Table 1.

Empirical parameter estimates from the competing risks model with a time-varying covariate (TVC) under low ($k_1=7$), medium ($k_1=3.5$), and high ($k_1=1$) familial dependence; permanent exposure (PE), exponential decay (ED) or Cox and Oakes (CO) models are considered for TVC.

TVC		True	$k_1=7,\tau =0.07$				True	$k_1=3.5,\tau =0.13$				True	$k_1=1,\tau =0.33$
model		value	Bias	ESE	ASE	ECP	value	Bias	ESE	ASE	ECP	value	Bias	ESE	ASE	ECP
PE	$\text{log}({\lambda }_1)$	−4.83	−0.01	0.06	0.06	0.95	−4.83	0.00	0.06	0.06	0.95	−4.83	0.00	0.06	0.06	0.94
	$\text{log}({\rho }_1)$	0.88	0.00	0.03	0.03	0.94	0.88	0.00	0.03	0.03	0.93	0.88	0.00	0.03	0.03	0.96
	$\text{log}({\lambda }_2)$	−4.96	−0.01	0.09	0.10	0.95	−4.96	−0.02	0.10	0.10	0.94	−4.96	−0.01	0.09	0.10	0.96
	$\text{log}({\rho }_2)$	1.12	0.00	0.07	0.07	0.95	1.12	0.00	0.07	0.07	0.95	1.12	0.00	0.06	0.07	0.96
	${\beta }_{1g\mathit{ene}}$	1.95	0.01	0.12	0.12	0.95	1.95	0.01	0.12	0.12	0.96	1.95	0.00	0.12	0.11	0.94
	${\beta }_{2g\mathit{ene}}$	1.19	0.03	0.23	0.23	0.96	1.19	0.03	0.24	0.23	0.95	1.19	0.02	0.22	0.24	0.96
	${\beta }_{1tvc}$	0.67	0.01	0.11	0.11	0.95	0.67	0.00	0.10	0.11	0.96	0.67	0.00	0.11	0.11	0.96
	$\text{log}(k_1)$	1.95	0.24	1.08	0.85	0.92	1.25	0.13	0.69	0.48	0.95	0.00	0.02	0.25	0.25	0.95
	$\text{log}(k_2)$	1.06	0.62	2.17	1.38	0.80	1.06	0.72	2.20	1.41	0.84	1.06	0.61	2.05	1.46	0.86
ED	$\text{log}({\lambda }_1)$	−4.83	−0.01	0.05	0.06	0.96	−4.83	0.00	0.06	0.06	0.95	−4.83	0.00	0.06	0.06	0.96
	$\text{log}({\rho }_1)$	0.83	0.00	0.03	0.03	0.96	0.83	0.00	0.03	0.03	0.95	0.83	0.00	0.03	0.03	0.96
	$\text{log}({\lambda }_2)$	−4.96	0.00	0.09	0.09	0.95	−4.96	−0.01	0.09	0.09	0.96	−4.96	−0.01	0.09	0.09	0.95
	$\text{log}({\rho }_2)$	1.08	0.00	0.06	0.06	0.95	1.08	0.00	0.06	0.06	0.95	1.08	0.00	0.06	0.06	0.95
	${\beta }_{1g\mathit{ene}}$	1.86	0.00	0.12	0.12	0.96	1.86	0.01	0.11	0.12	0.95	1.86	0.01	0.11	0.11	0.94
	${\beta }_{2g\mathit{ene}}$	1.22	0.01	0.20	0.21	0.95	1.22	0.03	0.22	0.21	0.96	1.22	0.02	0.21	0.22	0.96
	${\beta }_{1tvc}$	1.87	0.03	0.25	0.25	0.94	1.87	−0.01	0.25	0.25	0.95	1.87	0.03	0.24	0.24	0.94
	$\text{log}(\eta )$	−1.28	0.02	0.32	0.31	0.94	−1.28	0.00	0.32	0.31	0.94	−1.28	0.03	0.30	0.30	0.94
	$\text{log}(k_1)$	1.95	0.23	0.99	0.88	0.93	1.25	0.08	0.49	0.48	0.97	0.00	0.02	0.23	0.24	0.96
	$\text{log}(k_2)$	1.18	0.51	2.04	1.18	0.85	1.18	0.53	1.70	1.26	0.84	1.18	0.48	1.47	1.28	0.84
CO	$\text{log}({\lambda }_1)$	−4.83	0.00	0.05	0.06	0.95	−4.83	0.00	0.05	0.06	0.94	−4.83	0.00	0.05	0.06	0.96
	$\text{log}({\rho }_1)$	0.83	0.00	0.03	0.03	0.94	0.83	0.00	0.03	0.03	0.96	0.83	0.00	0.03	0.03	0.97
	$\text{log}({\lambda }_2)$	−4.96	0.00	0.07	0.09	0.95	−4.96	0.00	0.07	0.09	0.97	−4.96	0.00	0.08	0.09	0.95
	$\text{log}({\rho }_2)$	1.07	0.00	0.05	0.06	0.96	1.07	0.00	0.05	0.06	0.97	1.07	0.00	0.05	0.06	0.96
	${\beta }_{1g\mathit{ene}}$	2.08	0.01	0.10	0.12	0.94	2.08	0.01	0.10	0.12	0.95	2.08	0.01	0.09	0.11	0.96
	${\beta }_{2g\mathit{ene}}$	1.57	0.00	0.17	0.21	0.98	1.57	0.00	0.17	0.21	0.94	1.57	0.01	0.16	0.21	0.97
	${\beta }_{1tvc}$	1.52	0.04	0.32	0.42	0.96	1.52	0.04	0.33	0.42	0.94	1.52	0.02	0.32	0.42	0.96
	$\text{log}(\eta )$	−0.18	−0.02	0.50	0.58	0.90	−0.18	0.01	0.50	0.60	0.91	−0.18	−0.03	0.48	0.62	0.91
	η 0	0.21	−0.02	0.12	0.14	0.95	0.21	−0.01	0.12	0.14	0.96	0.21	−0.02	0.12	0.14	0.95
	$\text{log}(k_1)$	1.95	0.20	0.74	0.86	0.91	1.25	0.10	0.39	0.46	0.96	0.00	0.02	0.18	0.22	0.97
	$\text{log}(k_2)$	1.26	0.38	1.15	1.39	0.86	1.26	0.35	0.98	1.40	0.90	1.26	0.36	1.10	1.32	0.87

For each scenario, the mean bias, empirical standard error (ESE), average standard error (ASE), and estimated 95% coverage probability (ECP) are obtained from 500 replicates each with n = 500 families. λ_j and ρ_j are baseline hazard parameters for event $j,j=1,2$; β_jgene is the regression coefficient of a time-invariant covariate for event j; ${\beta }_{1tvc}$, η, and η₀ are parameters to describe TVC effects; k_j is the frailty parameter for event j.