# Life table

 In Opasnet many pages being worked on and are in different classes of progression. Thus the information on those pages should be regarded with consideration. The progression class of this page has been assessed: This page is checked The content has been checked and the references are in place. An equivalent to a manuscript to be sent to a scientific journal. The content and quality of this page is/was being curated by the project that produced the page.

Life table method is for estimating mortality of a population in time.

## Question

How should the life expectancy be measured for an individual or a population?

The life expectancy Y is

$Y = \sum_i (S_i t_i + (S_{i-1} - S_i)(t_i/2)),$

where Si is survival at the end of time period i (with duration t) assuming that those who died, lived half of the period on average.

## Rationale

Often the exposure-response relationships are estimated from log-linear models:

ln p(x) = α + β x,


where p(x) is the probability of event at exposure level x, exp(α) is the background risk, and β is the slope for the exposure-response function (ERF). ERF is assumed to be exponential.

Relative risk (RR) between two exposure levels x0 and x is

RR = p(x)/p(x0).


Therefore,

ln(RR) = α + β x - (α + β x0) <=> β = ln(RR)/(x - x0)


The life table is a table where the a) survival of and b) the years lived by a population are followed over the lifetime of the individuals in the population.

Assuming a constant rate of mortality (k) for a given time period, the survival S is

S = S0 exp(-kt),


where t is the length of the time period. If we look at the survival conditional that the population is alive in the beginning, S0 (survival in the beginning) equals 1.

As we previously showed, the mortality rate can be expressed as k = exp(α + β x), and thus the survival over one time period with a constant mortality rate is

S = S0 exp( -exp(α + β x) t),


and the survival over several consequent time periods (from the beginning up to time period i) is (assuming that changes in exposure are reflected in mortality with only a small delay).

S = Πi (exp( -exp(αi + β xi) ti)).


The life years lived Y is

Y = Σi (Siti + (Si-1 - Si)(1/2 ti)),


assuming that those who died, lived half of the period on average.

### Management

You can use this model: Impact calculation tool.ANA.