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:

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 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:

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