# Lesson 7: Overview of Life Tables and Survival Rates

### Course Objective

Acquire skills to use life tables and calculate survival rates

### Expected Outcome

Ability to use alternative sources of data (life tables and census reports) to construct survival rates

The cohort component projection method projects the population into the future by age (usually 5-year age groups) and sex. Survival rates are used to calculate the number of people that will be alive at a future date in time. This lesson provides information on alternative ways to calculate survival rates. It begins with a discussion of life tables, since survival rates are derived from life tables. It demonstrates how to calculate rates for ages birth to 85 plus. It also discusses ways to use census data to compute survival rates when life tables are not available.

## 7.1 Taking Demographic Projections

Survival rates are used extensively in demographic projection techniques. Survival rates are derived from life tables or census data, and are used to calculate the number of people that will be alive in the future. In many cases, planners can obtain survival rates from a national or regional statistics office, or from life tables. If survival rates or life tables are not available, the rates may be computed from a model life table or census data.

Life tables
Life tables are used to measure mortality, survivorship, and the life expectancy of a population at varying ages.

There are several types of life tables. A generation or cohort life table is a life history of the mortality experiences of an actual cohort of individuals. The cohort begins at birth and their mortality experiences are recorded through the death of the last member of that cohort. For example, demographers use the table to trace the mortality experiences of a cohort or group of individuals born in 1910 and record the mortality experiences of each member until the last one dies. In most cases, generation life tables are used to study historic periods.

Current or period life tables
Period life tables are based on the mortality experience of a hypothetical cohort of newborn babies, usually 100,000 newborns, who are subject to the age-specific mortality rates on which the table is based. It traces the cohort of newborn babies throughout their lifetime under the assumption that they are subject to the age-specific mortality rates of a region or country.

There are two types of current life tables:

• Unabridged, for single years of life
• Abridged, for 5-year cohorts of life

In many countries, life tables are based on an average of age-specific death rates for a 3-year time period, generally around a census taking. In many cases, the life tables are prepared every 10 years. For example, a country or state would collect age-specific death rates for 1999, 2000, and 2001. The census for year 2000 would be used for the base population.

## 7.2 Columns of a Life Table

Table 7-1 provides an example of an abridged life table.

 Table 7-1Example of an Abridged Life Table Age IntervalColumn 1 nQxColumn 2 lxColumn 3 ndxColumn 4 LxColumn 5 TxColumn 6 eColumn 7 00-011-55-1010-1515-2020-2525-3030-3535-4040-4545-5050-5555-6060-6565-7070-7575-8080-8585-9090-9595+ 0.025920.00420.002320.002010.004430.006110.006320.006540.010980.017650.027650.043870.059870.096540.136540.187650.254390.378870.478980.579081 100000974089699996774965799615195564949609433993303916568912285212801117237762494507673785323511122505156 259240922519542858760462110361647253439105102773498821172712915143411126170945156 97408387996483869482897480757477820474800471695466516458282445610426061400553361884312472253837189263117557612502578116548 689285567954476407451592358254406864959928448210840073083535613306909726108152165205173914413385919767076642354103992211351035784232916548 68.9369.7666.0661.2156.3351.5846.9042.2037.4832.8928.4824.2920.4116.7113.4910.638.085.844.413.463.21

The columns of the life table include:

Column 1
Age interval, x to x+n: Age interval between exact ages for each row of the life table

Column 2
nQx: The proportion of the population in each age interval that are alive at the beginning of the interval, and dead before reaching the end of the interval. The proportion is computed from the observed mortality rates of an actual population and is used to derive the remaining columns of the life table.

Column 3
lx: The number of persons alive at the beginning of the age interval

Column 4
ndx: The number of persons dying during the age interval

Column 5
Lx: The total number of person-years in the stationary population for each age interval. It can be viewed as the average population size between birthdays, taking into account the distribution of deaths throughout the year.

Column 6
Tx: This column records the stationary population in the indicated age interval and all subsequent intervals. It is the cumulative sum of the nLx values. It can be viewed as the total number of person-years that would be lived for a particular age cohort if the cohort were to progress through the remainder of the life table.

Columns 5 and 6 represent a hypothetical stationary population which has experienced:

1. No migration
2. Constant age-specific number of births each year
3. An increase by a constant number of births each year and decrease by the same constant number of deaths each year
4. Stationary age structure size — In each age group, the number of person-years lived is always the same as that of the original life table cohort.

When a person dies or enters the next higher age interval, their place is immediately taken by someone entering from the next lower age interval. The number of persons in the age interval remains the same. The values in the Lx and Tx columns are based on the assumption that an additional 100,000 persons are added to the table annually and are subject to the mortality rates computed in the nQx column. The population is considered stationary because the total population and the number of people in each age interval do not change.

Column 7
e: This column indicates the average remaining lifetime for a given age group.

## 7.3 Calculating Survival Rates

Life tables are used to calculate survival rates. For population projections, 5-year survival rates are computed. For estimates of net migration, 10-year survival rates are calculated. Calculations of survival rates rely on two columns in the life table, Lx and Tx.

