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from Epidemiological Bulletin, Vol. 24 No. 4, December 2003
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Box 1: Formulas to calculate the life table*
|
|
nMx = dx
/ Px |
| * Nota: el subíndice derecho representa el punto inicial del intervalo. El subíndice izquierdo representa la amplitud del intervalo. |
EXACT AGE (x). This column presents the
lower limit of each age interval (usually 5-year periods), beginning with 0
and incrementing to 1, 5, 10, 15 and so on until the last, open interval is
reached. As mentioned before, the first and second age groups are usually under
1 and 1-4, therefore the values of the second and third rows
of this column are 0 and 1. This reflects the importance and specific interest
in mortality among children under 1, known as infant mortality rate (a).
Further, it is preferable to separate the calculation for age 0, and occasionally
for age 1, from the age groups 1-4 or 2-4, due to the lack of homogeneity of
mortality in this interval. Since the first stratum is a one-year age group,
the following stratum from 1 to 4 is a 4-year age group. When adequate statistics
are available, it is better to calculate directly the probabilities of death
in the first and second years of life, using infant birth and death statistics.(3)
For a final, open interval, the most commonly used is 85 years
and over, although it can vary depending on the life expectancy of the country.
WIDTH (IN YEARS) OF THE AGE INTERVAL (n).
Usually, the first value is 1 (interval 0, 1), the second 4 (interval 1, 5)
and the remaining values are 5 (5-year intervals), with the exception of the
last value that normally is represented with the sign + indicating an open interval.
NUMBER OF DEATHS RECORDED IN THE INTERVAL
(dx). This column presents the number of subjects dying
in that age group during the year corresponding to the life table.
NUMBER OF SUBJECTS IN THAT AGE GROUP (Px).
These numbers indicate the size of the corresponding age groups in the population
under study, during the year considered.
AVERAGE NUMBER OF YEARS LIVED BY THOSE WHO DIE BETWEEN AGES x AND x+n, CALLED "SEPARATION FACTOR" (nax). Although it is necessary in its calculation, this factor is not typically presented as a column of the life table. Each person living in the interval (x, x+n) has lived x complete years plus some fraction of the interval (x, x+n). In a complete life table, a value of 0.5 (i.e. half of one year) is valid from the age of 5. For a simpler calculation, it is also assumed that those who die in the 5-year age intervals of an abridged life table live on average 2.5 years.(2) However, this is not necessarily the best value for the separation factor, because the value of this fraction depends on the mortality pattern over the entire interval and not the mortality rate for any single year. In addition, since a large proportion of infant deaths occur in the first weeks of life, this value is much smaller in the 0-1 age group and in the age group 1-4. Calculation of the separation factor is easy if the date of birth and date of death are available. When they are not, values from model life tables, such as those tabulated by Coale and Demeny, shown in Table 1, can be utilized for 1a0 and 4a1.
|
Table 1: Separation fctors for ages 0 and 1-4
|
|||||||
|
Separation factor for age 0
|
Separation factor for age 1-4
|
||||||
|
Zones
|
Men
|
Women
|
Both sexes
|
Men
|
Women
|
Both sexes
|
|
|
Infant Mortality Rate > 0.100
|
North (1) |
0.33
|
0.35
|
0.3500
|
1.558
|
1.570
|
1.5700
|
| East (2) |
0.29
|
0.31
|
0.3100
|
1.313
|
1.324
|
1.3240
|
|
| South (3) |
0.33
|
0.35
|
0.3500
|
1.240
|
1.239
|
1.2390
|
|
| West (4) |
0.33
|
0.35
|
0.3500
|
1.352
|
1.361
|
1.3610
|
|
|
Infant Mortality Rate <0.100
|
North (1) |
0.0425
|
0.05
|
0.0500
|
1.859
|
1.733
|
1.7330
|
| East (2) |
0.0025
|
0.01
|
0.0100
|
1.614
|
1.487
|
1.4870
|
|
| South (3) |
0.0425
|
0.05
|
0.0500
|
1.541
|
1.402
|
1.4020
|
|
| West (4) |
0.0425
|
0.05
|
0.0500
|
1.653
|
1.524
|
1.5240
|
|
| (1) Iceland, Norway and Switzerland; (2) Austria, Czechoslovakia, North-central Italy, Poland and Hungary; (3) South Italy, Portugal and Spain; (4) Rest of the World. | |||||||
CENTRAL MORTALITY RATE (MORTALITY RATE)
(nMx). This column results from
dividing the deaths in the x, x+n interval (column dx)
by the number of people in this age group (column Px).
