Comparison analysis of separate and joint models in case of time-to-death event of HIV/AIDS patients under ART follow-up

Figures & data

Table 1 Posterior point estimates of separate and joint models in scenario I

Separate model: lognormal					Bayesian joint model: lognormal
Parameter	Mean	SD	SE	95% CI	Mean	SD	SE	95% CI
For longitudinal
βvt	0.2547	0.0309	0.0604	(0.192, 0.314)	0.2364	0.0252	0.0045	(0.174, 0.296)
βvts	−0.0831	0.0051	0.0930	(−0.08, −0.09)	−0.0805	0.0040	0.0001	(−0.091, −0.07)
σ^2u1	0.0039	0.0029	0.0058	(0.003, 0.005)	0.0038	0.0002	0.0001	(0.003, 0.005)
σ^2u2	0.33	0.041	0.0012	(0.26, 0.43)	0.3188	0.0400	0.0011	(0.248, 0.409)
σ^2CD4	0.0678	0.0022	0.0425	(0.063, 0.072)	0.0067	0.0015	0.0004	(0.063, 0.072)
For survival
α0	0.3632	0.0398	0.0248	(0.292, 0.439)	0.2591	0.0205	0.0025	(0.201, 0.326)
αkart	0.1425	0.0263	0.0107	(0.0913, 0.19)	0.1107	0.0005	0.0005	(0.079, 0.139)
αcu	0.1583	0.0294	0.0133	(0.101, 0.214)	0.0386	0.0012	0.0003	(0.005, 0.083)
αois	0.0726	0.0786	0.0352	(0.056, 0.088)	0.0157	0.0040	0.0000	(0.004, 0.026)
αtb	0.1908	0.0124	0.0720	(0.167, 0.215)	0.058	0.0008	0.0005	(0.041, 0.075)
r					0.0506	0.0009	0.0001	(0.046, 0.055)
σ^2t	0.4734	0.0544	0.0267	(0.376,0.589)	0.0675	0.0021	0.0001	(0.063, 0.072)
Separate model: log-logistic					Bayesian joint model: log-logistic
Parameter	Mean	SD	SE	95% CI	Mean	SD	SE	95% CI
For longitudinal
βvt	0.2530	0.0306	0.0550	(0.192, 0.313)	0.2406	0.0031	0.0003	(0.179, 0.30)
βvts	−0.0832	0.0051	0.0107	(−0.09, −0.07)	−0.0808	0.0051	0.0001	(−0.09, −0.07)
σ^2u1	0.0039	0.0029	0.0005	(0.003, 0.004)	0.0039	0.0003	0.0000	(0.003, 0.004)
σ^2u2	0.3200	0.0400	0.0011	(0.2547, 0.42)	0.3205	0.0415	0.0013	(0.248, 0.411)
σ^2CD4	0.0678	0.0021	0.0005	(0.064, 0.072)	0.0675	0.0021	0.0000	(0.063, 0.072)
For survival
α0	0.0079	0.0314	0.0112	(−0.056, 0.065)	0.0533	0.0319	0.0006	(−0.01, 0.116)
αkart	0.0012	0.0311	0.0101	(−0.059, 0.062)	0.0977	0.0296	0.0005	(0.039, 0.156)
αcu	0.0039	0.0320	0.0116	(−0.062, 0.061)	−0.0201	0.0303	0.0005	(−0.08, 0.041)
αois	0.0131	0.0318	0.0106	(−0.05, 0.0745)	0.0347	0.0187	0.0005	(0.004, 0.069)
αtb	0.0107	0.0317	0.0096	(−0.049, 0.069)	0.0769	0.0264	0.0007	(0.024, 0.127)
r					0.1851	0.0044	0.0001	(0.176, 0.197)
σ^2t	0.0311	0.0014	0.0019	(0.0284, 0.034)	0.0524	0.0026	0.0003	(0.047, 0.057)
Separate model: Weibull					Bayesian joint model: Weibull
Parameter	Mean	SD	SE	95% CI	Mean	SD	SE	95% CI
For longitudinal
βvt	0.2535	0.0309	0.0068	(0.193, 0.31)	0.2307	0.0307	0.0005	(0.169, 0.289)
βvts	−0.0802	0.0060	0.0092	(−0.09, −0.07)	−0.0789	0.005	0.0001	(−0.080, −0.063)
σ^2u1	0.0039	0.0003	0.0004	(0.003, 0.004)	0.0038	0.0001	0.0000	(0.003, 0.004)
σ^2u2	0.324	0.04	0.0013	(0.26, 0.43)	0.3220	0.042	0.0008	(0.248, 0.414)
σ^2CD4	0.0678	0.0022	0.0248	(0.063, 0.072)	0.1617	0.0013	0.0003	(0.136, 0.189)
For survival
α0	−0.85,7	0.026,	0.0034	(−0.90, −0.801)	−0.0881	0.0314	0.0005	(−0.150, −0.027)
αkart	−0.3447	0.0233	0.0022	(−0.39, −0.299)	−0.1008	0.0283	0.0000	(−0.150, −0.044)
αcu	−0.3398	0.0274	0.0029	(−0.39, −0.286)	−0.1027	0.0257	0.0005	(−0.120, −0.006)
αois	−0.3494	0.0185	0.0021	(−0.380, −0.313)	−0.0591	0.0208	0.0004	(−0.100, −0.018)
αtb	−0.0799	0.0308	0.0029	(−0.840, −0.750)	0.0064	0.0305	0.0004	(0.006, 0.065)
r					−0.2249	0.0056	0.0001	(−0.236, −0.214)
σ2	0.02779	0.0396	0.0028	(0.006, 0.06)	0.01935	0.0318	0.0004	(0.015, 0.065)

