Introducing a new estimator and test for the weighted all-cause hazard ratio

Ozga, Ann-Kathrin; Rauch, Geraldine

doi:10.1186/s12874-019-0765-1

BMC Medical Research Methodology

Table 1 Investigated simulation scenarios

From: Introducing a new estimator and test for the weighted all-cause hazard ratio

Scenario	\(\lambda ^{I}_{EP_{1}}(t)\)	\(\lambda ^{C}_{EP_{1}}(t)\)	\(\lambda ^{I}_{EP_{2}}(t)\)	\({\lambda ^{C}_{EP_{2}}(t)}\)	Description	Assumptions for original parametric estimator ^∗	Assumptions for new non-parametric estimator ^#
1	0.24	0.4	0.24t	0.8t	Weibull distributed;	\(\checkmark \)	✗
					PH assumption only fulfilled for components;
					unequal cause-specific baseline hazards
2	0.192t^−0.2	0.28t^−0.3	0.084t^−0.3	0.32t^−0.2	Weibull distributed;	✗	✗
					PH assumption not fulfilled for components and composite;
					unequal cause-specific baseline hazards
3	0.24t	0.8t	1.2t²	0.72t²	Weibull distributed;	\(\checkmark \)	✗
					PH assumption only fulfilled for components;
					unequal cause-specific baseline hazards
4	0.2t^−0.4	0.3t^−0.3	0.1t	0.1t^1.5	Weibull distributed;	✗	✗
					PH assumption not fulfilled for components and composite;
					unequal cause-specific baseline hazards
5	0.5t^−0.4	0.9t^−0.5	0.25	0.22t^0.1	Weibull distributed;	✗	✗
					PH assumption not fulfilled for components and composite;
					unequal cause-specific baseline hazards
6	0.24	0.24	0.24t	0.24t	Weibull distributed;	\(\checkmark \)	✗
					PH assumption fulfilled for components and composite;
					unequal cause-specific baseline hazards
7	0.05	0.1	1	0.5	Weibull distributed;	\(\checkmark \)	\(\checkmark \)
					PH assumption fulfilled for components and composite;
					equal cause-specific baseline hazards
8	0.42e^0.7t−0.42	0.7e^0.7t−0.7	0.21e^0.7t−0.21	0.7e^0.7t−0.7	Gompertz distributed;	✗	\(\checkmark \)
					PH assumption fulfilled for components and composite;
					equal cause-specific baseline hazards
9	0.42e^2t−0.42	0.7e^2t−0.7	0.21e^0.7t−0.21	0.7e^0.7t−0.7	Gompertz distributed;	✗	✗
					PH assumption only fulfilled for components;
					unequal cause-specific baseline hazards
10	0.42e^0.7t	0.7e^0.8t	0.21e^0.8t	0.7e^0.6t	Gompertz distributed;	✗	✗
					PH assumption not fulfilled for components and composite;
					unequal cause-specific baseline hazards

\(\lambda ^{I}_{EP_{1}}(t),\lambda ^{C}_{EP_{1}}(t),\lambda ^{I}_{EP_{2}}(t),\lambda ^{C}_{EP_{2}}(t)\): cause-specific hazard functions for EP₁ and EP₂ in the intervention and the control group, respectively; PH: proportional hazards;
^*It is assumed that the Weibull model used to estimate the cause-specific hazards is the correct one and the PH assumption is fulfilled for the components;
^#It is assumed that the cause-specific baseline hazards are equal.

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