A progressive three-state model to estimate time to cancer: a likelihood-based approach

Akwiwu, Eddymurphy U.; Klausch, Thomas; Jodal, Henriette C.; Carvalho, Beatriz; Løberg, Magnus; Kalager, Mette; Berkhof, Johannes; Coupé, Veerle M.H.

doi:10.1186/s12874-022-01645-2

BMC Medical Research Methodology

Table 4 Summary of performance measures for different model specifications under Scenario I, based on 1000 MC simulation runs with sample sizes n= 1000 and 5000

From: A progressive three-state model to estimate time to cancer: a likelihood-based approach

			*n=1000*						*n=5000*
Model	PAR	True	MCM	RMSE	RB^a	CV	CR	AW	MCM	RMSE	RB^a	CV	CR	AW
M1	λ₁	0.03	0.03	0.002	0.3	0.05	0.955	0.01	0.03	0.001	0.0	0.01	0.958	0.00
	λ₂	0.04	0.04	0.005	0.1	0.14	0.960	0.02	0.04	0.003	0.0	0.06	0.955	0.01
M2	α₀	− 3.60	− 3.60	0.062	0.0	− 0.02	0.951	0.24	− 3.60	0.027	0.0	−0.01	0.951	0.11
	α₁	− 1.00	−1.00	0.059	0.2	− 0.06	0.942	0.23	− 1.00	0.026	0.2	− 0.03	0.944	0.10
	λ₂	0.04	0.04	0.006	− 0.1	0.15	0.944	0.02	0.04	0.003	0.3	0.07	0.949	0.01
M3	λ₁	0.04	0.03	0.002	− 0.1	0.05	0.955	0.01	0.04	0.001	0.0	0.02	0.946	0.00
	β₀	− 3.20	− 3.22	0.142	0.7	− 0.04	0.967	0.57	− 3.20	0.063	0.2	−0.02	0.952	0.25
	β₁	− 0.10	−0.10	0.152	− 1.9	− 1.52	0.946	0.56	− 0.10	0.062	2.5	− 0.62	0.952	0.25
M4	α₀	− 3.50	− 3.50	0.056	0.0	− 0.02	0.944	0.22	− 3.50	0.025	0.0	−0.01	0.944	0.10
	α₁	− 0.80	−0.80	0.056	0.2	− 0.07	0.944	0.21	− 0.80	0.024	0.2	− 0.03	0.953	0.10
	β₀	− 4.50	− 4.58	0.394	1.7	− 0.09	0.949	1.45	− 4.51	0.155	0.1	−0.03	0.956	0.63
	β₁	− 1.50	−1.56	0.298	3.7	− 0.20	0.948	1.13	− 1.50	0.123	0.2	− 0.08	0.957	0.49
M5	λ₁	0.03	0.03	0.001	0.1	0.05	0.958	0.01	0.03	0.001	0.1	0.01	0.946	0.00
	κ₂	2.00	2.08	0.486	4.0	0.24	0.963	1.82	2.00	0.195	0.2	0.13	0.946	0.77
	θ₂	10.00	10.51	2.966	5.1	0.29	0.928	8.41	10.10	0.768	1.0	0.12	0.959	3.00
M6	α₀	− 4.00	− 4.00	0.084	0.1	− 0.04	0.951	0.34	− 4.00	0.038	0.0	−0.01	0.954	0.15
	α₁	− 2.00	−2.01	0.084	0.3	− 0.03	0.950	0.33	− 2.00	0.038	0.1	− 0.01	0.944	0.15
	κ₂	4.00	4.29	3.327	7.3	0.77	0.959	3.82	4.03	0.355	0.8	0.11	0.956	1.40
	θ₂	8.50	8.56	0.630	0.7	0.14	0.949	2.35	8.51	0.260	0.2	0.04	0.954	1.01
M7	λ₁	0.04	0.04	0.002	− 0.2	0.04	0.950	0.01	0.04	0.001	0.0	0.01	0.950	0.00
	κ₂	4.10	5.67	6.456	38.3	1.51	0.928	9.21	4.25	0.612	3.5	0.11	0.963	2.26
	β₀	2.50	2.51	0.177	0.5	0.13	0.933	0.68	2.50	0.075	− 0.1	0.04	0.936	0.29
	β₁	3.50	4.75	5.065	35.7	1.38	0.933	6.70	3.59	0.437	2.7	0.13	0.963	1.63
M8	α₀	− 3.50	− 3.50	0.060	0.1	− 0.02	0.945	0.23	− 3.50	0.026	0.0	−0.01	0.958	0.10
	α₁	− 1.00	−1.00	0.058	0.2	− 0.06	0.952	0.22	− 1.00	0.026	0.2	− 0.03	0.958	0.10
	κ₂	1.80	1.84	0.408	2.1	0.23	0.955	1.52	1.81	0.171	0.7	0.09	0.945	0.66
	β₀	2.50	2.54	0.227	1.7	0.09	0.946	0.86	2.51	0.090	0.2	0.04	0.954	0.35
	β₁	0.10	0.10	0.184	2.9	1.84	0.956	0.72	0.11	0.082	− 0.5	0.81	0.954	0.31
M9	κ₁	0.40	0.40	0.029	0.2	0.07	0.954	0.11	0.40	0.013	0.0	0.03	0.950	0.05
	θ₁	65.00	66.42	11.453	2.2	0.17	0.956	44.21	65.34	4.993	0.5	0.08	0.951	19.09
	λ₂	0.04	0.04	0.005	0.6	0.14	0.948	0.02	0.04	0.002	0.3	0.06	0.946	0.01
M10	κ₁	2.00	2.02	0.110	0.8	0.05	0.952	0.41	2.00	0.047	0.2	0.02	0.953	0.18
	α₀	3.50	3.50	0.056	0.0	0.02	0.955	0.23	3.50	0.025	0.0	0.01	0.962	0.10
