A parametric approach to relaxing the independence assumption in relative survival analysis

Reuben Adatorwovor; Aurelien Latouche; Jason P. Fine

doi:10.1515/ijb-2021-0016

Published by De Gruyter January 24, 2022

A parametric approach to relaxing the independence assumption in relative survival analysis

Reuben Adatorwovor , Aurelien Latouche and Jason P. Fine

From the journal The International Journal of Biostatistics

https://doi.org/10.1515/ijb-2021-0016

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Abstract

With known cause of death (CoD), competing risk survival methods are applicable in estimating disease-specific survival. Relative survival analysis may be used to estimate disease-specific survival when cause of death is either unknown or subject to misspecification and not reliable for practical usage. This method is popular for population-based cancer survival studies using registry data and does not require CoD information. The standard estimator is the ratio of all-cause survival in the cancer cohort group to the known expected survival from a general reference population. Disease-specific death competes with other causes of mortality, potentially creating dependence among the CoD. The standard ratio estimate is only valid when death from disease and death from other causes are independent. To relax the independence assumption, we formulate dependence using a copula-based model. Likelihood-based parametric method is used to fit the distribution of disease-specific death without CoD information, where the copula is assumed known and the distribution of other cause of mortality is derived from the reference population. We propose a sensitivity analysis, where the analysis is conducted across a range of assumed dependence structures. We demonstrate the utility of our method through simulation studies and an application to French breast cancer data.

Keywords: competing risks; copula; dependence modelling; net survival; relative survival

Corresponding author: Reuben Adatorwovor, University of Kentucky, Lexington, USA, E-mail: radatorwovor@uky.edu

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

Table 5:

Gumbel model: estimated parameters of the Weibull model for T ₁ across samples sizes (N), dependence levels (τ _ken) and levels of censoring (C) treating T ₂ as a competing event and vice versa.

τ _ken	C		0.10					0.50
	N	η ̂	Mean	Bias^a	ModB^a	EMP^a	CP	Mean	Bias^a	ModB^a	EMP^a	CP
0.00	1000	λ 1 ̂	0.182	−0.080	0.090	0.090	0.940	0.182	−0.290	0.150	0.170	0.928
		α 1 ̂	1.610	0.790	1.420	1.560	0.938	1.611	1.980	2.620	2.720	0.950
	5000	λ 1 ̂	0.182	−0.050	0.020	0.020	0.948	0.182	0.050	0.030	0.030	0.958
		α 1 ̂	1.610	0.520	0.280	0.280	0.954	1.610	0.290	0.520	0.530	0.956
	1000	λ 2 ̂	0.748	5.980	9.940	10.270	0.936	0.746	3.650	14.410	15.320	0.922
		α 2 ̂	0.694	0.840	6.790	7.020	0.944	0.697	4.070	8.490	0.010	0.948
	5000	λ 2 ̂	0.743	0.630	1.870	1.640	0.962	0.743	1.000	2.700	2.460	0.968
		α 2 ̂	0.693	0.100	1.340	1.210	0.962	0.693	0.280	1.680	1.530	0.958
0.25	1000	λ 1 ̂	0.182	−0.480	0.080	0.080	0.948	0.182	−0.450	0.140	0.130	0.954
		α 1 ̂	1.610	1.490	1.310	1.340	0.952	1.613	3.960	2.480	2.840	0.934
	5000	λ 1 ̂	0.182	−0.050	0.020	0.020	0.956	0.182	−0.130	0.030	0.030	0.940
		α 1 ̂	1.609	−0.250	0.260	0.240	0.956	1.610	0.590	0.500	0.460	0.950
	1000	λ 2 ̂	0.753	10.920	14.700	14.430	0.938	0.760	18.430	20.460	20.810	0.946
		α 2 ̂	0.690	−3.170	8.240	7.930	0.954	0.687	−5.930	10.080	9.900	0.944
	5000	λ 2 ̂	0.741	−0.950	2.690	2.530	0.950	0.742	0.260	3.620	3.580	0.964
		α 2 ̂	0.695	1.940	1.610	1.480	0.958	0.695	2.260	1.960	1.940	0.948
0.50	1000	λ 1 ̂	0.182	−0.030	0.070	0.070	0.956	0.182	−0.260	0.120	0.120	0.954
		α 1 ̂	1.611	2.270	1.270	1.300	0.948	1.613	3.840	2.380	2.710	0.928
	5000	λ 1 ̂	0.182	0.010	0.010	0.020	0.946	1.824	0.040	0.020	0.030	0.956
		α 1 ̂	1.609	−0.340	0.250	0.240	0.954	1.610	0.450	0.480	0.510	0.932
	1000	λ 2 ̂	0.759	17.440	19.080	20.140	0.932	0.767	25.540	25.050	25.330	0.932
		α 2 ̂	0.688	−5.180	9.440	9.910	0.944	0.684	−9.170	11.210	11.510	0.940
	5000	λ 2 ̂	0.740	−1.580	3.360	3.380	0.944	0.744	1.620	4.340	4.660	0.946
		α 2 ̂	0.695	1.720	1.820	1.870	0.936	0.692	−0.750	2.150	2.270	0.944
0.75	1000	λ 1 ̂	0.182	−0.200	0.060	0.070	0.956	0.182	−0.260	0.100	0.100	0.948
		α 1 ̂	1.610	0.490	1.060	1.520	0.936	1.612	2.660	2.090	2.370	0.942
	5000	λ 1 ̂	0.182	0.050	0.010	0.010	0.948	0.182	−0.020	0.020	0.020	0.952
		α 1 ̂	1.609	−0.010	0.210	0.210	0.944	1.609	−0.120	0.420	0.450	0.948
	1000	λ 2 ̂	0.760	17.780	20.190	20.360	0.938	0.766	24.200	26.040	25.790	0.952
		α 2 ̂	0.689	−4.600	9.870	10.440	0.946	0.685	−7.510	11.580	11.590	0.956
	5000	λ 2 ̂	0.742	−0.190	3.540	3.770	0.946	0.743	0.830	4.440	4.540	0.950
		α 2 ̂	0.694	0.550	1.900	1.990	0.930	0.693	0.350	2.210	2.280	0.944

