In Which It Is Mathematically Proven That the Doctors Were Right After All

A Medical Quest

In Which It Is Mathematically Proven That the Doctors Were Right After All

A Medical Quest

In Which It Is Mathematically Proven That the Doctors Were Right After All
Health and medicine explained.
June 15 2004 1:41 PM

A Medical Quest


Illustration by Robert Neubecker

When I enter David Laibson's office in Harvard's economics department, he is on the phone saying things like, "This one is a .4 chance" and "She is no more than a .5 chance." He is going down a list of prospective graduate students, each of whom has a rating based on the chances of his or her accepting an offer from Harvard. This is why I came to see David: He really thinks this way. One of the little-known facts about economics is that it's actually a study of how people make decisions—the problem is, for most of the history of the dismal science, economists have based their suppositions on the blatantly absurd assumption that people will always act rationally and in their own best interest. David is a behavioral economist, which means that he tries to figure out how people really act. In his popular Harvard course "Psychology and Economics," he and his co-teacher, Andrei Shleifer, discuss things like lotteries, the Milgram obedience experiment, and how the media cooks the news (although they happen to be all wrong about journalism) and try to talk about the way people make decisions—and why their decisions are often wrong. I hope that because David has studied decision-making mistakes, he can help me avoid them.

Instead, David first does what anyone would do: He tries to find a way to avoid making a decision. He tells me about two NYU economists who have done a study showing that people don't really want to know what will happen to them, especially when it's not clear what to do with the knowledge. I tell him there are in fact two other studies, specifically concerning the BRCA mutations, which show that women do want to know whether they have the mutation. "They just say they want to know," David responds. So, I finally tell him I have a BRCA mutation, which elicits the predictable sympathetic look from David. (I am not making light; it was obvious to me his sadness and sympathy were sincere—it's just that I still haven't figured out what I, a healthy woman with more knowledge and options than most women in a similar situation, should do with that kind of sympathy.) Then he suggests we make a spreadsheet.


In the left column of the spreadsheet David puts ages, year by year, starting at 37 (which happens to be his age as well as mine). Across the top, he places the options: "Oophorectomy," "Mastectomy," "Oophorectomy and Mastectomy" and "Do Nothing." Now we devise the formulas for figuring out the value of life under each possible decision for each possible year. "Let's normalize not being alive as having a utility of 0," suggests David. "Further, let's normalize a year of healthy life to have a utility of 100. Should we assume that life without breasts is equally good?" Sure, I say, provided the surgery goes well—as we know, people adjust very quickly to radical change. What would be the value of life with cancer? That's harder to calculate, since the value gets higher as probability gets lower: The longer a woman lives after the initial diagnosis, the more likely it is that her life will once again approach normalcy. We consider equating a diagnosis of cancer with death, which has a value of 0, but then David decides to calculate the values more precisely—or as precisely as algebra will allow what he calls the "pseudoscience" of behavioral economics, to become.

He switches from his computer screen to the huge dry-erase board opposite his desk and gives up on the spreadsheet. Here is roughly what we come up with (where we is the sum of David's genius and my clumsily conveyed supply of information). If you cannot stomach equations, now is a good time to skip to the conclusion of this piece.

V(G) = 100 + B [(.98)V(G) + (.02)V(illness)]

Translated, that says:


The value of a year of life in what we consider a "good state"—apparent health with none of the possible disadvantages of surgery—is equal to 100 (this is the utility we have assigned to a year of normal healthy life). The value of an entire life that begins in the good state is 100 plus "the discount factor" B [beta] multiplied by the value of the state that starts next year, if I survive. B reflects the normal mortality risk for young people, which is about ¼ of 1 percent. The value of life in the next period is in brackets. The first term is the 98 percent likelihood of staying in the "good state," and the second term is the 2 percent likelihood of a diagnosis of cancer.

Now we have to decide what the value of life with cancer would be. To prevent, eh, reader attrition, I'll skip the calculation that leads us to the utility of 70 for life with cancer (compared to 100 for health and 0 for death). Then

V(illness) = 70 + B[(.8)V(illness)+(.2)X0] = (skipping the algebra here) = 347

What that means is that while life—actual life—with cancer might have a utility flow of 70 utils per year, which is not all that far from normal healthy life, the likelihood of death makes the value of life with cancer just 347. Plugging this value into the first equation, we find out that life in the "good state" has a value of 4752, even allowing for the 2 percent annual risk of breast cancer.


I already know that if I got a mastectomy, the utility of one year of life would be essentially the same because I would adjust. The value of life, however, would actually increase because the risk of breast cancer would be cut by 90 percent. Still, after the age of 40 the risk of ovarian cancer would kick in, also to the tune of about 2 percent a year, bringing the value of my now breastless life right back down to 1500. What if I got an oophorectomy? Sure, that would cut my ovarian cancer risk by 95 percent. But it might make me instantly old, susceptible to heart disease, high blood pressure, osteoporosis, and so forth. I suggest David look at my collection of all the age-specific risk statistics of all these diseases. David's solution is far more elegant: He just changes our "discount factor" (B') to reflect the mortality risk of an older person—someone over 65. The assumption here is that the mortality risk is a good enough expression of all the disadvantages of life after menopause. That mortality risk is about 1.5 percent a year, rather than a quarter of a percent, as it is for young people. The utility of life in a state we termed "Good/old" turned out to be 95 (as compared to 100 for good/young). And the value?

V(G/old) = 95+B'[V(G/old)] = 9500.

Huh? Does that mean it is better to be old and cancer-risk-free than young and at risk?

"Hey," says David. "Being old isn't so bad. I can't do many of the things I could do when I was 20, and I don't miss them."

I think about this. The truth is, physically, I don't feel that different than when I was 20. True, to achieve this state of well-being I have to drink a lot less, smoke not at all, and work out a lot, but these don't count as costs. (On the other hand, when I was 20, I could actually follow this kind of math with pleasure.)

"You know what?" continues David. "Having a 1 percent chance of dying from being old just doesn't compare to having a 2 percent chance of getting cancer." Sure it does. It's only half as bad.

"It seems the doctors are right," concludes David: Surgery is the solution. We stare at each other for a minute; this is an outcome neither of us expected. I, for one, had been almost sure we'd prove that the absurd-sounding recommendation of a preventive oophorectomy was wrong.