Kendall’s tau for hierarchical data

This paper is concerned with hierarchical data having three levels. The level 1 units are nested in the level 2 units or subclusters which are themselves nested in the level 3 clusters. The model for this data is assumed to fulfill some symmetry assumptions. The level 1 units within each subcluster are exchangeable and a permutation of the subclusters belonging to the same cluster leaves the model unchanged. We are interested in measuring the dependence associated to clusters and subclusters respectively. Two exchangeable Kendall’s tau are proposed as non parametric measures of these two associations and estimators for these measures are proposed. Their asymptotic properties are then investigated under the proposed hierarchical model for the data. These statistics are then used to estimate the intra-class correlation coefficients for data drawn from elliptical hierarchical distributions. Hypothesis tests for the cluster and subcluster effects based on the proposed estimators are developed and their performances are assessed using Pitman efficiencies and a Monte Carlo study.