## Expected residual life time and years of life lost

In demography the actuarial estimator of the survival function has been used for centuries, typically with one year time intervals. This estimator was also the default in medical science and epidemiology until the advent of computers, when the Kaplan‐Meier estimator and later the Cox model became the de facto standard for analysis of time‐to‐event data, with survival curves reported as the well‐known “noisy” step curves. The expected lifetime (at birth) is computed as the area under the survival curve. The expected residual lifetime (ERL) given survival until age a, say, is computed as the area under the conditional survival curve S(x)/S(a) for a < x < ∞.

This is usually computed using approximations based on 5‐year age‐classes since the output from programs based on non‐parametric models is cumbersome to handle.

I will argue that a representation based on a smooth parametric model is both sensible from a biological point of view and practical from a computational point of view, particularly when more complex functionals of mortality rates are needed. This has implications for calculating the years of life lost (YLL) to a disease, a sensible definition of which is the difference in life expectancy between people with and without disease.

There are, however, three different ways defining the survival curve for people without disease (1) Use the mortality among non‐diseased persons; (2) Use the total population mortality; and (3) Use an illness‐death model to compute the survival of someone without disease, taking into account that they may contract the disease and incur a higher mortality. For common diseases like diabetes (prevalence of 15%‐20% in older ages) these three approaches can produce very different results. Using results based on incidence and mortality rates for the entire Danish population in the period 1996‐2015 I will illustrate the difference between the methods and the corresponding additions of the functions ERL and YRL to the Epi package for R.