Fallacy of composition and life expectancy

1/08/2015 10:19:00 AM
Bill Gardner has a nice post on what the International Journal of Epidemiology calls "lagged selection bias" in how we analyze life expectancy, where we incorrectly take the difference in life expectancy for a certain sub-population and claim that life expectancy has "increased" or "decreased" for that group. This procedure is not valid. I learned about this concept in the context of Labor Economics, where the more common term is "fallacy of composition." These are the same concepts.

Gardner's post mentioned studies of female mortality, which had reported that women's life expectancy was lower in some US counties than 20 years earlier, and concludes that life expectancy has fallen for women in these counties. Because of the fallacy of composition, this conclusion does not logically follow. To drive that point home, here's a numerical example. Suppose there are two counties: Healthypeopleshire and Unhealthypeopleshire. We perform surveys over two periods to calculate life expectancy in each county each period. In the first survey, there is one resident of Healthypeopleshire with a life expectancy of 90 years, and two residents of Unhealthypeopleshire with life expectancies of 70 and 56 years respectively.
Residents of each county, with their life expectancies indicated. Our survey, however, we see only the average life expectancy for each county which is 90 for Healthypeopleshire and 63 for Unhealthypeopleshire.
After a vigorous public health campaign, life expectancy rises dramatically for all individuals in period 2:
Ann and Brad both saw their life expectancies rise by 5 years, while Chris's rose by 4 years. Nevertheless, life expectancy fell in both Healthypeopleshire and Unhealthypeopleshire.
Incidentally, one effect of our public health campaign is that Brad left Unhealthypeopleshire to go live the ways of the healthy people in Healthypeopleshire. Because Brad has a lower expectancy than the other residents of Healthypeopleshire, but a higher expectancy of the other residents of Unhealthypeopleshire, our survey actually shows that life expectancy dropped in both counties, by 5 and 3 years respectively, even though it increased dramatically overall!