## Introduction

The mid-year population estimates published by the Office for National Statistics (ONS) include estimates by single year of age to 89, with a final category for ages 90 and over.

Until 2007, population estimates for single years of age beyond 90 were calculated for England and Wales by ONS (and previously Government Actuary’s Department) for use in compiling national interim life tables and in producing the national population projections.

These estimates were made available for research purposes but were not officially published.

Interest in population estimates at the oldest ages by single year of age has increased as life expectancy increases and the number of centenarians grows.

In recognition of this, ONS began to publish these estimates in 2007 as experimental statistics.

These estimates are now published as National Statistics (since 2011).

The statistics provide estimates of the population aged 90 and over by single year of age within the UK and within England and Wales as a whole.

They are constructed using the Kannisto-Thatcher (KT) model of population at advanced ages. They are consistent with the mid-year population estimates and death registrations statistics.

The estimates are produced by the Population Statistics Division within ONS.

The estimates for Scotland and Northern Ireland, which feed into the UK estimates, are produced and published by National Records of Scotland (NRS) and the Northern Ireland Statistics and Research Agency (NISRA).

The estimates are published annually; the tables provide mid-year population estimates by sex and single year of age for 90 to 104 and for the 105 and over group.

Quality reports for the mid-year population estimates and estimates of the very old are available on the ONS website.

## Methodology

To produce single year of age estimates of the population aged 90 and over, ONS, NRS and NISRA use the KT method.

These estimates are produced using a modified form of the survival ratios proposed by Kannisto-Thatcher.

The population at a given age is estimated by looking at the ratio of the number of survivors of a cohort still alive to the number of that cohort who died in the last few years.

By making an assumption about the highest age at which everyone in a given cohort will have died, it is then possible to produce an algorithm using these survival ratios that will give estimates of the numbers of people alive at earlier ages for each cohort.

That is, the KT method uses “age-at-death” data to build up distribution profiles of the numbers of elderly people in previous years.

For example, if someone dies in 2012 aged 105, then this means that they were alive and aged 104 in 2011 and 103 in 2010, etc.

By collating “age-at-death” data for a series of years, it becomes possible to make an estimate of the number of people of a given age alive in any particular year and so create age distribution profiles, assuming that migration at these oldest ages is minimal.

To make estimates for the current year, it is not possible to use death data, as we are interested in the population who are currently or very recently alive.

So the KT method uses an average of the last 5 years of age-at-death information to produce an estimate of the number of survivors for the most current year.

The estimates are then constrained to the official mid-year population estimate of the total number in the population aged 90 and over for that year.

## Notation

The method as used here can be expressed as follows.

For all years prior to the year that you are currently calculating (where P is population, D is Deaths age x at the beginning of the year, x is age and t is year):

For the year under consideration (where c is the correction factor, S is the survival ratio and T is the “current” year):

Where S is calculated as:

In addition, it is assumed that no one survives beyond age 120, and c is calculated such that:

One consequence of this method is that each year the estimates for earlier years become more accurate as more death data becomes available to inform the age profiles.

Once all the members of a given cohort have died, the dates of birth and death recorded at their death registration give enough information to reconstruct the numbers who were alive at each date in the past, at least for ages where international migration can be ignored.

By this method, known as the method of “extinct generations” or “extinct cohorts”, improved estimates of past population numbers can be made retrospectively.

For more information on the Kannisto-Thatcher methodology, see Thatcher R et al. (2002) “The survivor ratio method for estimating numbers at high ages”.