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"This is a personal story of statistics..."
THE MEDIAN
ISN'T THE MESSAGE
by Stephen Jay Gould
Born in 1941, Stephen Jay Gould was a geologist, zoologist, paleontologist and evolutionary biologist at Harvard. He was also one of the most noted, prolific and best-selling scientific writers of our day. He was diagnosed in 1982 with abdominal mesothelioma, a rare and very deadly form of cancer associated with exposure to asbestos. This is his story. It was first published in Discover magazine in June 1985 and was reprinted here at Phoenix5 with his kind permission. He beat the cancer for 20 years, finally passing on May 20, 2002, giving all of us a valuable lesson in beating the odds.
In 1982, I learned I was suffering from a rare and serious cancer. After surgery, I asked my doctor what the best technical literature on the cancer was. She told me, with a touch of diplomacy, that there was nothing really worth reading. I soon realized why she had offered that humane advice: my cancer is incurable, with a median mortality of eight months after discovery.
The problem may be briefly stated: What does "median mortality of eight months" signify in our vernacular? I suspect that most people, without training in statistics, would read such a statement as "I will probably be dead in eight months" the very conclusion that must be avoided, since it isn't so.
My life has recently intersected, in a most personal way, two of Mark
Twain's famous quips. One I shall defer to the end of this essay. The
other (sometimes attributed to Disraeli), identifies three species of
mendacity, each worse than the one before - lies, damned lies, and
statistics.
Consider the standard example of stretching the truth with numbers - a
case quite relevant to my story. Statistics recognizes different measures
of an "average," or central tendency. The mean is our usual
concept of an overall average - add up the items and divide them by the
number of sharers (100 candy bars collected for five kids next Halloween
will yield 20 for each in a just world). The median, a different
measure of central tendency, is the half-way point. If I line up five
kids by height, the median child is shorter than two and taller than the
other two (who might have trouble getting their mean share of the candy).
A politician in power might say with pride, "The mean income of our
citizens is $15,000 per year." The leader of the opposition might retort,
"But half our citizens make less than $10,000 per year." Both are right,
but neither cites a statistic with impassive objectivity. The first
invokes a mean, the second a median. (Means are higher than medians in
such cases because one millionaire may outweigh hundreds of poor people
in setting a mean; but he can balance only one mendicant in calculating a
median).
The larger issue that creates a common distrust or contempt for statistics
is more troubling. Many people make an unfortunate and invalid separation
between heart and mind, or feeling and intellect. In some contemporary
traditions, abetted by attitudes stereotypically centered on Southern
California, feelings are exalted as more "real" and the only proper basis
for action - if it feels good, do it - while intellect gets short shrift
as a hang-up of outmoded elitism. Statistics, in this absurd dichotomy,
often become the symbol of the enemy. As Hilaire Belloc wrote, "Statistics
are the triumph of the quantitative method, and the quantitative method is
the victory of sterility and death."
This is a personal story of statistics, properly interpreted, as
profoundly nurturant and life-giving. It declares holy war on the
downgrading of intellect by telling a small story about the utility of
dry, academic knowledge about science. Heart and head are focal points of
one body, one personality.
In July 1982, I learned that I was suffering from abdominal mesothelioma,
a rare and serious cancer usually associated with exposure to asbestos.
When I revived after surgery, I asked my first question of my doctor and
chemotherapist: "What is the best technical literature about
mesothelioma?" She replied, with a touch of diplomacy (the only departure
she has ever made from direct frankness), that the medical literature
contained nothing really worth reading.
Of course, trying to keep an intellectual away from literature works
about as well as recommending chastity to Homo sapiens, the
sexiest primate of all. As soon as I could walk, I made a beeline for
Harvard's Countway medical library and punched mesothelioma into the
computer's bibliographic search program. An hour later, surrounded by the
latest literature on abdominal mesothelioma, I realized with a gulp why
my doctor had offered that humane advice. The literature couldn't have
been more brutally clear: mesothelioma is incurable, with a median
mortality of only eight months after discovery. I sat stunned for about
fifteen minutes, then smiled and said to myself: so that's why they
didn't give me anything to read. Then my mind started to work again,
thank goodness.
If a little learning could ever be a dangerous thing, I had encountered a
classic example. Attitude clearly matters in fighting cancer. We don't
know why (from my old-style materialistic perspective, I suspect that
mental states feed back upon the immune system). But match people with the
same cancer for age, class, health, socioeconomic status, and, in general,
those with positive attitudes, with a strong will and purpose for living,
with commitment to struggle, with an active response to aiding their own
treatment and not just a passive acceptance of anything doctors say, tend
to live longer. A few months later I asked Sir Peter Medawar, my personal
scientific guru and a Nobelist in immunology, what the best prescription
for success against cancer might be. "A sanguine personality," he replied.
Fortunately (since one can't reconstruct oneself at short notice and for a
definite purpose), I am, if anything, even-tempered and confident in just
this manner.