η ̂ : estimated parameters, ModB: model-based variance, EMP: empirical variance, CP: 95% coverage probability. ^a: ×10⁻³.

A.1 Simulation results for Gumbel and Clayton Copula models

We simulated data to mimic the French breast cancer data set for sample sizes; 1000, and 5000 with 500 replications. The latent failure times for T _j ∼ Weibull(α _j, λ _j) with probability density function defined in Section 3. The parameters for the Weibull distribution for T ₁ were λ ₁ = 0.182 and α ₁ = 1.609, while those for T ₂ were λ ₂ = 0.742 and α ₂ = 0.693. In the estimation of λ ₁, α ₁ for T ₁, λ ₂, α ₂ are assumed known for T ₂, and vice versa for estimation of λ ₂, α ₂. Noninformative censoring times were generated from a uniform distribution (0, γ), where γ was chosen for 10, 30 and 50% censoring. We consider the Clayton copula with Kendall’s tau, τ k e n = θ θ + 2 = 0 , 0.25 , 0.50 , 0.75 . Initial parameter values were randomly chosen from uniform distributions, with multiple starting values as described in Section 3. The simulation results based on the Clayton copula are presented in Table 6 below.

Table 6:

Clayton Model: Estimated parameters of the Weibull model for T ₁ across samples sizes (N), dependence levels (τ _ken) and levels of censoring (C) treating T ₂ as a competing event and vice versa.

