Clone of The force of mortality

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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.

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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 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 analyses: all are based on the analysis of the hazard curve, or force of mortality.

Table showing how survival curves are got from knowing annual hazards: women in the UK, 2004-2006
Age at start of year Hazard: the % of people who are alive at the start of the year, but who die during the year Survival: the number of people alive at the start of the year, out of 100,000 born Number of people dying during the year
0 0.455% 100,000 455
1 0.038% 99545 38
2 0.019% 99507 19
3 0.015% 99488 15
4 0.010% 99473 10
... .... ... ...
95 23.5% 9673 2273
96 25.4% 7400 1880
97 27.5% 5520 1518
98 29.2% 4002 1169
99 30.7% 2833 870

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. Moving on to the next line, of the 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 at a specific age, say for people who have already reached 54, we 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 that the hazards do not change in the future, which is very unlikely to be the case. Survival curves should be adjusted to take account of (hopefully) improving population health as years go by.

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. But this is only approximate, and more accurate methods can be used.

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