PROBABILITY OF DYING BETWEEN THE AGES x AND x+n
(nqx). The probabilities of dying
are calculated based on the age-specific mortality rates for each age group.
This column should be interpreted as the probability of dying between the two
ages for the subject that has survived up to age x. For the last age group of
the table, where death is unavoidable, the probability of dying is 1. For the
other age groups, the calculation is more complicated (see Box 1).
PROBABILITY OF SURVIVAL BETWEEN THE AGES x AND
x+n (npx). This column
is the complement to 1 of nqx
, and therefore it is sometimes not included in the life table. It should be
interpreted as the probability of an individual who reaches age x to reach the
exact age x+n alive.
SURVIVORS TO EXACT AGE x (nlx).
l0 is the initial number of newborns composing the generation,
who are destined to die through the process of mortality followed by the life
table. It is called the radix of the table and has a value of 100,000 (or 10^k).
DEATHS BETWEEN THE EXACT AGES x AND x+n (ndx).
In order to obtain ndx, lx
is multiplied by nqx.
NUMBER OF YEARS LIVED BY THE TOTAL OF THE COHORT
OF 100,000 BIRTHS IN THE INTERVAL x, x+n (nLx).
Each member of the cohort that survives the interval x, x+n contributes n years
to L, while each member who dies in the interval x and x+n contributes the average
number of years lived by those which die in this period, i.e. the separation
factor of deaths mentioned previously. For the last, open group, Lw
is used.
TOTAL YEARS LIVED AFTER EXACT AGE x (Tx).
This number is essential for the calculation of life expectancy. It indicates
the total number of years lived by the survivors lx between the anniversary
x and the extinction of the whole generation. The value T0
is the total number of years lived by the cohort until the death of its last
component.
LIFE EXPECTANCY AT AGE x (nex).
Among all the indicators provided by the life table, the most widely used is
life expectancy (ex), which represents the average number
of years remaining to be lived by survivors to age x. As a result, life expectancy
at birth (e0) is the average number of years lived by
a generation of newborns under given mortality conditions. This synthetic indicator
is one of the most widely used to compare the general level of mortality between
countries and over time.(2)
Life expectancy always decreases from the first row of the table
to the last, with the exception of the second row (1-4), which can be greater
than the first (0-1) in countries with very high infant mortality.(4) For a
given population, life expectancy is greater in women that in men and the overall
life expectancy should be approximately between the two. Exceptions to this
rule could arise in countries with high fertility and high maternal mortality,
or in populations in which, for cultural reasons, the nutritional and general
living conditions of women are markedly worse than those of men.
Applications
The life table is a widely-used statistical table in demographic, social and
health studies. The principal objective of a life table is to calculate life
expectancy, at birth and at other ages. However life tables provide other interesting
demographic data. Since the life table measures the probability of death (or
some other end point) at each designated time interval, it thus provides the
survival curve for a cohort of individuals. It is common to use the life table
method to compare survival curves for two patient cohorts receiving different
therapies in evaluating the differences or effectiveness of these therapies.
It also allows calculating the survival ratio. This ratio, usually presented
for a 5-year period (5Px = 5Lx+5 / 5Lx ), represents the survival between 2
age groups, i.e. the average chance that a person in an age group will survive
5 more years to the next age group. It is used in particular for making population
projections.
Example
Box 2 presents data on deaths and population in Brazil in 2000. These data allow
calculating the life table. The calculation starts with nMx.
