Saturday, February 7, 2009

The Scales of Justice

Dear Dr. Math,
I don't mean to be crass, but what are the chances that Ruth Bader Ginsburg is going to die during Barack Obama's presidency? I read that the average lifespan of an American is 78 years, and she's only 75 but now she has cancer. Also, some of the other judges are old, too. How many appointments is he probably going to have to make in the next 4 years?

Dear ScotusLover,

With the recent news about Justice Ginsburg being diagnosed with pancreatic cancer, this is a topic on the minds of a lot of people. Of course, we're all hoping for the best for her, but the issue of Supreme Court appointments has ramifications far beyond our wishes for her health. When one vote can make the difference in who can stick what in whose what or whether little old ladies deserve equal pay for equal work or even who the president was for the last 8 years, the news that one of the justices has a potentially deadly illness gets everyone's attention.

First off, we should dispense with the whole "average lifespan" argument, which is largely irrelevant here. The statistics you've probably seen quoted are the expected (in the sense of average, or mean) lifespan of someone born today. It includes the effect of a fair number of people dying young. As someone gets older, his/her expected lifespan increases, because we have to incorporate into the calculation the fact that he/she is still alive. The relevant numbers for these things can be found in what are called life tables, which are tools that actuaries use to figure out what your grandmother's life insurance premiums should be, etc. However, these are still just averages over large swaths of the population and they don't take into consideration any more particular information we might have about someone.

So, in Justice Ginsburg's case, the more relevant number is the mortality rate for her particular form of pancreatic cancer, given the stage at which it was diagnosed. And unfortunately, the numbers are not particularly good. One number that the news media seems to have latched onto is that only 5% of people diagnosed with pancreatic cancer survive for 5 years after the diagnosis. Again, I'm not a doctor, but from Googling around I've discovered that a big part of the reason pancreatic cancer is so deadly is that it frequently goes undetected until it's already in a fairly advanced stage. So, the fact that her cancer was caught relatively early should work to her favor. I think the right number to be considering here is the survival rate for pancreatic cancer in its earliest stage, which is something like 35% after 5 years.

It's important to remember, also, that these statistics just reflect the potential of dying from the disease (assuming they're measuring relative survival rates). That is, the number 35% is supposed to represent the percentage of people diagnosed with cancer who are still alive after 5 years given that they would have been alive anyway. Of course, it's a difficult thing to measure, but it means we should include the fact that Ruth is 75 years old and that just being 75 itself has a 5 year survival rate of 83% (for American females). To determine that overall survival rate, I took the number of women alive at age 80 and divided by the number alive at age 75.

In the final tally, then, my best guess for Justice Ginsburg's chances of living out the next 5 years would be the chance that any 75-year-old woman would live 5 more years times the relative survival rate for someone with an early diagnosis of pancreatic cancer, that is, (.83)*(.35) = .29, or 29%. There's no accounting for determination, the will to live, or hatred of Antonin Scalia, however. The lesson to draw from all of this is that the more information you have about a particular person, the more precisely you can fine-tune the analysis of his/her situation, but also the less data you have to draw conclusions from. Really, what we'd like to know is the 5 year survival rate for being Ruth Bader Ginsburg, but there's only been 1 known case in history.

To answer your other question about likely Supreme Court appointments, I could go through each remaining justice and compile mortality tables for each based on his particular lifestyle and risk factors, but maybe I should leave that as an exercise for the reader. Personally, I find all this kind of creepy. I will say that Justice Stevens is 88, and the 5 year survival rate for an 88-year-old American male is 36%, or about the same as Justice Ginsburg's cancer diagnosis. Of course, there can be many reasons for a justice to leave the court besides death, as well. Of the 101 Supreme Court justices who have left the court, only 50 have done so by dying. The remaining 51 resigned or retired, presumably before they died, unless they were pulling a Jeremy Bentham. It appears that having an extremely silly name is not a risk factor.

A rough estimate for the average rate of appointments can be gotten by dividing the total number of appointments, 110, by the age of the Supreme Court, which is 220 years young (happy birthday, Supreme Court!). So that's a rate of about half a justice per year, meaning Barack Obama will have to appoint an average of 2 justices for each term he serves as president.


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