In mathematical terms:
Obviously, this method assumes that there is no migration after age x for the cohort in question. This assumption may be reasonable for advanced ages only, so the method should not be used to estimate population size at younger ages. By convention, we use the method of extinct generations to estimate population sizes for ages 80 and above only.
The extinct generations method can only be applied to cohorts which have reached age w (the highest age possible). This leaves a missing triangle in the lexis diagram. The intercensal survival method is used to fill in part of this missing triangle, but often census counts aggregate data at the oldest ages. The survival ratio method provides a means of calculating population estimates for cohorts which are two old to be observed in two consecutive censuses, but which are not yet extinct by the final year of data.
The basic assumption of the method is that survival ratios (the ratio of the population aged x to the number of deaths to the cohort in k previous years) remain constant over time*. In mathematical terms:
Then an estimate of the population age x on January 1 of year y+1 can be obtained as follows:
Deaths can then be cumulated back from as in the method of extinct generations to get an estimate for . The survival ratio method is then applied again to estimate as follows:
Working back iteratively in this way, the triangle left missing by the extinct cohorts method can be filled in.
A variation of the method is to calculate survival ratios by pooling data over the last few cohorts instead of basing estimates on the experience of a single cohort.
All survival ratio estimates on BMD are calculated by pooling the previous five cohorts and setting .
*This is a slight modification of the method described by A. R. Thatcher in "Overview of Methods for Estimating Population Numbers at High Ages from Data on Deaths". Thatcher defines a survival ratio as the number of survivors to age x in year y divided by the number of deaths to the cohort in k previous years.