Approaches to representing mortality (cont.)
Mathematical models (cont.)
Fitting mathematical expressions
From the preceding discussion, it should be clear that – with the possible exception of the Heligman-Pollard functional form – most mathematical representations of mortality do not offer sufficient flexibility in terms of fitting a mortality curve to observed data over the entire age range.
Fitting a Gompertz curve to older age data or to extend a life table
The Gompertz curve is still often used to derive values of lx where no empirical data exist – there is little point in making it more complex by using a Makeham curve. (Why? A: The additional parameter, A, in the Makeham law is designed to better capture the effect of younger age mortality – at older ages, the Makeham and Gompertz curves are virtually indistinguishable.)
From
| n px | = | exp | ( | - | n ∫ 0 |
BC x+t dt | ) | = | exp | ( |
|
.C x (C n - 1) | ) |
Thus equation has two unknowns, so we can use the last three values of an enumerated age structure to estimate the population in the open interval. For example, consider a life table that has tabulated values of l(x) as follows: l(65) = 0.625; l(70)=0.5; l(75)=0.375. If we wished to extend this life table to ages older than 75, we could do so by assuming that mortality after aged 65 followed a Gompertz function. We note that
| 5 p65 | = |
|
= |
|
= | exp | ( |
|
.C 65(C 5 - 1) | ) |
| 5 p70 | = |
|
= |
|
= | exp | ( |
|
.C 70(C 5 - 1) | ) |
Taking the natural logs of both sides, and dividing by the expression for 5 p70 results in the equation
|
= |
|
= | C -5 |
To five decimal places, what is the value of C?
The correct answer is 1.05212
Please attempt the answer.
Well done, 1.05212 is correct.
Sorry, that is not correct. Please try again.
Substituting this into the expression for 5 p65 and solving for B then requires the solution to
| 5 p65 | = |
|
= | exp | ( |
|
.1.05212 65 (1.05212 5 - 1) | ) |
The value of B to five decimal places, is ?
The correct answer is 0.00144
Please attempt the answer.
Well done, 0.00144 is correct.
Sorry, that is not correct. Please try again.
The Gompertz curve that defines the pattern of older age mortality in this population is thus
μx = 0.00144 (1.05212x) and hence
| l(x) | = | exp | ( |
|
(1.05212 x - 1) | ) |
The final step is to note that this mortality function implies that mortality before age 65 should have resulted in
| l(65) | = | exp | ( |
|
(1.05212 x - 1) | ) | = | 0.47616 |
Our data, however, has l(65) = 0.625, so we need to multiply the derived l(x) function up by a constant, k, to ensure equivalence of the values of l(65). This constant is simply the ratio of the observed value of l(65) to that implied by the Gompertz curve above. In other words,
| k | = |
|
= | 1.31258 |
Our final derived life table is therefore
| Age x | Original values | Fitted values |
|---|---|---|
| 65 | 0.625 | 0.62500 |
| 70 | 0.5 | 0.50018 |
| 75 | 0.375 | 0.37530 |
| 80 | 0.25915 | |
| 85 | 0.16078 | |
| 90 | 0.08688 | |
| etc |
(The slight difference between the fitted values and the original values arises from the rounding of the values of B, C and k. Working with a spreadsheet or calculator with full precision results in an exact equivalence.)