The Intraclass correlation is used as a measure of association when studying the reliability of raters. Shrout and Fleiss (1979) outline 6 different estimates, that depend upon the particular experimental design. All are implemented and given confidence limits.
ICC(x,missing=TRUE,alpha=.05)a matrix or dataframe of ratings
if TRUE, remove missing data -- work on complete cases only
The alpha level for significance for finding the confidence intervals
A matrix of 6 rows and 8 columns, including the ICCs, F test, p values, and confidence limits
The anova summary table
The anova statistics
Mean Square Within based upon the anova
Shrout and Fleiss (1979) consider six cases of reliability of ratings done by k raters on n targets.
ICC1: Each target is rated by a different judge and the judges are selected at random. (This is a one-way ANOVA fixed effects model and is found by (MSB- MSW)/(MSB+ (nr-1)*MSW))
ICC2: A random sample of k judges rate each target. The measure is one of absolute agreement in the ratings. Found as (MSB- MSE)/(MSB + (nr-1)*MSE + nr*(MSJ-MSE)/nc)
ICC3: A fixed set of k judges rate each target. There is no generalization to a larger population of judges. (MSB - MSE)/(MSB+ (nr-1)*MSE)
Then, for each of these cases, is reliability to be estimated for a single rating or for the average of k ratings? (The 1 rating case is equivalent to the average intercorrelation, the k rating case to the Spearman Brown adjusted reliability.)
ICC1 is sensitive to differences in means between raters and is a measure of absolute agreement.
ICC2 and ICC3 remove mean differences between judges, but are sensitive to interactions of raters by judges. The difference between ICC2 and ICC3 is whether raters are seen as fixed or random effects.
ICC1k, ICC2k, ICC3K reflect the means of k raters.
The intraclass correlation is used if raters are all of the same ``class". That is, there is no logical way of distinguishing them. Examples include correlations between pairs of twins, correlations between raters. If the variables are logically distinguishable (e.g., different items on a test), then the more typical coefficient is based upon the inter-class correlation (e.g., a Pearson r) and a statistic such as alpha or omega might be used.
Shrout, Patrick E. and Fleiss, Joseph L. Intraclass correlations: uses in assessing rater reliability. Psychological Bulletin, 1979, 86, 420-3428.
McGraw, Kenneth O. and Wong, S. P. (1996), Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1, 30-46. + errata on page 390.
Revelle, W. (in prep) An introduction to psychometric theory with applications in R. Springer. (working draft available at http://personality-project.org/r/book/
# NOT RUN {
sf <- matrix(c(9, 2, 5, 8,
6, 1, 3, 2,
8, 4, 6, 8,
7, 1, 2, 6,
10, 5, 6, 9,
6, 2, 4, 7),ncol=4,byrow=TRUE)
colnames(sf) <- paste("J",1:4,sep="")
rownames(sf) <- paste("S",1:6,sep="")
sf #example from Shrout and Fleiss (1979)
ICC(sf)
# }
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