Appendix 1

Stages in the construction of a relational life table by the Brass method

1. Assemble your observed life table measures. You need at least two measures to allow a fit in this system, and they should be well spaced by age – for instance one or more in the child ages and one or more in adult ages. The more estimates you have the better. Estimates should all be of the type "proportion dead by age x", which is the same as “under-x mortality”. A single estimate alone cannot be fitted by this technique but could simply be matched with an appropriate empirical table. A set of estimates that are all child or all adult estimates can be fitted but this is unlikely to be representative.*
2. Choose an appropriate standard life table. This may be a table that is accepted as representative for the country but it must be considered reliable – having been constructed from high quality data. If this is not available the standard should be chosen from one of the two Empirical Model Life Table collections. Take one or two child-level estimates and one or two adult estimates and find the best match from the patterns available – resulting in a table that seems appropriate and of a similar level of mortality. The parameters to match may be in the form of _nm_x, _nq_x or l_x values, depending on what is available to you.

Note that this selection stage does not need to be very precise – we merely need a table that fits the observed data approximately. The selected standard does NOT need to be from the same part of the world as your data, although if there are two appropriate standards and one is a geographic match also you might accept this table. If the estimates are “all over the place” i.e. wildly diverging and very difficult to match with any known pattern then the recommendation is to stick to one of the general or default patterns – that is the West pattern in the Coale-Demeny system or the General Pattern in the New UN system.

3. Enter the observed l_x-type estimates into a spreadsheet. Take the age-equivalent l_x values from the standard and enter them alongside. Perform a logit transformation for each series of values.
4. Plot the series graphically, with the standard logit values on the x axis. Produce a “best-fit” line for the points. This can usually be done automatically on a spreadsheet graph – if so you want a display of the intercept and slope. Alternatively you can find a best fit line using linear regression and produce predicted values which can be plotted on the graph.
5. Observe the series and fitted line – are there any points so seriously out of line that they may need to be ignored? Is a tolerably straight line apparent? Do the points drift away from linearity at the extremes?
6. If the points follow the best-fit line quite well (some minor scatter is inevitable) then, congratulations, you have your fitted life table! Enter all the l_x values from the standard onto the spreadsheet, produce the logits, then produce a new set of logits by applying the intercept (alpha) and slope (beta) parameters:-
$$Newfittedlogitset=standardlogitset*beta+alpha$$
7. Transform the fitted logits back to l_x values using the anti-logit formula. You have fitted a relational model! Other columns of the life table can be constructed as necessary.
8. Looking back at point 5, what if there are notable deviations from linearity? These deviations tend to fall into different groupings:
1. The points do not follow the line at all well and are widely scattered – the observed data is probably wildly inaccurate – just do what you can but do not expect too much and report the fit as unreliable due to data problems!
2. The points seem to form a coherent series but drift systematically away from the line at the extremes such that a curved line would be more suitable – usually a sign of a poorly selected standard – try a standard with a different pattern.
3. The fit to the line is quite good except for one or two very deviant points – a very common situation. The relational theory is that points deviant from the line are erroneous and would be better replaced with a point on the line. But if any points are deviant then ignoring them would re-parameterize the line. One has to make a value judgement about whether any points are sufficiently deviant as to discount them. This can be rather subjective but is helped if there is some other information that can be used. A common situation is that deviant points in infancy are a known result of underestimation of infant deaths – if there is supporting evidence that this is occurring then ignoring an infant point may be justified. Similarly for old-age points where there is a common problem with age exaggeration. For other areas of the graph, if surrounding points are well fitted but one is not it may be clear that this is erroneous.
9. If 1 or more points need to be discounted then re-fit the line without them and use the new alpha and beta to produce the new life table, as in 6 & 7 above. But do not discard points lightly.

* Note: It is quite common that only two points are available. As long as these are well-spaced e.g. a child point and an adult point, the fitting can still proceed and simply joining the two points will be sufficient. If the points are not well-spaced, or if there is only a single point to work with, then relational fitting is not sensible and the best that can be done is to match as closely as possible to a “default” empirical model – such as one from the West or General patterns.

See also http://demographicestimation.iussp.org/content/introduction-model-life-tables