Appendix A — Life table methods
Most of the content in this appendix is taken directly from Preston et al. (2001), but I have reproduced it here for reference and for completeness.
A.1 Period life tables
Calculating life expectancy for a cohort is possible, but you have to wait until every member of the cohort has died. Instead, demographers use period (or “current”) life tables, which consider what would happen to a hypothetical cohort that are subjected to the deaths rates in each age group at an exact period in time. Life tables can be constructed using discrete age bands starting at age \(x\) and ending at age \(x+n\). We supply the age-specific death rates, \({}_{n}m_{x}\), and the average person-years lived by those dying in the interval, \({}_{n}a_{x}\), and the life table calculates the mean age at death – the life expectancy, \(e_x\).
The probability of dying, \({}_{n}q_{x}\), is defined as the ratio of the number of people who died in the age interval, \({}_{n}d_{x}\), to the number who survived to age \(x\), \(l_x\): \[ {}_{n}q_{x} = \frac{{}_{n}d_{x}}{l_x}. \tag{A.1}\]
The age-specific death rate is defined as the ratio of the number of people who died in the age interval to the total number of person-years lived, \({}_{n}L_{x}\), which is the weighted sum of the number of person-years lived (\(n\)) by those who survived, which, in turn, is the difference between those who survived to age \(x\) and those who died in the interval (\(l_x - {}_{n}d_{x}\)), and the number of person-years lived on average (\({}_{n}a_{x}\)) by those who died (\({}_{n}d_{x}\)): \[
{}_{n}m_{x} = \frac{{}_{n}d_{x}}{n \cdot (l_x - {}_{n}d_{x}) + {}_{n}a_{x} \cdot {}_{n}d_{x}}.
\tag{A.2}\] We assume the denominator of Equation A.2 can be approximated by the mid-year population, \({}_{n}P_{x}\), which leads us to recover the expression for the cross-sectional, empirical death rate in Equation 4.1. By rearranging the denominator to make the number of survivors the subject, we obtain \[
l_x = \frac{1}{n} \left({}_{n}P_{x} + (n - {}_{n}a_{x} \cdot {}_{n}d_{x})\right).
\tag{A.3}\] We can substitute this expression into Equation A.1 and divide by \({}_{n}P_{x}\) to obtain
\[
{}_{n}q_{x} = \frac{n \cdot {}_{n}m_{x}}{1 + (n - {}_{n}a_{x}) {}_{n}m_{x}}.
\tag{A.4}\] This expression, although unintuitive, allows us to convert from \({}_{n}m_{x}\) to \({}_{n}q_{x}\) with only the parameter \({}_{n}a_{x}\).
In the period life table, we start with a hypothetical cohort of size \(l_0 = 100,000\) and sequentially apply the probability of surviving in each age group, \({}_{n}p_{x} = 1 - {}_{n}q_{x}\), to calculate the number of survivors as \[ l_{x+n} = l_x \cdot {}_{n}p_{x}. \tag{A.5}\]
The number of person-years lived is the sum of the number of survivors weighted by the band width and number of people who died (\({}_{n}d_{x} = l_{x} \cdot {}_{n}q_{x}\)) weighted by \({}_{n}a_{x}\) \[ {}_{n}L_{x} = n \cdot l_x + {}_{n}a_{x} \cdot l_{x} \cdot {}_{n}q_{x}. \tag{A.6}\]
The open interval \({}_{\infty}q_{x} = 1\), as nobody is immortal. Using Equation A.1, it follows that the number of deaths in this interval is equal to the number who survived to the final age group, i.e. \({{}_\infty}d_{x} = l_x\). Since the death rate from Equation A.2 can be rewritten using the number of person-years lived, \({}_{n}L_{x}\), as the denominator and we can substitute the number of deaths with the number surviving to the final age group, we can obtain an expression for the number of person-years lived in the open-ended age interval \[ {}_{\infty}L_{x} = \frac{{}_{\infty}d_{x}}{{}_{\infty}m_{x}} = \frac{l_x}{{}_{\infty}m_{x}}. \tag{A.7}\]
The total number of person-years lived above \(x\) is \[ T_{x} = \sum^{\infty}_{x = a} {}_{n}L_{x}. \tag{A.8}\]
Then, life expectancy is given by dividing the number of person-years lived by the number of people who will live them \[ e_x = \frac{T_x}{l_x}. \tag{A.9}\]
Throughout the thesis, I only consider life expectancy at birth.
