Parameter estimation using time-dependent Weibull proportional hazards model for survival analysis with partly interval censored data

doi:10.12122/j.issn.1673-4254.2024.12.23

Abstract

Abstract:

Objective To assess the validity and effectiveness of parameter estimation using a time-dependent Weibull proportional hazards model for survival analysis containing partly interval censored data and explore the impact of different covariates on the results of analysis. Methods We established a time-dependent Weibull proportional hazards model using the Weibull distribution as the baseline hazard function of the model which incorporated time-varying covariates. Maximum likelihood estimation was employed to estimate the model parameters, which were obtained by optimization of the likelihood function. Results and Conclusion Numerical simulation results showed that with higher proportions of precise observations across different sample sizes and parameter settings, the proposed model resulted in improved accuracy of parameter estimation with coverage probabilities approximating the theoretical expectation of 95%. As the sample sizes increased, the parameter biases of the model tended to decrease. Experiments with empirical data further validated the effectiveness of the model. Compared with the failure time data for each precisely observed individual, additional interval-censored data helped to obtain more effective estimates of the regression parameters. Comparison with the Cox model that included time-varying covariates further demonstrated the effectiveness of the time-dependent Weibull proportional hazards model for parameter estimation in survival analysis with partly interval censored data.

Key words: partly interval-censored data, time-dependent covariates, Weibull proportional hazards model, likelihood function, maximum likelihood estimation

Shuying WANG, Xinyu LIU, Rundong LI, Yang LI. Parameter estimation using time-dependent Weibull proportional hazards model for survival analysis with partly interval censored data[J]. Journal of Southern Medical University, 2024, 44(12): 2461-2468.

Figures/Tables 7

Tab.1 Parameter estimation results for truth setting (1) with independent covariates

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 1.1$	0.0227	0.1405	0.1399	0.952	0.0138	0.1201	0.1199	0.955
	$γ = 1.2$	0.0499	0.1187	0.1130	0.936	0.0353	0.0936	0.0901	0.940
	$α 1 = 0.6$	0.0312	0.2129	0.1998	0.943	0.0195	0.1738	0.1675	0.947
	$α 2 = 1$	0.0458	0.2307	0.2138	0.934	0.0318	0.1861	0.1781	0.940
	$α 3 = 1$	0.0377	0.2392	0.2393	0.942	0.0344	0.2111	0.1999	0.936
400	$λ = 1.1$	0.0140	0.0954	0.0982	0.956	0.0081	0.0900	0.0846	0.942
	$γ = 1.2$	0.0191	0.0785	0.0769	0.935	0.0177	0.0631	0.0626	0.955
	$α 1 = 0.6$	0.0142	0.1351	0.1372	0.956	0.0039	0.1205	0.1167	0.941
	$α 2 = 1$	0.0213	0.1503	0.1466	0.939	0.0196	0.1230	0.1241	0.947
	$α 3 = 1$	0.0187	0.1696	0.1651	0.948	0.0160	0.1460	0.1395	0.947
600	$λ = 1.1$	0.0087	0.0815	0.0796	0.950	0.0084	0.1121	0.1102	0.943
	$γ = 1.2$	0.0145	0.0623	0.0627	0.946	0.0106	0.0494	0.0508	0.960
	$α 1 = 0.6$	0.0070	0.1151	0.1114	0.943	0.0068	0.1939	0.1863	0.941
	$α 2 = 1$	0.0149	0.1216	0.1193	0.952	0.0043	0.0998	0.1015	0.953
	$α 3 = 1$	0.0155	0.1325	0.1343	0.947	0.0063	0.1166	0.1144	0.955

Tab.1 Parameter estimation results for truth setting (1) with independent covariates

