Simulated maximum likelihood estimation in joint models for multiple longitudinal markers and recurrent events of multiple types, in the presence of a terminal event

Figures & data

Table 1. Number of recurrences within patients for all 4 recurrent infections.

	Frequency
Infection type	0	1	2	3	4	5	6	7	8	9	10
Cytomegalovirus	244	87	21	4	0	0	1
Upper respiratory	224	83	31	13	4	1	1
Wound	260	71	17	4	3	1	0	0	0	1
Urinary tract	199	61	44	13	13	11	6	5	3	1	1

Table 2. Simulation results for the SML estimator and CSML estimator using a small (Z=10,000) Sobol point set.

			SML estimator (${\mathit{\theta }}_{\mathrm{SML}}$)					CSML estimator (${\mathit{\theta }}_{\mathrm{CSML}}$)
				$E\left[\right|\mathit{\theta }-\hat{\mathit{\theta }}\left|\right]$	$E\left[\right(\mathit{\theta }-\hat{\mathit{\theta }}{)}^2]$	$E\left[\mathrm{var}\right(\hat{\mathit{\theta }}\left)\right]$	Coverage		$E\left[\right|\mathit{\theta }-\hat{\mathit{\theta }}\left|\right]$	$E\left[\right(\mathit{\theta }-\hat{\mathit{\theta }}{)}^2]$	$E\left[\mathrm{var}\right(\hat{\mathit{\theta }}\left)\right]$	Coverage
	Parameter	$\mathit{\theta }$	$E\left[\hat{\mathit{\theta }}\right]$	$\times 1000$	$\times 1000$	$\times 1000$	95% CI	$E\left[\hat{\mathit{\theta }}\right]$	$\times 1000$	$\times 1000$	$\times 1000$	95% CI
Random effects markers and recurrent events	$\log \left({\alpha }_{1,1}^{v_1}\right)$	−0.450	−0.444	48.932	3.755	3.144	0.92	−0.465	50.798	3.880	3.144	0.91
	${\alpha }_{1,2}^{v_1}$	0.000	0.004	17.213	0.443	0.145	0.69	0.003	14.699	0.330	0.145	0.78
	$\log \left({\alpha }_{2,2}^{v_1}\right)$	−1.750	−1.726	64.693	5.914	2.674	0.77	−1.775	58.908	5.053	2.674	0.84
	$\log \left({\alpha }_{1,1}^{v_2}\right)$	−0.600	−0.600	52.482	4.555	3.760	0.92	−0.618	53.271	4.292	3.760	0.93
	${\alpha }_{1,2}^{v_2}$	0.020	0.025	20.355	0.664	0.176	0.71	0.022	15.155	0.371	0.176	0.81
	$\log \left({\alpha }_{2,2}^{v_2}\right)$	−1.600	−1.577	54.291	4.405	2.339	0.81	−1.618	57.767	5.429	2.339	0.81
	$\log \left({\alpha }_{1,1}^w\right)$	−0.450	−0.475	78.031	8.884	8.172	0.95	−0.481	73.983	8.138	8.172	0.93
