# Calculating Life Expectancy

The starting point for calculating life expectancies is the age-specific death rates of the population members. A very simple model of age-specific mortality uses the Gompertz function, although these days more sophisticated methods are used.

In cases where the amount of data is relatively small, the most common methods are to fit the data to a mathematical formula, such as an extension of the Gompertz function, or to look at an established mortality table previously derived for a larger population and make a simple adjustment to it (e.g. multiply by a constant factor) to fit the data.

With a large amount of data, one looks at the mortality rates actually experienced at each age, and applies smoothing (e.g. by cubic splines) to iron out any apparently random statistical fluctuations from one year of age to the next.

While the data required is easily identified in the case of humans, the computation of life expectancy of industrial products and wild animals involves more indirect techniques. The life expectancy and demography of wild animals are often estimated by capturing, marking and recapturing them. The life of a product, more often termed shelf life is also computed using similar methods. In the case of long-lived components such as those used in critical applications, such as in aircraft methods such as accelerated aging are used to model the life expectancy of a component.

The age-specific death rates are calculated separately for separate groups of data which are believed to have different mortality rates (e.g. males and females, and perhaps smokers and non-smokers if data is available separately for those groups) and are then used to calculate a life table, from which one can calculate the probability of surviving to each age. In actuarial notation the probability of surviving from age ''x'' to age ''x+n'' is denoted $\backslash ,\_np\_x\backslash !$ and the probability of dying during age ''x'' (i.e. between ages ''x'' and ''x''+1) is denoted $q\_x\backslash !$. For example, if 10% of a group of people alive at their 90th birthday die before their 91st birthday, then the age-specific death probability at age 90 would be 10%. Note that this is a probability rather than a mortality rate.

The life expectancy at age ''x'', denoted $\backslash ,e\_x\backslash !$, is then calculated by adding up the probabilities to survive to every age. This is the expected number of complete years lived (one may think of it as the number of birthdays they celebrate).

- $e\_x\; =\backslash sum\_\{t=1\}^\{\backslash infty\}\backslash ,\_tp\_x\; =\; \backslash sum\_\{t=0\}^\{\backslash infty\}t\; \backslash ,\_tp\_x\; q\_\{x+t\}$

Because age is rounded down to the last birthday, on average people live half a year beyond their final birthday, so half a year is added to the life expectancy to calculate the full life expectancy. (This is denoted by $\backslash ,e\_x\backslash !$ with a circle over the "''e''".)

Life expectancy is by definition an arithmetic mean. It can also be calculated by integrating the survival curve from ages 0 to positive infinity (or equivalently to the maximum lifespan, sometimes called 'omega'). For an extinct or completed cohort (all people born in year 1850, for example), of course, it can simply be calculated by averaging the ages at death. For cohorts with some survivors, it is estimated by using mortality experience in recent years. These estimates are called period cohort life expectancies.

It is important to note that this statistic is usually based on past mortality experience, and assumes that the same age-specific mortality rates will continue into the future. Thus such life expectancy figures need to be adjusted for temporal trends before calculating how long a currently living individual of a particular age is expected to live. Period life expectancy remains a commonly used statistic to summarize the current health status of a population.

However for some purposes, such as pensions calculations, it is usual to adjust the life table used, thus assuming that age-specific death rates will continue to decrease over the years, as they have done in the past. This is often done by simply extrapolating past trends; however some models do exist to account for the evolution of mortality (e.g., the Lee-Carter model).

As discussed above, on an individual basis, there are a number of factors that have been shown to correlate with a longer life. Factors that are associated with variations in life expectancy include family history, marital status, economic status, physique, exercise, diet, drug use including smoking and alcohol consumption, disposition, education, environment, sleep, climate, and health care. which uses the singular value decomposition on a set of transformed age-specific mortality rates to reduce their dimensionality to a single time series, forecasts that time series, and then recovers a full set of age-specific mortality rates from that forecasted value. Software for this approach include Professor Rob J. Hyndman's R package and UC Berkeley's LCFIT system.

## Further Reading

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