  Clustered data are common in biomedical research. Observations in the same
cluster are often more similar to each other than to observations from other
clusters. The intraclass correlation coefficient (ICC), first introduced by R.
A. Fisher, is frequently used to measure this degree of similarity. However,
the ICC is sensitive to extreme values and skewed distributions, and depends on
the scale of the data. It is also not applicable to ordered categorical data.
We define the rank ICC as a natural extension of Fisher's ICC to the rank
scale, and describe its corresponding population parameter. The rank ICC is
simply interpreted as the rank correlation between a random pair of
observations from the same cluster. We also extend the definition when the
underlying distribution has more than two hierarchies. We describe estimation
and inference procedures, show the asymptotic properties of our estimator,
conduct simulations to evaluate its performance, and illustrate our method in
three real data examples with skewed data, count data, and three-level ordered
categorical data.
