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Abridged Life Table

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This function provides a current life table (actuarial table) that displays the survival experience of a given population in abridged form.

The table is constructed by the following definitions of Greenwood (1922) and Chiang (1984):

- where qi hat is the probability that an individual will die in the ith interval, n_i is the length of the interval, M_i is the death rate in the interval (i.e. the number of individuals dying in the interval [D_i] divided by the mid-year population [P_i], which is the number of years lived in the interval by those alive at the start of the interval, i.e. it is the person-time denominator for the rate), and ai is the fraction of the last age interval of life.

To explain a_i: When a person dies at a certain age they have lived only a fraction of the interval in which their age at death sits, the average of all of these fractions of the interval for all people dying in the interval is call the fraction of the last age interval of life, a_i. Infant deaths tend to occur early in the first year of life (which is the usual first age interval for abridged life tables). The ai value for this interval is around 0.1 in developed countries and higher where infant mortality rates are higher. The values for young childhood intervals are around 0.4 and for adult intervals are around 0.5. The proper values for ai can be calculated from the full death records. If the full records are not available then the WHO guidelines are to use the following ai values for the first interval given the following infant mortality rates:

The rest of the calculations proceed using the following formulae on a theoretical standard starting population of 100,000 (the radix value) living at the start. In other words, we are constructing an artificial cohort of 100,000 and overlaying current mortality experience on them in order to work out life expectancies.

- where w is the number of intervals, d_i is the number out of the artificial cohort dying in the ith interval, l_i is the number out of the artificial cohort alive at the start of the interval, L_i is the number of years lived in the interval by the artificial cohort, T_i is the total number of years lived by those individuals from the artificial cohort attaining the age that starts the interval, and e_i is the observed expectation of life at the age that starts the interval.

Note that the value for the last interval length is not important, since this is calculated as an open interval as above. When preparing your data you will therefore have one less row in the interval column than in the columns for mid-year population in the interval and the deaths in the interval. The conventional interval pattern is:

- which is extended to 90 nowadays.

Standard errors and confidence intervals for q and e are calculated using the formulae given by Chiang (1984):

- where s squared e hat alpha is the variance of the expectation of life at the age of the start of the interval alpha, and s squared q hat i is the variance of the probability of death for the ith interval.

If you want to test whether or not the probability of death in one age interval is statistically significantly different from another interval, or compare the probability of death in a given age interval from two different populations (e.g. male vs. female), then you can use the following formulae:

- where z is a standard normal test statistic and SE is the standard error of the difference between the two (ith vs. jth) probabilities of death that you are comparing.

Comparison of two expectation of life statistics can be made in a similar way to the above, but the standard error for the difference between two e statistics is simply the square root of the sum of the squared standard errors of the e statistics being compared.

Adjusting life expectancy for a given utility

You can specify a weighting variable for utility to be applied to each interval. This is used, for example in the calculation of health adjusted life expectancy (HALE) by assuming that there is more health utility (sometimes defined by absence of disability) in some periods of life than in others. Wolfson (1996) describes the principles of health adjusted life expectancy.

StatsDirect simply multiplies Ti (the total number of years lived by those individuals from the artificial cohort attaining the age that starts the interval) by the given ith utility weight, then divides as usual by li (the number out of the artificial cohort alive at the start of the interval) in order to compute adjusted life expectancy.

Data preparation

Prepare your data in four columns as follows:

1. Length of age interval (w-1 rows corresponding to w intervals as described above)

2. Mid-year population, or number of years lived in the interval by those alive at its start (w rows)

3. Deaths in interval (w rows)

4. Fraction (a) of last age interval of life (w-1 rows)

5. (Utility weight [optional], e.g. proportion of the interval of life spent without disability in a given population)

If the fraction 'a' is not provided then it is assumed to be 0.1 for the infant interval, 0.4 for the early childhood interval and 0.5 for all other intervals. You should endeavour to supply the best estimate of 'a' possible.

Example

From Chiang (1984, p141): The total population of California in 1970.

Test workbook (Survival worksheet: Interval, Population, Deaths, Fraction a).

Abridged life table

Median expectation of life (age at which half of original cohort survives) = 75.876035

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