. 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 3.

Features of the covariance structure under random effects models for longitudinal correlated data with 3-time points.

	Random intercept for subject and eye within subject (equation 3)	Random intercept and slope for subject; random intercept for eye within subject (equation 6)	Random intercept and slope for subject; random slope for eye within subject (equation 6)
Vector of random effects (transpose)	(u_i, v_i0, v_i1)	(u_i, w_i, v_i0, v_i1)	(u_i, w_i, z_i0, z_i1)
Covariance matrix for random effects (G)	$(\begin{matrix} σ_{u}^{2} & 0 & 0 \\ 0 & σ_{v}^{2} & 0 \\ 0 & 0 & σ_{v}^{2} \end{matrix})$	$(\begin{matrix} σ_{u}^{2} & σ_{uw} & 0 & 0 \\ σ_{uw} & σ_{w}^{2} & 0 & 0 \\ 0 & 0 & σ_{v}^{2} & 0 \\ 0 & 0 & 0 & σ_{v}^{2} \end{matrix})$	$(\begin{matrix} σ_{u}^{2} & σ_{uw} & 0 & 0 \\ σ_{uw} & σ_{w}^{2} & 0 & 0 \\ 0 & 0 & σ_{z}^{2} & 0 \\ 0 & 0 & 0 & σ_{z}^{2} \end{matrix})$
Variance of outcome in an eye at visit k	$σ_{u}^{2} + σ_{v}^{2} + σ_{e}^{2}$	$σ_{u}^{2} + 2 t_{k} σ_{uw} + t_{k}^{2} σ_{w}^{2} + σ_{v}^{2} + σ_{e}^{2}$	$σ_{u}^{2} + 2 t_{k} σ_{uw} + t_{k}^{2} σ_{w}^{2} + t_{k}^{2} σ_{z}^{2} + σ_{e}^{2}$
Inter-eye correlation at one visit k	$\frac{σ_{u}^{2}}{σ_{u}^{2} + σ_{v}^{2} + σ_{e}^{2}}$	$\frac{σ_{u}^{2} + 2 t_{k} σ_{uw} + t_{k}^{2} σ_{w}^{2}}{σ_{u}^{2} + 2 t_{k} σ_{uw} + t_{k}^{2} σ_{w}^{2} + σ_{v}^{2} + σ_{e}^{2}}$	$\frac{σ_{u}^{2} + 2 t_{k} σ_{uw} + t_{k}^{2} σ_{w}^{2}}{σ_{u}^{2} + 2 t_{k} σ_{uw} + t_{k}^{2} σ_{w}^{2} + σ_{v}^{2} + σ_{e}^{2}}$
Longitudinal correlation between visits k₁ and k₂	$\frac{σ_{u}^{2} + σ_{v}^{2}}{σ_{u}^{2} + σ_{v}^{2} + σ_{e}^{2}}$	$\frac{σ_{u}^{2} + (t_{k_{1}} + t_{k 2}) σ_{uw} + t_{k_{1}} t_{k_{2}} σ_{w}^{2} + σ_{v}^{2}}{\sqrt{(σ_{u}^{2} + 2 t_{k_{1}} σ_{uw} + t_{k_{1}}^{2} σ_{w}^{2} + σ_{v}^{2} + σ_{e}^{2}) (σ_{u}^{2} + 2 t_{k_{2}} σ_{uw} + t_{k_{2}}^{2} σ_{w}^{2} + σ_{v}^{2} + σ_{e}^{2})}}$	$\frac{σ_{u}^{2} + (t_{k_{1}} + t_{k_{2}}) σ_{uw} + t_{k_{1}} t_{k_{2}} σ_{w}^{2} + t_{k_{1}} t_{k_{2}} σ_{z}^{2}}{\sqrt{(σ_{u}^{2} + 2 t_{k_{1}} σ_{uw} + t_{k_{1}}^{2} σ_{w}^{2} + t_{k_{1}}^{2} σ_{z}^{2} + σ_{e}^{2}) (σ_{u}^{2} + 2 t_{k_{2}} σ_{uw} + t_{k_{2}}^{2} σ_{w}^{2} + t_{k_{2}}^{2} σ_{z}^{2} + σ_{e}^{2})}}$