In this module of chapter 3 on mortality, module 3.5,
I will describe mortality models used in demography,
including why they are important
and seven of the best known ones.
There are three principal justifications for postulating
an analytical form for mortality and survival functions.
First, many phenomena in the physical sciences
can be explained efficiently by simple formulae,
therefore some scientists have suggested that survival
in both humans and in non-human species
is governed by simple laws.
Second, it is easier to communicate a function
with a few parameters than it is to communicate a life table
with several hundred parameters and probabilities.
Finally, a simple analytical survival function
is easily estimated using a few parameters
from the original determination of the mortality data.
Here, I will describe and briefly discuss a number of
the more prominent, better known, and/or useful
mortality models in demography.
I will name the model, show the mortality formula
and the survival equation, and make brief comments for each.
The first model in this collection of models
is the de Moivre model.
I include this model, not because it is realistic
or used by many demographers,
but because it was one of the first models
to attempt to describe age-specific mortality.
Mortality rate in the de Moivre model equals the inverse of
the difference between the maximal and the current age.
It tends to unity as age approaches a maximum.
The survival schedule is a linearly decreasing
function of age from 1.0 at age zero,
to zero at the oldest age.
The advantage of this model is it’s simple,
it’s transparent, is easily understood,
requires only a single parameter,
and produces a linear survival function.
The assumption that survival is linear with age
is often applied over short age intervals.
Next is the Gompertz Model.
The assumption of this important model is that
mortality beyond the age of sexual maturity
(or another predetermined age)
is an exponentially increasing function of age.
The mortality trajectory for the Gompertz model is exponential,
and its survival trajectory is sigmoidal.
This model contains two parameters.
Initial mortality rate, a, which denotes mortality
at the youngest age class in the specified age interval.
The exponential rate of increase in death rate, b,
is the second parameter, and it denotes
the age-specific slope of the mortality function
and is often referred to as the “Gompertz” parameter.
The Gompertz model provides two useful formulae.
The mortality doubling time, denoted MDT,
defined as the time required for the mortality
to increase by twofold.
This is the formula.
The estimated maximum life span is the second formula,
denoted T(max), and is defined as the age when
a population subject to Gompertzian mortality rates
has diminished to one survivor.
In the following graphs, I show the effect of
different values of the two parameters, a and b,
on both the mortality and survival trajectories.
We have age on the x-axis of these graphs.
The first shows the effects on mortality
and survival trajectories of parameter a
in the top two graphs.
This graphic shows mortality schedules
that differ in initial level due to the values of a,
with the black line on top representing
the schedule with the highest initial mortality,
and the gray line at the bottom representing
the schedule with the lowest initial mortality.
Mortality in each of these schedules is increasing
at exactly the same exponential rate, b.
The corresponding survival curves, as the outcome of
these different mortality levels, are shown here,
with the shortest lived in black and the longest lived in gray.
The survival differences are scaled to the right
as survival increases with the reduction in mortality.
Now we see in the bottom set of graphs, mortality schedules
that differ in their rates of increase with age
due to differences in the Gompertz parameter, b.
Note the starting point in the initial mortality level
(i.e., parameter a) is identical in all three of these schedules.
The schedule with the highest rate of increase
in mortality with age is shown in black,
and the lowest is shown in gray.
We see in the survival differences
that the different values of the b-parameter,
that controls the rate of increase in mortality,
have much greater impact on survival,
with, for example, the lowest survival in the black line,
and the highest survival to the right with the gray line.
Clearly, similar changes in the magnitude
of each of these two parameters, a and b,
have substantially different survival outcomes,
changes in the former much less consequential
than changes in the latter.
The next model to be considered is the Makeham model,
also known as the Gompertz-Makeham model.
It purportedly represents an improvement
in the Gompertz model.
That is, Makeham found that overall mortality levels
could be better represented if a constant term
was added to the Gompertz formula to account for
causes of mortality not dependent on age
(i.e., accidental deaths).
The Makeham survival model includes
the product of the constant, c, and age x.
The fourth model is the exponential model,
which is effectively the Gompertz-Makeham model
minus the Gompertz component.
This model accounts for only accidental deaths, c,
and thus assumes age-independent mortality.
Because mortality is constant with age in this model,
its plot is simply a horizontal line intercepting the y-axis at c
and extending to the right.
The survival function for this model
decreases exponentially with age.
Whereas the de Moivre, Gompertz, and Makeham models
were derived in an actuarial context,
the next model we consider, called the Weibull Model,
was developed in the context of reliability engineering.
This model is a generalization of the exponential model,
but unlike the exponential model, the Weibull model
does not assume a constant hazard rate
and thus has broader application.
This model has two parameters.
Parameter a determines the location along the age axis,
and parameter n determines the shape of the distribution curve.
Here, I will show different values of these two parameters
and how they affect the respective location
and shape of the death distribution, dx.
Here, we have age on the x-axis
and the distribution of deaths, dx, on the y-axis.
Different values of the location parameter, a,
shifts the mean and the range of the death distribution
along the age axis but does not change the overall shape.
In contrast, the shape parameter, n, can shift the location
but more importantly changes the shape
ranging from narrow to broad distribution.
So the combination of parameters a and n
in the Weibull model have the potential to allow
this model to be fitted to virtually any observed
death distribution, whether it’s in humans,
as well as in non-human species.
This adaptability is one of the main strengths of this model.
A sixth model in this mortality model overview
is the logistic model.
This model was first introduced to demography
by Pearl in 1920, who used the model to estimate
not mortality, but the ceiling of the U.S. population.
This model assumption is that the mortality
reaches a plateau.
Therefore the rate of increase in mortality slows
at the advanced ages and eventually levels off.
This creates a long tail in survival.
Both g and n, the parameters in the model,
control the shape and location.
The last model I consider is the three-parameter model
of mortality introduced by Siler as a competing hazards model.
Siler added a third component to the Gompertz-Makeham model
to represent the early-life segment of humans.
Siler called his model a competing hazards model
precisely because he interpreted its three components
as sets of risks that compete simultaneously throughout life.
Parameters a1 and b1 control the scale
and rate of change in infant mortality.
Parameter c is age-independent mortality.
And parameters a2 and b2 control the mortality scale
and trajectory in young, middle, and old age.
I will finish this module by showing graphs
of the age-specific mortality model trajectories
for the different models or groups of models
as well as the trajectories of survival and deaths.
The first and simplest is the exponential model,
with constant mortality, as you see in the qx,
as a flat line, and that results in a geometrically
decreasing survival and death trajectories.
The next group of mortality models includes the Gompertz,
the Gompertz-Makeham, and the Weibell, that shows
the exponentially increasing mortality trajectory of qx
and age patterns of survival and death.
The shapes and details of these will depend
upon the particular parameter values used
in each of these models.
The next model is the Siler mortality model.
The unique and important feature of this model
is the early infant mortality that, in turn,
shows up as the early drop in survival
and the early bump in deaths.
The last model is the Logistic-type model,
in which mortality slows and plateaus at the oldest ages.
This plateau in mortality results in a long tail
in both the survival and the dx schedules
at the oldest ages.
This completes module 3.5 on mortality models.
There are a number of other mortality models
besides the ones I covered here that are referenced
in the Princeton book on Biodemography, as well as
any number of the mainstream books on demography.