Australian Government, Australian Government Actuary

2. Construction of the Australian Life Tables 2010‑12

There are three main elements in the process of constructing the Australian Life Tables. The first is the derivation of the exposed‑to‑risk and crude mortality rates from the information provided by the Australian Bureau of Statistics (ABS). The second is the graduation of the crude rates and associated statistical testing of the quality of the graduation. The final task is the calculation of the Life Table functions. This chapter concludes with a discussion of the methodology used to estimate the mortality improvement factors.

2.1 Calculation of exposed‑to‑risk and crude mortality rates

The calculation of mortality rates requires a measure of both the number of deaths and the population which was at risk of dying — the exposed‑to‑risk — over the same period. The raw data used for these calculations was provided by the ABS and comprised the following:

  1. Estimates of the numbers of males and females resident in Australia at each age last birthday up to 115 years and over, as at 30 June 2011. These estimates are based on the 2011 Census of Population and Housing adjusted for under‑enumeration and the lapse of time between 30 June and 9 August 2011 (the night on which the Census was taken). They differ from the published official estimates of Australian resident population which contain further adjustments to exclude overseas visitors temporarily in Australia and include Australian residents who are temporarily absent.
  2. The numbers of deaths occurring inside Australia for each month from January 2010 to December 2012, classified by sex and age last birthday at the time of death. This data covers all registrations of deaths to the end of 2013 and is considered to be effectively a complete record of all deaths occurring over the three year period.
  3. The number of registered births classified by sex in each month from January 2006 to December 2012.
  4. The number of deaths of those aged 3 years or less in each month from January 2006 to December 2012, classified by sex and age last birthday, with deaths of those aged less than one year classified by detailed duration.
  5. The numbers of persons moving into and out of Australia in each month from January 2010 to December 2012 for those aged 4 or more, and from January 2006 to December 2012 for those aged less than 4, grouped by age last birthday and sex.

Appendix B includes some selected summary information on the population, number of deaths and population movements, while Appendix C provides the detailed estimates of the population at each age last birthday at 30 June 2011, and the number of deaths at each age occurring over the three years 2010 to 2012.

The ABS conducts a five‑yearly Census of Population and Housing. Adjusted population estimates based on a particular Census will usually differ from those produced by updating the results of the previous Census for population change (that is, for births, deaths and migration) during the following five years. The difference between an estimate based on the results of a particular Census and that produced by updating results from the previous Census is called intercensal discrepancy. It is caused by unattributable errors in either or both of the start and finish population estimates, together with any errors in the estimates of births, deaths or migration in the intervening period.

The Australian Life Tables are based on the most recent Census population estimates. This is consistent with the view of the ABS that the best available estimate of the population at 30 June of the Census year is the one based on that year’s Census, not the one carried forward from the previous period. Intercensal discrepancy can, however, affect the comparability of reported mortality rates, and consequently life expectancies and improvement factors.

The crude mortality rates are calculated by dividing the number of deaths at a particular age by the exposed‑to‑risk for that age. It is essential, then, that the measure of the exposed‑to‑risk and the number of deaths should refer to the same population. Effectively, this means that a person in the population should be included in the denominator (that is, counted in the exposed‑to‑risk) only if their death would have been included in the numerator had they died.

The deaths used in deriving these Tables are those which occurred in Australia during 2010-12, regardless of usual place of residence. The appropriate exposed‑to‑risk is, therefore, exposure of people actually present in Australia at any time during the three year period. The official population estimates published by the ABS (Australian Demographic Statistics, ABS Catalogue No 3101.0) are intended to measure the population usually resident in Australia and accordingly include adjustments to remove the effect of short‑term movements, which are not appropriate for these Tables. Adjustment does, however, need to be made to the exposed‑to‑risk to take account of those persons who, as a result of death or international movement, are not present in Australia for the full three year period.

The base estimate of the exposed‑to‑risk at age x, which assumes that all those present on Census night contribute a full three years to the exposed‑to‑risk, was taken to be:

where is the population inside Australia aged x last birthday as measured in the 2011 Census adjusted only for under‑enumeration and the lapse of time from 30 June to Census night.

This estimate was then modified to reduce exposure for those who arrived in Australia between January 2010 and June 2011, or who died or left Australia between July 2011 and December 2012. Similarly, exposure was increased to take account of those who arrived between July 2011 and December 2012 or who died or left Australia between January 2010 and June 2011.

