11.5 The current life table and life expectancy
A current (period) life table applies the age-specific death rates \(m_x\) of one calendar period to a hypothetical cohort, and summarises them as life expectancy: the average length of life of a cohort that experienced those rates throughout. For an age interval of length \(n\), the death rate \(m_x\) (deaths per person-year) must be converted to a probability of dying in the interval.
Proposition 11.5.1. If deaths in the interval occur on average half-way through it, the probability of dying in the interval for a person alive at its start is \[ q_x = \frac {n\,m_x}{1 + n\,m_x/2} . \] If instead the hazard is constant over the interval, \(q_x = 1 - e^{-n m_x}\).
Proof. Of \(l\) people alive at the start, \(d = l q_x\) die in the interval. Survivors contribute \(n\) person-years each and those who die \(n/2\) on average, so the person-years are \(n(l - d) + nd/2 = n(l - d/2)\). Then \(m_x = d/\{n(l - d/2)\}\), so \(nm_x(l - d/2) = d\), and dividing by \(l\), \(nm_x(1 - q_x/2) = q_x\), which rearranges to the result. The constant-hazard form is Theorem 3.3.3. □
With \(q_x\) for each interval, the survivors \(l_x\) of a hypothetical cohort follow by multiplication, the person-years lived in each interval are \(L_x = n(l_x + l_{x+n})/2\), and the life expectancy at age \(x\) is \(e_x = \sum _{y \geq x} L_y/l_x\). Life expectancy at birth is a summary of a period’s age-specific death rates that is free of the age structure of the population, a form of standardisation (Problem 11.6.5).
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