ICC {DescTools}  R Documentation 
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, type = c("all", "ICC1", "ICC2", "ICC3", "ICC1k", "ICC2k", "ICC3k"),
conf.level = NA, na.rm = FALSE)
## S3 method for class 'ICC'
print(x, digits = 3, ...)
x 

type 
one out of "all", "ICC1", "ICC2", "ICC3", "ICC1k", "ICC2k", "ICC3k". See details. 
conf.level 
confidence level of the interval. If set to 
na.rm 
logical, indicating whether 
digits 
number of digits to use in printing 
... 
further arguments to be passed to or from methods. 
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 oneway ANOVA fixed effects model and is found by (MSB MSW)/(MSB+ (nr1)*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 + (nr1)*MSE + nr*(MSJMSE)/nc)  
ICC3  A fixed set of k judges rate each target. There is no generalization to a larger population 
of judges. (MSB  MSE)/(MSB+ (nr1)*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 interclass correlation (e.g., a Pearson r) and a statistic such as alpha or omega might be used.
if method is set to "all", then the result will be
results 
A matrix of 6 rows and 8 columns, including the ICCs, F test, p values, and confidence limits 
summary 
The anova summary table 
stats 
The anova statistics 
MSW 
Mean Square Within based upon the anova 
if a specific type has been defined, the function will first check, whether no confidence intervals are requested:
if so, the result will be the estimate as numeric value
else a named numeric vector with 3 elements
ICCx 
estimate (name is the selected type of coefficient) 
lwr.ci 
lower confidence interval 
upr.ci 
upper confidence interval 
The results for the lower and upper Bounds for ICC(2,k) do not match those of SPSS 9 or 10, but do match the definitions of Shrout and Fleiss. SPSS seems to have been using the formula in McGraw and Wong, but not the errata on p 390. They seem to have fixed it in more recent releases (15).
William Revelle <revelle@northwestern.edu>, some editorial amendments Andri Signorell <andri@signorell.net>
Shrout, P. E., Fleiss, J. L. (1979) Intraclass correlations: uses in assessing rater reliability. Psychological Bulletin, 86, 4203428.
McGraw, K. O., Wong, S. P. (1996) Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1, 3046. + errata on page 390.
Revelle, W. (in prep) An introduction to psychometric theory with applications in R Springer. (working draft available at http://personalityproject.org/r/book/
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,
dimnames=list(paste("S", 1:6, sep=""), paste("J", 1:4, sep=""))
)
sf #example from Shrout and Fleiss (1979)
ICC(sf)