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 1.1$	0.0227	0.1405	0.1399	0.952	0.0138	0.1201	0.1199	0.955
	$γ = 1.2$	0.0499	0.1187	0.1130	0.936	0.0353	0.0936	0.0901	0.940
	$α 1 = 0.6$	0.0312	0.2129	0.1998	0.943	0.0195	0.1738	0.1675	0.947
	$α 2 = 1$	0.0458	0.2307	0.2138	0.934	0.0318	0.1861	0.1781	0.940
	$α 3 = 1$	0.0377	0.2392	0.2393	0.942	0.0344	0.2111	0.1999	0.936
400	$λ = 1.1$	0.0140	0.0954	0.0982	0.956	0.0081	0.0900	0.0846	0.942
	$γ = 1.2$	0.0191	0.0785	0.0769	0.935	0.0177	0.0631	0.0626	0.955
	$α 1 = 0.6$	0.0142	0.1351	0.1372	0.956	0.0039	0.1205	0.1167	0.941
	$α 2 = 1$	0.0213	0.1503	0.1466	0.939	0.0196	0.1230	0.1241	0.947
	$α 3 = 1$	0.0187	0.1696	0.1651	0.948	0.0160	0.1460	0.1395	0.947
600	$λ = 1.1$	0.0087	0.0815	0.0796	0.950	0.0084	0.1121	0.1102	0.943
	$γ = 1.2$	0.0145	0.0623	0.0627	0.946	0.0106	0.0494	0.0508	0.960
	$α 1 = 0.6$	0.0070	0.1151	0.1114	0.943	0.0068	0.1939	0.1863	0.941
	$α 2 = 1$	0.0149	0.1216	0.1193	0.952	0.0043	0.0998	0.1015	0.953
	$α 3 = 1$	0.0155	0.1325	0.1343	0.947	0.0063	0.1166	0.1144	0.955

Tab.2 Parameter estimation results for truth setting (2) with independent covariates

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 0.5$	0.0147	0.0898	0.0868	0.952	0.0087	0.0837	0.0810	0.942
	$γ = 0.8$	0.0275	0.0794	0.0764	0.938	0.0215	0.0634	0.0623	0.943
	$α 1 = - 0.7$	0.0335	0.1823	0.1805	0.951	0.0254	0.1617	0.1645	0.947
	$α 2 = 0.4$	0.0260	0.1799	0.1737	0.944	0.0212	0.1658	0.1597	0.945
	$α 3 = 0.7$	0.0207	0.2102	0.2020	0.940	0.0147	0.1833	0.1858	0.956
400	$λ = 0.5$	0.0057	0.0600	0.0607	0.956	0.0036	0.0558	0.0571	0.959
	$γ = 0.8$	0.0089	0.0532	0.0525	0.952	0.0082	0.0436	0.0433	0.949
	$α 1 = - 0.7$	0.0147	0.1237	0.1241	0.951	-0.0108	0.1134	0.1147	0.953
	$α 2 = 0.4$	0.0131	0.1235	0.1200	0.955	0.0024	0.1164	0.1116	0.940
	$α 3 = 0.7$	0.0136	0.1409	0.1400	0.954	0.0092	0.1253	0.1300	0.960
600	$λ = 0.5$	0.0035	0.0496	0.0490	0.951	0.0031	0.0485	0.0465	0.944
	$γ = 0.8$	0.0087	0.0445	0.0428	0.945	0.0070	0.0364	0.0352	0.946
	$α 1 = - 0.7$	-0.0065	0.0998	0.1009	0.956	-0.0059	0.1005	0.0933	0.943
	$α 2 = 0.4$	0.0037	0.0992	0.0973	0.940	-0.0022	0.0897	0.0905	0.938
	$α 3 = 0.7$	0.0090	0.1152	0.1137	0.943	0.0075	0.1104	0.1057	0.937

Tab.2 Parameter estimation results for truth setting (2) with independent covariates

