joineRML: a joint model and software package for time-to-event and multivariate longitudinal outcomes

Table 4 Fitted multivariate and separate univariate joint models to the PBC data

	joineRML (NR)			joineRML (GN)			Bootstrap		joineR
	Estimate	SE	95% CI^d	Estimate	SE	95% CI^d	SE	95% CI^e	Estimate	SE	95% CI^e
β _0,1	0.5541	0.0858	(0.3859, 0.7223)	0.5549	0.0846	(0.3892, 0.7207)	0.0800	(0.4264, 0.7435)	0.5545	0.0838	(0.3802, 0.7031)
β _1,1	0.2009	0.0201	(0.1616, 0.2402)	0.2008	0.0201	(0.1614, 0.2402)	0.0204	(0.1669, 0.2468)	0.1808	0.0209	(0.1430, 0.2324)
β _0,2	3.5549	0.0356	(3.4850, 3.6248)	3.5546	0.0357	(3.4846, 3.6245)	0.0255	(3.4972, 3.5904)	3.5437	0.0333	(3.4418, 3.6095)
β _1,2	−0.1245	0.0101	(−0.1444, −0.1047)	−0.1246	0.0101	(−0.1444, −0.1047)	0.0120	(−0.1489, −0.1063)	−0.0997	0.0113	(−0.1256, −0.0773)
β _0,3	0.8304	0.0212	(0.7888, 0.8719)	0.8301	0.0210	(0.7888, 0.8713)	0.0196	(0.7953, 0.8638)	0.8233	0.0220	(0.7818, 0.8677)
β _1,3	−0.0577	0.0062	(−0.0699, −0.0456)	−0.0577	0.0062	(−0.0698, −0.0455)	0.0057	(−0.0698, −0.0486)	−0.0447	0.0052	(−0.0555, −0.0362)
γ _v	0.0462	0.0151	(0.0166, 0.0759)	0.0462	0.0152	(0.0165, 0.0759)	0.0173	(0.0198, 0.0880)	0.0575^a	0.0123^a	(0.0314, 0.0760)^a
									0.0413^b	0.0150^b	(0.0113, 0.0714)^b
									0.0424^c	0.0157^c	(0.0146, 0.0724)^c
γ _bil	0.8181	0.2046	(0.4171, 1.2191)	0.8187	0.2036	(0.4197, 1.2177)	0.2153	(0.5172, 1.4021)	1.2182	0.1654	(0.9800, 1.5331)
γ _alb	−1.7060	0.6181	(−2.9173, −0.4946)	−1.6973	0.6163	(−2.9053, −0.4893)	0.7562	(−3.3862, −0.5188)	−3.0770	0.6052	(−4.7133, −2.1987)
γ _pro	−2.2085	1.6070	(−5.3582, 0.9412)	−2.2148	1.6133	(−5.3768, 0.9472)	1.6094	(−5.3050, 0.6723)	−7.2078	1.2640	(−10.5247, −5.2616)

Notation: SE = standard error; CI = confidence interval; NR = one-step Newton-Raphson update for γ; GN = one-step Gauss-Newton-like update for γ
^aSeparate model fit for serBilir
^bSeparate model fit for albumin
^cSeparate model fit for prothrombin
^dSEs are calculated from the inverse profile empirical information matrix, and confidence intervals are based on normal approximations of the type \(\hat {\boldsymbol {\theta }} \pm 1.96 \text {SE}(\hat {\boldsymbol {\theta }})\), where \(\hat {\boldsymbol {\theta }}\) denote the estimated maximum likelihood estimates
^eSEs and confidence intervals are derived from B=100 bootstrap samples, with confidence intervals based on the 2.5% and 97.5% percentiles. NB. one model failed to converge using joineRML within the maximum number of MC iterations, and so SEs and CIs are based on 99 bootstrap samples only