. Author manuscript; available in PMC: 2021 May 26.

Published in final edited form as: Ophthalmic Epidemiol. 2020 Aug 2;28(1):3–20. doi: 10.1080/09286586.2020.1786590

Table 2.

The features of covariance structure of UN@UN, UN@CS and UN@AR for fixed effects model of the longitudinal correlated data with 3-time points.

	UN@UN	UN@CS	UN@AR
Inter-eye covariance matrix	$(\begin{matrix} σ_{L}^{2} & σ_{L R} \\ σ_{L R} & σ_{R}^{2} \end{matrix})$	$(\begin{matrix} σ_{L}^{2} & σ_{L R} \\ σ_{L R} & σ_{R}^{2} \end{matrix})$	$(\begin{matrix} σ_{L}^{2} & σ_{L R} \\ σ_{L R} & σ_{R}^{2} \end{matrix})$
Longitudinal covariance matrix	$(\begin{matrix} 1 & λ_{12} & λ_{13} \\ λ_{22} & λ_{23} \\ λ_{33} \end{matrix})$	$(\begin{matrix} 1 & λ & λ \\ 1 & λ \\ 1 \end{matrix})$	$(\begin{matrix} 1 & λ & λ^{2} \\ 1 & λ \\ 1 \end{matrix})$
Product of inter-eye and longitudinal covariance matrix	$\begin{matrix} σ_{L}^{2} & σ_{L}^{2} λ_{12} & σ_{L}^{2} λ_{13} & σ_{L R} & σ_{L R} λ_{12} & σ_{L R} λ_{13} \\ σ_{L}^{2} λ_{12} & σ_{L}^{2} λ_{22} & σ_{L}^{2} λ_{23} & σ_{L R} λ_{12} & σ_{L R} λ_{22} & σ_{L R} λ_{23} \\ σ_{L}^{2} λ_{13} & σ_{L}^{2} λ_{23} & σ_{L}^{2} λ_{33} & σ_{L R} λ_{13} & σ_{L R} λ_{23} & σ_{L R} λ_{33} \\ σ_{L R} & σ_{L R} λ_{12} & σ_{L R} λ_{13} & σ_{R}^{2} & σ_{R}^{2} λ_{12} & σ_{R}^{2} λ_{13} \\ σ_{L R} λ_{12} & σ_{L R} λ_{22} & σ_{L R} λ_{23} & σ_{R}^{2} λ_{12} & σ_{R}^{2} λ_{22} & σ_{R}^{2} λ_{23} \\ σ_{L R} λ_{13} & σ_{L R} λ_{23} & σ_{L R} λ_{33} & σ_{R}^{2} λ_{13} & σ_{R}^{2} λ_{23} & σ_{R}^{2} λ_{33} \end{matrix}$	$\begin{matrix} σ_{L}^{2} & σ_{L}^{2} λ & σ_{L}^{2} λ & σ_{L R} & σ_{L R} λ & σ_{L R} λ \\ σ_{L}^{2} λ & σ_{L}^{2} & σ_{L}^{2} λ & σ_{L R} λ & σ_{L R} & σ_{L R} λ \\ σ_{L}^{2} λ & σ_{L}^{2} λ & σ_{L}^{2} & σ_{L R} λ & σ_{L R} λ & σ_{L R} \\ σ_{L R} & σ_{L R} λ & σ_{L R} λ & σ_{R}^{2} & σ_{R}^{2} λ & σ_{R}^{2} λ \\ σ_{L R} λ & σ_{L R} & σ_{L R} λ & σ_{R}^{2} λ & σ_{R}^{2} λ & σ_{R}^{2} λ \\ σ_{L R} λ & σ_{L R} λ & σ_{L R} & σ_{R}^{2} λ & σ_{R}^{2} λ & σ_{R}^{2} \end{matrix}$	$\begin{matrix} σ_{L}^{2} & σ_{L}^{2} λ & σ_{L}^{2} λ & σ_{L R} & σ_{L R} λ & σ_{L R} λ \\ σ_{L}^{2} λ & σ_{L}^{2} & σ_{L}^{2} λ & σ_{L R} λ & σ_{L R} & σ_{L R} λ \\ σ_{L}^{2} λ^{2} & σ_{L}^{2} λ & σ_{L}^{2} & σ_{L R} λ^{2} & σ_{L R} λ & σ_{L R} \\ σ_{L R} & σ_{L R} λ & σ_{L R} λ^{2} & σ_{R}^{2} & σ_{R}^{2} λ & σ_{R}^{2} λ \\ σ_{L R} λ & σ_{L R} & σ_{L R} λ & σ_{R}^{2} λ & σ_{R}^{2} & σ_{R}^{2} λ \\ σ_{L R} λ^{2} & σ_{L R} λ & σ_{L R} & σ_{R}^{2} λ^{2} & σ_{R}^{2} λ & σ_{R}^{2} \end{matrix}$
Variance of outcome for the left (right) eyes at visit k	$σ_{L}^{2} λ_{k k} (σ_{R}^{2} λ_{k k})$	$σ_{L}^{2} (σ_{R}^{2})$	$σ_{L}^{2} (σ_{R}^{2})$
Inter-eye correlation at one visit k	$\frac{σ_{L R}}{σ_{L} σ_{R}}$	$\frac{σ_{L R}}{σ_{L} σ_{R}}$	$\frac{σ_{L R}}{σ_{L} σ_{R}}$
Longitudinal correlation between visits k₁ and k₂	$\frac{λ_{k_{1} k_{2}}}{\sqrt{λ_{k_{1} k_{1}} λ_{k_{2} k_{2}}}}$	$λ$	$λ^{\| k_{1} - k_{2} \|}$
Cross correlation between one eye at visit k₁ and the fellow eye at another visit k₂	$(\frac{σ_{L R}}{σ_{L} σ_{R}}) \frac{λ_{k_{1} k_{2}}}{\sqrt{λ_{k_{1} k_{1}} λ_{k_{2} k_{2}}}}$	$(\frac{σ_{L R}}{σ_{L} σ_{R}}) λ$	$(\frac{σ_{L R}}{σ_{L} σ_{R}}) λ^{\| k_{1} - k_{2} \|}$
Total number of parameters to estimate	8	4	4

UN = unstructured, CS = compound symmetry, AR = autoregressive