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 0.5$	0.0147	0.0898	0.0868	0.952	0.0087	0.0837	0.0810	0.942
	$γ = 0.8$	0.0275	0.0794	0.0764	0.938	0.0215	0.0634	0.0623	0.943
	$α 1 = - 0.7$	0.0335	0.1823	0.1805	0.951	0.0254	0.1617	0.1645	0.947
	$α 2 = 0.4$	0.0260	0.1799	0.1737	0.944	0.0212	0.1658	0.1597	0.945
	$α 3 = 0.7$	0.0207	0.2102	0.2020	0.940	0.0147	0.1833	0.1858	0.956
400	$λ = 0.5$	0.0057	0.0600	0.0607	0.956	0.0036	0.0558	0.0571	0.959
	$γ = 0.8$	0.0089	0.0532	0.0525	0.952	0.0082	0.0436	0.0433	0.949
	$α 1 = - 0.7$	0.0147	0.1237	0.1241	0.951	-0.0108	0.1134	0.1147	0.953
	$α 2 = 0.4$	0.0131	0.1235	0.1200	0.955	0.0024	0.1164	0.1116	0.940
	$α 3 = 0.7$	0.0136	0.1409	0.1400	0.954	0.0092	0.1253	0.1300	0.960
600	$λ = 0.5$	0.0035	0.0496	0.0490	0.951	0.0031	0.0485	0.0465	0.944
	$γ = 0.8$	0.0087	0.0445	0.0428	0.945	0.0070	0.0364	0.0352	0.946
	$α 1 = - 0.7$	-0.0065	0.0998	0.1009	0.956	-0.0059	0.1005	0.0933	0.943
	$α 2 = 0.4$	0.0037	0.0992	0.0973	0.940	-0.0022	0.0897	0.0905	0.938
	$α 3 = 0.7$	0.0090	0.1152	0.1137	0.943	0.0075	0.1104	0.1057	0.937

Tab.3 Partial time-dependent covariate parameter estimates under truth setting (1)

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 1.1$	0.0213	0.1685	0.1610	0.930	0.0142	0.1389	0.1315	0.937
	$γ = 1.2$	0.0475	0.1433	0.1405	0.950	0.0310	0.1124	0.1079	0.935
	$α = 0.6$	0.0421	0.3103	0.2970	0.943	0.0348	0.2283	0.2267	0.953
	$β 1 = - 0.2$	-0.0084	0.2119	0.1949	0.936	-0.0072	0.1616	0.1602	0.953
	$β 2 = 0.2$	0.0061	0.2305	0.2231	0.944	0.0034	0.1981	0.1843	0.942
400	$λ = 1.1$	0.0124	0.1160	0.1125	0.948	0.0067	0.0935	0.0925	0.951
	$γ = 1.2$	0.0224	0.0968	0.0968	0.956	0.0096	0.0733	0.0744	0.957
	$α = 0.6$	0.0203	0.2082	0.2024	0.947	0.0136	0.1541	0.1561	0.951
	$β 1 = - 0.2$	-0.0082	0.1374	0.1346	0.935	-0.0037	0.1089	0.1120	0.955
	$β 2 = 0.2$	-0.0078	0.1528	0.1546	0.953	0.0052	0.1336	0.1288	0.945
600	$λ = 1.1$	0.0060	0.0932	0.0908	0.951	0.0049	0.0767	0.0751	0.946
	$γ = 1.2$	0.0147	0.0787	0.0781	0.951	0.0086	0.0614	0.0607	0.950
	$α = 0.6$	0.0127	0.1578	0.1631	0.959	0.0068	0.1247	0.1264	0.965
	$β 1 = - 0.2$	-0.0048	0.1100	0.1090	0.953	-0.0026	0.0931	0.0912	0.945
	$β 2 = 0.2$	0.0072	0.1288	0.1256	0.934	-0.0025	0.1086	0.1048	0.934

Tab.3 Partial time-dependent covariate parameter estimates under truth setting (1)

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 1.1$	0.0213	0.1685	0.1610	0.930	0.0142	0.1389	0.1315	0.937
	$γ = 1.2$	0.0475	0.1433	0.1405	0.950	0.0310	0.1124	0.1079	0.935
	$α = 0.6$	0.0421	0.3103	0.2970	0.943	0.0348	0.2283	0.2267	0.953
	$β 1 = - 0.2$	-0.0084	0.2119	0.1949	0.936	-0.0072	0.1616	0.1602	0.953
	$β 2 = 0.2$	0.0061	0.2305	0.2231	0.944	0.0034	0.1981	0.1843	0.942
400	$λ = 1.1$	0.0124	0.1160	0.1125	0.948	0.0067	0.0935	0.0925	0.951
	$γ = 1.2$	0.0224	0.0968	0.0968	0.956	0.0096	0.0733	0.0744	0.957
	$α = 0.6$	0.0203	0.2082	0.2024	0.947	0.0136	0.1541	0.1561	0.951
	$β 1 = - 0.2$	-0.0082	0.1374	0.1346	0.935	-0.0037	0.1089	0.1120	0.955
	$β 2 = 0.2$	-0.0078	0.1528	0.1546	0.953	0.0052	0.1336	0.1288	0.945
600	$λ = 1.1$	0.0060	0.0932	0.0908	0.951	0.0049	0.0767	0.0751	0.946
	$γ = 1.2$	0.0147	0.0787	0.0781	0.951	0.0086	0.0614	0.0607	0.950
	$α = 0.6$	0.0127	0.1578	0.1631	0.959	0.0068	0.1247	0.1264	0.965
	$β 1 = - 0.2$	-0.0048	0.1100	0.1090	0.953	-0.0026	0.0931	0.0912	0.945
	$β 2 = 0.2$	0.0072	0.1288	0.1256	0.934	-0.0025	0.1086	0.1048	0.934