|
Box 2: Example of calculation of a life table: Brazil,
2000
|
||||||||||||
| Data from the death registry and population census: |
Questions related to the interpretation of the values in the life
table:
|
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|
Age group
|
Deaths
(1) |
Population
(2) |
||||||||||
| 0-1 |
65,532
|
3,205,108*
|
||||||||||
| 1-4 |
11,271
|
13,084,650
|
||||||||||
| 5-9 |
5,366
|
16,533,114
|
||||||||||
| 10-14 |
6,294
|
17,406,984
|
||||||||||
| 15-19 |
19,255
|
17,847,032
|
||||||||||
| 20-24 |
26,620
|
16,500,057
|
||||||||||
| 25-29 |
25,404
|
14,534,868
|
||||||||||
| 30-34 |
28,162
|
13,533,472
|
||||||||||
| 35-39 |
33,578
|
12,953,294
|
||||||||||
| 40-44 |
39,855
|
10,942,252
|
||||||||||
| 45-49 |
45,880
|
9,106,099
|
||||||||||
| 50-54 |
52,276
|
7,139,958
|
||||||||||
| 55-59 |
58,078
|
5,425,966
|
||||||||||
| 60-64 |
72,044
|
4,553,017
|
||||||||||
| 65-69 |
81,641
|
3,365,780
|
||||||||||
| 70-74 |
93,339
|
2,588,020
|
||||||||||
| 75-79 |
90,927
|
1,602,984
|
||||||||||
| 80-84 |
80,847
|
857,170
|
||||||||||
| 85+ |
103,085
|
460,928
|
||||||||||
|
x
|
n
|
dx
|
Px
|
nax**
|
nMx
|
nqx
|
npx
|
nlx
|
ndx
|
nLx
|
nTx
|
nex
|
| 0-1 |
0
|
65,532 |
3,205,108*
|
0.05
|
0.02045
|
0.02006
|
0.97994
|
100,000
|
2,006
|
98,095
|
7,196,592
|
71.97
|
| 1-4 |
1
|
11,271 |
13,084,650
|
1.524
|
0.00086
|
0.00344
|
0.99656
|
97,994
|
337
|
391,143
|
7,098,498
|
72.44
|
| 5-9 |
4
|
5,366 |
16,533,114
|
2.5
|
0.00032
|
0.00162
|
0.99838
|
97,657
|
158
|
487,891
|
6,707,355
|
68.68
|
| 10-14 |
5
|
6,294 |
17,406,984
|
2.5
|
0.00036
|
0.00181
|
0.99819
|
97,499
|
176
|
487,055
|
6,219,463
|
63.79
|
| 15-19 |
5
|
19,255 |
17,847,032
|
2.5
|
0.00108
|
0.00538
|
0.99462
|
97,323
|
524
|
485,306
|
5,732,408
|
58.90
|
| 20-24 |
5
|
26,620 |
16,500,057
|
2.5
|
0.00161
|
0.00803
|
0.99197
|
96,799
|
778
|
482,053
|
5,247,103
|
54.21
|
| 25-29 |
5
|
25,404 |
14,534,868
|
2.5
|
0.00175
|
0.00870
|
0.99130
|
96,022
|
835
|
478,020
|
4,765,050
|
49.62
|
| 30-34 |
5
|
28,162 |
13,533,472
|
2.5
|
0.00208
|
0.01035
|
0.98965
|
95,186
|
985
|
473,468
|
4,287,030
|
45.04
|
| 35-39 |
5
|
33,578 |
12,953,294
|
2.5
|
0.00259
|
0.01288
|
0.98712
|
94,201
|
1,213
|
467,972
|
3,813,563
|
40.48
|
| 40-44 |
5
|
39,855 |
10.942,252
|
2.5
|
0.00364
|
0.01805
|
0.98195
|
92,988
|
1,678
|
460,744
|
3,345,591
|
35.98
|
| 45-49 |
5
|
45,880 |
9,106,099
|
2.5
|
0.00504
|
0.02488
|
0.97512
|
91,310
|
2,272
|
450,869
|
2,884,847
|
31.59
|
| 50-54 |
5
|
52,276 |
7,139,958
|
2.5
|
0.00732
|
0.03595
|
0.96405
|
89,038
|
3,201
|
437,188
|
2,433,978
|
27.34
|
| 55-59 |
5
|
58,078 |
5,425,966
|
2.5
|
0.01070
|
0.05212
|
0.94788
|
85,837
|
4,474
|
418,000
|
1,996,790
|
23.26
|
| 60-64 |
5
|
72,044 |
4,553,017
|
2.5
|
0.01582
|
0.07611
|
0.92389
|
81,363
|
6,192
|
391,334
|
1,578,790
|
19.40
|
| 65-69 |
5
|
81,641 |
3,365,780
|
2.5
|
0.02426
|
0.11435
|
0.88565
|
75,171
|
8,596
|
354,365
|
1,187,456
|
15.80
|
| 70-74 |
5
|
93,339 |
2,588,020
|
2.5
|
0.03607
|
0.16541
|
0.83459
|
66,575
|
11,012
|
305,345
|
833,091
|
12.51
|
| 75-79 |
5
|
90,927 |
1,602,984
|
2.5
|
0.05672
|
0.24839
|
0.75161
|
55,563
|
13,801
|
243,310
|
527,746
|
9.50
|
| 80-84 |
5
|
80,847 |
857,170
|
2.5
|
0.09432
|
0.38161
|
0.61839
|
41,761
|
15,937
|
168,965
|
284,436
|
6.91
|
| 85+ |
+
|
103,085 |
460,928
|
0.22365
|
1.00000
|
0.00000
|
25,825
|
25,825
|
115,471
|
115,471
|
4.47
|
|
| * Number of live births ** These values of the separation factor were selected because the infant mortality rate in Brazil is less than 0.1 (i.e. less than 100 deaths per 1,000 live births) and in the Coale y Demeny classification of countries, Brazil is part of the West group (see table 1) |
