Standard Expected Years of Life Lost as a Measure of Disease Burden: An Investigation of Its Presentation, Meaning and Interpretation

Abstract:

Standard expected years of life lost (SEYLL) is a measure of the years that might have been lived from the time of death. Although the global burden of disease project attempted to standardize this measure with respect to standard model life tables, authors use alternative standards. Manipulations of SEYLL – discounting, age-weighting, age-standardizing – are also variously applied, and SEYLL may be presented as a total, or per living person, or per death. Further, whenever a death occurs there is potential living time lost, and SEYLL is never zero, or even close to zero, so that reference values for “acceptable” SEYLL are needed in order to assess whether an observed SEYLL really is an adverse contributor to the burden of disease.

In this chapter, the way that SEYLL is presented in the literature is discussed and the effects of different manipulations of SEYLL are investigated, in particular, their effect on rank order of causes of death. Reference values, or norms, for SEYLL measures are discussed and these provide a new way to present, and rank by cause of death, total SEYLL by adjusting for normative values.

Appendix

1.1 Continuous Age Analysis of Norm of SEYLL _d

Although age is most often recorded in age bands or to the nearest year, it is an underlying continuous measure. If a person dies at age a and e(a) is a continuous measure of the expected remaining years of life a person reaching the age might be expected to live then the expected SEYLL when a person dies is, by analogy to equation (2):

$$SEYLL_{\rm{d}} = \int {e\left( a \right)f\left( a \right)da} $$

((A1))

where f(a) is the distribution of age at death and the integral is from zero to an indefinite upper value (formally infinity ∞).

The probability density of life left, t, for someone alive at age a is the function f(t)/S(a), where S(a) is the survivor function, and so e(a) can be expressed

$$e\left( a \right) = S\left( a \right)^{ - 1} \int_{\rm{a}} {\left( {t - a} \right)f\left( t \right)dt = S\left( a \right)^{ - 1} \int_{\rm{a}} {t\;f\left( t \right)dt - a} }$$

((A2))

with integration over t from a to ∞

Differentiating (A2) with respect to a leads to the differential of e(a), say e(a)′, satisfying e(a)′ = h(a)e(a)-1, where h(a) = f(a)/S(a) is the hazard function, from which

$$h\left( a \right) = e\left( a \right)^{ - 1} \left[ {e\left( a \right)^\prime + 1} \right] $$

((A3))

Consequently, replacing f(a) in equation (A1) by f(a) = h(a)S(a) with h(a) substituted by h(a) = e(a) ^-1 (e(a)′ + 1) from (A3) gives the norm for SEYLL _d

$$ SEYLL_{{\rm{norm}}} = \int {\left[ {e\left( a \right)^\prime + 1} \right]} S\left( a \right)da $$

((A4))

with integration from 0 to ∞. Further, the hazard h(a) and survivor function S(a) are mathematically linked by the well known

$$ S\left( a \right) = \exp \left[ { - \int {h\left( t \right)dt} } \right] $$

((A5))

with integration over t from 0 to a, so that if h(a) is replaced by formula (A3), S(a) can be expressed in just terms of e(a).

Together equations (A3), (A4) and (A5) specify SEYLL _norm in terms only of e(a). Unfortunately replacing even simple expressions for e(a) leads to intractable solutions and the Model Life Tables (Coale and Guo, 1989, 1990) are based on smoothed Gompetz functions that defy direct analysis. However, SEYLL _norm can be approximated assuming e _a, for annual age increments, are points on the e(a) function and putting e(a)′ ≈ e _{a + 1} -e _a and using quadrature for numerical integration.