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Table 1 Two-way mixed-effect analysis of variance (ANOVA) model

From: Assessing test–retest reliability of patient-reported outcome measures using intraclass correlation coefficients: recommendations for selecting and documenting the analytical formula

Case 3 model of McGraw and Wong [6, p34]: Two-way mixed model with interaction
\({{\varvec{x}}_{{\varvec{i}}{\varvec{j}}}}=\varvec{\mu}+{{\varvec{p}}_{\varvec{i}}}+{{\varvec{t}}_{\varvec{j}}}+{\varvec{p}}{{\varvec{t}}_{{\varvec{i}}{\varvec{j}}}}+{{\varvec{e}}_{{\varvec{i}}{\varvec{j}}}}\), where
\(\varvec{\mu}\): grand mean
\({{\varvec{p}}_{\varvec{i}}}:\) difference due to patient i (i = 1, …, n), \({{\varvec{p}}_{{\varvec{i}}~}}\sim ~{\mathbf{Normal}}(0,~~\varvec{\sigma}_{{\varvec{p}}}^{2})\)
\({{\varvec{t}}_{\varvec{j}}}:\) difference due to time point j (j = 1, …, k), \(\mathop \sum \limits_{{{\varvec{j}}=1}}^{{\varvec{k}}} {{\varvec{t}}_{\varvec{j}}}=0\)
\({\varvec{p}}{{\varvec{t}}_{{\varvec{i}}{\varvec{j}}}}:\) interaction between patient i and time point j, \(\mathop \sum \limits_{{{\varvec{j}}=1}}^{{\varvec{k}}} {\varvec{p}}{{\varvec{t}}_{{\varvec{i}}{\varvec{j}}}}=0\) and \({\varvec{p}}{{\varvec{t}}_{{\varvec{i}}{\varvec{j}}}}~\sim ~{\mathbf{Normal}}(0,~~\varvec{\sigma}_{{{\varvec{p}}{\varvec{t}}}}^{2})\)
\({{\varvec{e}}_{{\varvec{i}}{\varvec{j}}}}\): random error, \({{\varvec{e}}_{{\varvec{i}}{\varvec{j}}}}~\sim ~{\mathbf{Normal}}(0,~~\varvec{\sigma}_{{\varvec{e}}}^{2})\)
Source of variance df MS Expected components in MS
Between patients n − 1 MS P \(k\sigma _{p}^{2}+\sigma _{e}^{2}\)
Within patients    
Between time points k − 1 MS T \(n\mathop \sum \limits_{{j=1}}^{k} t_{j}^{2}/(k - 1)+\frac{k}{{k - 1}}\sigma _{{pt}}^{2}+\sigma _{e}^{2}\)
Error (p × t) (n − 1)(k − 1) MS E \(\frac{k}{{k - 1}}\sigma _{{pt}}^{2}+\sigma _{e}^{2}\)
ICC (A, 1) of McGraw and Wong [6, p 35] =\(\frac{{\varvec{\sigma}_{{\varvec{p}}}^{2} - \varvec{\sigma}_{{{\varvec{p}}{\varvec{t}}}}^{2}/({\varvec{k}} - 1)}}{{\varvec{\sigma}_{{\varvec{p}}}^{2}+\varvec{\sigma}_{{{\varvec{p}}{\varvec{t}}}}^{2}+\varvec{\sigma}_{{\varvec{e}}}^{2}+\mathop \sum \nolimits_{{{\varvec{j}}=1}}^{{\varvec{k}}} {\varvec{t}}_{{\varvec{j}}}^{2}/({\varvec{k}} - 1)}}~=\frac{{{\varvec{M}}{{\varvec{S}}_{\varvec{P}}} - {\varvec{M}}{{\varvec{S}}_{\varvec{E}}}}}{{{\varvec{M}}{{\varvec{S}}_{\varvec{P}}}+({\varvec{k}} - 1){\varvec{M}}{{\varvec{S}}_{\varvec{E}}}+({\varvec{k}}/{\varvec{n}})({\varvec{M}}{{\varvec{S}}_{\varvec{T}}} - {\varvec{M}}{{\varvec{S}}_{\varvec{E}}})}}\)
  1. A absolute agreement, E, e error, k number of time points, MS mean squares, n number of patients in the test–retest evaluation, P, p patients, T, t time points
  2. In a typical test–retest assessment with two time points, k is 2 in the above ANOVA model and ICC (A, 1) formula. SAS Proc GLM and Proc Mixed can be used to generate the components needed to compute the intraclass correlation coefficient (ICC). Programming information is available upon request to the corresponding author, and a publicly available macro for computing ICCs in the notational system of Shrout and Fleiss can be found at the SAS website http://support.sas.com/kb/25/031.html
  3. The confidence interval formula of ICC (A, 1) for case 3 model of McGraw and Wong [6] can be found on page 41 of the original paper