Tab.4 Partial time-dependent covariate parameter estimates under truth setting (2)

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 0.6$	0.0167	0.1318	0.1251	0.948	0.0095	0.1161	0.1082	0.942
	$γ = 0.8$	0.0227	0.1187	0.1134	0.946	0.0189	0.0979	0.0933	0.957
	$α = 1$	0.0281	0.2226	0.2235	0.954	0.0229	0.2076	0.1999	0.940
	$β 1 = - 2$	-0.0507	0.2549	0.2579	0.955	-0.0476	0.2416	0.2355	0.953
	$β 2 = - 2$	-0.0476	0.2883	0.2752	0.951	-0.0354	0.2531	0.2537	0.949
400	$λ = 0.6$	0.0107	0.0891	0.0854	0.946	0.0063	0.0775	0.0750	0.942
	$γ = 0.8$	0.0159	0.0783	0.0792	0.951	0.0125	0.0654	0.0652	0.954
	$α = 1$	0.0109	0.1590	0.1561	0.943	0.0098	0.1395	0.1395	0.951
	$β 1 = - 2$	-0.0323	0.1790	0.1795	0.952	-0.0218	0.1660	0.1633	0.945
	$β 2 = - 2$	-0.0372	0.1960	0.1924	0.951	-0.0244	0.1790	0.1771	0.944
600	$λ = 0.6$	0.0068	0.0704	0.0693	0.956	-0.0001	0.0600	0.0607	0.950
	$γ = 0.8$	0.0088	0.0615	0.0640	0.960	0.0081	0.0541	0.0530	0.951
	$α = 1$	0.0066	0.1239	0.1266	0.955	0.0061	0.1214	0.1135	0.933
	$β 1 = - 2$	-0.0259	0.1456	0.1454	0.954	-0.0176	0.1339	0.1330	0.951
	$β 2 = - 2$	-0.0256	0.1586	0.1558	0.952	-0.0119	0.1411	0.1440	0.961

Tab.4 Partial time-dependent covariate parameter estimates under truth setting (2)

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 0.6$	0.0167	0.1318	0.1251	0.948	0.0095	0.1161	0.1082	0.942
	$γ = 0.8$	0.0227	0.1187	0.1134	0.946	0.0189	0.0979	0.0933	0.957
	$α = 1$	0.0281	0.2226	0.2235	0.954	0.0229	0.2076	0.1999	0.940
	$β 1 = - 2$	-0.0507	0.2549	0.2579	0.955	-0.0476	0.2416	0.2355	0.953
	$β 2 = - 2$	-0.0476	0.2883	0.2752	0.951	-0.0354	0.2531	0.2537	0.949
400	$λ = 0.6$	0.0107	0.0891	0.0854	0.946	0.0063	0.0775	0.0750	0.942
	$γ = 0.8$	0.0159	0.0783	0.0792	0.951	0.0125	0.0654	0.0652	0.954
	$α = 1$	0.0109	0.1590	0.1561	0.943	0.0098	0.1395	0.1395	0.951
	$β 1 = - 2$	-0.0323	0.1790	0.1795	0.952	-0.0218	0.1660	0.1633	0.945
	$β 2 = - 2$	-0.0372	0.1960	0.1924	0.951	-0.0244	0.1790	0.1771	0.944
600	$λ = 0.6$	0.0068	0.0704	0.0693	0.956	-0.0001	0.0600	0.0607	0.950
	$γ = 0.8$	0.0088	0.0615	0.0640	0.960	0.0081	0.0541	0.0530	0.951
	$α = 1$	0.0066	0.1239	0.1266	0.955	0.0061	0.1214	0.1135	0.933
	$β 1 = - 2$	-0.0259	0.1456	0.1454	0.954	-0.0176	0.1339	0.1330	0.951
	$β 2 = - 2$	-0.0256	0.1586	0.1558	0.952	-0.0119	0.1411	0.1440	0.961