||||||||||||
| (1) PAHO. Technical Information System: Regional Mortality
Database. AIS; Washington, D.C.; 2003. (2) United Nations Population Division. World Population Prospects: The 2002 Revision. New York; 2003. |
||||||||||||
Figure 1 shows nqx and nMx. They are presented on a logarithmic scale because the magnitude of the range of these two indicators is such that it cannot be visualized on a single graph with an arithmetic scale. The two curves are parallel, except in the extreme ages where they coincide or start to join. In effect, the probability of dying consistently overestimates the mortality rate, except in the group of children less than 1 year of age, where nMx is above nqx. The two curves have the characteristic J shape, decreasing until the 5-9 interval, where they start to increase slightly until the 10-14 age group, then more rapidly until the 15 to 20 age group, and then regularly until they start joining at the 85-89 group.
|
Figure 1: nMx
y nqx, Brazil, 2000 (logarithmic
scale)
|
![]() |
Conclusion
Life tables present the mortality and survival experience of a whole population
and permit evaluation of its effect on specific groups and over different periods.
It is a simple instrument that is easily constructed with data collected routinely.
It is important to keep in mind that life tables are constructed based on population data from censuses and mortality registries, and therefore that the quality limitations of the latter will also affect , to different degrees, the validity of the estimations from the life table.
(a) Technical note: In a strict sense, the infant mortality rate is not equal to the under-one mortality rate, because they have different denominators. The first one is live births, and the second children under one year of age, which is more difficult to determine.
References:
(1) Pan American Health Organization. Area of Health Analysis and Information
Systems. Techiques for the Measurement of the Impact of Mortality: Years of
Potential Life Lost. Epidemiological Bulletin. 24(2):1-4; 2003
(2) United States Bureau of the Census. Shryock H, Siegel JS et al. The Methods
and materials of Demography, Second Printing (rev.). Washington, DC: United
States Government Printing Office; 1973.
(3) Livi-Bacci M. Introducción a la demografía. Barcelona: Ed
Ariel; 1993.
(4) Chang CJ. Life Tables and Mortality Analysis. Geneva: World Health Organization;
1980.
(5) Grundy EMD. Populations and population dynamics. In: Detels R, Holland WW,
McEwen JMc and Omenn GS Eds. Oxford textbook of Public Health, vol 1. The Scope
of Public Health. London: Oxford University Press; 1997.
(6) United States Census Bureau. Population Analysis Spreadsheets (PAS) [Internet
page]. Available at: http://www.census.gov/ipc/www/pas.html.
Accessed on 5 December 2003.
(7) Xunta de Galicia, Consellería de Sanidade e Servicios Sociais. Organización
Panamericana de la Salud. Area de Análisis de Situación y Sistemas
de Información. Análisis Epidemiológico de Datos Tabulados
(Epidat), version 3.0 [Computer Program for Windows]; [To be published]
(8) Coale, Ansley J, Demeny P. Regional Model Life Tables and Stable Populations,
Princeton University Press, 1966.
Source: Prepared by Dr. Enrique Vázquez from PAHOs Area of Health Situation Analysis and Information Systems (AIS) in the PAHO/WHO Argentina, Dr. Francisco Camaño (Universidad de Santiago de Compostela, Spain), Mr. John Silvi and Ms. Anne Roca (AIS - Washington, D. C.).
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Epidemiological Bulletin, Vol. 24 No. 4, December
2003



