Age-Specific Death Rates are Maximum Likelihood Estimators of the
Force of Mortality Assumed Constant in the Age Interval
Griffith Feeney
19 August 2003
The age-specific death rate for any suitably defined two-dimensional set in the age-time
plane is defined as the number of deaths occurring in this set divided by the number of
person years lived by members of the exposed population in this set.
Let the individuals in question be numbered i  1,2,..., n ; let xi and yi denote,
respectively, the ages at which the i -th individual comes into and leaves exposure; and
let S and D denote, respectively, the index sets of the individuals who survive and the
individuals who die.
The likelihood for the i-th individual, assuming a constant force of mortality m in the
given region, is
yi
exp[   dx if i is in S
xi
and
yi
 exp[   dx if i is in D
xi
The first expression is simply the probability of survival over the period of exposure for
the i -th individual. The second expression is the probability of survival to the exact age
at death times the instantaneous probability of death at that age. These expressions
assume that  does not depend on time or age within the age-time set for which the rate
is defined.
The likelihood for all individuals is the product



  e   ( yi  xi )   e   ( yi  xi )  .
 iS
 iD

We want to choose  so as to minimize this quantity. To do so we proceed in the usual
way, by minimizing the log likelihood, which is
 ( y
iS
i
 xi )   log ( e   ( yi  xi ) ) 
iD
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 ( y
i
 xi )   log(  )     ( yi  xi ) 
 ( y
i
 xi )  D log(  )     ( yi  xi ) 
iS
iS
 ( y
iAll
i
iD
iD
iD
 xi )  D log(  )  PYL  D log(  )
To find the minimizing  we take the derivative of this log likelihood with respect to 
and equate it to zero,
 PYL 
D

 0,
for which the solution is

D
.
PYL
The right hand side is by definition the occurrence-exposure rate. Thus we have shown
that the occurrence-exposure rate is a maximum likelihood estimate of the force of
mortality  when  is assumed constant over the times and ages at which exposure
occurs.
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