Using the abridged life table presented in Table 7-1, calculate 5-year survival rates as shown in Equation 7-1.

Equation 7-1
5-year Survival Rate

To calculate a rate to survive women ages 25–29 into the next 5-year age cohort (30–34), use the following numbers from the Lx column in Table 7-1, as shown in the following example.

Example of Equation 7-1
Surviving One Age Group into the Next Year Age Cohort

This process is repeated for most age groups; the first and last age groups are exceptions. Slight modifications are required to survive these two groups into the next age group.

Equation 7-2 provides an example of survival rates for those in the first age interval, 0–4. Note that the initial size of the first cohort is multiplied by 5 in the denominator. Why? There is no earlier value for Lx for the denominator. The number 100,000 is multiplied by 5 because, hypothetically, 100,000 new born babies are added to the table each year for a 5-year period.

Equation 7-2
Surviving the youngest age cohort

For the last age cohort (75–85+),use the Tx column to create a 5-year survival rate as shown in Equation 7-3. The value of Tx represents the number of survivors in a particular age group and all older age groups.

Equation 7-3
Surviving the oldest age cohort

## 7.4 Using Model Life Tables

If life tables are not available for a particular country, use model life tables to obtain survival rates, preferably regional model life tables.

Model life table
A model life table is derived from life tables and mortality experiences of a number of countries. They are primarily used to assist countries that do not have vital statistics systems to record deaths. Using regression analysis, the United Nations published its first set of model life tables in 1955. The tables were based on life tables from 158 countries. In 1966, Coale and Demeny introduced regional model life tables. The authors used 326 life tables to develop 200 regional model life tables.

Model life tables include the 5-year survival rates in the Px column. Chart 7-1 describes the steps to use the regional model life tables produced by Coale and Demeny. Their book is available at most libraries of the United Nations' Fund for Population Activities (UNFP).

 Using Regional Model Life TablesChart 7-1 Step 1: Selecting the region There are four sets of regional models to choose from. Select the region that most closely matches the characteristics of the country of interest. EAST MODELS: (1878–1920) Historic data for Austria, Germany, Czechoslovakia, Northern Italy, and Poland. Characteristics: high mortality in infancy and increasingly high rates over age 50. NORTH MODELS: (1851–1955) Norway, Sweden and Iceland. Characteristics: Low infant mortality and high adult 45+ mortality from tuberculosis. SOUTH MODELS: (1876–1958) Spain, Portugal, and Southern Italy. Characteristics: High childhood mortality 0–4, low mortality ages 40–60, and high mortality 65+. WEST MODELS: (1881–1959) Australia, Belguim, Canada, Denmark, Israel, Japan, Sweden, England, Wales, Finland, France, Netherlands, New Zealand, Northern Ireland, Scotland, White South Africa, Taiwan, and the United States. Characteristics: residual collection of the other countries not included in the other model tables. Step 2: Selecting an appropriate table The selection of tables within a region should be based on the infant mortality rate (IMR) of the country/locale and the life expectancy of males and females. The 1000 m column at age 0 depicts the infant mortality rate for the regional model life table. For the selected region, use the 1000 mx column at age 0 to identify a table that closely matches the infant mortality rate of your the country/locale. The table must also have a similar life expectancy to that of the country of interest. To compare the life expectancy of males and females in a country to that of the regional model life table, use the ex column at age 0. Once tables for males and females have been identified, use the 5-year survival rates in column Px, as shown in Equation 7-2 (use pop-up box).

Coale, Demeny, and Vaughan (pages 29–36) provide a detailed description of the table columns and indicate additional ways the regional model life tables can be used in developing countries.

If copies of the regional model life tables are not available, use the model life tables produced by the United Nations. The infant mortality and life expectancy rates are used to select UN model life tables. Once appropriate tables have been selected, calculate survival rates using the Lx and Tx columns.

## 7.5 Calculating National Inter-Census Survival Rates

As mentioned earlier, it is best to obtain survival rates from the central or regional statistics office in the country of interest. If survival rates, life tables, or model life tables are not available, use national census information to calculate census survival rates. National census information must be used in the calculations. Why? Regions, districts, cities and towns are heavily influenced by migration. Population movements will not drastically influence the population size of a country, as such a small percentage of the population is comprised on international migrants.

National census survival rates represent the ratio of the population in a given age group from one census period to the population in the same age group in the prior census (Shryock and Siegel, 1973, p. 454). The basic assumptions of this method are:

• No migration during the inter-census period
• No abnormal influence on mortality
• The census information is accurate

Note: This method requires that accurate census data be obtained for two points in time, usually the last two census takings. Countries experiencing a high incidence of AIDS (an abnormal influence on mortality) and/or high levels of international migration may not wish to use this approach to calculate national survival rates.

Table 7-2 demonstrates how to set up the information for the calculation.

