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. Uses either aov or lmer depending upon options. lmer allows for missing values.

`ICC(x,missing=TRUE,alpha=.05,lmer=TRUE,check.keys=FALSE)`

x

a matrix or dataframe of ratings

missing

if TRUE, remove missing data -- work on complete cases only (aov only)

alpha

The alpha level for significance for finding the confidence intervals

lmer

Should we use the lmer function from lme4? This handles missing data and gives variance components as well. TRUE by default.

check.keys

If TRUE reverse those items that do not correlate with total score. This is not done by default.

A matrix of 6 rows and 8 columns, including the ICCs, F test, p values, and confidence limits

The anova summary table or the lmer summary table

The anova statistics (converted from lmer if using lmer)

Mean Square Within based upon the anova

The variance decomposition if using the lmer option

Shrout and Fleiss (1979) consider six cases of reliability of ratings done by k raters on n targets. McGraw and Wong (1996) consider 10, 6 of which are identical to Shrout and Fleiss and 4 are conceptually different but use the same equations as the 6 in Shrout and Fleiss.

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. alpha and ICC3k are identical.

Where the data are laid out in terms of Rows (subjects) and Columns (rater or tests), the various ICCs are found by the ratio of various estimates of variance components. In all cases, subjects are taken as varying at random, and the residual variance is also random. The distinction between models 2 and 3 is whether the judges (items/tests) are seen as random or fixed. A further distinction is whether the emphasis is upon absolute agreement of the judges, or merely consistency.

As discussed by Liljequist et al. (2019), McGraw and Wong lay out 5 models which use just three forms of the ICC equations.

Model 1 is a one way model with

ICC1: Each target is rated by a different judge and the judges are selected at random. $$ ICC(1,1) = \rho_{1,1} = \frac{\sigma^2_r}{\sigma^2_r + \sigma^2_w} $$ (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. $$ ICC(2,1) = \rho_{2,1} = \frac{\sigma^2_r}{\sigma^2_r + \sigma^2_c +\sigma^2_{rc} + \sigma^2_e} $$

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. $$ ICC(3,1) = \rho_{3,1} = \frac{\sigma^2_r}{\sigma^2_r + \sigma^2_c + \sigma^2_e} $$

(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.

If using the lmer option, then missing data are allowed. In addition the lme object returns the variance decomposition. (This is simliar to `testRetest`

which works on the items from two occasions.

The check.keys option by default reverses items that are negatively correlated with total score. A message is issued.

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.

Liljequist David, Elfving Britt and Skavberg Kirsti (2019) Intraclass correlation-A discussion and demonstration of basic features. PLoS ONE 14(7): e0219854. https://doi.org/10.1371/journal.

Revelle, W. (in prep) An introduction to psychometric theory with applications in R. Springer. (working draft available at https://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,lmer=FALSE) #just use the aov procedure
#data(sai)
sai <- psychTools::sai
sai.xray <- subset(sai,(sai$study=="XRAY") & (sai$time==1))
xray.icc <- ICC(sai.xray[-c(1:3)],lmer=TRUE,check.keys=TRUE)
xray.icc
xray.icc$lme #show the variance components as well
# }
```

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