Appendix 1
Closing the life table
Closing the life table is the term applied to inserting the final estimate in the nLx column, which is the estimate of the person-years of life expected in the last open-ended age group. In this session this age group has mostly been the 85+ age group but could, of course, be much older groups.
For this last open-ended age interval there is a difficulty "closing" the nLx column because the formula calls for an lx value for the next interval and there is no next interval.
Hence l85 is the survivors at exact age 85. The corresponding nqx value is often not stated, but if stated it is 1 – everybody dies in the last interval. Therefore this figure is not much use to us to calculate how many person-years that age group will survive. So what to do?
Suggested approaches are:
Imputation
Impute a value from another suitable life table. Model life tables (see PAPP103_S01) are useful for this. This would fix the life expectancy for this group but would introduce very little error into the table as a whole. Remember that demography is a very pragmatic discipline and we are not ashamed to borrow sensible values from other sources. This approach is rough and ready but quite suitable when working with data which is of suspect quality anyway – which is often the case with developing country or historical data. The approach is less suitable if your interest is specifically in survival at older ages or you are working at high levels of accuracy.
Use of a death rate
Although an nqx value for the final interval is of no use to us we may well have a death rate – an nmx value. If so we can utilise that.
We can make the assumption that:
nmx = ndx / nLx i.e. death rate = deaths over person-years
This is an assumption because we are actually mixing up a rate derived from empirical data with values from a contrived life table – but let’s accept the assumption.
Rearranging this gives:
nLx = ndx / nmx
For the last interval, at older ages, we could write:
L85+ = d85+ / m85+
and since the d85+ group in this case are the same as the l85+ group:
L85+ = l85+ / m85+
This allows us to insert a final nLx value and thus "close" the table and proceed to the Tx column.
Apply a trend
A third method, rather more refined, is to assume a rising trend in nqx values for the older ages and continue this trend by a mathematical model or graphical extrapolation to impute the deaths by each single year until no one remains at, say, age 120. From this trend estimate the person years. This is probably only really worthwhile if you are especially focusing on mortality at older ages and therefore want a high level of precision. However the assumption of extrapolated trend can still be subjective.