Assessing implicit hypotheses in life table construction

Abstract

Mortality figures are of capital importance for policy-making and public planning, as in forecasting financial provisions in public pension systems. General population life tables are constructed from aggregated statistics, an issue that usually entails adopting some (implicit) assumptions in their construction, such as the hypothesis of closed demographic system or the hypotheses of uniform distributions of death counts (and migration events) by age and calendar year. As microdata have become more abundant and reliable, these hypotheses could be assessed and more assumption-free estimators employed. Using a real database from Spain, we show that the above hypotheses are not appropriate in general and that the more efficient estimators proposed in this paper should be promoted, as differences persist depending on the estimator computed.

Keywords:

• Mortality tables
• death probabilities
• migration flows
• cohort-based estimators
• microdata
• point patterns
• efficiency

Acknowledgements

The authors wish to thank two anonymous referees for their valuable comments and suggestions and Instituto Nacional de Estadística for their first-rate assistance in responding to our request for the detailed statistics of deaths handled in this research. Thanks are also due to M. Hodkinson for reviewing the English of the paper. The usual disclaimer applies.

Notes

¹ Depending on the estimator used and/or the literature, the denominator of m_x is also computed/defined as the central exposed-to-risk population or the total number of ‘person-years’ at risk.

² For example, estimators of this family are currently used, although with great differences, in Spain (INE 2009) and in the UK (Hinde 1998; ONS 2010, 2012).

³ Under the hypothesis of uniform distribution of deaths by age and calendar year, the average lifespan of those dying within triangles BCF and CFG can be calculated as follows. Taking B as the origin of a Cartesian coordinate system and considering an arbitrary point P with Cartesian coordinates within BCF, it is not difficult to prove that the lifeline of P intersects BC in a point P’ with coordinates , from which it follows that the Euclidean distance from P to P′ is and that the sum of all the distances of those dying within BCF is . Dividing this sum by the Euclidean area of BCF gives us an average distance of in lifeline distance, which combined with the fact that a Euclidean distance of of a lifeline corresponds to a year yields of a year as average of the time living with age x for those dying within BCF. A similar reasoning leads to the number for those dying within CFG. An alternative proof can be found in Appendix A of Carstensen (2007).

⁴ Special treatments are also advisable for advanced ages, a matter of great interest because progressively more and more people are living to very high ages. In these cases, data are more scarce, and also less reliable (Cairns et al. 2016, forthcoming). Hence, researchers are forced to use models and to pool data from several calendar periods, countries and/or ages to reach reliable estimates (Society of Actuaries 2005). Although this topic deserves to be dealt with in more detail, we will not pursue this issue here for not obscuring, with the associated complexities, the focus of this research.

⁵ Although nowadays (almost) all statistical systems record at least the cohort for aged zero deaths, in order to be completely consistent with the hypotheses of this scenario we will consider that only statistics of the year of decease are recorded. Under these circumstances, in Spain they have been traditionally distributed 70% in the lower triangle and 30% in the upper one (Goerlich-Gisbert 2008). According to Pavía et al. (2012), these numbers could be reached ‘assuming that the number of deaths occurring during the first half of age zero is approximately four times the number of deaths registered during the second half’. This assumption together with the hypothesis of uniform distribution of deaths inside each half of the age also allows obtaining that deaths located in the lower triangle live on average of a year and deaths in the upper triangle of a year, from which the formulae follow.

⁶ In addition to the well-known concentration of deaths in the lower triangle during the first year of life, Wilmoth et al. (2007) point out that, ‘at any age, the distribution of deaths across the two triangles is affected by the relative size of the two cohorts (and sometimes by historical events as well)’, as, for instance, happens with cohorts born at beginnings or ends of major wars (Vallin 1973, pp. 39, 40).

⁷ Death Spanish microdata were provided by INE under request and on payment, migrant microdata were downloaded from http://www.ine.es/prodyser/micro_varires.htm, and Spanish Population Now-Cast estimates came from http://www.ine.es/jaxiBD/menu.do?L=1&divi=EPOB&his=0&type=db.

⁸ For males the exact values are 0.13% in both 2006 and 2007; and for females they are 0.20 and 0.14% in, respectively, 2006 and 2007. From age 96, the average discrepancies soar, reaching 5.51, 3.67, 4.57 and 3.26%, respectively.

⁹ According to the probability Laplace principle, the CSR assumption entails that the number of point events located in any arbitrary sub-region A of a target region D is proportional to its area.

¹⁰ Using the well-known result that the density function of a uniform bivariate variable in a surface S of area A is equal to for , a way to arrive at the above distributions is as follows. Taking B in Figure 1 as the origin of a Cartesian coordinate system, we have under the hypothesis of uniform distribution of deaths by age and calendar year that the density distribution functions of an event occurring respectively in the whole square, the lower triangle and the upper triangle can be expressed by: for 0 ≤ t₁, t₂ < 1, for and for 0 ≤ t₁ ≤ t₂ < 1. From which, observing that the years lived for those individuals dying aged x, last birthday in their last year of life is equal to t₂, we have after calculating the marginal distributions of t₂ the densities of interest: , and . The distributions of the time exposed to risk of dying for emigrants and immigrants are obtained in a similar fashion.

¹¹ In the case of immigrants we cannot talk of probability given that before immigration they are not members of the target population. Emigrants, on the other hand, are exits from the target population.

¹² We have considered data of three calendar years (2006–2008), have dealt independently with males and females and have studied the range of ages from 0 to 108 (yet some tests could have been performed up to age 112). In particular, we have completed 19,008 spatial tests, 14,388 functional tests, 6322 point parametric tests and 3488 tests to assess the hypothesis of closed demographic system.

¹³ The same picture is obtained scrutinizing in the supplementary material the Figures 1A–4A. Figures 1A–4A display the outputs of the tests for the whole database.

¹⁴ Figures 5A–12A in the supplementary material offer the results for the whole period covered by our database. These figures display the same overall picture.

¹⁵ This can be observed in Figures 21A–31A in the supplementary material.

¹⁶ This can be observed in Figures 37A and 38A and Figures 39A and 40A available in the supplementary graphical appendix, where respectively the crude and graduated estimated life tables have been displayed in the log-scale.

¹⁷ Graphically available in Figures 41A and 42A in the supplementary material.

¹⁸ This can be observed comparing Figure 45A vs. Figure 50A and Figure 47A vs. Figure 52A in the supplementary material.

¹⁹ See Figures 43A and 48A, for raw tables, and Figures 49A and 12 for graduated tables.

²⁰ See Figures 45A and 50A in the supplementary material.

²¹ Both tables are almost equal up to age 75, age from which the effect of the modelling strategy that HMD employs for older ages starts to be evident.

²² The INE table offers probabilities up to age 100 and the HMD table is delivered with just five decimal places.

²³ Table 4 displays numbers differentiated by gender for the sake of comparison. Currently, and because of the Test-Achats case, gender cannot be used in the EU to discriminate premiums and benefits under insurance contracts.