	${\alpha }_{1,2}^w$	0.070	0.073	67.724	7.044	5.232	0.90	0.066	60.925	5.788	5.232	0.93
	$\log \left({\alpha }_{2,2}^w\right)$	−0.600	−0.646	97.386	14.332	11.571	0.93	−0.637	89.675	12.091	11.571	0.95
Marker submodels	${\phi }_{10}$	0.500	0.497	40.373	2.461	1.832	0.91	0.500	38.476	2.225	1.832	0.93
	${\phi }_{11}$	0.050	0.049	14.220	0.324	0.143	0.79	0.050	11.622	0.209	0.143	0.91
	$\log \left({\alpha }_1\right)$	−0.900	−0.892	15.761	0.353	0.418	0.94	−0.901	15.370	0.381	0.418	0.94
	${\phi }_{20}$	1.000	1.000	33.813	1.768	1.468	0.91	1.002	32.250	1.600	1.468	0.94
	${\phi }_{21}$	0.000	−0.001	17.757	0.473	0.171	0.75	0.000	12.996	0.274	0.171	0.88
	$\log \left({\alpha }_2\right)$	−0.900	−0.892	16.460	0.353	0.415	0.96	−0.901	15.578	0.379	0.415	0.96
Recurrent event submodels	$\log \left(h_{01}\right)$	−1.200	−1.190	77.156	9.195	8.753	0.95	−1.200	76.475	8.923	8.753	0.95
	${\beta }_1$	−0.150	−0.150	89.106	12.297	11.407	0.95	−0.148	82.861	11.301	11.407	0.95
	${\gamma }_{11}$	0.100	0.100	37.417	2.260	2.204	0.95	0.099	35.840	2.136	2.204	0.94
	${\gamma }_{12}$	0.080	0.079	33.524	1.854	1.792	0.94	0.081	32.905	1.728	1.792	0.95
	$\log \left(h_{02}\right)$	−1.400	−1.386	73.057	8.651	8.408	0.93	−1.398	74.007	8.774	8.408	0.94
	${\beta }_2$	0.200	0.198	81.102	10.197	9.852	0.94	0.201	81.185	10.191	9.852	0.95
	${\gamma }_{21}$	0.090	0.090	38.374	2.279	1.999	0.93	0.090	37.831	2.202	1.999	0.94
	${\gamma }_{22}$	0.120	0.117	32.922	1.693	1.601	0.94	0.117	34.373	1.796	1.601	0.94
Terminal event submodel	$\log \left({\lambda }_0\right)$	−3.000	−3.023	119.121	22.664	24.998	0.96	−2.995	123.289	23.306	24.998	0.96
	${\beta }_0$	0.350	0.358	125.003	25.217	25.406	0.95	0.359	129.600	26.793	25.406	0.94
	${\gamma }_{01}$	0.150	0.154	58.680	5.572	6.080	0.97	0.153	61.984	6.095	6.080	0.95
	${\gamma }_{02}$	0.100	0.106	52.092	4.228	4.906	0.96	0.108	53.328	4.593	4.906	0.95
	${\chi }_1$	0.300	0.308	177.805	54.036	52.435	0.95	0.308	182.523	56.144	52.435	0.94
	${\chi }_2$	0.200	0.231	214.066	71.351	83.475	0.97	0.228	217.644	74.118	83.475	0.96

