Rank intraclass correlation for clustered data

Abstract

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.

Keywords: clustered data; intraclass correlation; rank association measures.

FIGURE 1

Parameters of rank ICC $\left({\gamma }_I\right)$ and Fisher’s ICC $\left({\rho }_I\right)$ as a function of the within-cluster correlation $(\rho )$ of $X_{ij}$ under normality (Scenario I) and after exponentiating the data (Scenario II).

FIGURE 2

Bias and coverage of 95% confidence intervals for our estimator of ${\gamma }_I$ at different true values of ${\gamma }_I$ and different numbers of clusters under Scenarios I (normality), II (exponentiated outcomes), and III (exponentiated cluster means). The number of observations per cluster was set at 30.

FIGURE 3

Bias and coverage of 95% confidence intervals for our estimator of ${\gamma }_I$ at different true values of ${\gamma }_I$ and different cluster sizes under Scenarios I (normality), II (exponentiated outcomes), and III (exponentiated cluster means). The number of clusters was set at 200. “2–50” means the cluster size follows a uniform distribution from 2 to 50, “2/30” means half of the clusters have size 2 and half have 30.

FIGURE 4

Bias and coverage of 95% confidence intervals for our estimator of ${\gamma }_I$ at different true positive and negative values of ${\gamma }_I$ when the cluster size in the population was 2. The number of clusters was set at 200.

FIGURE 5

Root mean squared error (RMSE), bias, and empirical SE of estimates obtained by the four weighting approaches for our estimator of ${\gamma }_I$. “Equal clusters” refers to assigning equal weights to clusters, “Equal obs” refers to assigning equal weights to observations, “ESS” refers to the iterative weighting approach based on the effective sample size, and “Combination” refers to the iterative weighting approach based on the linear combination of equal weights for clusters and equal weights for observations. We set the tolerance of the two iterative approaches to 0.00001.

FIGURE 6

Parameters of rank ICC $\left({\gamma }_I\right)$ as a function of the within-cluster correlation $(\rho )$ of $X_{ij}$ when data are continuous or discretized into ordered categorical variables with 3,5, or 10 levels.

FIGURE 7

Bias and coverage of 95% confidence intervals for our estimators of ${\gamma }_{I2}$ and ${\gamma }_{I3}$ at different true values of ${\gamma }_{I2}$ and ${\gamma }_{I3}$ and different numbers of level-3 units. The number of level-2 units in a level-3 unit was set at 15. The number of level-1 units in a level-2 unit was set at 2.

FIGURE 8

Scatter plot of the first and second uACR measurements of each person in the example of albumin-creatinine ratio.

FIGURE 9

Histogram of numbers of seizures of children with untreated epilepsy from the 60 primary healthcare centers in the example of status epilepticus.

FIGURE 10

Scatter plot of PHQ-9 scores of male and female partners enrolled in the clustered randomized clinical trial in the example of Patient Health Questionaire-9 score.