Difference between revisions of "Life table"
(life table equations added) 
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where t is the length of the time period. If we look at the survival conditional that the population is alive in the beginning, S<sub>0</sub> (survival in the beginning) equals 1.  where t is the length of the time period. If we look at the survival conditional that the population is alive in the beginning, S<sub>0</sub> (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 time period i with constant rate is  +  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 
S<sub>i</sub> = S<sub>0</sub> exp( exp(α<sub>i</sub> + β x) t<sub>i</sub>),  S<sub>i</sub> = S<sub>0</sub> exp( exp(α<sub>i</sub> + β x) t<sub>i</sub>),  
−  and the survival over several consequent time periods is (assuming that changes in exposure are reflected in mortality with only a small delay).  +  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 = Π<sub>i</sub> (exp( exp(α<sub>i</sub> + β x<sub>i</sub>) t<sub>i</sub>)).  S = Π<sub>i</sub> (exp( exp(α<sub>i</sub> + β x<sub>i</sub>) t<sub>i</sub>)).  
−  The life years lived is  +  The life years lived Y is 
Y = Σ<sub>i</sub> (S<sub>i</sub>t<sub>i</sub> + (1S<sub>i</sub>)(1/2 t<sub>i</sub>)),  Y = Σ<sub>i</sub> (S<sub>i</sub>t<sub>i</sub> + (1S<sub>i</sub>)(1/2 t<sub>i</sub>)), 
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Life table method is for estimating mortality of a population in time.
Contents
Scope
Definition
Procedure
Often the exposureresponse relationships are estimated from loglinear 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 exposureresponse function (ERF). ERF is assumed to be exponential.
Relative risk (RR) between two exposure levels x_{0} and x is
RR = p(x)/p(x_{0}).
Therefore,
ln(RR) = α + β x  (α + β x_{0}) <=> β = ln(RR)/(x  x_{0})
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 = S_{0} 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, S_{0} (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
S_{i} = S_{0} exp( exp(α_{i} + β x) t_{i}),
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 = Π_{i} (exp( exp(α_{i} + β x_{i}) t_{i})).
The life years lived Y is
Y = Σ_{i} (S_{i}t_{i} + (1S_{i})(1/2 t_{i})),
assuming that those who died, lived half of the period on average.