variable between two members within a group, for instance between two children of one family. This is in contrast to Pearson's Correlation, where the variables of interest are modeled as two distinct traits, with the mean and variance of each being estimated separately. In the intraclass correlation, the trait's mean and variance are derived from pooled estimates across all members of all groups. Because of this, the intraclass correlation gives the proportion of variance attributable to between group differences. In the example of siblings nested in families, the intraclass correlation gives the proportion of variance accounted for by family membership, while the Pearson gives the proportion of shared variance between the two members of a pair without respect to group (family) membership. You might think of it as the equivalent of a matched-sample t-test.

There are several measures of ICC and they may yield different values for the same data set.

Early definition
Consider a data set with two groups represented as two columns of a data matrix $$X(N' \times 2)$$ then the intraclass correlation r is computed from
 * $$\bar{x} = \frac{1}{2N'} \sum_{n=1}^{N'} (x_{n,1} + x_{n,2}) $$,
 * $$s^2 = \frac{1}{2N} \left\{ \sum_{n=1}^{N} ( x_{n,1} - \bar{x})^2 + \sum_{n=1}^{N} ( x_{n,2} - \bar{x})^2 \right\} $$,
 * $$r = \frac{1}{Ns^2} \sum_{n=1}^{N} ( x_{n,1} - \bar{x}) ( x_{n,2} - \bar{x}) $$,

where N is the degrees of freedom (Note that the precise form of the formula differ between versions of Fisher's book: The 1954 edition uses $$N'$$ in places where the 1925 edition uses $$N$$). This form is not the same as the interclass correlation. For the data set with two groups the intraclass correlation r will be confined to the interval [-1, +1].

The intraclass correlation is also defined for data sets with more than two groups, e.g., for three groups it is computed as
 * $$\bar{x} = \frac{1}{3 N'} \sum_{n=1}^{N'} (x_{n,1} + x_{n,2} + x_{n,3}) $$,
 * $$s^2 = \frac{1}{3 N} \left\{ \sum_{n=1}^{N} ( x_{n,1} - \bar{x})^2 + \sum_{n=1}^{N} ( x_{n,2} - \bar{x})^2 + \sum_{n=1}^{N} ( x_{n,3} - \bar{x})^2\right\} $$,
 * $$r = \frac{1}{3Ns^2} \sum_{n=1}^{N} \left\{ ( x_{n,1} - \bar{x})( x_{n,2} - \bar{x}) + (x_{n,1} - \bar{x})( x_{n,3} - \bar{x})+( x_{n,2} - \bar{x})( x_{n,3} - \bar{x}) \right\} $$.

(Also this form differs between editions of Fisher's book)

As the number of groups grow, the number of terms in the form will grow exponentially, but another form has been suggested that does not require so many computations
 * $$K\sum_{k=1}^{K} ( \bar{x}_k - \bar{x})^2 = Ns^2 \left\{1+(K-1) r \right\}$$,

where K is the number of groups. This form is usually attributed to Harris. The left term is non-negative, consequently the intraclass correlation must be
 * $$r \geq -1 /(K-1)$$.

"Modern" ICCs
Beginning with Ronald Fisher the intraclass correlation has been regarded within the framework of analysis of variance (ANOVA). Different ICCs arise with different ANOVA models, e.g., one-way analysis or two-way analysis, and they may produce marked different results. An article by McGraw and Wong lists these variations.

Yet another measure that has been regarded as an intraclass correlation coefficient is the concordance correlation coefficient.

Output from the SPSS program may be used to compute an intraclass correlation.

The open-source R-Project may also be used to compute the intraclass correlation (package 'psy').

Other methods of measuring agreement/correlation
The Cohen's kappa statistic is used for correlation of paired inter-observer agreement. The Fleiss kappa can be used to compare multiple observers' agreement.