	α₁	3.50	3.52	0.173	0.7	0.05	0.948	0.65	3.51	0.073	0.2	0.02	0.943	0.29
	λ₂	0.04	0.04	0.006	0.4	0.14	0.953	0.02	0.04	0.003	0.2	0.06	0.947	0.01
M11	κ₁	4.00	4.01	0.197	0.3	0.05	0.942	0.76	4.00	0.088	0.1	0.02	0.954	0.34
	θ₁	20.00	20.00	0.300	0.0	0.02	0.936	1.11	20.00	0.130	0.0	0.01	0.939	0.50
	β₀	− 3.50	− 3.55	0.264	1.5	− 0.07	0.960	1.00	− 3.51	0.115	0.2	−0.03	0.941	0.44
	β₁	− 1.50	−1.54	0.234	2.7	− 0.15	0.957	0.89	− 1.51	0.102	0.4	− 0.07	0.949	0.39
M12	κ₁	0.50	0.50	0.039	0.7	0.08	0.961	0.16	0.50	0.018	0.1	0.04	0.944	0.07
	α₀	5.00	5.00	0.247	0.1	0.05	0.954	0.98	5.00	0.112	0.1	0.02	0.941	0.44
	α₁	− 2.00	−2.01	0.092	0.3	− 0.05	0.957	0.37	− 2.00	0.043	0.0	− 0.02	0.945	0.16
	β₀	− 1.00	− 0.96	0.273	− 3.5	− 0.27	0.949	1.02	− 1.00	0.112	− 0.4	−0.11	0.952	0.44
	β₁	5.50	5.67	0.789	3.1	0.14	0.962	2.92	5.53	0.314	0.6	0.06	0.954	1.24
M13	κ₁	1.50	1.50	0.073	0.0	0.05	0.952	0.29	1.50	0.032	−0.1	0.02	0.956	0.13
	θ₁	23.00	23.04	0.880	0.2	0.04	0.940	3.32	23.01	0.375	0.0	0.02	0.945	1.47
	κ₂	0.80	0.81	0.260	0.9	0.32	0.959	1.01	0.81	0.117	0.8	0.15	0.952	0.45
	θ₂^b	28.00	−	−	−	−	−	−	31.01	14.638	10.8	0.51	0.905	43.25
M14	κ₁	2.00	2.01	0.115	0.6	0.06	0.946	0.44	2.00	0.049	0.0	0.02	0.952	0.20
	α₀	3.50	3.50	0.061	0.1	0.02	0.936	0.24	3.50	0.027	0.0	0.01	0.949	0.11
	α₁	4.50	4.54	0.232	0.9	0.05	0.949	0.90	4.50	0.100	0.0	0.02	0.955	0.40
	κ₂	2.50	2.55	0.391	2.1	0.15	0.951	1.47	2.52	0.164	0.8	0.07	0.952	0.64
	θ₂	10.00	10.07	0.952	0.7	0.09	0.936	3.46	10.00	0.372	0.0	0.04	0.955	1.46
M15	κ₁	1.50	1.50	0.080	0.1	0.05	0.948	0.30	1.50	0.035	0.0	0.02	0.953	0.14
	θ₁	25.00	25.05	1.011	0.2	0.04	0.951	3.97	25.02	0.448	0.1	0.02	0.952	1.77
	κ₂	1.50	1.58	0.489	5.1	0.32	0.958	1.79	1.52	0.196	1.3	0.13	0.962	0.75
	β₀	3.00	3.10	0.536	3.4	0.18	0.920	1.96	3.01	0.197	0.4	0.07	0.947	0.77
	β₁	1.50	1.56	0.307	4.3	0.20	0.966	1.07	1.51	0.117	0.7	0.08	0.941	0.44

PAR, parameter; MCM, Monte Carlo means; RMSE, root mean squared error; RB, relative bias % ; CV, coefficient of variation; CR, coverage rate of a Wald-based 95% confidence interval; AW, average confidence interval width.
Note: λ₁ and λ₂ represent the exponential rate parameters of the first and second transition times, respectively; α₀ and α₁ represent the regression intercept and regression coefficient of the covariate w∼N(0,1) for the first transition time; β₀ and β₁ represent the regression intercept and regression coefficient of the covariate w∼N(0,1) for the second transition time; κ₁ and κ₂ represent the Weibull shape parameters of the first and second transition times, respectively; θ₁ and θ₂ represent the Weibull scale parameters of the first and second transition times, respectively.
^aThe negative signs correspond to underestimation (overestimation) for positive (negative) true values while the positive signs correspond to overestimation (underestimation) for positive (negative) true values. The 0.0 values are due to approximation.
^bThe estimates were extremely large due to the small sample size and the small proportion of CRCs

Back to article page

ISSN: 1471-2288

Contact us

General enquiries: journalsubmissions@springernature.com