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 midyear 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 persontime 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:
Infant mortality rate per 1000  a_{i} 
< 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, 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 midyear 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:

Length of age interval (w1 rows corresponding to w intervals as described above)

Midyear population, or number of years lived in the interval by those alive at its start (w rows)

Deaths in interval (w rows)

Fraction (a) of last age interval of life (w1 rows)

(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