Semiparametric estimation for accelerated failure time mixture cure model allowing non-curable competing risk

Figures & data

Table 1. Estimates of parameters for three baseline survivals.

	n=100					n=500					n=1000
Par.	Bias	MSE	SE	ESD	CP	Bias	MSE	SE	ESD	CP	Bias	MSE	SE	ESD	CP
Baseline-Weibull AFT mixture cure model
${\alpha }_1$	0.0553	0.0488	0.2017	0.2140	0.960	0.0186	0.0091	0.0912	0.0938	0.957	0.0012	0.0047	0.0627	0.0688	0.956
${\alpha }_2$	0.0798	0.0556	0.2194	0.2219	0.916	0.0593	0.0136	0.1000	0.1003	0.976	0.0580	0.0086	0.0708	0.0723	0.962
${\alpha }_3$	0.0319	0.1681	0.3957	0.4088	0.962	−0.0480	0.0332	0.1776	0.1757	0.915	−0.0479	0.0180	0.1117	0.1253	0.956
${\theta }_1$	0.0184	0.0789	0.2694	0.2803	0.968	0.0128	0.0144	0.1201	0.1194	0.971	−0.0039	0.0071	0.0829	0.0843	0.953
${\theta }_2$	0.0626	0.0537	0.2248	0.2231	0.940	0.0657	0.0125	0.0913	0.0906	0.910	0.0688	0.0087	0.0616	0.0633	0.957
${\theta }_3$	0.0154	0.1744	0.4066	0.4174	0.958	−0.0549	0.0343	0.1750	0.1769	0.927	−0.0566	0.0183	0.1216	0.1227	0.933
${\beta }_{11}$	−0.0030	0.0077	0.0786	0.0878	0.968	−0.0061	0.0007	0.0239	0.0258	0.969	−0.0041	0.0003	0.0164	0.0160	0.931
${\beta }_{12}$	−0.0023	0.0270	0.1429	0.1644	0.974	0.0008	0.0026	0.0503	0.0511	0.971	−0.0034	0.0010	0.0326	0.0317	0.938
${\beta }_{21}$	0.0359	0.0214	0.0766	0.1418	0.982	0.0099	0.0004	0.0161	0.0175	0.922	0.0087	0.0002	0.0108	0.0105	0.944
${\beta }_{22}$	0.0348	0.0538	0.1349	0.2293	0.988	0.0186	0.0078	0.0843	0.0863	0.947	0.0190	0.0020	0.0409	0.0411	0.936
Baseline-Lognormal AFT mixture cure model
${\alpha }_1$	0.0173	0.0473	0.2041	0.2167	0.960	-0.0571	0.0123	0.0810	0.0951	0.932	−0.0713	0.0101	0.0580	0.0711	0.952
${\alpha }_2$	0.0411	0.0493	0.2113	0.2182	0.940	0.0451	0.0108	0.0950	0.0934	0.910	0.0463	0.0067	0.0656	0.0677	0.948
${\alpha }_3$	0.0216	0.1643	0.3750	0.4047	0.970	−0.0360	0.0290	0.1511	0.1664	0.950	−0.0506	0.0173	0.1050	0.1213	0.950
${\theta }_1$	−0.0378	0.0905	0.2877	0.2985	0.964	−0.0962	0.0260	0.1168	0.1293	0.914	−0.1170	0.0222	0.0825	0.0924	0.984
${\theta }_2$	0.0041	0.0519	0.2106	0.2278	0.978	0.0382	0.0095	0.0953	0.0899	0.922	0.0424	0.0058	0.0641	0.0630	0.988
${\theta }_3$	0.0061	0.1820	0.4046	0.4266	0.960	−0.0361	0.0315	0.1631	0.1737	0.968	−0.0382	0.0164	0.1172	0.1223	0.950
${\beta }_{11}$	0.0076	0.0268	0.1478	0.1636	0.966	−0.0114	0.0029	0.0506	0.0525	0.954	−0.0160	0.0015	0.0328	0.0346	0.942
${\beta }_{12}$	0.0185	0.0848	0.2864	0.2907	0.938	−0.0128	0.0105	0.0998	0.1016	0.946	−0.0159	0.0047	0.0646	0.0667	0.944
${\beta }_{21}$	0.0270	0.0636	0.1766	0.2507	0.982	0.0109	0.0036	0.0581	0.0589	0.946	0.0104	0.0015	0.0358	0.0377	0.958
${\beta }_{22}$	0.0095	0.1519	0.3073	0.3896	0.966	0.0152	0.0148	0.1097	0.1206	0.954	0.0078	0.0066	0.0704	0.0809	0.958
Baseline-Weilbull-Lognormal AFT mixture cure model
${\alpha }_1$	0.0049	0.0467	0.2039	0.2161	0.922	−0.0403	0.0106	0.0879	0.0946	0.931	−0.0485	0.0073	0.0617	0.0701	0.906
${\alpha }_2$	0.0924	0.0528	0.2097	0.2104	0.922	0.0443	0.0111	0.0949	0.0957	0.916	0.0471	0.0072	0.0677	0.0703	0.914
${\alpha }_3$	0.0536	0.1509	0.4017	0.3847	0.922	−0.0510	0.0311	0.1659	0.1687	0.945	−0.0610	0.0188	0.1149	0.1226	0.936
${\theta }_1$	−0.0471	0.0925	0.2816	0.3004	0.922	−0.0710	0.0211	0.1195	0.1268	0.934	−0.0855	0.0154	0.0868	0.0901	0.954
${\theta }_2$	0.0611	0.0515	0.2230	0.2185	0.910	0.0482	0.0106	0.0885	0.0912	0.916	0.0478	0.0063	0.0603	0.0635	0.978
${\theta }_3$	0.0432	0.1656	0.4533	0.4047	0.912	−0.0535	0.0335	0.1735	0.1751	0.942	−0.0484	0.0174	0.1250	0.1227	0.912
${\beta }_{11}$	0.0058	0.0069	0.0729	0.0830	0.923	−0.0036	0.0007	0.0247	0.0254	0.965	−0.0051	0.0003	0.0156	0.0157	0.942
${\beta }_{12}$	−0.0004	0.0222	0.1365	0.1491	0.923	0.0011	0.0024	0.0489	0.0494	0.958	−0.0036	0.0010	0.0313	0.0310	0.956
${\beta }_{21}$	0.0689	0.1691	0.4272	0.4054	0.923	−0.0002	0.0035	0.0589	0.0594	0.947	−0.0023	0.0015	0.0403	0.0382	0.928
${\beta }_{22}$	0.0400	0.3310	0.3687	0.5740	0.923	0.0020	0.0143	0.1099	0.1196	0.947	−0.0016	0.0049	0.0705	0.0702	0.940

