Despite enjoying smoking a pipe, my grandfather lived to be 101 years old. Perhaps for this reason the emphasis placed on the increase in the number of very old people as a factor shaping population health has never seemed particularly interesting to me, although I must own up to a passion for passive pipe smoking. Mellow Virginia, in particular. Counting the number of people who reach 100 years of age is more difficult than you might think. The older people get the more difficult it is to know how old they really are but the Office of National Statistics (ONS) estimate that in 2010 there were 12,640 centenarians in the UK in 2010. The number of centenarians is likely to rise in future and several newspapers (The Telegraph and The Guardian) reported last week that a third of people born today will live to be 100 years old.
The calculation of life expectancy is an alternative to age-standardisation as a way of summarising the overall level of population health. For example, over the last 20 years (1990 to 2010) life expectancy at birth for men has increased from 72.9 years to 78.5 years for men while for women the increase has been from 78.5 years to 82.4 years. Life expectancy is calculated from what is termed a ‘lifetable’. The lifetable is one of the oldest techniques in statistics having been first used in the 17th century for the purpose of detecting outbreaks of epidemics, mainly the plague. There are two types of lifetable, a cohort and a period lifetable, and both can be used to calculate life expectancy. In this article we will look briefly at the difference between a cohort and a period lifetable and assess how much confidence we should have in predictions of the future number of centenarians.
The data on which a lifetable is based consists of the number of deaths in a year and an estimate of the mid-year population, both by single year of age. The key quantity used in calculating life expectancy from either type of lifetable is the probability of dying between successive ages. Once we know this we can calculate the probability of a person of any age today being alive at any future age as the product of the probability of not dying over the series of intervening successive one year age intervals. To calculate life expectancy at any age we simply add up the total lifetime left in person-years that we expect for individuals of that age and then divide by the number of people of that age in the population. For example, if there were 100,000 people aged 50 this year and we calculate that they will live for a total of 3,500,000 person-years in the future, then life expectancy at age 50 would be 35 years (or 3,500,000/100,000).
Period and cohort lifetables make different assumptions about future death rates. The period lifetable is constructed using the mortality rate at each age in a single year. That is, the life expectancy of someone who is age 45 in 2012 is calculated using the death rates at ages of 46 years and over that were observed in 2012. The period lifetable therefore tells you what life expectancy would be if death rates remained unchanged. The cohort life table, in contrast, is constructed using a forecast of future death rates. That is, life expectancy for someone who is 45 years old in 2012 is calculated from forecasts of the death rate at age 46 in 2013, age 47 in 2014, age 48 in 2015, and so on. In a previous article we have seen that death rates have fallen notably over the last 40 years. Because the cohort lifetable takes into account future changes in death rates, life expectancy calculated from a cohort life table is considered more accurate than that calculated from a period lifetable. It will also be higher if it is assumed that death rates will continue to fall in the future.
The report by ONS on which last week’s stories were based used results from cohort life tables. Because how death rates will change in the future is uncertain, the report produced three different estimates of life expectancy. The 'high life expectancy' and 'prinicpal' forecasts both assumed that death rates would continue to fall but at different rates while the 'low life expectancy' forecast assumed that the fall in death rates would slow down and then stabilise sometime around 2035.
The figure above shows the number of people who are forecast to reach 100 years of age by their age in 2012. The ‘principal’ forecast predicts that slightly more than 150,000 women who are born this year (or around 1/3 of births) will reach 100 years of age. On the other hand, the 'low life expectancy' forecast predicts that only around 35,000 women born this year will reach 100 years of age while the 'high life expectancy' forecast predicts that nearly 280,000 women born this year will still be alive in 100 years’ time.
In short, the calculation of life expectancy is very dependent on the accuracy of the assumptions concerning how death rates will change in the future. Predictions are hard to make, especially when they are about the future.