Figure 15 compares the Census population count with the exposed‑to‑risk after all adjustments have been made. It can be seen that the exposed‑to‑risk formula smooths to some extent the fluctuations from age to age apparent in the unadjusted population count. The peak resulting from the high birth rates in 1971 remains clearly visible, as does the baby boomer cohort who were in their 50s and early 60s at the time of the Census. The impact of net inward migration over recent years can be seen in the fact that the exposed‑to‑risk sits above the Census population count for most of the prime working ages from 20 to 40, particularly for females.

Figure 15: Comparison of census population count and
exposed‑to‑risk

Males

Females

For ages 2 and above, the crude central rate of mortality at age x, , was in most cases calculated by dividing the deaths at age x during 2010, 2011 and 2012 by the relevant exposed‑to‑risk. An exception was made for ages 4 to 16 inclusive. The very small number of deaths now seen at these ages has increased the potential for random fluctuations to result in dramatically different smoothed mortality rates from one set of Tables to the next. In order to avoid giving undue weight to random variation, we have combined the experience from 2005-07 and 2010-12 over these ages. The deaths data from 2005-07 has been adjusted to take account of the average level of mortality improvement over these ages before combining with the 2010-12 experience.

The exposed‑to‑risk for ages 0 and 1 was derived more directly by keeping a count of those at each age for each month of the three year period using monthly birth, death and movement records from 2006 to 2012. Because of the rapid fall in the force of mortality, , over the first few weeks of life, , rather than , was calculated for age zero. The formulae used are available on request.

2.2 Graduation of the crude mortality rates

Figure 16 shows the crude mortality rates. The crude central rates of mortality, even when calculated over three years of experience, exhibit considerable fluctuation from one age to the next, particularly among the very young and very old where the number of deaths is typically low. From a first principles perspective, however, there is no reason to suppose that these fluctuations are anything other than a reflection of the random nature of the underlying mortality process. Hence, when constructing a life table to represent the mortality experience of a population, it is customary to graduate the crude rates to obtain a curve that progresses smoothly with age.

Figure 16: Crude central mortality rates

As with previous Life Tables, a combination of manual graduation and fitted cubic splines was used. Cubic splines were fitted over all but the two youngest ages and the very top of the age distribution. At the oldest ages, there is little exposure and few deaths and a different approach is required. This is discussed below.

The method of cubic splines involves fitting a series of cubic polynomials to the crude rates of mortality. These polynomials are constrained to be not only continuous at the 'knots' where they join, but also to have equal first and second derivatives at those points. This constraint, of itself, is insufficient to ensure that the fitted curve is smooth in the sense of having a low rate of change of curvature. A large number of knots or closely spaced knots would allow the curve to follow the random fluctuations in the crude rates. At the same time, large intervals between the knots can reduce the fitted curve's fidelity to the observed results. The choice of the number and location of knots, therefore, involves a balance between achieving a smooth curve and deriving fitted rates that are reasonably consistent with the observed mortality rates.

For any given choice of knots, the criterion used to arrive at the cubic spline was that the following weighted sum of squares (an approximate χ2 variable) should be minimised:

where:

is the number of observed deaths aged x in the three years 2010, 2011 and 2012;

is the central exposed‑to‑risk at age x;

is the graduated value of the central mortality rate at age x, produced by the cubic spline;

is a preliminary value of obtained by minimising a sum of squares similar to that above, but with as the denominator;

is the lowest age of the range to which the cubic spline is to be fitted; and

is the highest age of the range to which the cubic spline is to be fitted.

As in the 2005-07 Life Tables, the knots were initially selected based on observation of the crude data. A computer program was then used to modify the location of the knots to improve the fidelity of the graduated rates to the data, and a series of statistical tests were performed on the rates to assess the adequacy of the fit. A process of trial‑and‑error was followed whereby a variety of initial knots was input into the program to produce alternative sets of graduated rates. The knots used in the graduation adopted are shown below.