Hence the dilemma for humane doctors: since attitude matters so
critically, should such a somber conclusion be advertised, especially
since few people have sufficient understanding of statistics to evaluate
what the statements really mean? From years of experience with the
small-scale evolution of Bahamian land snails treated quantitatively, I
have developed this technical knowledge - and I am convinced that it
played a major role in saving my life. Knowledge is indeed power, in
Bacon's proverb.
The problem may be briefly stated: What does "median mortality of eight
months" signify in our vernacular? I suspect that most people, without
training in statistics, would read such a statement as "I will probably
be dead in eight months" - the very conclusion that must be avoided,
since it isn't so, and since attitude matters so much.
I was not, of course, overjoyed, but I didn't read the statement in this
vernacular way either. My technical training enjoined a different
perspective on "eight months median mortality." The point is a subtle
one, but profound - for it embodies the distinctive way of thinking in my
own field of evolutionary biology and natural history.
We still carry the historical baggage of a Platonic heritage that seeks
sharp essences and definite boundaries. (Thus we hope to find an
unambiguous "beginning of life" or "definition of death," although nature
often comes to us as irreducible continua.) This Platonic heritage, with
its emphasis in clear distinctions and separated immutable entities, leads
us to view statistical measures of central tendency wrongly, indeed
opposite to the appropriate interpretation in our actual world of
variation, shadings, and continua. In short, we view means and medians as
the hard "realities," and the variation that permits their calculation as
a set of transient and imperfect measurements of this hidden essence. If
the median is the reality and variation around the median just a device
for its calculation, the "I will probably be dead in eight months" may
pass as a reasonable interpretation.
But all evolutionary biologists know that variation itself is nature's
only irreducible essence. Variation is the hard reality, not a set of
imperfect measures for a central tendency. Means and medians are the
abstractions. Therefore, I looked at the mesothelioma statistics quite
differently - and not only because I am an optimist who tends to see the
doughnut instead of the hole, but primarily because I know that variation
itself is the reality. I had to place myself amidst the variation.
When I learned about the eight-month median, my first intellectual
reaction was: fine, half the people will live longer; now what are my
chances of being in that half. I read for a furious and nervous hour and
concluded, with relief: damned good. I possessed every one of the
characteristics conferring a probability of longer life: I was young; my
disease had been recognized in a relatively early stage; I would receive
the nation's best medical treatment; I had the world to live for; I knew
how to read the data properly and not despair.
Another technical point then added even more solace. I immediately
recognized that the distribution of variation about the eight-month
median would almost surely be what statisticians call "right skewed." (In
a symmetrical distribution, the profile of variation to the left of the
central tendency is a mirror image of variation to the right. In skewed
distributions, variation to one side of the central tendency is more
stretched out - left skewed if extended to the left, right skewed if
stretched out to the right.) The distribution of variation had to be
right skewed, I reasoned. After all, the left of the distribution contains
an irrevocable lower boundary of zero (since mesothelioma can only be
identified at death or before). Thus, there isn't much room for the
distribution's lower (or left) half - it must be scrunched up between
zero and eight months. But the upper (or right) half can extend out for
years and years, even if nobody ultimately survives. The distribution
must be right skewed, and I needed to know how long the extended tail ran
- for I had already concluded that my favorable profile made me a good
candidate for that part of the curve.
The distribution was indeed, strongly right skewed, with a long tail
(however small) that extended for several years above the eight month
median. I saw no reason why I shouldn't be in that small tail, and I
breathed a very long sigh of relief. My technical knowledge had helped. I
had read the graph correctly. I had asked the right question and found
the answers. I had obtained, in all probability, the most precious of all
possible gifts in the circumstances - substantial time. I didn't have to
stop and immediately follow Isaiah's injunction to Hezekiah - set thine
house in order for thou shalt die, and not live. I would have time to
think, to plan, and to fight.
One final point about statistical distributions. They apply only to a
prescribed set of circumstances - in this case to survival with
mesothelioma under conventional modes of treatment. If circumstances
change, the distribution may alter. I was placed on an experimental
protocol of treatment and, if fortune holds, will be in the first cohort
of a new distribution with high median and a right tail extending to
death by natural causes at advanced old age.
It has become, in my view, a bit too trendy to regard the acceptance of
death as something tantamount to intrinsic dignity. Of course I agree
with the preacher of Ecclesiastes that there is a time to love and a time
to die - and when my skein runs out I hope to face the end calmly and in
my own way. For most situations, however, I prefer the more martial view
that death is the ultimate enemy - and I find nothing reproachable in
those who rage mightily against the dying of the light.
The swords of battle are numerous, and none more effective than humor. My
death was announced at a meeting of my colleagues in Scotland, and I
almost experienced the delicious pleasure of reading my obituary penned
by one of my best friends (the so-and-so got suspicious and checked; he
too is a statistician, and didn't expect to find me so far out on the
right tail). Still, the incident provided my first good laugh after the
diagnosis. Just think, I almost got to repeat Mark Twain's most famous
line of all: the reports of my death are greatly exaggerated.
Biographical and bibliographical information on Dr. Gould can be found at http://prelectur.stanford.edu/lecturers/gould/ and
http://www.annonline.com/interviews/961009/biography.html.
[Links worked when this was posted to P5.]
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