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Life table method is for estimating mortality of a population in time.
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).
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 a time period i with a constant mortality rate is
Si = S0 exp( -exp(αi + β x) ti),
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 + (1-Si)(1/2 ti)),
assuming that those who died, lived half of the period on average.