A.1.1 The very young ages and the very old ages
On average, it is a good approximation to assume deaths occur halfway through the age interval: \({}_{n}a_{x} = n / 2\). But for younger ages, particularly at lower levels of mortality, the majority of infant deaths lie further towards the earliest stages of infancy. Coale and Demeny used regression on a series of international datasets to recommend suitable values for \({}_{1}a_{0}\) and \({}_{4}a_{1}\) instead of the midpoint (Coale et al., 1983).
The start of the open-ended age group can be many years away from some of the ages at death, particularly in ageing populations. In order to produce reliable estimates of death rates at older ages, I used the Kannisto-Thatcher method to expand the terminal age group (\(\geq 85\) years) of the life table and adjust \({}_{n}a_{x}\) above 70 years (Thatcher et al., 2002). The Kannisto-Thatcher method assumes the probability of dying is a logistic function of age. The logit-transformed probability of dying above 70 years is regressed upon age. The resulting curve is extrapolated through to 129 years before calculating the number of survivors in the cohort following the adjusted probability of dying to estimate \({}_{n}a_{x}\) above 70 years.
A.2 Probability of dying
The probability of dying from a specific cause of death, \(i\), is calculated as in Equation A.4. Equally, we can calculate the probability of dying by subtracting the probability of surviving in each age group through to that age from unity, i.e. \(1 - \prod_x {}_{n}p^i_{x}\). Note, even for the smallest death rates, \({}_{\infty}q^i_{x} = 1\) – if you live to infinity, you’ll die of it eventually.
A.3 Cause-specific decomposition of differences in life expectancy
Using quantities generated from the life tables of two populations as above, Arriaga (1984) proposed a method to calculate the age-specific contributions to the difference in life expectancy between these populations as \[ {}_{n}\Delta_{x} = \frac{l^1_x}{l^1_0} \left( \frac{{}_{n}L^2_{x}}{l^2_x} - \frac{{}_{n}L^1_{x}}{l^1_x} \right) + \frac{T^2_{x+n}}{l^1_0} \left( \frac{l^1_x}{l^2_x} - \frac{l^1_{x+n}}{l^2_{x+n}} \right). \tag{A.10}\] The first term on the right hand side corresponds to the “direct effect” on the life expectancy difference between the two populations in the average number of person-years lived by the survivors to that age group (\({}_{n}L_{x} / l_x\)). The second term represents the “indirect effect” on the number of survivors caused by the mortality changes within an age group.
We then assume the age- and cause-specific contributions are proportional to the difference in cause-specific death rates between the two populations: \[ {}_{n}\Delta^i_{x} = {}_{n}\Delta_{x} \cdot \frac{{}_{n}m^i_{x}(2) - {}_{n}m^i_{x}(1)}{{}_{n}m_{x}(2) - {}_{n}m_{x}(1)} \tag{A.11}\]
Arriaga showed the sum of the age- and cause-specific contributions are equal to the difference in life expectancy, \[ e_0(2) - e_0(1) = \sum_x {}_{n}\Delta_{x} = \sum_x \sum_i {}_{n}\Delta^i_{x}. \tag{A.12}\]
So, we can collapse over age groups to get the cause-specific contributions to life expectancy as \(\sum_x {}_{n}\Delta^i_{x}\).
A.4 Mean age at death
The mean age at death among those who died from a specific cause of death was calculated as \[ \text{mean age at death} = \frac{\sum_x {}_{n}d_{x} \cdot {}_{n}a_{x}}{\sum_x {}_{n}d_{x}}, \tag{A.13}\] where \({}_{n}d_{x}\) is the number of deaths in an age band, calculated as the product of the death rate and the population.