After decades in which Cancer screening was promoted as an
unmitigated good, as the best — perhaps only — way for people to protect
themselves from the ravages of a frightening disease, a pronounced
shift is under way.
Now expert groups are proposing less
screening for prostate, breast, and cervical Cancer and have emphasized
that screening comes with harms as well as benefits. Two years ago, the
influential United States Preventive Services Task Force, which
evaluates evidence and publishes screening guidelines, said that women
in their 40s do not appear to benefit from mammograms and that women
ages 50 to 74 should consider having them every two years instead of
every year.
This year the group said the widely used
P.S.A. screening test for prostate Cancer does not save lives and causes
enormous harm. It also concluded that most women should have Pap tests
for cervical Cancer every three years instead of every year.
What changed?
The
answer, for the most part, is that more information became available.
New clinical trials were completed, as were analyses of other sorts of
medical data. Researchers studied the risks and costs of screening more
rigorously than ever before.
Two recent clinical trials of
prostate Cancer screening cast doubt on whether many lives — or any —
are saved. And it said that screening often leads to what can be
disabling treatments for men whose Cancer otherwise would never have
harmed them.
A new analysis of mammography concluded that
while mammograms find Cancer in 138,000 women each year, as many as
120,000 to 134,000 of those women either have Cancers that are already
lethal or have Cancers that grow so slowly they do not need to be
treated.
But these concepts are difficult for many to
swallow. Specialists like urologists, radiologists and oncologists, who
see patients who are sick and dying from Cancer, often resist the idea
of doing less screening. General practitioners, who may agree with the
new guidelines, worry about getting involved in long conversations with
patients trying to explain why they might reconsider having a mammogram
every year or a P.S.A. test at all.
When the initial
recommendations came out, there was a burst of comments from women who
felt their lives were saved by early Cancer detection and many were
younger than the new age recommendation. I feel the same way about the
P.S.A. results since that test led to an early discovery of my Cancer,
which I then had treated aggressively. Of course, it is this very
aggressive response to these tests that the doctors are warning against.
Some
of the discussion is around economic issues and cost management —
remember the death panels? But much of the discussion is neither due to a
cost savings criteria nor a desire of doctors to increase their income
from treatments that may not be needed or just to protect themselves
from malpractice law suits. The problems of screening and what you do
with test results are very difficult to resolve.
The
current issue of The New England Journal of Medicine, for example, has
an article by two prostate Cancer specialists who note that one recent
study concludes that $5.2 million must be spent on screening to prevent
one prostate Cancer death. And, add the authors, that figure is not
inclusive. The true cost is undoubtedly even greater. Of course, if you
were the one Cancer death that was prevented by the spending, I think
you would probably agree the money was well worth it.
One
key factor is that prostate Cancer, at least typically, is a very slow
growing Cancer that occurs in older men. Therefore the probabilities are
pretty good that men who have discovered they have this Cancer will
likely die of some other ailment before the Cancer becomes terminal.
Also, the risk of surgery and other aggressive treatments for people of
advanced age may be greater than the risk from the Cancer.
In
fact, even the biopsy procedure usually performed after a positive
P.S.A. test is itself somewhat dangerous as well as an uncomfortable
thing that many doctors would like to spare their patients from going
through. In some cases there were several negative biopsies performed
which then can complicate surgery if it is ultimately required due to
the scaring from the biopsy tests. So it is complicated to decide what
action to take. The key is that you don’t know the outcome until after
the procedure is performed, and then it is too late. Once you’ve done
it, it is done and can’t be changed.
So good analysis
requires we use statistical inferences. But be careful applying
statistics to individual cases. Probability is very useful if you are
trying to determine the best course of action when there are unknowns.
Given 1000 men with a positive P.S.A. test, what is the best statistical
course of action. What are the odds.
But for an
individual who knows he or she has Cancer, such as myself, the odds of
Cancer in the general population doesn't mean much. The odds are 100%
that I have Cancer.
So what does it mean to have a
positive P.S.A. result and you don't have any other information? Does
that guarantee you have Cancer? What are the odds? Should you have a
biopsy performed? Again, what are the odds?
