TABLE 2.

Glossary of notation.

Notation	Meaning
L,I,T,N	Number of outcomes, clusters, periods, subjects per cluster-period (balanced design)
${\mathbf{\text{Σ}}}_{\mathbf{b}}$,${\mathbf{\text{Σ}}}_{\mathbf{s}}$ ,${\mathbf{\text{Σ}}}_{\mathbf{ϵ}}$ ,${\mathbf{\text{Σ}}}_{\mathbf{\gamma }}$	cluster, cluster-period, error, within-subject (closed-cohort) variance components
I u ,J u	u × u identity matrix, u × u matrix of ones
A⊗ B	Kronecker product of the matrices A and B
A ○ B	Hadamard (element-wise) product of the matrices A and B
U, V, W	${\sum }_{i=1}^I{\sum }_{j=1}^T{X}_{ij}$, ${{\sum }_{i=1}^I\left({\sum }_{j=1}^T{X}_{ij}\right)}^2$, ${{\sum }_{j=1}^T\left({\sum }_{j=1}^I{X}_{ij}\right)}^2$
${\mathbf{\Lambda }}_{\mathit{Y}}$	$\mathrm{diag}\left({\sigma }_{\mathit{Y}1}^2,\dots ,{\sigma }_{\mathit{Y}L}^2\right)$
${\mathbf{\sigma }}_{ϵ}$	${\left({\sigma }_{ϵ1},\dots ,{\sigma }_{ϵL}\right)}^{\top }$
$\mathbf{\omega }$	${\left({\sigma }_{\mathit{Y}1}{\left(1-{\rho }_0^1\right)}^{1/2},\dots ,{\sigma }_{\mathit{Y}L}{\left(1-{\rho }_0^L\right)}^{1/2}\right)}^{\top }$ or ${\left({\sigma }_{\mathit{Y}1}{\left(1-{\rho }_0^1+{\rho }_1^1-{\rho }_2^1\right)}^{1/2},\dots ,{\sigma }_{\mathit{Y}L}{\left(1-{\rho }_0^L+{\rho }_1^L-{\rho }_2^L\right)}^{1/2}\right)}^{\top }$ (closed-cohort)
λ 1	1 − ρ0 or 1 − ρ0 + ρ1 − ρ2 (Closed-cohort)
λ 2	$1+(N-1){\rho }_0-N{\rho }_1$
λ 3	$1+(N-1){\rho }_0+(T-1)N{\rho }_1\text{ or }1+(N-1)\left({\rho }_0-{\rho }_1\right)-{\rho }_2$(closed-cohort)
λ 4	$1+(N-1){\rho }_0+(T-1)(N-1){\rho }_1+(T-1){\rho }_2$
τ 2	$(N-1){\rho }_{00}-N{\rho }_{11}+{\rho }_2$
τ 3	$(N-1){\rho }_{00}-N{\rho }_{11}+{\rho }_2+TN{\rho }_{11}\text{ or }{\rho }_{2.0}-{\rho }_{2.1}+(N-1)\left({\rho }_{00}-{\rho }_{11}\right)$(closed-cohort)
τ 4	${\rho }_{2,0}-{\rho }_{2,1}+(N-1)\left({\rho }_{00}-{\rho }_{11}\right)+T\left({\rho }_{2,1}+(N-1){\rho }_{11}\right)$
${\mathbf{\Gamma }}_0,{\mathbf{\Gamma }}_1$	$\left(\begin{array}{cccc}{\rho }_0^1& {\rho }_0^{12}& \dots & {\rho }_0^{1L}\\ {\rho }_0^{12}& {\rho }_0^2& \dots & {\rho }_0^{2L}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_0^{1L}& {\rho }_0^{2L}& \dots & {\rho }_0^L\end{array}\right),\hspace{1em}\left(\begin{array}{cccc}{\rho }_1^1& {\rho }_1^{12}& \dots & {\rho }_1^{1L}\\ {\rho }_1^{12}& {\rho }_1^2& \dots & {\rho }_1^{2L}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_1^{1L}& {\rho }_1^{2L}& \dots & {\rho }_1^L\end{array}\right)$
${\mathbf{\Gamma }}_2$	$\left(\begin{array}{cccc}1& {\rho }_2^{12}& \dots & {\rho }_2^{1L}\\ {\rho }_2^{12}& 1& \dots & {\rho }_2^{2L}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_2^{1L}& {\rho }_2^{2L}& \dots & 1\end{array}\right)\text{ or }\left(\begin{array}{cccc}1& {\rho }_{2,0}^{12}& \dots & {\rho }_{2,0}^{1L}\\ {\rho }_{2,0}^{12}& 1& \dots & {\rho }_{2,0}^{2L}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_{2,0}^{1L}& {\rho }_{2,0}^{2L}& \dots & 1\end{array}\right)$(closed-cohort)
${\mathbf{\Gamma }}_{2^{\prime }}$	$\left(\begin{array}{cccc}{\rho }_2^1& {\rho }_{2,1}^{12}& \cdots & {\rho }_{2,1}^{1L}\\ {\rho }_{2,1}^{12}& {\rho }_2^2& \cdots & {\rho }_{2,1}^{2L}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_{2,1}^{1L}& {\rho }_{2,1}^{2L}& \cdots & {\rho }_2^L\end{array}\right)$