## The accumulation of small changes

It is often easy to be fooled into thinking that a small change is of little importance.  Small changes can persist over time, and sometimes it is only in retrospect that one realises just how big the accumulated change is.  For example, in a post last year we looked at the radical change in survival to age 90 in recent memory.

The subject has come up again with the media coverage surrounding a recent press release about centenarians from the Department of Work and Pensions (DWP) in the UK. The press release does not contain a new projection of mortality rates, but it does illustrate a neat point about projections over long periods of time.  The DWP results contain a table stating that a newly born female in the United Kingdom has a probability of 33.0% of becoming a centenarian, i.e. of living to age 100 or more.  On the face of it this seems hard to credit, particularly when the same table gives the probability of a 97-year-old woman surviving to age 100 as 35.5%.  How can it possibly be that a baby can have almost the same probability of surviving a hundred years as an elderly woman has of surviving three years?

An initial reaction might be that there is something wrong with the survival calculation from birth — if we use the 2007-09 interim life table for the UK population, the probability of surviving from age zero to age 100 is just 2.2%, which is nowhere near 33.0%.  Leaving aside some kind of calculation error, the next question is what sort of mortality-improvement rate could produce such a huge change, and is this where a mistake might lie?

We can take the mortality rates in the 2007-09 interim life table and apply a constant annual rate of improvement, then back-solve to find the rate of improvement which can raise the survival probability from 2.2% to 33.0%.  As it happens, the answer is the surprisingly modest rate of 1.3% per annum.  The steady accumulation of small improvements leads to a radical change in survival probability.

One way of reconciling this with our intuition is to split the interval of a hundred years into two equal parts.  Using the interim life tables without any improvements, a newly born female baby in the UK has a 97.0% chance of surviving to age 50.  We are intuitively comfortable with the idea of low mortality in infancy and young adulthood, since this is what we know and experience now, so this figure of 97.0% does not violate any expectations.

The next step is to consider the following fifty years between ages 50 and 100.  By this time society will hopefully have seen many decades of improvements in standards of living and medical technology. Intuitively we understand that mortality rates could have fallen substantially, especially over the time it would take our new-born infant to reach the age of 96 (say).  It does not stretch credulity that, in nearly a century's time, the mortality rate of a 96-year-old is just one quarter of what it is at the time of writing.  This is exactly what results when the 2007-09 interim life table is projected with constant mortality improvements of 1.3% per annum.

The DWP results are based on ONS projections, which are themselves not particularly revolutionary or challenging.  However, the steady accumulation of a small change each year leads to a revolution in human survival rates, and the DWP results show what such a demographic revolution could look like.

Assume we have a random variable, $$X$$, with expected value ... Read more