Figure 1. Estimated CIFs with Weibull distributions (solid lines), their 95% pointwise confidence intervals (dashed and dotted lines), and the true CIFs (dotdash lines). (a) $n=100,F_1$. (b) $n=100,F_2$. (c) $n=500,F_1$. (d) $n=500,F_2$. (e) $n=1000,F_1$. (f) $n=1000,F_2$.

Figure 2. Estimated CIFs with lognormal distributions (solid lines), their 95% pointwise confidence intervals (dashed and dotted lines), and the true CIFs dotdash lines). (a) $n=100,F_1$. (b) $n=100,F_2$. (c) $n=500,F_1$. (d) $n=500,F_2$. (e) $n=1000,F_1$. (f) $n=1000,F_2$.

Figure 3. Estimated CIFs with Weibull and lognormal distributions (solid lines), their 95% pointwise confidence intervals (dashed and dotted lines), and the true CIFs (dotdash lines). (a) $n=100,F_1$. (b) $n=100,F_2$. (c) $n=500,F_1$. (d) $n=500,F_2$. (e) $n=1000,F_1$. (f) $n=1000,F_2$.

Table 2. Results for colorectal cancer clinical trial data.

	Proposed model
Parameters	Coef	Sd	P
Uncure part
Intercept	0.6968	0.1460	<0.001
Chem	0.1329	0.3021	0.660
Charlson(0)	1.7539	0.3044	<0.001
Charlson(1−2)	1.5675	0.6664	0.019
Competing part
Intercept	0.5339	0.2392	0.026
Chem	−0.2515	0.4433	0.570
Charlson(0)	2.2674	0.4442	<0.001
Charlson(1−2)	1.7019	0.9262	0.066
PH regression for rehospitalisation
Chem	−0.0991	0.0465	0.033
Charlson(0)	−0.0810	0.0479	0.091
Charlson(1−2)	−1.6663	0.3322	<0.001
PH regression for death
Chem	−0.0991	0.0482	0.040
Charlson(0)	0.0548	0.0581	0.346
Charlson(1−2)	2.2774	0.2298	<0.001

Figure 4. Proposed model.