Modeling cause-of-death mortality using hierarchical Archimedean copula

ABSTRACT

Studying changes in cause-specific (or competing risks) mortality rates may provide significant insights for the insurance business as well as the pension systems, as they provide more information than the aggregate mortality data. However, the forecasting of cause-specific mortality rates requires new tools to capture the dependence among the competing causes. This paper introduces a class of hierarchical Archimedean copula (HAC) models for cause-specific mortality data. The approach extends the standard Archimedean copula models by allowing for asymmetric dependence among competing risks, while preserving closed-form expressions for mortality forecasts. Moreover, the HAC model allows for a convenient analysis of the impact of hypothetical reduction, or elimination, of mortality of one or more causes on the life expectancy. Using US cohort mortality data, we analyze the historical mortality patterns of different causes of death, provide an explanation for the ‘failure’ of the War on Cancer, and evaluate the impact on life expectancy of hypothetical scenarios where cancer mortality is reduced or eliminated. We find that accounting for longevity improvement across cohorts can alter the results found in existing studies that are focused on one single cohort.

KEYWORDS:

• Competing risks
• hierarchical Archimedean copula
• war on cancer
• longevity risk

Acknowledgements

The authors are grateful to the editor and the anonymous reviewer for their insightful comments. All errors are our responsibility.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Yang Lu http://orcid.org/0000-0002-7141-1048

Notes

1 Source: Scor Global Life America Publication Scor View, September 2015. The percentage is calculated in terms of the amount of the policies.

2 Indeed $m(m-1)/2$ parameters are needed for a m-cause model.

3 For technical details related to the existence of J, we refer to Tsiatis (1975).

4 Also referred to as the force of mortality in the actuarial literature.

5 In the limiting case where $\theta =0$, we have independence among different causes.

6 In the case where the mortality data are observed in discrete ages, the integration is replaced by a simple summation.

7 The Kendall's tau of a bivariate survivor copula $\mathbf{S}$ is the correlation between U and V , where U, V have uniform margins and the joint survivor function $\mathbf{S}$.

8 Indeed, it can be checked that function $t\mapsto t^{\alpha }$ is m-monotonous , thus the composite function $t\mapsto \psi \left(t^{\alpha }\right)$ is still m-monotonous.

9 The data is downloaded from: www.nber.org/data/vital-statistics-mortality-data-multiple-cause-of-death.html

10 The cause of death is coded using the International Classification of Diseases (ICD). There are about 10,000 different causes of death under the current ICD system called ICD 10, which has been used by CDC since the year 2003. For former years, the CDC has used ICD 9 (from 1999 until 2002), ICD 8 (from 1979 to 1998), ICD 7 (from 1973 to 1997).

11 The analysis of a larger number of causes of death is also possible, and the advantage of the hierarchical Archimedean copula model may become more important. However, in order to facilitate the comparison with the existing literature, we only consider five causes.

12 Source: http://www.cdc.gov/men/lcod/.

13 See the official website http://www.mortality.org

14 Constraint (27(27) $\sum _t{\kappa }_{j,t}=0,\mathrm{and}\frac{{\kappa }_{j,T}-{\kappa }_{j,1}}{T}=d,\mathrm{for}j=1,\dots ,m,$(27) ) is different from the traditional mortality literature, which typically assumes $\sum _t{\kappa }_{j,t}=0$ and $\sum _x{\beta }_{j,x,t}=1$ for each j (see e.g. Lee and Carter 1992), but it facilitates the comparison between mortality patterns for different causes. Indeed, under this normalization, the ${\kappa }_{j,t}$'s have the same average trend for all causes, and thus the comparison of the mortality improvement for different causes amounts to comparing only the ${\beta }_{j,x}$'s. In particular, the larger the latter, the faster the improvement.

15 We have also considered Gumbel copula, but the results are similar and thus not reported in this paper. A computer program is available from the authors upon request.

16 Results with other values of θ exhibit similar qualitative patterns, and are available from the authors upon request.

17 These individuals are hypothetical, because such individuals could have already died between time 0 and t due to an other cause $k\ne j$.

18 Of course, these conclusions are based on the Clayton and Frank copula assumption. As the copula function is not identified, other copula specification might lead to different conclusions.

19 See for instance the last column for hierarchical Clayton copula.

20 See for instance the first column for hierarchical Clayton copula.

21 We have computed the period life expectancy, rather than, as in the previous analysis, the cohort life expectancy, in order to facilitate the comparison with the existing results of the literature.