“Either candidate can be expected to serve two full terms, without age or health being an issue,” said John M. Bragg, the firm chairman.

John Aravosis points out that Bragg's numbers don't add up:

The actuarial (sic) says McCain has 8.4 "healthy" years left. He mistakenly drew the conclusion that McCain would therefore be healthy through two terms in office, thinking that eight years from now is the end of McCain's term. That's incorrect. The end of the term is in 8 years 5.5 months from now. Statistically, that means McCain won't make it through two terms as a "healthy" president. By healthy, he means “the person does not require the care provided by an assisted living facility and is free of Alzheimer’s disease.”

By my back-of-the-envelope calculations, that means McCain wouldn't even be able to complete two full terms in office. His time runs out on December 30th or so of 2016, a good three weeks before the new president would be sworn in. So that means, statistically speaking, before the end of two terms, we should expect President McCain to have to be moved to a retirement home because he'll no longer be able to care for himself, and we should expect that he'll already be suffering from Alzheimer's.

Aravosis is correct in noting that 8.4 years from now does not give the new President (whoever it is) two full terms. His explanation of what that means is a little sloppy, but the end result is most likely correct. The issue is what it means that "we should expect" McCain (not) to be healthy and alive by the end of the two terms. I assume that Aravosis means "it is more likely than not", but that doesn't follow so simply from the published statistic.

Mathematical expectation is the theoretical mean, not median. What it means is the following: if we had a thousand men of the same age and other relevant characteristics as John McCain, we follow them all until they die and note how long each lives, and then we sum up all of their times from now until death, and divide the result by 1000 (the number of them), we will get something very close to 8 years, 146 days.

It does not mean that 500 of them will live longer than 8.4 years and the other 500 shorter. That would follow only if the remaining lifetime were distributed symmetrically, for example, if it followed the bell-shaped normal distribution. But that is not the case: all remaining lifetimes must be greater than zero, but there is no reason some of them couldn't be 30 years or more. The expectation (8.4 years) is obviously not in the middle of the range, so the distribution is skewed (asymmetric).

Now it turns out that, in most distributions of this kind (with a "tail" on the right), the median is less than the mean. If the median healthy survival is less than 8.4 years, it would mean that fewer than 500 of the initial 1000 "McCain clones" would have more than 8.4 years of healthy life - and, obviously, even fewer would have 8.46 or so years needed to complete two terms in office. So Aravosis made a correct statement, after all.

If Aravosis is correct, Bragg must be incorrect. It is not possible to say, based on Bragg's own numbers, that McCain "can be expected to serve two full terms, without age or health being an issue". Because the median is almost certainly lower than the mean, it would not be true even if the expected healthy life were somewhat longer, say 8.5 years, extending beyond two terms. It would still imply that McCain is more likely than not to die or become seriously impaired before the end of the second term.

But this error in arithmetic is not the most bothersome part of Bragg's statement. After all, if all the critique amounted to were nitpicking about chances being 49% rather than 51%, it would be rather academic. I see a far more serious problem in Bragg's choice of risk measure.

Suppose for a moment that Bragg's arithmetic was right, and that McCain could be expected, albeit barely so, to be alive and healthy on January 20, 2017. Suppose his chances are 55%. That's a generous "improvement" of Bragg's estimate, but it still means a 45% chance he would be dead or incapacitated. Would most people be comfortable with a president facing those odds? Would most people, upon hearing that number, characterize a potential McCain presidency as "without age or health being an issue"? I doubt that. From the risk management point of view, Bragg has cited an almost useless statistic.

John M. Bragg is not just any actuary. He was President of the Society of Actuaries. That, however, was a long time ago, when the US President was named Gerald Ford.

## 2 comments:

I doubt if there are any actuarial tables that take McCain's experiences as a POW into account. I would think it would have a big negative effect on his life expectancy. But I think Obama is more likely to die in office, due to assassination.

There is certainly more actuarial data on the longevity effects of POW and related experiences than there can be on assassination risk.

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