Intraclass Correlation

E. C. Hedberg

doi:10.4135/9781483381411

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Intraclass Correlation

By: E. C. Hedberg
In:The SAGE Encyclopedia of Communication Research Methods
Chapter DOI:https://doi.org/10.4135/9781483381411.n277
Subject:Communication and Media Studies, Sociology

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When data are clustered (i.e., unit observations nested within clusters), researchers often wish to know how units within clusters are correlated. For example, it may be useful for rater reliability studies to know how the ratings of judges (units) are correlated within targets (clusters). Another example from survey research is how individuals’ responses (units) are correlated within neighborhoods (clusters). These questions can be answered with intraclass correlation (ICC or ρ) coefficients, which are ratios of the variance associated with the cluster variable in question to the total variance. These ratios can be interpreted as a correlation coefficient (i.e., how units are correlated within clusters).

In addition to answering substantive questions, ICCs are important parameters in a priori power calculations for experiments or precision estimates for surveys where the sampling design involves clusters. This entry primarily focuses on the one-way random effects model, as it is the most common. For simplicity, this entry also makes the assumption of balanced data (i.e., all clusters have the same number of cases).

Overview

In 1925, R. A. Fisher called the correlation of units within a cluster an intraclass correlation. This parameter is different from the more familiar interclass correlation because the ICC makes the assumption that the observations within a cluster share a common mean and standard deviation. To illustrate the difference, consider a population of 2 units per j = 1,2,3,…m clusters where each cluster is a row with two variables, one for each i = 1,2 units in the cluster, y1 and y2, with means

1

{\bar{y}}_{1} = \frac{Σ y_{1}}{m}

and

2

{\bar{y}}_{2} = \frac{Σ y_{2}}{m},

and standard deviations

3

s_{1} = \sqrt{\frac{{\sum (y_{1} - {\bar{y}}_{1})}^{2}}{m}}

and

4

s_{2} = \sqrt{\frac{\sum {(y_{2} - {\bar{y}}_{2})}^{2}}{m}} .

The estimate of the interclass correlation is defined as

5

r_{interclass} = \frac{Σ (y_{1} - {\bar{y}}_{1}) (y_{2} - {\bar{y}}_{2})}{m s_{1} s_{2}} .

However, the ICC relies on a common mean

6

{\bar{y}}_{c} = \frac{Σ (y_{1} + y_{2})}{2 m},

...

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