Modern ICC Definitions: Simpler Formula But Positive Bias
Beginning with Ronald Fisher, the intraclass correlation has been regarded within the framework of analysis of variance (ANOVA), and more recently in the framework of random effects models. A number of ICC estimators have been proposed. Most of the estimators can be defined in terms of the random effects model
where Yij is the ith observation in the jth group, μ is an unobserved overall mean, αj is an unobserved random effect shared by all values in group j, and εij is an unobserved noise term. For the model to be identified, the αj and εij are assumed to have expected value zero and to be uncorrelated with each other. Also, the αj are assumed to be identically distributed, and the εij are assumed to be identically distributed. The variance of αj is denoted σα2 and the variance of εij is denoted σε2.
The population ICC in this framework is
An advantage of this ANOVA framework is that different groups can have different numbers of data values, which is difficult to handle using the earlier ICC statistics. Note also that this ICC is always non-negative, allowing it to be interpreted as the proportion of total variance that is "between groups." This ICC can be generalized to allow for covariate effects, in which case the ICC is interpreted as capturing the within-class similarity of the covariate-adjusted data values.
This expression can never be negative (unlike Fisher's original formula) and therefore, in samples from a population which has an ICC of 0, the ICCs in the samples will be higher than the ICC of the population.
A number of different ICC statistics have been proposed, not all of which estimate the same population parameter. There has been considerable debate about which ICC statistics are appropriate for a given use, since they may produce markedly different results for the same data.
Read more about this topic: Intraclass Correlation
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