—from Epidemiological Bulletin, Vol. 24 No. 4, December 2003—

Life Tables: A Technique to Summarize Mortality and Survival

Introduction
In a previous article of the Epidemiological Bulletin on years of potential life lost (YPLL)(1), emphasis was put on the importance of the age of death as a variable in mortality analysis. A concept closely linked to an individual’s age at death is that of survival. While the YPLL consider the years of life lost as a result of premature death, another descriptive technique used in mortality analysis considers the years lived by individuals in a population before their death. This technique is that of mortality tables, more commonly known as life tables. It is used in public health to essentially measure mortality, but also in demographic, actuarial and other studies to examine longevity, fertility, migration, population growth and projections of population size and in studies of length of working life and length of disability-free life.(2)

In essence, life tables describe the process of extinction of a population experiencing the mortality observed at a given time, until the last of its components has died. A characteristic of life tables is that they end with the death of the last individual, and the fundamental difference between different life tables is the speed at which this end is reached.(3) Life tables can be calculated for a whole population or for a specific population subgroup (e.g., females, males, or Hispanics). In its simplest form, the entire table is generated from age specific mortality rates and the resulting values are used to measure mortality, survivorship and life expectancy, the most frequently used indicator provided by the life table. In other applications, the mortality rates are combined with demographic data to build a more complex model that measures the combined effect of mortality and changes in one or more socioeconomic characteristic.(2) One of the main advantages of life tables is that they do not reflect the effects of the age distribution of an actual population and do not require the use of a standard population for comparative analysis of levels of mortality in different populations.(2)

There are two classic forms of life tables: the cohort (or generation) and the current (or period) tables. Cohort life tables consist of monitoring a population longitudinally from a determining event (e.g., a birth cohort or a treatment cohort in a clinical trial) until all the individuals die or until the observation period is discontinued. Its use in the description of the survival of the whole population presents a series of practical difficulties, the most noteworthy being the large population needed to calculate a life table; the follow-up time required; and the losses due to migrations or other causes. The cohort table is usually used in survival analysis of clinical trials, which are carried out on smaller population samples and over a shorter period of time.

Current life tables provide a transversal view of mortality and survival experiences at all ages of a population during a short period of time, usually a year. They depend directly on the age-specific mortality rates for the year for which they are constructed. Thus, in a current life table the mortality experience of a population during a given year is applied to a hypothetical cohort of 10,000, 100,000 live births or in general 10k individuals. Although the calculation is based on a “fictitious” population size, life tables reflect the “real” mortality experience of the population and are a very useful tool to compare mortality data at the international level and to assess mortality trends at the national level.(4,5)

The complete (or unabridged) life table is constructed using every single year of age from birth to the last applicable age. However, the abbreviated (abridged) life tables are more often used, in which each age is presented in groups, usually of children under 1 year, children 1 to 4 years, and 5-year age groups for the remainder of the ages until the final age interval, which remains open. The use of abbreviated tables expanded because mortality data are usually available and sufficiently accurate in the form of rates for 5-year age groups and not for each individual age. In all cases, it is assumed that deaths are distributed evenly throughout each age interval.

In addition to their general use, life tables can serve to study the impact of a cause or group of causes of death through the so-called cause-elimination or multiple-decrement life tables. They involve constructing a life table with all deaths and another one eliminating the cause or causes of interest. Upon comparing the two tables, the impact of the eliminated deaths can be observed in the different indicators of the life table.(4) The years of life expectancy lost (YLEL) are based on a similar concept and will be presented in a future issue of the Epidemiological Bulletin.

Limitations of life tables
Life table estimates have all the disadvantages of any statistical measure based on population censuses and vital records. Data on ages and mortality registries may be incomplete or biased. Infant mortality weighs heavily on life expectancy, which means that under-reporting of this indicator, a habitual fact in many countries, can have an important effect on the results of the tables. The same can be said about the procedure used in closing the final, open interval of the mortality table (e.g, 85 and more, 90 and more) and the information inaccuracies existing in these age intervals. Also, important differences in specific age/sex groups with high mortality may be overlooked, since this would have little effect on the overall life expectancy.
Constructing life tables for small populations, at the local or subregional level, is generally not recommended, since migratory movements affect the population structure more than at the regional or national levels. In these cases, a very small number of deaths can be obtained, which may produce imprecise calculations of the table’s columns.

Construction and interpretation of a life table
The construction of a life table is a simple process. It involves following a few routine steps that are repeated for each age group, which can be enormously facilitated by the use of a spreadsheet such as the one proposed by the United States Bureau of Census (6) or any other software offering this tool, such as Epidat 3.0.(7) The different components usually included in a life table are presented below, as well as their interpretation.(3, 4) The formulas to calculate them are presented in box 1.

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.

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.

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 PAHO’s 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