Tab.5 Partial time-dependent covariate parameter estimates under truth setting (3)

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 0.4$	0.0289	0.1344	0.1256	0.948	0.0278	0.1217	0.1177	0.951
	$γ = 0.5$	0.0140	0.0647	0.0636	0.946	0.0135	0.0520	0.0514	0.957
	$α = - 0.3$	-0.0095	0.0625	0.0624	0.957	-0.0086	0.0498	0.0500	0.961
	$β 1 = 0.5$	0.0207	0.1835	0.1749	0.937	0.0089	0.1750	0.1620	0.931
	$β 2 = 0.7$	0.0220	0.2064	0.1996	0.950	0.0148	0.1924	0.1860	0.940
400	$λ = 0.4$	0.0119	0.0874	0.0843	0.958	0.0054	0.0821	0.0789	0.935
	$γ = 0.5$	0.0074	0.0453	0.0445	0.943	0.0070	0.0375	0.0359	0.938
	$α = - 0.3$	-0.0055	0.0446	0.0436	0.945	-0.0050	0.0364	0.0349	0.945
	$β 1 = 0.5$	0.0064	0.1232	0.1215	0.946	0.0026	0.1170	0.1134	0.947
	$β 2 = 0.7$	0.0134	0.1373	0.1391	0.955	0.0134	0.1338	0.1306	0.942
600	$λ = 0.4$	0.0072	0.0686	0.0677	0.935	0.0047	0.0653	0.0643	0.948
	$γ = 0.5$	0.0058	0.0358	0.0362	0.946	0.0051	0.0298	0.0293	0.955
	$α = - 0.3$	-0.0044	0.0347	0.0354	0.951	-0.004	0.0292	0.0284	0.943
	$β 1 = 0.5$	0.0024	0.0987	0.0986	0.951	0.0025	0.0942	0.0922	0.951
	$β 2 = 0.7$	0.0114	0.1123	0.1131	0.945	0.0099	0.1061	0.1063	0.943

Tab.5 Partial time-dependent covariate parameter estimates under truth setting (3)

n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
n	Parameter	$p t = 0.4$				$p t = 0.75$
200	$λ = 0.4$	0.0289	0.1344	0.1256	0.948	0.0278	0.1217	0.1177	0.951
	$γ = 0.5$	0.0140	0.0647	0.0636	0.946	0.0135	0.0520	0.0514	0.957
	$α = - 0.3$	-0.0095	0.0625	0.0624	0.957	-0.0086	0.0498	0.0500	0.961
	$β 1 = 0.5$	0.0207	0.1835	0.1749	0.937	0.0089	0.1750	0.1620	0.931
	$β 2 = 0.7$	0.0220	0.2064	0.1996	0.950	0.0148	0.1924	0.1860	0.940
400	$λ = 0.4$	0.0119	0.0874	0.0843	0.958	0.0054	0.0821	0.0789	0.935
	$γ = 0.5$	0.0074	0.0453	0.0445	0.943	0.0070	0.0375	0.0359	0.938
	$α = - 0.3$	-0.0055	0.0446	0.0436	0.945	-0.0050	0.0364	0.0349	0.945
	$β 1 = 0.5$	0.0064	0.1232	0.1215	0.946	0.0026	0.1170	0.1134	0.947
	$β 2 = 0.7$	0.0134	0.1373	0.1391	0.955	0.0134	0.1338	0.1306	0.942
600	$λ = 0.4$	0.0072	0.0686	0.0677	0.935	0.0047	0.0653	0.0643	0.948
	$γ = 0.5$	0.0058	0.0358	0.0362	0.946	0.0051	0.0298	0.0293	0.955
	$α = - 0.3$	-0.0044	0.0347	0.0354	0.951	-0.004	0.0292	0.0284	0.943
	$β 1 = 0.5$	0.0024	0.0987	0.0986	0.951	0.0025	0.0942	0.0922	0.951
	$β 2 = 0.7$	0.0114	0.1123	0.1131	0.945	0.0099	0.1061	0.1063	0.943

