Table 3.

Equivalent ICC Forms Between Shrout and Fleiss (1979) and McGraw and Wong (1996)

McGraw and Wong (1996) Conventiona	Shrout and Fleiss (1979) Conventionb	Formulas for Calculating ICCc
One-way random effects, absolute agreement, single rater/measurement	ICC (1,1)	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{W}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\left(k+1\right){\mathrm{M}\mathrm{S}}_{\mathrm{W}}}$
Two-way random effects, consistency, single rater/measurement	–	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\left(k-1\right){\mathrm{M}\mathrm{S}}_{\mathrm{E}}}$
Two-way random effects, absolute agreement, single rater/measurement	ICC (2,1)	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\left(k-1\right){\mathrm{M}\mathrm{S}}_{\mathrm{E}}+\frac{k}{n}\left({\mathrm{M}\mathrm{S}}_{\mathrm{C}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}\right)}$
Two-way mixed effects, consistency, single rater/measurement	ICC (3,1)	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\left(k-1\right){\mathrm{M}\mathrm{S}}_{\mathrm{E}}}$
Two-way mixed effects, absolute agreement, single rater/measurement	–	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\left(k-1\right){\mathrm{M}\mathrm{S}}_{\mathrm{E}}+\frac{k}{n}\left({\mathrm{M}\mathrm{S}}_{\mathrm{C}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}\right)}$
One-way random effects, absolute agreement, multiple raters/measurements	ICC (1,k)	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{W}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}}$
Two-way random effects, consistency, multiple raters/measurements	–	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}}$
Two-way random effects, absolute agreement, multiple raters/measurements	ICC (2,k)	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{C}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{n}}$
Two-way mixed effects, consistency, multiple raters/measurements	ICC (3,k)	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}}$
Two-way mixed effects, absolute agreement, multiple raters/measurements	–	$\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{{\mathrm{M}\mathrm{S}}_{\mathrm{R}}+\frac{{\mathrm{M}\mathrm{S}}_{\mathrm{C}}-{\mathrm{M}\mathrm{S}}_{\mathrm{E}}}{n}}$

ICC, intraclass correlation coefficients.

^aMcGraw and Wong¹⁸ defined 10 forms of ICC based on the model (1-way random effects, 2-way random effects, or 2-way fixed effects), the type (single rater/measurement or the mean of k raters/measurements), and the definition of relationship considered to be important (consistency or absolute agreement). In SPSS, ICC calculation is based on the terminology of McGraw and Wong.

^bShrout and Fleiss¹⁹ defined 6 forms of ICC, and they are presented as 2 numbers in parentheses [eg, ICC (2,1)]. The first number refers to the model (1, 2, or 3), and the second number refers to the type, which is either a single rater/measurement (1) or the mean of k raters/measurements (k).

^cThis column is intended for researchers only. MS_R = mean square for rows; MS_W = mean square for residual sources of variance; MS_{E =} mean square for error; MS_C = mean square for columns; n = number of subjects; k = number of raters/measurements.