Table 3. Simulation results for the SML estimator and CSML estimator using a large (Z=60,000 ) Sobol point set.

			SML estimator (${\mathit{\theta }}_{\mathrm{SML}}$)					CSML estimator (${\mathit{\theta }}_{\mathrm{CSML}}$)
				$E\left[\right|\mathit{\theta }-\hat{\mathit{\theta }}\left|\right]$	$E\left[\right(\mathit{\theta }-\hat{\mathit{\theta }}{)}^2]$	$E\left[\mathrm{var}\right(\hat{\mathit{\theta }}\left)\right]$	Coverage		$E\left[\right|\mathit{\theta }-\hat{\mathit{\theta }}\left|\right]$	$E\left[\right(\mathit{\theta }-\hat{\mathit{\theta }}{)}^2]$	$E\left[\mathrm{var}\right(\hat{\mathit{\theta }}\left)\right]$	Coverage
	Parameter	$\mathit{\theta }$	$E\left[\hat{\mathit{\theta }}\right]$	$\times 1000$	$\times 1000$	$\times 1000$	95% CI	$E\left[\hat{\mathit{\theta }}\right]$	$\times 1000$	$\times 1000$	$\times 1000$	95% CI
Random effects markers and recurrent events	$\log \left({\alpha }_{1,1}^{v_1}\right)$	−0.450	−0.448	46.195	3.344	3.089	0.93	−0.457	46.838	3.348	3.089	0.92
	${\alpha }_{1,2}^{v_1}$	0.000	0.002	14.769	0.346	0.137	0.77	0.002	13.661	0.282	0.137	0.83
	$\log \left({\alpha }_{2,2}^{v_1}\right)$	−1.750	−1.749	48.546	3.704	2.421	0.86	−1.762	49.165	3.632	2.421	0.88
	$\log \left({\alpha }_{1,1}^{v_2}\right)$	−0.600	−0.599	44.929	3.277	3.650	0.96	−0.606	45.104	3.296	3.650	0.95
	${\alpha }_{1,2}^{v_2}$	0.020	0.023	16.828	0.438	0.166	0.76	0.023	15.636	0.370	0.166	0.80
	$\log \left({\alpha }_{2,2}^{v_2}\right)$	−1.600	−1.596	48.511	3.710	2.154	0.86	−1.608	48.121	3.632	2.154	0.86
	$\log \left({\alpha }_{1,1}^w\right)$	−0.450	−0.452	71.970	8.117	7.946	0.95	−0.462	69.453	7.568	7.946	0.95
	${\alpha }_{1,2}^w$	0.070	0.066	56.076	5.112	5.247	0.94	0.066	56.001	4.928	5.247	0.95
	$\log \left({\alpha }_{2,2}^w\right)$	−0.600	−0.623	82.866	10.742	11.515	0.96	−0.630	82.519	10.728	11.515	0.94
Marker submodels	${\phi }_{10}$	0.500	0.500	33.786	1.862	1.780	0.95	0.500	32.806	1.751	1.780	0.95
	${\phi }_{11}$	0.050	0.051	11.276	0.207	0.131	0.87	0.051	10.559	0.186	0.131	0.90
	$\log \left({\alpha }_1\right)$	−0.900	−0.898	15.314	0.363	0.405	0.97	−0.898	15.440	0.368	0.405	0.97
	${\phi }_{20}$	1.000	1.000	30.694	1.446	1.431	0.95	1.000	31.023	1.454	1.431	0.96
	${\phi }_{21}$	0.000	−0.000	13.936	0.309	0.157	0.85	−0.001	12.518	0.261	0.157	0.87
	$\log \left({\alpha }_2\right)$	−0.900	−0.898	15.523	0.375	0.401	0.95	−0.898	15.599	0.378	0.401	0.95
Recurrent event submodels	$\log \left(h_{01}\right)$	−1.200	−1.199	67.432	7.276	9.083	0.98	−1.202	68.101	7.397	9.083	0.97
	${\beta }_1$	−0.150	−0.156	84.043	11.095	11.800	0.96	−0.152	83.149	11.174	11.800	0.94
	${\gamma }_{11}$	0.100	0.101	36.762	2.089	2.263	0.95	0.101	38.116	2.208	2.263	0.94
	${\gamma }_{12}$	0.080	0.080	31.466	1.573	1.855	0.97	0.079	31.790	1.599	1.855	0.97
	$\log \left(h_{02}\right)$	−1.400	−1.401	71.542	8.276	8.621	0.95	−1.402	71.496	8.258	8.621	0.94
	${\beta }_2$	0.200	0.202	78.657	9.586	10.107	0.95	0.202	78.441	9.585	10.107	0.96
	${\gamma }_{21}$	0.090	0.090	36.839	2.163	2.074	0.94	0.091	36.417	2.108	2.074	0.95
	${\gamma }_{22}$	0.120	0.120	32.058	1.613	1.654	0.94	0.120	32.547	1.624	1.654	0.95
Terminal event submodel	$\log \left({\lambda }_0\right)$	−3.000	−3.023	121.437	22.222	25.026	0.96	−3.012	120.290	22.151	25.026	0.96
	${\beta }_0$	0.350	0.365	121.682	23.075	25.420	0.95	0.365	123.127	23.515	25.420	0.95
	${\gamma }_{01}$	0.150	0.152	60.637	5.587	6.155	0.96	0.152	60.798	5.740	6.155	0.96
	${\gamma }_{02}$	0.100	0.103	51.556	4.211	4.912	0.96	0.103	52.131	4.312	4.912	0.96
	${\chi }_1$	0.300	0.313	173.087	49.005	48.407	0.96	0.315	175.603	49.292	48.407	0.96
	${\chi }_2$	0.200	0.211	219.150	78.011	77.404	0.95	0.214	216.717	77.137	77.404	0.95

Table 4. Simulation results for the GHQ estimator.

