  Clustered data are common in practice. Clustering arises when subjects are
measured repeatedly, or subjects are nested in groups (e.g., households,
schools). It is often of interest to evaluate the correlation between two
variables with clustered data. There are three commonly used Pearson
correlation coefficients (total, between-, and within-cluster), which together
provide an enriched perspective of the correlation. However, these Pearson
correlation coefficients are sensitive to extreme values and skewed
distributions. They also depend on the scale of the data and are not applicable
to ordered categorical data. Current non-parametric measures for clustered data
are only for the total correlation. Here we define population parameters for
the between- and within-cluster Spearman rank correlations. The definitions are
natural extensions of the Pearson between- and within-cluster correlations to
the rank scale. We show that the total Spearman rank correlation approximates a
weighted sum of the between- and within-cluster Spearman rank correlations,
where the weights are functions of rank intraclass correlations of the two
random variables. We also discuss the equivalence between the within-cluster
Spearman rank correlation and the covariate-adjusted partial Spearman rank
correlation. Furthermore, we describe estimation and inference for the three
Spearman rank correlations, conduct simulations to evaluate the performance of
our estimators, and illustrate their use with data from a longitudinal
biomarker study and a clustered randomized trial.
