Appendix 1
This section contains technical details of alternative approaches to the conversion of nmx to qx.
1. Our recommended method for the conversion of an nmx rate to a nqx rate – necessary for starting the life table – is to use Chiang’s formula with nax proportions available from known sources or suggested by a model life table (see section 5). If nax proportions are not known they should be imputed as approximations.
A sensible scheme is given in Appendix Table 1.
Appendix Table 1: Approximate values of nax proportion used for convenience for first 5 years of life in high mortality and low mortality countries.
| nax | High mortality* | Low mortality* |
|---|---|---|
| 1a0 | 0.3 | 0.1 |
| 1a1 | 0.4 | 0.5 |
| 1a2 and above |
0.5 | 0.5 |
| 4a1 | 0.4 | 0.4 |
| 5a0 | 0.4 | 0.1 |
| 5a5 and above |
0.5 | 0.5 |
| * where high mortality taken arbitrarily as IMR > 50 per thousand | ||
A refinement on this rather arbitrary sort of allocation was suggested by demographers Coale and Demeny (1983). It provides a sort of ready reckoner, based on regression coefficients and model life tables, to calculate values for 1a0 and 4a1. An adapted version is provided by kind permission of Preston et al.
Appendix Table 2: Suggested values of nax , in years, for first 5 years of life.
| nax | Males | Females |
|---|---|---|
| 1a0 If 1m0 >= .107 |
.330 | .350 |
| 1a0 If 1m0 < .107 |
.045 + 2.684 * 1m0 | .053 + 2.800 * 1m0 |
| 4a1 If 1m0 >= .107 |
1.352 | 1.361 |
| 4a1 If 1m0 < .107 |
1.651 - 2.816 * 1m0 | 1.522 – 1.518 * 1m0 |
| * In this table high mortality is taken arbitrarily as IMR >= 107 per thousand. | ||
Although Table 2 is the result of careful analyses of empirical data it does rather suggest that the choice of nax should be carried out at high levels of precision. The very little difference that values of nax calculated to several places of decimal, compared to the simple values of Table 1 suggests that working at high levels of precision is unnecessary where data quality is questionable. What is important is that some sensible notice is taken of nax values for ages under 5, but high precision is usually not available or necessary in these circumstances.
2. Reed-Merrell method
These demographers produced formulae (and tabulations) for the direct conversion of nmx to nqx. The formulae vary depending on which age interval is chosen. The formulae can be found online and are easily programmed into a calculator or spreadsheet. They are not presented here because they produce results very little different from Chiang’s formula with any sensible imputed nax values. They were derived exclusively from historical US data.
3. "Instantaneous" production of lx column directly from nmx values
If a constant instantaneous death rate (force of mortality) is assumed across any age group then a negative exponential function of that rate will capture the reduction in the lx column from one age group to another. The nmx function is assumed to be the best estimate of the force of mortality and so the jump from nmx to lx can be made without the intermediary nqx and npx steps by means of the following formula:
This will give very similar results to Chiang’s formula when nmx values are small and can therefore be a useful short cut. However where nmx values are large, as in older ages (increasingly over age 70) the concept of constant death rate is unrealistic and results drift away from those where a known (or sensibly imputed) nax is used.
This means that the formula will be less useful at older ages and in populations with extremely high young age mortality.