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 ICC
print(x, digits = 3, ...)

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

- x
\(n \times m\) matrix or dataframe, k subjects (in rows) m raters (in columns).

- type
one out of "all", "ICC1", "ICC2", "ICC3", "ICC1k", "ICC2k", "ICC3k". See details.

- conf.level
confidence level of the interval. If set to

`NA`

(which is the default) no confidence intervals will be calculated.- na.rm
logical, indicating whether

`NA`

values should be stripped before the computation proceeds. If set to`TRUE`

only the complete cases of the ratings will be used. Defaults to`FALSE`

.- digits
number of digits to use in printing

- ...
further arguments to be passed to or from methods.

William Revelle <revelle@northwestern.edu>, some editorial amendments Andri Signorell <andri@signorell.net>

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, P. E., Fleiss, J. L. (1979) Intraclass correlations: uses in assessing rater reliability. * Psychological Bulletin*, 86, 420-3428.

McGraw, K. O., 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/

```
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)
```

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