			One-step estimator (${\mathit{\theta }}_{\mathrm{SML}}$)
				$E\left[\right|\mathit{\theta }-\hat{\mathit{\theta }}\left|\right]$	$E\left[\right(\mathit{\theta }-\hat{\mathit{\theta }}{)}^2]$	$E\left[\mathrm{var}\right(\hat{\mathit{\theta }}\left)\right]$	Coverage
	Parameter	$\mathit{\theta }$	$E\left[\hat{\mathit{\theta }}\right]$	$\times 1000$	$\times 1000$	$\times 1000$	95% CI
Random effects markers and recurrent events	$\log \left({\alpha }_{1,1}^{v_1}\right)$	−0.450	−0.691	240.688	3.890	1.420	0.01
	${\sigma }_{1,2}^{v_1}$	0.000	0.006	9.311	0.097	0.014	0.48
	$\log \left({\alpha }_{2,2}^{v_1}\right)$	−1.750	−1.872	126.505	7.123	0.226	0.12
	$\log \left({\alpha }_{1,1}^{v_2}\right)$	−0.600	−0.871	270.994	5.345	1.976	0.00
	${\sigma }_{1,2}^{v_2}$	0.020	0.027	10.090	0.128	0.013	0.45
	$\log \left({\alpha }_{2,2}^{v_2}\right)$	−1.600	−1.712	117.162	6.560	0.148	0.09
	$\log \left({\alpha }_{1,1}^w\right)$	−0.450	−0.471	68.567	6.735	7.640	0.97
	${\sigma }_{1,2}^w$	0.070	0.074	48.803	3.668	5.341	0.98
	$\log \left({\alpha }_{2,2}^w\right)$	−0.600	−0.623	76.353	8.869	11.012	0.97
Marker submodels	${\phi }_{10}$	0.500	0.500	44.758	3.139	0.513	0.56
	${\phi }_{11}$	0.050	0.051	20.260	0.614	0.019	0.25
	${\alpha }_1$	−0.900	−0.822	77.831	0.365	0.387	0.02
	${\phi }_{20}$	1.000	1.000	34.632	1.828	0.478	0.68
	${\phi }_{21}$	0.000	−0.002	22.020	0.727	0.017	0.21
	${\alpha }_2$	−0.900	−0.818	81.864	0.369	0.384	0.01
Recurrent event submodels	$h_{01}$	−1.200	−1.198	71.763	7.943	8.878	0.96
	${\beta }_1$	−0.150	−0.148	84.129	10.845	11.295	0.97
	${\gamma }_{11}$	0.100	0.100	36.403	2.102	2.243	0.94
	${\gamma }_{12}$	0.080	0.079	32.851	1.766	1.848	0.95
	$h_{02}$	−1.400	−1.399	68.683	7.596	8.694	0.96
	${\beta }_2$	0.200	0.202	77.269	9.199	10.081	0.96
	${\gamma }_{21}$	0.090	0.091	34.764	1.873	2.113	0.97
	${\gamma }_{22}$	0.120	0.119	30.242	1.422	1.707	0.98
Terminal event submodel	${\lambda }_0$	−3.000	−3.014	119.664	21.680	24.895	0.97
	${\beta }_0$	0.350	0.352	126.241	24.282	25.434	0.96
	${\gamma }_{01}$	0.150	0.151	63.075	6.474	6.227	0.92
	${\gamma }_{02}$	0.100	0.102	55.525	4.785	4.987	0.96
	${\chi }_1$	0.300	0.313	171.530	45.619	49.391	0.96
	${\chi }_2$	0.200	0.200	206.844	71.781	76.290	0.95

Table 5. Joint model fixed parameter estimates (Est.) and standard errors (Std. err.) for the post kidney transplant data.