Males: 7 12 16 18 20 32 53 54 61 66 77 90 95
Females: 7 9 14 17 19 27 31 54 58 61 75 91 101

The cubic splines were fitted from ages 2 to 105. In general, a larger number of knots is required at and near the ages where mortality undergoes a marked transition. For males, knots at ages 16, 18 and 20 enabled the construction of a graduated curve that captured the behaviour of mortality rates at the edge of the accident ‘cliff’. Similarly, for females, knots were needed at ages 14, 17 and 19 to capture the sharp increase and subsequent flattening in mortality rates over this age range.

The 2006 Census was the first to record individual ages for those aged 100 or more. It also asked for date of birth which allowed the internal consistency of the records to be checked. As result, both the quality and volume of data at very old ages improved and this process has continued with the 2011 Census where a high priority was placed on data integrity for centenarians. Nonetheless, the data remains scanty and an alternative approach is required for graduation at the very oldest ages.

The rates for these ages were constructed by extrapolating the trend of the crude rates from ages where there were sufficient deaths to make the crude rates meaningful. The trend result was determined by fitting a Makeham curve of the form:

where A, B and C are constants.

For males, the function was fitted to the crude rates from age 88 to age 104, while for females the data covered ages from 90 to 106. It should be noted that the results are quite sensitive to the age range chosen and there is necessarily a degree of judgement involved. For the current Tables, data on death registrations over the past 25 years was used to estimate mortality rates based on the extinct generations methodology and this influenced the selection of the age ranges used for fitting. The fitted Makeham curve was then used to extrapolate the graduated rates from age 99 for males and 100 for females.

As has been the case for the last six Tables, the raw mortality rates for males and females cross at a very old age. The 1990‑92 Tables maintained the apparent crossover as a genuine feature, resulting in male mortality rates falling below the female rates from age 103. Since that time, the crossover in both the raw and graduated rates has varied within a fairly narrow range. The following table summarises the experience.

Life Tables Crossover in crude rates Crossover in graduated rates

1990-92

100 103

1995-97

96 98

2000-02

96 103

2005-07

99 100

2010-12

100 103

1 The male crude rates cross below female rates for the first time at age 100, but female rates are lower at ages 104 and 105. Male rates are lower at all subsequent ages.

A negligible percentage of death registrations in 2010-12 did not include the age at death (less than 0.001 per cent for all ages), and consequently no adjustments were considered necessary to the graduated rates.

A number of tests were applied to the graduated rates to assess the suitability of the graduation. These tests indicated that the deviations between the crude rates and graduated rates were consistent with the hypothesis that the observed deaths represented a random sample from an underlying mortality distribution following the smoothed rates. Appendix D provides a comparison between the actual and expected number of deaths at each age.

2.3 Calculation of life table functions

As noted above, the function graduated over all but the very youngest ages was the central rate of mortality, . The formulae adopted for calculating the functions included in the Life Tables were as follows:

, the radix of the Life Table, was chosen to be 100,000.

All of the Life Table entries can be calculated from using the formulae above with the exception of , , and . These figures cannot be calculated using the standard formulae because of the rapid decline in mortality over the first year of life. Details of the calculations of , , and can be provided on request.

2.4 Estimation of mortality improvement factors

As noted in Section 1.5, a slightly different methodology has been adopted for estimating mortality improvement factors for the current publication. In previous Tables, the improvement factor at any given age has been calculated using the following formula:

where

Ix is the rate of improvement at age x;

qx(t) is the mortality rate at age x in the current Tables; and

qx(t ‑ n) is the mortality rate reported for age x in the Tables n years previously.

This measure depends only upon the mortality rates at the beginning and end of the period and gives no weight to the experience over the intervening period. As a result, this methodology can yield results that do not reflect the general pattern of mortality improvement over the period.

The alternative methodology which has been adopted for these Tables is to fit a polynomial to the mortality rates over the period of interest (either 25 years or the full history of the Life Tables) and use the fitted values for estimating the mortality improvement. The results produced on this basis are not dissimilar to those generated by the previous approach of calculating the annual percentage change between the mortality rate at the start and end of the period, but it ensures that factors reflect the experience over the whole period, not just the end points.

Figure 17 illustrates this process for a male aged 4 looking at improvement over the last 25 years. In this case, a cubic polynomial has been fitted to the six data points and the values for from the fitted function used to estimate the constant annual improvement that would give rise to the same results.

Figure 17: Estimating mortality improvement for a male aged 4