Life tables from a set of rates (cont.)
Converting nmx into nqx
The two following formulae are used for converting nmx rates to nqx rates. These need some explanation.
Figure 5: The two nmx to nqx rates conversions formulae
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... (1) Chiang's formula |
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... (2) Simplified formula |
where:
- q = probability of dying (or failing)
- m = central death (or failure) rate
- x = age at start of interval
- n = width of interval
- and nax present as a proportion (see p7 on this section)
Formula (1) is the formula that we recommend. We like to call it Chiang's Formula but it is not universally known by this name.
Formula (2) is a simplified version of the same formula which applies under some conditions and is very commonly seen in life table texts. The difference between them will become clear later.
Either formula is easily programmable in a spreadsheet, statistical program, or on a programmable calculator.
The various notations are given above. NB. the symbol m is conventionally given to central rates, especially age-specific death rates. Sometimes upper case "M" is used, sometimes lower case "m". Although there is sometimes a rationale for upper or lower case it is not consistently applied and, at least for the time being, either form is acceptable.
The component of the formula which now needs some discussion is nax. In Table 4 on the previous page you can see that an nax column is supplied. Because the heading has a prefix and a suffix you should now understand that this factor applies across an interval.
Note: Various schemes for the conversion of nmx to nqx and lx are mentioned in Appendix 1 to this session, including the exponential formula for conversion of nmx to lx.