# LIFE EXPECTANCY AND LIFE TABLES

A life table converts a set of age-specific mortality rates into a survival curve, from which summary statistics, such as life expectancy, can be derived. The procedure was developed first for humans, primarily for the purpose of calculating premiums for life insurance and annuities. Later the same approach was used to study the survival of patients, other living species, and inanimate objects.

Crude life tables were produced by the Roman Aemilius Macer in Rome in 225 c.e., and by John Graunt and William Petty in the seventeenth century. The astronomer Edmund Halley, in 1693, was the first to employ correct mathematical methods to calculate a life table, using vital statistics collated by Caspar Neumann of Breslau. Figure 1 shows Halley's Breslau life table, William Farr's life table for England and Wales 150 years later, and a life table for Canada 300 years later.

A life table is easy to calculate if the mortality rates are known for each year of age. Starting with an arbitrary large number, say 1,000, of newborn infants, the number surviving to age one year can be estimated from the mortality rate in the first year of life. Then the number surviving to the age two years can be estimated using the mortality rate in the second year of life, and so on.

In practice, calculations are complicated by the fact that the mortality rates for single years of age cannot be estimated precisely, even in large

Figure 1

populations, and some form of smoothing is required. The sharp decline in mortality during infancy and childhood, and the small numbers in extreme old age, also create problems. However the simple method can be used to calculate abridged life tables, using mortality rates for broader age groups, which are more readily available. Such tables are usually sufficient for public health purposes.

Figure 2 shows the distribution of the ages at death implied by the English and Canadian life tables of Figure 1. In the earlier table the distribution has two peaks, one in the early childhood and the other in the 7074 age group. In the later table the deaths in childhood have shifted to old age, producing a single peak at 8084 years. The arithmetic mean of the distribution of ages at death is called life expectancy (i.e., expectation of life at birth), and is widely used as an indicator of the health of the population. The expectation of life at birth is forty-five years for the English life table and eighty-one years for the Canadian life table.

The vast majority of published life tables are period life tables, which are based on mortality rates over a limited time period. Since mortality

Figure 2

changes over time, no actual population experiences the survival depicted in a period life table. Such a table represents, instead, a hypothetical, or synthetic, cohort. Comparing values of life expectancy from different period life tables is really equivalent to comparing age-standardized mortality rates, since reciprocal of life expectancy is a form of age-standardized mortality rate.

True cohort, or generation, life tables require age-specific mortality rates covering nearly 100 years. Table 1 shows the life expectancy for two completed cohorts. Life expectancy is consistently greater for females than for males, and the difference has widened over the sixty-year period between the two groups. Information on the cause of death can be incorporated into the life table calculations in two ways. First, it is possible to calculate the number, out of those alive at a given age, who will die subsequently from a particular cause. Second, the gain in life expectancy which would be obtained by eliminating a particular cause of death can be calculated.

As life expectancy increases toward its natural upper limit, life tables become less useful as indices of the health of a population. Beginning in the 1960s procedures were developed to incorporate information on disability into life tables. The simplest approach is to multiply the number living at a

Table 1

 Mean and Median Survival (Years) in Two Canadian Cohort Life Tables Cohort Date of Birth 1831 1891 Males Femals Males Femals source: Statistics Canada. Report on the Demographic Situation in Canada 1992. Current Demographic Analysis. Ottawa: Minister of Industry, Science and Technology, 1992; p. 150 Mean Survival (e0) 4 0 42 49 54 Median Survival 45 48 62 68

given age, derived from a current life table, by the proportion of the population at that age who are found, in a health survey, to be free of disability. The sum of these products over the life span gives the disability-free life expectation. In such calculations people are considered to be either disabled or not disabled, and the definition is somewhat arbitrary. The method can be expanded to incorporate different levels of disability, which are weighted to give the disability-adjusted life expectancy.

Gerry B. Hill

## Bibliography

Benjamin, B. (1968). Demographic Analysis. London: George Allen and Unwin, Ltd.

Chiang, C. L. (1984). The Life Table and Its Applications. Malabar, FL: Robert E. Krieger Publishing Company.

Colvez, A., and Blanchet, M. (1983). "Potential Gains in the Life Expectancy Free of Disability: A Tool for Health Planning." International Journal of Epidemiology 12:224, 229.

Preston, S. H.; Keyfitz, N.; and Schoen, R. (1972). Causes of Death. Life Tables for National Populations. New York and London: Seminar Press.

Statistics Canada (1992). Report on the Demographic Situation in Canada 1992. Current Demographic Analysis. Ottawa: Minister of Industry, Science and Technology.

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