Table 2 Posterior point estimates of separate and joint models in case of scenario II

Separate model: lognormal					Bayesian joint model: lognormal
Parameter	Mean	SD	SE	95% CI	Mean	SD	SE	95% CI
For longitudinal
βvt	0.2547	0.0309	0.0061	(0.193, 0.255)	0.1300	0.0330	0.0005	(0.066, 0.194)
βvts	0.0831	0.0050	0.0009	(0.093, 0.053)	0.0380	0.0009	0.0003	(0.055, 0.020)
σ^2u1	0.0037	0.0003	0.0006	(0.003, 0.005)	0.0050	0.0003	0.0000	(0.004, 0.006)
σ^2u2	0.3200	0.0400	0.0011	(0.254, 0.420)	0.3200	0.0390	0.0009	(0.249, 0.410)
σ^2CD1	0.0678	0.0022	0.0004	(0.064, 0.072)	0.2340	0.0214	0.0002	(0.190, 0.278)
For survival
α0	0.3632	0.0398	0.0025	(0.292, 0.439)	0.2620	0.0347	0.0007	(0.193, 0.330)
αkart	0.1425	0.0263	0.0011	(0.091, 0.193)	0.1150	0.0147	0.0000	(0.086, 0.144)
αcu	0.1583	0.0294	0.0013	(0.101, 0.214)	0.0490	0.0090	0.0000	(0.031, 0.068)
αois	0.0726	0.0079	0.0035	(0.056, 0.088)	0.0150	0.0057	0.0001	(0.004, 0.026)
αtb	0.1908	0.0124	0.0072	(0.167, 0.220)	0.0560	0.0092	0.0005	(0.038, 0.742)
r1					0.0500	0.0003	0.0000	(0.045, 0.056)
r2					0.099	0.0130	0.0002	(0.073, 0.1256)
σ^2t	0.4734	0.05441	0.002675	(0.367, 0.564)	1.616	0.1590	0.0018	(1.324, 1.948)
Separate model: log-logistic					Bayesian joint model: log-logistic
Parameter	Mean	SD	SE	95% CI	Mean	SD	SE	95% CI
For longitudinal
βvt	0.0111	0.0316	0.0061	(0.195, 0.314)	0.1494	0.0032	0.0001	(0.085, 0.213)
βvts	0.083	0.0053	0.001	(0.093, 0.063)	0.0290	0.0089	0.0002	(0.047, 0.012)
σ^2u1	0.0039	0.0029	0.0001	(0.003, 0.005)	0.0049	0.0003	0.0000	(0.004, 0.005)
σ^2u2	0.320	0.0430	0.0008	(0.253, 0.420)	0.3328	0.0446	0.001	(0.255, 0.430)
σ^2CD4	0.0678	0.0022	0.0005	(0.064, 0.072)	0.2300	0.0210	0.0001	(0.195, 0.280)
For survival
α0	0.0056	0.0326	0.0008	(0.056, 0.067)	0.0570	0.032	0.0002	(0.005, 0.120)
αkart	0.0039	0.0331	0.0012	(0.062, 0.067)	0.0998	0.029	0.000	(0.041, 0.157)
αcu	0.0022	0.0308	0.0011	(0.060, 0.062)	0.0160	0.030	0.0002	(0.076, 0.044)
αois	0.0128	0.0317	0.0010	(0.046, 0.076)	0.0404	0.019	0.0002	(0.002, 0.077)
αtb	0.0131	0.0323	0.0011	(0.049, 0.076)	0.0590	0.025	0.0003	(0.008, 0.108)
r1					0.2060	0.0056	0.0001	(0.195, 0.217)
r2					0.0831	0.0288	0.0003	(0.026, 0.139)
σ^2t	0.03109	0.001291	0.0002	(0.024, 0.042)	0.0540	0.0028	0.0001	(0.049, 0.061)
Separate model: Weibull					Bayesian joint model: Weibull
Parameter	Mean	SD	SE	95% CI	Mean	SD	SE	95% CI
For longitudinal
βvt	0.0829	0.0051	0.0094	(0.194, 0.314)	0.1466	0.033	0.0004	(0.082, 0.211)
βvts	0.0800	0.0050	0.0094	(0.09, 0.070)	0.0250	0.009	0.0002	(0.043, 0.007)
σ^2u1	0.0039	0.0003	0.0051	(0.0004, 0.005)	0.0049	0.0004	0.0000	(0.004, 0.006)
σ^2u2	0.3278	0.0430	0.0009	(0.253, 0.421)	0.3500	0.0470	0.0001	(0.267, 0.450)
σ^2CD1	0.0678	0.0022	0.0035	(0.064, 0.072)	0.2300	0.0214	0.0002	(0.193, 0.277)
For survival
α0	0.2530	0.0305	0.0029	(0.194, 0.311)	0.089	0.0310	0.0001	(0.151, 0.028)
αkart	0.0410	0.0110	0.0054	(0.063, 0.012)	0.0970	0.0280	0.0001	(0.152, 0.042)
αcu	0.1190	0.0171	0.0010	(0.086, 0.154)	0.0690	0.0210	0.0001	(0.111, 0.028)
αois	0.0072	0.0036	0.0016	(0.0002, 0.014)	0.1160	0.0206	0.0004	(.157, 0.077)
αtb	0.0195	0.0060	0.0029	(0.007, 0.030)	0.1080	0.0260	0.0001	(0.159, 0.057)
r1					0.2400	0.0069	0.0000	(0.250, 0.230)
r2					0.1186	0.0290	0.0002	(0.175, 0.062)
σ^2t	0.034	0.0205	0.0407	(0.005, 0.057)	0.1283	0.0319	0.0004	(0.067, 0.191)

