# Abridged Life Table

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, ni is the length of the interval, Mi is the death rate in the interval (i.e. the number of individuals dying in the interval [Di] divided by the mid-year population [Pi], 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 ai: 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, ai. 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:

 Infant mortality rate per 1000 ai < 20 0.09 20 - 40 0.15 40 - 60 0.23 > 60 0.30

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, di is the number out of the artificial cohort dying in the ith interval, li is the number out of the artificial cohort alive at the start of the interval, Li is the number of years lived in the interval by the artificial cohort, Ti is the total number of years lived by those individuals from the artificial cohort attaining the age that starts the interval, and ei 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:

 Interval length Interval 1 0 to 1 4 1 to 4 5 5 to 9 5 10 to 14 5 15 to 19 5 20 to 24 5 25 to 29 5 30 to 34 5 35 to 39 5 40 to 44 5 45 to 49 5 50 to 54 5 55 to 59 5 60 to 64 5 65 to 69 5 70 to 74 5 75 to 79 5 80 to 84 85 up

- 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

 Interval Population Deaths Death rate 0 to 1 340483 6234 0.018309 1 to 4 1302198 1049 0.000806 5 to 9 1918117 723 0.000377 10 to 14 1963681 735 0.000374 15 to 19 1817379 2054 0.00113 20 to 24 1740966 2702 0.001552 25 to 29 1457614 2071 0.001421 30 to 34 1219389 1964 0.001611 35 to 39 1149999 2588 0.00225 40 to 44 1208550 4114 0.003404 45 to 49 1245903 6722 0.005395 50 to 54 1083852 8948 0.008256 55 to 59 933244 11942 0.012796 60 to 64 770770 14309 0.018565 65 to 69 620805 17088 0.027526 70 to 74 484431 19149 0.039529 75 to 79 342097 21325 0.062336 80 to 84 210953 20129 0.095419 85 up 142691 22483 0.157564

 Interval Probability of dying [qx] SE of qx 95% CI for qx 0 to 1 0.018009 0.000226 0.017566 to 0.018452 1 to 4 0.003216 0.000099 0.003022 to 0.00341 5 to 9 0.001883 0.00007 0.001746 to 0.00202 10 to 14 0.00187 0.000069 0.001735 to 0.002005 15 to 19 0.005638 0.000124 0.005395 to 0.005881 20 to 24 0.007729 0.000148 0.007439 to 0.00802 25 to 29 0.007079 0.000155 0.006776 to 0.007383 30 to 34 0.008022 0.00018 0.007669 to 0.008376 35 to 39 0.011193 0.000219 0.010764 to 0.011622 40 to 44 0.016888 0.000261 0.016376 to 0.0174 45 to 49 0.026639 0.000321 0.02601 to 0.027267 50 to 54 0.040493 0.000419 0.039671 to 0.041315 55 to 59 0.062075 0.00055 0.060997 to 0.063153 60 to 64 0.088863 0.000709 0.087474 to 0.090253 65 to 69 0.128933 0.000921 0.127129 to 0.130737 70 to 74 0.180519 0.001181 0.178204 to 0.182833 75 to 79 0.270386 0.001582 0.267286 to 0.273486 80 to 84 0.385206 0.002129 0.381034 to 0.389379 85 up 1 * * to *

 Interval Living at start [lx] Dying [dx] Fraction of last interval of life [ax] 0 to 1 100000 1801 0.09 1 to 4 98199 316 0.41 5 to 9 97883 184 0.44 10 to 14 97699 183 0.54 15 to 19 97516 550 0.59 20 to 24 96966 749 0.49 25 to 29 96217 681 0.51 30 to 34 95536 766 0.52 35 to 39 94769 1061 0.53 40 to 44 93709 1583 0.54 45 to 49 92126 2454 0.53 50 to 54 89672 3631 0.53 55 to 59 86041 5341 0.52 60 to 64 80700 7171 0.52 65 to 69 73529 9480 0.51 70 to 74 64048 11562 0.52 75 to 79 52486 14192 0.51 80 to 84 38295 14751 0.5 85 up 23543 23543 *

 Interval Years in interval [Lx] Years beyond start of interval [Tx] 0 to 1 98361 7195231 1 to 4 392051 7096870 5 to 9 488900 6704819 10 to 14 488075 6215919 15 to 19 486454 5727844 20 to 24 482921 5241390 25 to 29 479416 4758468 30 to 34 475840 4279052 35 to 39 471354 3803213 40 to 44 464903 3331858 45 to 49 454863 2866955 50 to 54 439827 2412091 55 to 59 417386 1972264 60 to 64 386289 1554878 65 to 69 344417 1168590 70 to 74 292493 824173 75 to 79 227663 531680 80 to 84 154596 304017 85 up 149421 149421

 Interval Expectation of life [ex] SE of ex 95% CI for ex 0 to 1 71.952313 0.037362 71.879085 to 72.025541 1 to 4 72.270232 0.034115 72.203367 to 72.337097 5 to 9 68.498121 0.033492 68.432478 to 68.563764 10 to 14 63.623174 0.033231 63.558043 to 63.688305 15 to 19 58.737306 0.033025 58.672578 to 58.802034 20 to 24 54.053615 0.032466 53.989981 to 54.117248 25 to 29 49.45559 0.031785 49.393293 to 49.517888 30 to 34 44.790023 0.031151 44.728969 to 44.851077 35 to 39 40.131217 0.030436 40.071563 to 40.190871 40 to 44 35.555493 0.029616 35.497446 to 35.613539 45 to 49 31.119893 0.028788 31.06347 to 31.176317 50 to 54 26.899049 0.027963 26.844242 to 26.953856 55 to 59 22.922407 0.02697 22.869548 to 22.975266 60 to 64 19.267406 0.025794 19.216851 to 19.31796 65 to 69 15.892984 0.024469 15.845026 to 15.940942 70 to 74 12.867973 0.022957 12.822978 to 12.912969 75 to 79 10.129843 0.021419 10.087862 to 10.171824 80 to 84 7.938844 0.018833 7.901931 to 7.975756 85 up 6.346617 * * to *

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