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An excel spreadsheet template for calculating life expectancy is available.
Life
tables conceptually trace a cohort of newborn babies through their entire life
under the
Age
Interval (x to x+n)
The
period of life between two exact ages; e.g.
“20 to 25” means the 5year interval between the 20^{th} and 25^{th}
birthdays.
Age
Specific Mortality Rate (_{n}M_{x})
The
age specific mortality rate , _{n}M_{x = n}D_{x}/_{n}P_{x
}where _{n}D_{x }is the number of_{ }deaths
occurring to persons aged n to x+n, and _{n}P_{x }is the number
of persons aged x to x+n alive at the midpoint of the period under
consideration. Mortality rates are
usually presented as deaths per x persons per year, where x is a convenient
population base, e.g. 10,000. If a
longer period is used to count the number of deaths (e.g. 5 years as in the life
table template) then an adjustment must accordingly be made to calculate the
annual age specific mortality rate by dividing the derived rate by the period of
observation years.
Probability
of Dying (_{n}q_{x})
To
estimate the exact probability of dying between age x and x+n (_{n}q_{x})
the deaths to persons aged x to x+n must be related to the true population at
risk. The midperiod population
estimate must be adjusted by adding half the number of deaths occurring over the
period to it. This assumes that
deaths are evenly spread throughout the time period and across the period of
life under consideration.
In
general for age groups; _{n}q_{x}
= 2 x n(_{n}M_{x})
The
assumption that deaths are spread evenly across the time period of life under
consideration is particularly unrealistic in the case of infants, where the
majority of deaths occur within the first few days of life.
To calculate the probability of dying for infants therefore, the infant
mortality rate is used, as the number of livebirths is an exact estimate of the
population at risk of dying. Therefore
for infant deaths we have;
_{1}q_{0} =
_{1}D_{0}/B where
B = number of livebirths over period of consideration.
Another
exception occurs in the final openended age group (85+ in the lifetable
templates). As everybody within
this age group must die, the probability of dying is equal to 1,
i.e. _{¥}q_{n
}= 1.000.
Persons
Alive (l_{x})
The
number of persons living at the beginning of the indicated age interval
(x) out of the total number of births assumed as the radix of the life table.
The
number of persons alive at the beginning of an age interval (l_{x+n}) is
equal to the number alive at the beginning of the previous age interval (l_{x}),
minus the numbers of persons dying within that previous age interval
over the time period considered (_{n}d_{x});
i.e. l_{x+n}
= l_{x}  _{n}d_{x}
Persons
Dying (_{n}d_{x})
The
number of persons dying within the indicated age interval (x to x+n) out of the
total number of births assumed in the table.
The
number of persons dying within a particular age interval (_{n}d_{x})
is equal to the number persons alive at the beginning of that age interval (l_{x})
multiplied by the probability of dying within that age interval (_{n}q_{x}),
i.e. _{n}d_{x}
= l_{x} x _{n}q_{x}
PersonYears
Lived in Age Interval (_{n}L_{x})
The
number of personyears that would be lived within the indicated age interval (x
to x+n) by the assumed cohort of 100,000 births. Again, assuming that deaths are spread equally across the age
intervals and also across the period of consideration, then the number of
personyears lived in an age interval is calculated as the mean of the
populations at the beginning and end of each age interval, multiplied by the
length of each age interval in years
i.e. _{ n}L_{x} =
n/2 x (l_{x} + l_{x+n})
A
difficulty arises with the estimation of _{1}L_{0},that is the
average number of infants alive who have not yet reached their first birthday.
Again the problem is that it cannot be assumed that infant deaths occur
uniformly throughout the 0 to 1 age interval.
For this reason ‘separation factors’ are used to weight the average
of l_{0} and l_{1} as follows;
_{1}L_{0} = 0.3l_{0} + 0.7l_{1 }_{ }
An additional problem lies with _{¥}L_{85}, the average number of persons alive over age 85. This is approximated from _{¥}L_{85} = _{¥}d_{85}/_{¥}M_{85. }_{ }
Person
Years Lived From Age x (T_{x})
The
total number of personyears that would be lived after the beginning of the
indicated age interval by the assumed cohort of 100,000 births.
This is calculated by simply cumulating the _{n}L_{x}
column from the oldest to the youngest age.
Life
Expectancy (e_{x})
The
average remaining lifetime (in years) for a person who survives to the beginning
of the indicated age interval. Calculated
by dividing the total number of personyears lived from age x (T_{x}) by
the number of persons alive at age x (l_{x})
i.e. e_{x} = T_{x}/l_{x}
robert.benington@bristol.gov.uk Injury Prevention Manager / Avonsafe Coordinator 01179222630 Public Health, 2nd Floor, Amelia Court, Pipe Lane, Bristol, BS1 5AA 