Tab.6 Parameter estimation results of the Cox model with time-varying covariates and the time-dependent Weibull proportional hazards model

Model	n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
Model	n		$p t = 0.4$				$p t = 0.75$
Cox model with time-varying covariates	200	$α = 0.2$	0.0226	0.2795	0.2636	0.939	0.0258	0.2295	0.2201	0.931
		$β 1 = 1.2$	0.0648	0.2041	0.2006	0.937	0.0417	0.1727	0.1684	0.945
		$β 2 = - 1$	-0.0572	0.2283	0.2140	0.934	-0.0367	0.1907	0.1802	0.938
	400	$α = 0.2$	0.0164	0.1842	0.1816	0.952	0.0170	0.1552	0.1550	0.952
		$β 1 = 1.2$	0.0341	0.1391	0.1388	0.943	0.0258	0.1204	0.1174	0.944
		$β 2 = - 1$	-0.0254	0.1519	0.1488	0.944	-0.0233	0.1287	0.1264	0.941
Time-dependent weibull proportional hazards model	200	$α = 0.2$	0.0123	0.1401	0.1348	0.950	0.0124	0.1115	0.1088	0.945
		$β 1 = 1.2$	0.0447	0.2017	0.1993	0.956	0.0394	0.1862	0.1844	0.956
		$β 2 = - 1$	-0.0344	0.2242	0.2135	0.944	-0.0285	0.2105	0.1998	0.941
	400	$α = 0.2$	0.0050	0.0919	0.0932	0.948	0.0088	0.0788	0.0761	0.958
		$β 1 = 1.2$	0.0211	0.1431	0.1383	0.944	0.0215	0.1283	0.1287	0.951
		$β 2 = - 1$	-0.0165	0.1518	0.1488	0.939	-0.0096	0.1477	0.1399	0.940

Tab.6 Parameter estimation results of the Cox model with time-varying covariates and the time-dependent Weibull proportional hazards model

Model	n	Parameter	BIAS	SSE	ESE	CP	BIAS	SSE	ESE	CP
Model	n		$p t = 0.4$				$p t = 0.75$
Cox model with time-varying covariates	200	$α = 0.2$	0.0226	0.2795	0.2636	0.939	0.0258	0.2295	0.2201	0.931
		$β 1 = 1.2$	0.0648	0.2041	0.2006	0.937	0.0417	0.1727	0.1684	0.945
		$β 2 = - 1$	-0.0572	0.2283	0.2140	0.934	-0.0367	0.1907	0.1802	0.938
	400	$α = 0.2$	0.0164	0.1842	0.1816	0.952	0.0170	0.1552	0.1550	0.952
		$β 1 = 1.2$	0.0341	0.1391	0.1388	0.943	0.0258	0.1204	0.1174	0.944
		$β 2 = - 1$	-0.0254	0.1519	0.1488	0.944	-0.0233	0.1287	0.1264	0.941
Time-dependent weibull proportional hazards model	200	$α = 0.2$	0.0123	0.1401	0.1348	0.950	0.0124	0.1115	0.1088	0.945
		$β 1 = 1.2$	0.0447	0.2017	0.1993	0.956	0.0394	0.1862	0.1844	0.956
		$β 2 = - 1$	-0.0344	0.2242	0.2135	0.944	-0.0285	0.2105	0.1998	0.941
	400	$α = 0.2$	0.0050	0.0919	0.0932	0.948	0.0088	0.0788	0.0761	0.958
		$β 1 = 1.2$	0.0211	0.1431	0.1383	0.944	0.0215	0.1283	0.1287	0.951
		$β 2 = - 1$	-0.0165	0.1518	0.1488	0.939	-0.0096	0.1477	0.1399	0.940

Tab.7 Estimates for Danish diabetes data obtained using the time-dependent Weibull proportional hazards model

Parameter	Scale parameter	Shape parameter	Gender	Scale parameter	Shape parameter	Gender
Parameter	Only the exact data			Partly interval censored data
Estimate	0.0532	2.8784	-0.1439	0.0546	2.8263	-0.1293
ESE	0.0012	0.0855	0.0838	0.0012	0.0795	0.0777
P	<0.0001	<0.0001	0.0428	<0.0001	<0.0001	0.0480

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