Table 2.

Performance indicators of conventional model and Model(1) using Maximum likelihood estimation on grouped data by different α

Scenario	Method	true value α	AB						SD						CVR
			α	π60	λ	γ	β	δ	α	π60	λ	γ	β	δ	α	π60	λ	γ	β	δ
BREASTa	conventional	2		− 0.281	− 0.042	0.011	−0.911	− 0.012		0.046	0.005	0.029	0.097	0.100		0%	0%	96%	0%	95%
	Model(1)	2	−0.005	−0.004	0.000	0.002	−0.011	−0.005	0.115	0.043	0.012	0.048	0.199	0.157	93%	93%	93%	95%	93%	94%
	conventional	1.5		−0.173	−0.030	− 0.028	−0.627	0.140		0.036	0.006	0.033	0.093	0.092		0%	0%	88%	0%	64%
	Model(1)	1.5	−0.001	−0.003	0.000	0.002	−0.007	−0.004	0.099	0.038	0.011	0.046	0.184	0.146	94%	93%	95%	95%	94%	94%
	conventional	1.2		−0.075	− 0.016	− 0.029	− 0.314	0.142		0.027	0.006	0.037	0.104	0.093		8%	30%	89%	18%	71%
	Model(1)	1.2	0.000	−0.003	0.000	0.002	−0.007	− 0.003	0.090	0.036	0.010	0.045	0.174	0.139	93%	94%	96%	96%	94%	94%
	conventional	1		−0.001	0.000	0.000	−0.002	0.000		0.018	0.007	0.040	0.097	0.106		95%	95%	95%	95%	94%
	Model(1)	1	−0.001	−0.003	0.000	0.002	−0.009	−0.001	0.084	0.034	0.010	0.044	0.169	0.134	93%	94%	95%	96%	94%	93%
	conventional	0.8		0.058	0.017	0.048	0.299	−0.221		0.011	0.008	0.042	0.084	0.106		2%	40%	79%	4%	41%
	Model(1)	0.8	−0.001	− 0.002	0.000	0.002	−0.007	−0.002	0.077	0.033	0.009	0.043	0.161	0.129	94%	92%	95%	95%	93%	93%
LUNGb	conventional	2		−0.010	− 0.012	− 0.032	− 0.131	0.017		0.004	0.013	0.014	0.061	0.023		22%	85%	35%	45%	86%
	Model(1)	2	−0.006	0.002	− 0.009	− 0.001	0.003	0.003	0.206	0.004	0.014	0.015	0.065	0.023	96%	92%	89%	95%	95%	95%
	conventional	1.5		−0.004	−0.011	− 0.016	− 0.061	0.011		0.004	0.013	0.014	0.059	0.023		84%	88%	76%	84%	92%
	Model(1)	1.5	− 0.004	0.002	− 0.009	− 0.001	0.003	0.003	0.165	0.004	0.013	0.014	0.063	0.023	96%	92%	89%	94%	95%	94%
	conventional	1.2		0.000	− 0.010	− 0.007	− 0.022	0.006		0.004	0.013	0.014	0.058	0.023		96%	89%	91%	95%	93%
	Model(1)	1.2	0.000	0.002	−0.009	0.000	0.004	0.003	0.149	0.004	0.013	0.014	0.062	0.023	95%	92%	90%	94%	94%	93%
	conventional	1		0.002	− 0.009	0.000	0.002	0.002		0.004	0.015	0.015	0.061	0.026		92%	98%	98%	95%	97%
	Model(1)	1	0.001	0.002	−0.009	0.000	0.004	0.003	0.132	0.004	0.013	0.014	0.061	0.023	96%	91%	90%	94%	94%	94%
	conventional	0.8		0.005	−0.008	0.006	0.028	0.000		0.004	0.013	0.013	0.057	0.022		75%	90%	92%	92%	94%
	Model(1)	0.8	−0.002	0.002	−0.009	0.000	0.003	0.003	0.121	0.004	0.013	0.014	0.060	0.022	95%	91%	90%	94%	94%	93%

Absolute Bias (AB) = Mean(estimates - true value); Standard Deviation (SD) = standard deviation over the set of 1000 estimates; Coverage (CVR) = the proportion of the time that the estimated 95% confidence intervals contained the true value. Estimated over 1000 simulation runs; Sample size 10,000; follow-up 15 years

^aTrue values: π₆₀ = 0.7, λ =0.1, γ =1.1, β = − 0.15 and δ = 0. ^bTrue values: π₆₀ = 0.1, λ =0.9, γ =0.8, β = − 0.75 and δ = − 0.3