The six intraclass correlations of Shrout and Fleiss (1979), with their F tests and confidence intervals.
Equivalent to psych::ICC(x, alpha, lmer) in R. Only complete data are accepted.
x is an n × k matrix (an array of rows): the ratings of the k raters for each of the n subjects. The forms are
ICC1 = (MSB − MSW)/(MSB + (k − 1)MSW) (one-way), ICC2 = (MSB − MSE)/(MSB + (k − 1)MSE + k(MSJ − MSE)/n) (two-way
random raters, absolute agreement), ICC3 = (MSB − MSE)/(MSB + (k − 1)MSE) (two-way fixed raters, consistency), and
their averages over k raters ICC1k, ICC2k, ICC3k. With lmer: true (the default, as psych) the mean squares are
rebuilt from the REML estimates of the variance components of the two-way random model, which are not negative;
where both ANOVA components are not negative they equal the ANOVA mean squares. psych fits that model with lme4,
whose optimiser stops early (results about 1e-5 relative off); here the REML estimates are computed exactly, in
closed form.
ICC1, ICC3, ICC1k and ICC3k agree with icc (irr's forms); the interval of ICC2k is psych's (ICC2's bounds
transformed by Spearman-Brown), not irr's (McGraw and Wong's average-agreement interval).
Where the residual mean square is 0 (ratings that differ between raters by constants), the F statistics of ICC2 and
ICC3 are Infinity with p-value 0, and ICC3, ICC3k and their bounds are 1.
Parameters
x: readonly (readonlynumber[])[]
An n × k matrix of ratings (array of rows), without missing values.
TypeError if x is not a matrix of numbers; RangeError for missing or infinite ratings (psych's REML path
accepts missing ratings; here they are an error), fewer than 2 subjects or raters, ratings that are all equal or
identical for every subject, or a level out of range.
The six intraclass correlations of Shrout and Fleiss (1979), with their F tests and confidence intervals. Equivalent to
psych::ICC(x, alpha, lmer)in R. Only complete data are accepted.xis an n × k matrix (an array of rows): the ratings of the k raters for each of the n subjects. The forms are ICC1 = (MSB − MSW)/(MSB + (k − 1)MSW) (one-way), ICC2 = (MSB − MSE)/(MSB + (k − 1)MSE + k(MSJ − MSE)/n) (two-way random raters, absolute agreement), ICC3 = (MSB − MSE)/(MSB + (k − 1)MSE) (two-way fixed raters, consistency), and their averages over k raters ICC1k, ICC2k, ICC3k. Withlmer: true(the default, as psych) the mean squares are rebuilt from the REML estimates of the variance components of the two-way random model, which are not negative; where both ANOVA components are not negative they equal the ANOVA mean squares. psych fits that model with lme4, whose optimiser stops early (results about 1e-5 relative off); here the REML estimates are computed exactly, in closed form.ICC1, ICC3, ICC1k and ICC3k agree with icc (irr's forms); the interval of ICC2k is psych's (ICC2's bounds transformed by Spearman-Brown), not irr's (McGraw and Wong's average-agreement interval).
Where the residual mean square is 0 (ratings that differ between raters by constants), the F statistics of ICC2 and ICC3 are Infinity with p-value 0, and ICC3, ICC3k and their bounds are 1.