τ _ken	C		0.10					0.50
	N	η ̂	Mean	Bias^a	ModB^a	EMP^a	CP	Mean	Bias^a	ModB^a	EMP^a	CP
0.00	1000	λ 1 ̂	0.182	0.000	0.080	0.090	0.948	0.182	−0.340	0.140	0.160	0.930
		α 1 ̂	1.610	0.710	1.390	1.520	0.942	1.611	1.820	2.490	2.540	0.948
	5000	λ 1 ̂	0.182	−0.020	0.020	0.020	0.942	0.182	0.060	0.030	0.030	0.946
		α 1 ̂	1.610	0.520	0.280	0.280	0.956	1.610	0.710	0.500	0.510	0.952
	1000	λ 1 ̂	0.747	5.910	9.750	9.910	0.940	0.746	3.570	14.010	14.150	0.922
		α 1 ̂	0.694	0.780	6.720	6.950	0.948	0.696	3.840	8.360	8.590	0.956
	5000	λ 1 ̂	0.742	0.690	1.840	1.610	0.962	0.742	0.880	2.640	2.380	0.964
		α 1 ̂	0.693	0.050	1.330	1.200	0.962	0.693	0.320	1.650	1.490	0.956
0.25	1000	λ 1 ̂	0.182	−0.010	0.070	0.080	0.952	0.182	−0.670	0.130	0.110	0.964
		α 1 ̂	1.610	0.560	1.280	1.410	0.930	1.611	1.870	2.390	0.220	0.952
	5000	λ 1 ̂	0.182	−0.180	0.010	0.010	0.946	0.182	−0.120	0.020	0.020	0.940
		α 1 ̂	1.609	−0.310	0.230	0.230	0.950	1.609	−0.030	0.430	0.450	0.958
	1000	λ 1 ̂	0.751	9.200	16.630	17.220	0.924	0.749	7.170	21.000	1.610	0.914
		α 2 ̂	0.693	0.200	8.520	8.840	0.940	0.696	2.890	10.090	10.680	0.942
	5000	λ 2 ̂	0.742	0.460	2.980	2.690	0.952	0.744	2.080	3.840	3.690	0.940
		α 2 ̂	0.693	0.550	1.660	1.54	0.950	0.693	0.140	1.980	1.930	0.942
0.50	1000	λ 1 ̂	0.182	−0.090	0.060	0.050	0.956	0.182	−0.440	0.100	0.100	0.964
		α 1 ̂	1.608	−0.980	1.030	1.090	0.952	1.612	2.580	2.020	2.060	0.968
	5000	λ 1 ̂	0.182	−0.180	0.010	0.010	0.936	0.182	−0.180	0.020	0.020	0.940
		α 1 ̂	1.609	−0.190	0.200	0.210	0.952	1.610	0.240	0.400	0.410	0.948
	1000	λ 2 ̂	0.750	9.020	17.470	18.370	0.924	0.748	6.530	21.500	22.090	0.912
		α 2 ̂	0.693	0.390	8.850	9.390	0.928	0.696	3.120	10.350	11.080	0.930
	5000	λ ₂	0.742	0.070	3.170	2.870	0.948	0.744	2.400	3.970	3.760	0.944
		α ₂	0.693	0.460	1.730	1.620	0.950	0.693	−0.080	2.040	1.960	0.948
0.75	1000	λ 1 ̂	0.182	0.130	0.040	0.040	0.944	0.182	−0.050	0.080	0.080	0.937
		α 1 ̂	1.609	0.030	7e-04	0.860	0.924	1.610	1.120	1.410	1.450	0.947
	5000	λ 1 ̂	0.182	−0.120	0.010	0.010	0.940	0.182	−0.040	0.020	0.020	0.948
		α 1 ̂	1.609	0.040	0.140	0.160	0.936	1.610	0.780	0.270	0.320	0.926
	1000	λ 1 ̂	0.747	4.900	13.000	14.790	0.924	0.744	1.780	16.030	17.270	0.928
		α 1 ̂	0.694	1.460	8.370	9.630	0.924	0.697	4.230	9.660	10.980	0.932
	5000	λ 1 ̂	0.743	1.160	2.440	2.120	0.966	0.744	1.920	3.040	2.590	0.966
		α 1 ̂	0.693	−0.090	1.650	1.490	0.970	0.693	−0.470	1.910	1700	0.962

η ̂ : estimated parameters, ModB: model-based variance, EMP: empirical variance, CP: 95% coverage probability. ^a: ×10⁻³.

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Received: 2021-02-21

Revised: 2021-12-05

Accepted: 2021-12-22

Published Online: 2022-01-24

A parametric approach to relaxing the independence assumption in relative survival analysis

Abstract

Appendix A

A.1 Simulation results for Gumbel and Clayton Copula models

References

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