Table 1.

Alternative Covariance Structures for Group Effects in Dynamic Group Models, Shown for Four Time Points

Name	Structure for Σu	Description
Fully Banded (Toeplitz)A1,B1	${\sigma }_u^2\left[\begin{array}{cccc}\hfill 1\hfill & \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill a\hfill & \hfill 1\hfill & \hfill a\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill a\hfill & \hfill 1\hfill & \hfill a\hfill \\ \hfill c\hfill & \hfill b\hfill & \hfill a\hfill & \hfill 1\hfill \end{array}\right]$	One parameter is estimated for each time lag; the correlation of the group effects one time point apart is a, two time points apart is b, three time points apart is c, etc.
Stabilizing Banded (SB)B2	${\sigma }_u^2\left[\begin{array}{cccc}\hfill 1\hfill & \hfill a\hfill & \hfill b\hfill & \hfill b\hfill \\ \hfill a\hfill & \hfill 1\hfill & \hfill a\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill a\hfill & \hfill 1\hfill & \hfill a\hfill \\ \hfill b\hfill & \hfill b\hfill & \hfill a\hfill & \hfill 1\hfill \end{array}\right]$	Assumes the over-time correlation for the group effects stabilizes at a set value at a particular distance in time, here shown to stabilize at lag 2 to the value b.
Compound Symmetric (CS)B3	${\sigma }_u^2\left[\begin{array}{cccc}\hfill 1\hfill & \hfill a\hfill & \hfill a\hfill & \hfill a\hfill \\ \hfill a\hfill & \hfill 1\hfill & \hfill a\hfill & \hfill a\hfill \\ \hfill a\hfill & \hfill a\hfill & \hfill 1\hfill & \hfill a\hfill \\ \hfill a\hfill & \hfill a\hfill & \hfill a\hfill & \hfill 1\hfill \end{array}\right]$	Assumes that the over-time correlation for the group effects, a, is the same at every time lag.
First-Order Autoregressive (AR)A3	${\sigma }_u^2\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \rho \hfill & \hfill {\rho }^2\hfill & \hfill {\rho }^3\hfill \\ \hfill \rho \hfill & \hfill 1\hfill & \hfill \rho \hfill & \hfill {\rho }^2\hfill \\ \hfill {\rho }^2\hfill & \hfill \rho \hfill & \hfill 1\hfill & \hfill \rho \hfill \\ \hfill {\rho }^3\hfill & \hfill {\rho }^2\hfill & \hfill \rho \hfill & \hfill 1\hfill \end{array}\right]$	Assumes that the over-time correlation for the group effects decays exponentially toward zero with the time lag, i.e., the autocorrelation is ρd where d is the distance in time between assessments.
First-Order Autoregressive Moving Average (ARMA)A2	${\sigma }_u^2\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \gamma \hfill & \hfill \gamma \rho \hfill & \hfill \gamma {\rho }^2\hfill \\ \hfill \gamma \hfill & \hfill 1\hfill & \hfill \gamma \hfill & \hfill \gamma \rho \hfill \\ \hfill \gamma \rho \hfill & \hfill \gamma \hfill & \hfill 1\hfill & \hfill \gamma \hfill \\ \hfill \gamma {\rho }^2\hfill & \hfill \gamma \rho \hfill & \hfill \gamma \hfill & \hfill 1\hfill \end{array}\right]$	As above, assumes that the over-time correlation for the group effects decays rapidly toward zero in accordance with the autoregressive and moving average parameters ρ and γ.

Note. Structures that share the same superscript letter (A or B) are nested in their covariance parameters. More restricted structures have higher numbers (e.g., the AR structure, A3, is nested within the ARMA structure, A2, as well as the Toeplitz structure, A1).