Figure 1 Empirical and fitted hazard rate estimates.

Table 3 Model comparison of separate and joint models for both scenarios

Scenario I
Separate models				Bayesian joint models
Model	Dbar	AIC	BIC	Dbar	AIC	BIC
Lognormal	15,377.2	16,293.8	16,325.8	14,956.0	15,868.6	15,903.5
Log-logistic	15,279.2	16,194.7	16,226.7	14,893.9	15,799.5	15,834.5
Weibull	16,015.4	16,932.6	16,964.6	15,093.9	16,004.1	16,039.1
Scenario II
Separate models				Bayesian joint models
Model	Dbar	AIC	BIC	Dbar	AIC	BIC
Lognormal	15,377.2	16,293.8	16,325.8	14,905.4	15,818.0	15,854.0
Log-logistic	15,279.2	16,194.7	16,226.7	14,898.6	15,805.2	15,841.2
Weibull	16,015.4	16,932.6	16,964.6	15,113.6	16,023.8	16,059.8

Abbreviations: Dbar, posterior mean of the deviance; AIC, Akaike information criteria; BIC, Bayesian information criteria; DIC, deviance information criteria.

Table 4 Result of model comparison of Bayesian joint models

Scenario I					Scenario II
Model	Dbar	Dhat	pD	DIC	Dbar	Dhat	pD	DIC
BJMLN	15,393.5	14,547.4	846.015	16,239.5	14,905.4	13,995.7	909.704	15,815.1
BJMLL	14,895.4	13,982.8	912.644	15,808.1	14,898.6	13,986.8	911.873	15,810.5
BJMW	15,538.7	14,697.3	841.388	16,380.1	15,113.6	14,204.0	909.554	16,023.1

Abbreviations: Dbar, posterior mean of the deviance; Dhat, deviance of the posterior means; PD, effective number of parameters; DIC, deviance information criteria; BJMLN, Bayesian Joint Model of Lognormal; BJMLL, Bayesian Joint Model of Log-Logistic; BJMW, Bayesian Joint Model of Weibull.