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, $${{\varvec{e}}_{{\varvec{i}}{\varvec{j}}}}~\sim ~{\mathbf{Normal}}(0,~~\varvec{\sigma}_{{\varvec{e}}}^{2})$$
Between patients n − 1 MS P $$k\sigma _{p}^{2}+\sigma _{e}^{2}$$
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] =$$\frac{{\varvec{\sigma}_{{\varvec{p}}}^{2} - \varvec{\sigma}_{{{\varvec{p}}{\varvec{t}}}}^{2}/({\varvec{k}} - 1)}}{{\varvec{\sigma}_{{\varvec{p}}}^{2}+\varvec{\sigma}_{{{\varvec{p}}{\varvec{t}}}}^{2}+\varvec{\sigma}_{{\varvec{e}}}^{2}+\mathop \sum \nolimits_{{{\varvec{j}}=1}}^{{\varvec{k}}} {\varvec{t}}_{{\varvec{j}}}^{2}/({\varvec{k}} - 1)}}~=\frac{{{\varvec{M}}{{\varvec{S}}_{\varvec{P}}} - {\varvec{M}}{{\varvec{S}}_{\varvec{E}}}}}{{{\varvec{M}}{{\varvec{S}}_{\varvec{P}}}+({\varvec{k}} - 1){\varvec{M}}{{\varvec{S}}_{\varvec{E}}}+({\varvec{k}}/{\varvec{n}})({\varvec{M}}{{\varvec{S}}_{\varvec{T}}} - {\varvec{M}}{{\varvec{S}}_{\varvec{E}}})}}$$