Estimating correlation between multivariate longitudinal data in the presence of heterogeneity

Table 1 Summary statistics of baseline covariates, the estimated regression coefficients, and the estimated parameters of variance (Var) and covariance (Cov) from the univariate and bivariate mixed models for longitudinal mean deviation (MD) and visual acuity (VA)

Variables	Mean ± SD or N (%)	Estimated fixed and random effects ± standard errors
		Univariate mixed model		Bivariate mixed model
		MD	VA	MD	VA
Fixed effects:
Intercept	-	−2.26 ± 0.34^#	50.75 ± 1.03^#	−2.23 ± 0.34^#	50.88 ± 1.00^#
Slope	-	−0.35 ± 0.04^#	−0.60 ± 0.11^#	−0.35 ± 0.04^#	−0.69 ± 0.11^#
Age (years)	65.7 ± 9.5	−0.49 ± 0.15^#	−3.45 ± 0.43^#	−0.43 ± 0.14^#	−3.07 ± 0.42^#
IOP (mmHg)	22.3 ± 6.2	−0.55 ± 0.16^#	−0.55 ± 0.47	−0.55 ± 0.16^#	−0.40 ± 0.45
CCT (μm)	558.4 ± 37.8	0.35 ± 0.15^*	−0.07 ± 0.45	0.35 ± 0.15^*	−0.22 ± 0.44
HCD	0.53 ± 0.19	0.20 ± 0.15	0.12 ± 0.44	0.20 ± 0.15	0.12 ± 0.44
Male	150 (56%)	0.11 ± 0.29	0.77 ± 0.87	0.06 ± 0.29	1.01 ± 0.84
African American	89 (33%)	−0.57 ± 0.31	−2.13 ± 0.94^*	−0.58 ± 0.31	−2.15 ± 0.91^*
Observation group	158 (59%)	−0.08 ± 0.32	−0.26 ± 0.94	−0.10 ± 0.32	−0.69 ± 0.92
Random effects:^&
Var(E_k)		2.03 ± 0.06^#	19.15 ± 0.96^#	2.02 ± 0.06^#	19.19 ± 0.96^#
Var(I_k)		5.05 ± 0.51^#	39.25 ± 4.69^#	5.09 ± 0.51^#	38.77 ± 4.63^#
Var(S_k)		0.29 ± 0.04^#	1.35 ± 0.25^#	0.29 ± 0.04^#	1.35 ± 0.24^#
Cov(I_k, S_k)		0.62 ± 0.10^#	−1.23 ± 0.83	0.63 ± 0.10^#	−0.73 ± 0.79
Cov(I₁, I₂)		-	-	-	3.88 ± 1.17^#
Cov(S₁, S₂)		-	-	-	0.37 ± 0.08^#
Cov(I₁, S₂)		-	-	-	1.16 ± 0.27^#
Cov(I₂, S₁)		-	-	-	0.67 ± 0.31^*

IOP intraocular pressure, CCT central corneal thickness, HCD horizontal cup-to-disc ratio
^&E_k, I_k, S_k: error term, random intercept, and random slope for MD (k = 1) and VA (k = 2)
^* p < 0.05, ^# p < 0.01

ISSN: 1471-2288