# Life Expectancy

Life expectancy is the average expected number of years of life remaining from a given age, in a given population, according to the current mortality experience (age-specific death rates) of persons in the same population. Life expectancy is calculated from a table called a "Life Table." It is most often expressed as the life expectancy from birth, but is also commonly expressed as life expectancy from age 65.

 Contents 1 Types of Life Expectancy Calculation 2 Abridged Life Table References

# Types of Life Expectancy Calculation

Life expectancy calculation is based on a "life table," a table of numbers that calculates various elements for a set of age ranges. The distinctions between types of life expectancy calculation depend on the characteristics of the life table and the inputs to it.

### Cohort Versus Period (or Current) Life Tables (Arias, 2008)

• A cohort (or generation) life table uses data from a particular birth cohort, for example, the age-specific death rates for all persons born in 1900, after no persons remain alive in the group.
• A period (or current) life table uses current death data, and represents what an hypothetical cohort would experience, given the mortality conditions in the current population. The current life table may be used to make statistical inferences and comparisons between the mortality experiences of different populations (SEPHO, 2005). The life table that is calculated in the NM-IBIS death data query system is calculated from a current life table.

### Complete Versus Abridged Life Tables (Anderson, 1999)

• A complete life table calculates life expectancy for every single year of age.
• An abridged life table calculates life expectancy for grouped age intervals, typically 5- or 10-year age groups. The abridged method is sometimes used when data are sparsely distributed by single years of age, or when single-year population estimates are not available (to compute age-specific death rates).
• Life expectancy that is calculated in the NM-IBIS death data query system is calculated from an abridged life table.

# Abridged Life Table

The life expectancy calculations in the NM-IBIS Life Expectancy Query Module, and in the example Life Table, below use a methodology developed by Chiang (1968). This methodology was used because it was demonstrated to produce better estimates of life expectancy for small populations (SEPHO, 2005). The SEPHO report also demonstrated that models with 5-year age bands to 85+ performed best, and that a population of 5,000 life years at risk produced an 'acceptable' 95% confidence interval of plus or minus 4 years of life expectancy.

x: Age interval
The period of life during the n-year interval staring with this age. For example, 10 represents the five-year interval 10-14.

n: Interval width
The width in years of the age interval.

a: Fraction of last age interval of life
Members of the hypothetical cohort who die during an age interval do not all do so at either the beginning or the end of the interval, but at various points through its length. The fraction of last age interval of life is the average fraction of that age interval that they survive before dying. This term is not shown in table 1.

Pop: Population years at risk
The population years at risk for the age interval in the study population. For example, for the 10-14 age interval in table 1, the sum of the mid-year population estimates of English males aged 10-14 years for the years 1998, 1999 and 2000 has been used, as three years' mortality data was analyzed to obtain the number of deaths.

D: Number of deaths in interval
The number of deaths observed in the age interval in the study population.

M: Annual death rate in interval
The average annual mortality rate for that age interval in the study population, based on the observed number of deaths. This is found by dividing the number of deaths in interval by the population years at risk.

q: Probability of dying in interval
The proportion of the hypothetical cohort alive at the beginning of the age interval who will die during the age interval. For example, in table 1 the probability of dying in the first year of life is 0.00628, that is for every 100,000 newborn babies 628 will die before their 1st birthday. This probability of death can be derived from the observed mortality rate (the annual death rate in interval), the interval width and the fraction of last age interval of life. Since the final age interval is open-ended the probability of dying in interval is by definition equal to 1.

p: Probability of surviving the interval
The proportion of the hypothetical cohort alive at the beginning of the age interval who will still be alive at the end of the age interval. Equal to 1-q.

l: Number alive at start of interval
The number of the hypothetical cohort alive at the start of the age interval. The size of the cohort at birth is arbitrary and is usually set to 100,000. The number alive at the start of subsequent age intervals can be calculated by subtracting the number dying in interval from the number alive at start of interval for the previous age interval.

d: Number dying in interval
The number of the hypothetical cohort dying in the age interval (that is, the expected number of deaths). It can be calculated by applying the probability of dying in interval to the number alive at start of interval. As the final age interval is openended all those alive at the start of the interval will die during it.

L: Number of years lived in interval
The number of person years lived during the age interval by the members of the cohort who are alive at the start of the interval. Those who survive the age interval each contribute to the whole interval width. Those who die during the interval each contribute on average a proportion of the interval width, determined by the fraction of last age interval of life.

T: Total number of years lived beyond start of interval
The total number of person years still to be lived by members of the cohort who are alive at the start of the age interval. It is the sum of the number of years lived in interval for the current age interval and all the subsequent intervals.

e: Observed expectation of life at start of interval
The average number of years that each of the members of the cohort alive at the start of the interval can expect to live. It is calculated by dividing the total number of years lived beyond start of interval by the number alive at start of interval. This is the Observed Expectation of Life or Life Expectancy. For example, in table 1 a newborn baby can expect to live for 75.4 years and one who reaches his 65th birthday can expect to live another 15.6 years.

### Sensitive to Infant Mortality

The calculation of life expectancy from birth is sensitive to infant mortality. For that reason, life expectancy from age 65 is often used to compare populations.

### 95% Confidence Interval

Calculation of 95% confidence interval for life expectancy is given in Chiang (1984). The variance of life expectancy (e) may be estimated as:

Where the variance of quantity 'p' is:

Then, assuming that life expectancy is normally distributed, the 95% confidence interval is found by multiplying the standard error by 1.96, where is standard error is the square root of the variance of life expectancy (e).

### Programming a Life Expectancy Calculator in SAS for NM-IBIS

A SAS program written by Zdeb and Dairman (1997) was adapted for use in the NM-IBIS Life Expectancy Query Module.

## References

1. Elizabeth Arias. United States Life Tables, 2008; vol 61 no.3. Hyattsville, Maryland: National Center for Health Statistics. September 24, 2012.

2. Robert Anderson. Method for Constructing Complete Annual U.S. Life Tables. National Center for Health Statistics, Series 2: Data Evaluation and Methods Research, No. 129; Hyattsville, Maryland: National Center for Health Statistics. December 1999.

3. SEPHO: South East Public Health Observatory. Technical Report: Calculating Life Expectancy in Small Areas. November 2005. ISBN 0-954-2971-4-8. Downloaded from http://www.sepho.org.uk/Download/Public/9847/1/Life%20Expectancy%20Nov%2005.pdf on 12/24/2012.

4. CL Chiang. The Life Table and Its Construction. In: Introduction to Stochastic Processes in Biostatistics. New York: John Wiley & Sons, 1968:189-214. As cited in SEPHO, 2005.

5. CL Chiang. The Life Table and Its Applications. Malabar, Florida: Robert E. Krieger Publ. Co.; 1984. As cited in Association of Public Health Epidemiologists in Ontario, "10 Life Table Template V1.2." Downloaded from http://www.apheo.ca/index.php?pid=223 on 12/24/2012.

6. Mike Zdeb and Matt Dairman, University at Albany-School of Public Health. Calculating and Illustrating the Probability of Developing Cancer Using SAS and SAS/Graph Software. SAS Users Group International Conference, March 16-19, 1997, San Diego, California; 1997. Downloaded from http://www2.sas.com/proceedings/sugi22/POSTERS/PAPER260.PDF on 12/24/2012.