Life table

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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?

Answer

The life expectancy Y is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = \sum_i (S_i t_i + (S_{i-1} - S_i)(t_i/2)),}

where S_i 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 = Π_i (exp( -exp(α_i + β x_i) t_i)).

The life years lived Y is

Y = Σ_i (S_it_i + (S_i-1 - S_i)(1/2 t_i)),

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

Management

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

Life table

Contents

Question

Answer

Rationale

Management

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