The force of mortality

How long are going to live? showed how the chances of dying each year depend on how old you are and what your health behaviours have been. Here we show how these annual 'hazards', survival curves and life expectancies can all be obtained from data on what proportion of people of each age die each year in the UK.

The animation above shows the 'hazard': this is known as a conditional probability, since it is the probability of dying each year, conditional on surviving till the start of that year. It shows a typical shape known as a 'bathtub': a fairly high early risk of death in the first year, then a period of very low risk, and then a long but steady increase in risk into old age. We can obtain an estimate of the current hazards in the country by seeing what percentage of, say, 2 year-old female children die each year. In 2006 this was 0.0186% (around 1 in 5400), and this percentage risk is shown in the second column of the Table below (rounded to 0.019%). This also shows that 0.455% (around 1 in 220) female babies die in their first year, and 25.4% (around 1 in 4) of 96-year old women die before their 97th birthday.

The Table forms part of what is known as a life table, which was first developed in the 17th century and forms the basis for life insurance, the study of demographics and medical survival analysis: all are based on the analysis of the hazard curve, or force of mortality.

We can get from hazard to survival curves by considering what we would expect to happen to 100,000 females born in 2006. We would expect 455 (0.455% of 100,000) to die in their first year of life, leaving 100,000 - 455 = 99,545: the third column of the Table shows the remaining number, and the final column shows the number dying each year. Of these 99,545, we expect 0.038% (39) to die before their second birthday, leaving 99,506, and so on. The 100,000 steadily decrease, until there are 2,833 expected to reach their 99th birthday, of whom 870 are expected to die before 100, leaving 2,833 - 870 = 1,967 (2%) reaching 100. The third column is therefore the survival curve showing the number out of 100,000 reaching each birthday, which is shown in the animation above (converted to a %). To get the survival curve starting a specific age, say for people who have already reached 54, then we just need look at the part of the table starting at 54, and rescale the third column so that it starts at 100,000.

Of course this all assumes the hazards do not change in the future, which of course is very unlikely to be the case, and so the survival curves should really be adjusted for whatever benefits to health can be expected in future.

Life expectancy

The final column of the Table shows the number of people expected to die at each age, and we can use this to get the average length of life, or life expectancy.

Start by assuming (unrealistically) that everyone dies on their birthday. Then out of 100,000 females born, 455 will live 0 years, 39 will live 1 year, 19 will live 2 years, etc, and so the average length of life is
$$ \frac{ (455 \times 0) + (39 \times 1) + (19 \times 2) + ...}{100,000} = 80.8.$$
Now it would be more realistic to assume that on average people die half-way between their birthdays, and so we should add 6 months onto this figure to get 81.3 years. This is only approximate and more accurate methods can be used.