Appendix A

Definitions and approximations

Definition of qx, the probability of dying within 12 months

For a given cohort or other group of individuals, we denote by the number who survive to reach the exact age x. (This is the traditional notation: the letter "l" originally stood for "lives". In life tables, is often taken as 100,000). Out of these individuals, the number who survive for 12 months, and so reach the exact age , is . The number who died within the 12 months is therefore

 (1)

and the proportion who died is

 (2)

If the numbers are sufficiently large, gives an estimate of the probability of dying within 12 months of reaching age x.

Alternatively, if a theoretical model provides an estimate that the probability of dying is , and if we know that the number who reached age x was , then we can estimate that the number of deaths will be . In the literature, is often described as the rate of mortality, or (in France) as the quotient of mortality.

Definition of , the central death rate

For a given population or cohort, the central death rate at age x during a given period of 12 months is found by dividing the number of people who died during this period while aged x (that is, after they had reached the exact age x but before reached the exact age ) by the average number who were living in that age group during the period.

The simplest case is for a stationary population, in which the number of people who leave the age group during the year (either by reaching age  or by dying) is exactly balanced by the number who enter the age group on reaching their x-th birthday. In this simple case, the number living in the age group is constant throughout the year, say . If the number of deaths at age x is , then the central death rate is given by

 (3)

For an observed population which is not stationary, can still be measured and can often be estimated as the average of the observed populations aged x at the beginning and end of the 12 months, or as the population at mid-year.

However, for cohorts with a known survival function we can do better than this. The average number living at age x during the year is

 (4)

and the central death rate is given by

 (5)

See Pollard (1973).

Definition of , the force of mortality

The central death rate is defined by (3) and (5) as the deaths in 12 months divided by the average population over 12 months. It is a measure of deaths per head of population per unit time, where in this case the unit of time is 12 months. We could repeat this calculation taking the period as 6 months instead of 12 months (but converting the result to an annual rate), and then over shorter and shorter periods, until we finally reach the instantaneous death rate . This is also known as the hazard rate.

In the notation of the calculus, the number of deaths between ages x and , in a cohort with the survival function , is The instantaneous death rate (i.e. deaths per head of population per unit time) is

 (6)

With defined in this way, the probability of dying between age x and age is .

We note that is a continuous function, which gives it certain advantages. For example, it can be used to calculate the probability of dying within periods of any length, not just multiples of 12 months.

Although can never exceed 1, both and can do so. Consider, for example, a cohort in which 8 members reach age x, of whom 7 die within 12 months.

The relationship between and

In a cohort, the central death rate relates to all deaths between the exact age x and the exact age . The force of mortality, being a continuous function, gives the instantaneous death rate at each separate age within this interval. More specifically, gives the instantaneous death rate at every age , where . From one point of view, can be regarded as a kind of average of all the instantaneous death rates in this interval.

It is therefore not surprising to find that is generally quite close to the instantaneous death rate (i.e., the force of mortality) in the middle of that interval. That is to say, there is an approximate relationship

 (7)

where the symbol means "approximately equals". This approximation is given by Pollard (1973), who gives a more formal derivation.

It is of interest to see the maximum possible error in the approximation (7) in two extreme cases. If increases exponentially with age, as in the Gompertz model then when and it can be shown that the error in (7) is at the most 0.8 per cent. On the other hand, when is constant, the error in (7) is zero. All the models considered in this Monograph fall between these two extremes.

The relationship between and

From the definitions (1) and (2), it follows that

 (8)

There is also a well-known exact equation, which can be proved by integrating (6), which states that

 (9)

If we combine (8) with (9), and assume that the integral of between x and will be close to the value at the mid-point of this interval, we obtain the approximations

 (10)

 (11)

The accuracy of these approximations can be examined in the same way as before, by considering extreme cases. When increases exponentially with age, as in the Gompertz model then when the error in (10) is only 0.04 per cent, irrespectively of the value of . When is constant, both (10) and (11) hold exactly. Thus both these approximations are highly accurate, in all the models considered in this Monograph.

Other approximations

By combining (7) with (10) and (11) we obtain

 (12)

 (13)

By similar methods, it is also possible to derive

 (14)

where

By a judicious choice and use of the approximations (7) and (10)-(14), it is possible to convert between , and at will.

Updated by V. Castanova, 1 March 1999