	Fixed effects
		CD8+ T Cells (cd8)		CD4+ T Cells (cd4)		B Cells (bcl)		Natural killer cells (nk)
Marker submodels		Est.	Std.err.	Est.	Std.err.	Est.	Std.err.	Est.	Std.err.
	${\phi }_0$	0.680	0.028	0.827	0.030	0.520	0.019	0.586	0.019
	${\phi }_1$	0.045	0.010	0.000	0.009	−0.007	0.005	0.018	0.006
	$\log \left(\alpha \right)$	−2.017	0.029	−1.954	0.025	−2.565	0.022	−2.304	0.024
	Random effects
		${\sigma }_{0,\mathrm{cd}8}$	${\sigma }_{1,\mathrm{cd}8}$	${\sigma }_{0,\mathrm{cd}4}$	${\sigma }_{1,\mathrm{cd}4}$	${\sigma }_{0,\mathrm{bcl}}$	${\sigma }_{1,\mathrm{bcl}}$	${\sigma }_{0,\mathrm{nk}}$	${\sigma }_{1,\mathrm{nk}}$
	${\sigma }_{0,\mathrm{cd}8}$	0.016	−0.003
	${\sigma }_{1,\mathrm{cd}8}$	−0.003	0.002
	${\sigma }_{0,\mathrm{cd}4}$			0.036	−0.007
	${\sigma }_{1,\mathrm{cd}4}$			−0.007	0.002
	${\sigma }_{0,\mathrm{bcl}}$					0.015	−0.001
	${\sigma }_{1,\mathrm{bcl}}$					−0.001	<0.001
	${\sigma }_{0,\mathrm{nk}}$							0.017	−0.003
	${\sigma }_{1,\mathrm{nk}}$							−0.003	0.001
	Fixed effects
		Cytomegalovirus (Cyto)		Upper respiratory (Resp)		Wound		Urinary tract (urinary)		Terminal event
Event submodels		Est.	Std.err.	Est.	Std.err.	Est.	Std.err.	Est.	Std.err.	Est.	Std.err.
	${\beta }_{\mathrm{female}}$	−0.054	0.214	−0.179	0.183	−0.174	0.293	0.799	0.265	−0.176	0.196
	${\beta }_{\mathrm{age}}$	0.006	0.009	−0.012	0.007	0.008	0.011	0.005	0.009	0.022	0.008
	${\beta }_{\mathrm{treatment}B}$	0.258	0.428	−0.034	0.298	0.791	0.492	0.084	0.528	0.046	0.445
	${\gamma }_{cd8}$	2.356	1.650	4.386	1.183	2.168	2.034	0.109	1.385	6.392	1.933
	${\gamma }_{cd4}$	−3.694	1.232	−4.414	0.997	0.384	1.674	−1.898	1.372	−7.050	1.843
	${\gamma }_{bcl}$	2.788	1.696	−0.426	1.663	−1.344	2.045	−0.297	1.796	4.011	2.226
	${\gamma }_{nk}$	−1.831	1.599	0.201	1.423	−1.968	2.123	1.960	1.537	1.597	1.908
	${\chi }_{\mathrm{Cyto}}$									0.016	>10
	${\chi }_{\mathrm{Resp}}$									0.375	>10
	${\chi }_{\mathrm{Wound}}$									−0.054	>10
	${\chi }_{\mathrm{Urinary}}$									0.085	5.347
	Frailties
		${\sigma }_{\mathrm{Cyto}}$	${\sigma }_{\mathrm{Resp}}$	${\sigma }_{\mathrm{Wound}}$	${\sigma }_{\mathrm{Urinary}}$
	${\sigma }_{\mathrm{Cyto}}$	<0.001	−0.002	−0.005	−0.005
	${\sigma }_{\mathrm{Resp}}$	−0.002	0.093	0.266	0.234
	${\sigma }_{\mathrm{Wound}}$	−0.005	0.266	0.833	0.426
	${\sigma }_{\mathrm{Urinary}}$	−0.005	0.234	0.426	1.489

Notes: The baseline hazard estimates of the (recurrent) event submodels are not shown. The intercept was based on a 48 year old man who was given treatment A at baseline and had cell counts of zero for all four longitudinal markers.

Figure 1. History and predicted probabilities of experiencing events for two patients. The dotted lines in the marker figures denote the fixed marker trajectories, that is, zero random effects. The continuous lines are the predicted marker trajectories. For the predictions, the probability of experiencing a particular infection is given. In addition, the sum of the infected no/yes probabilities is the probability that the patient has not dropped out of the study. (a) Patient without events and marker measurements until t=1 and (b) Patient with events and marker measurements until t=1.