Intraclass Correlation - Early ICC Definition: Unbiased But Complex Formula

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 (x_n,1, x_n,2), for n = 1, ..., N. The intraclass correlation r originally proposed by Ronald Fisher is

,

,

.

Later versions of this statistic used the proper 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 .

The intraclass correlation is also defined for data sets with groups having more than two values. For groups consisting of 3 values, it is defined as

,

,

.

As the number of values per groups grows, the number of cross-product terms in this expression grows rapidly. The equivalent form

where K is the number of data values per group, and is the sample mean of the nth group, is simpler to calculate. This form is usually attributed to Harris. The left term is non-negative, consequently the intraclass correlation must satisfy

.

For large K, this ICC is nearly equal to

which can be interpreted as the fraction of the total variance that is due to variation between groups. Ronald Fisher devotes an entire chapter to Intraclass correlation in his classic book Statistical Methods for Research Workers.

For data from a population that is completely noise, Fisher's formula produces ICC values that are distributed about 0, i.e. sometimes being negative. This is because Fisher designed the formula to be unbiased, and therefore its estimates are sometimes overestimates and sometimes underestimates. For small or 0 underlying values in the population, the ICC calculated from a sample may be negative.