Table 7-2Example of How to Calculate Census Survival Rates    Column 5 =Column 4 divided by Column 3 Column 6 =Square Root of Column 5 Ages Prior CensusColumn 1 Ages Recent CensusColumn 2 0-45-910-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970-7475+ 10-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970-7475-7980-8485+ 854046684900555790523890456765450900432980421123390879388012354678342123300987223456123451290876 8506547025625664325331184602134412324312234129323874323711233323213012322631231940229098780432 0.9960281.0257871.019141.0176141.007540.9785580.9959420.9805490.9911810.9564720.936960.8804780.8742000.8682780.0737020.885074 0.9980121.0128111.0095281.0087681.0037670.9892210.9979680.9902270.9955800.9779940.967960.9383380.9349860.931810.271480.940783

Columns 1 and 2 guide the placement of national census data. Data from the prior census (1990) goes in column 3, with census data for the last census year (2000) in column 4. The census survival rate is calculated by dividing Column 4 by Column 3 (year 2000 divided by year 1990). This produces a 10-year rate. To obtain a 5-year rate, take the square root of the 10-year rate. Please note that census survival rates are primarily used to estimate net migration. They should be used only in situations when access to life tables is not available to calculate survival rates. To improve the accuracy of the rates, use national census data for native-born residents. Exclude the foreign-born population in tables to help control the influence of international migration. At least two sets of census survival rates must be produced, one for males and one for females. Additional rates may be needed if other attributes of the population are to be included.

Discussion

Projections of the population by age and sex require the use of survival rates. This lesson provided information on alternative ways to calculate these rates. The next lesson will demonstrate how to use survival rates to calculate net migration and the projected size of a locale by 5-year age groups and sex.

## 7.6 Exercises

Use the information provided in Table 7-1 to answer the following questions.

1. Develop a 5-year survival rate to determine how many women ages 50–54 are expected to live to be 55–59 years of age. Which Lx numbers would be used?
1. Develop a 10-year survival rate to determine how many women ages 50–54 are expected to live to be 60–64 years of age.
1. Part 1: Calculate 10-year survival rates to estimate net migration. How would survival rates be calculated for the first three cohorts and the last age cohort? Note: Survival rates for births occurring from 1995 to 2000 and from 1990 to 1995 will be needed, in addition to a rate for those surviving from ages 0–4 to 10–14.

Part 2: Calculate a rate for those age 75+ in 1990 who will be age 85+ in year 2000.

The first survival rate is for the births that took place from 1995–2000. This is a 5-year period, calculate a 5-year rate for the births. Use the following numbers from the Lx column.

Next, develop a survival rate for the births that took place from 1990–1995.

Develop a 10-year rate to determine how many of those ages 0–4 in 1990 will be 10–14 years of age in the year 2000.

Next, develop a 10-year rate to determine how many of those ages 5–9 in 1990 will be 15–19 years of age in the year 2000.

The 10-year survival rate for the last age cohort relies on the Tx column. Develop a rate to determine how many of those ages 75+ in 1990 will be 85 + years in the year 2000.

Answer — Review Exercises: Question 1

When it is necessary to project population by age and sex to plan for different segments of the population.

Answer — Review Exercises: Question 2

1. It is highly dependent on obtaining reliable demographic information.
2. It assumes that the survival rates, birth rates, and estimates of migration are constant throughout the projection period, whether it's a 5-year or 20-year projection.
3. It does not consider the non-demographic factors that influence population growth or decline.

Answer — Review Exercises: Question 3

1. Compare projection results with another projection technique.
2. Look at old population pyramids. Do the age cohorts of the projected population move up the pyramid.
3. Consider external factors that could have alter the population composition.

### References

George W. Barclay, "The study of mortality," Techniques of Population Analysis (New York: John Wiley and Sons, 1958) 123–134.

Ansley J. Coale, Paul Demeny and Barbara Vaughan, 1983, "Uses of the Tables," Regional Model Life Tables and Stable Populations, 2nd ed. (New York: Academic Press, 1983) 29–36.

Donald J. Bogue, Kenneth Hinze and Michael White, Techniques of Estimating Net Migration (Chicago: Community and Family Study Center, University of Chicago, 1982).

Andrew M. Isserman, "The Right People, the Right Rates," Journal of the American Planning Association59.1(1993): 45–64.

Steve H. Murdock and David R. Ellis, Applied Demography: An Introduction to Basic Concepts, Methods, and Data (Boulder, CO: Westview Press, 1991).

James C. Raymondo, "Survival Rates: Census and Life Table Methods," Population Estimation and Projection(New York: Quorum Books, 1992) 43–60.

Henry S. Shryock and Jacob S. Siegel, "The Life Table," The Methods and Materials of Demography(Washington, D.C.: United States Bureau of the Census, 1973).

Note: The sample life table (Table X) was produced using an interactive life table provided by Simple Interactive Statistical Analysis and developed by Daan Uitenbroek, of the Netherlands. From the homepage click on Free Spreadsheets.

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