Now I was
trained as a mathematician, but not a statistician. However, I have
performed the role of statistical expert several times during my career
and my primary role the last five years at IBM was to perform
statistical analysis on system failures. So I know my way around “means”
and “medians.” In graduate school I specialized in Calculus, Analysis,
Differential Equations, and Abstract Algebra. The only statistics course
I took was in my undergraduate curriculum.
But I’ve
studied it a lot since then and taken courses both through IBM and
through the Society for Quality Assurance and the IEEE (Institute of
Electrical and Electronics Engineers). I’m currently taking a class
through Stanford University that has probability at its core. The
thing I’ve always noted about statistics is that the results sometimes,
don’t match common sense. It doesn’t make the statistical results
wrong; it just makes it hard to wrap our brains around the answer.
Let
me explain how you determine the ability of a given test to detect a
given condition. We are most familiar with using statistics to measure a
cause to an effect. Weathermen, for example, study various weather
parameters and predict the effect (weather forecast) based on the
causes. Mathematically we state that this way. P (effect | cause). That
is read as the probability (P) of the cause resulting in the effect.
(Note you read the notation a little backward.) But, in the case of the
results of a test being known, and the test is the measure of the
existence of some physical condition, then you go in the other
direction, P (cause | effect). What is the probability of a given
“cause” (or disease) if you have a certain “effect” such as a symptom or
a lab test result.
These are called “conditional
probabilities.” The conditional probability P (effect | cause)
quantifies the relationship in the causal direction, whereas P (cause |
effect) describes the diagnostic direction. In a task such as medical
diagnosis, we often have conditional probabilities on causal
relationships (that is, the doctor knows P (symptoms | disease) ) and
want to derive a diagnosis, P (disease | symptoms).
These
kinds of situations can be handled with a statistical method called
Bayesian analysis, which was developed by Rev. Thomas Bayes in the
eighteenth century. He was a mathematician and a Presbyterian minister.
(Ask your pastor if he has developed any mathematical formulas. You never
know.)
Let me just give you an example and teach you some
basic concepts of probability. Let’s assume that the probability of a
person having a specific Cancer is 0.01 or 1%. Now there are only two
possibilities, either a person has Cancer or they do not. The total of
all the possibilities in a probability must equal 1.0 or 100%. So, if
the probability of having this Cancer is 1%, then the probability of not
having it is 99%. It is a relatively rare disease.
Mathematically we will write this as:
P (C) = 0.01
P (¬ C) = 0.99 where “¬” is the symbol for “not.”
Now
let’s examine the conditional probabilities of the test itself. Assume
that, if you have Cancer, then you will get a positive test 90% of the
time. Then it stands to reason that, if you have Cancer, you will get a
negative result 10% of the time. The test isn’t perfect.
Writing this mathematically we get:
P (+ | C) = 0.9 where “+” means a positive test result, and
P (- | C) = 0.1 where “-“ means a negative test result.
Now
that is the probability for the “cause,” Cancer, producing a positive
test or “effect.” There are also probabilities for patients that don’t
have Cancer taking the test:
P (+ | ¬C) = 0.2 and, therefore,
P (- | ¬C) = 0.8.
Note
this implies that people who don’t have the Cancer will still get a
positive test about 20% of the time. Also note these results for people
who don’t have Cancer are different from the results of those that do.
Although
it isn’t a perfect test, it would seem that this is a fairly accurate
test since it will give a correct diagnostic for a person with Cancer
nine times out of ten. But to actually measure it accuracy as a
diagnostic, you have to calculate the joint probability. That is the
reversed view where you calculate the probability of the “cause” or
Cancer if you have a positive diagnostic test which is the “result.”
Remember,
you can get a positive result if you have Cancer, but also if you don’t
have Cancer. So we have to compare the probabilities.
Suppose you get a positive test result. Do you have Cancer for certain? No, that would not be logical.
Is it 90% likely you have Cancer? Surprise! NO.
Consider
P (+, C) which is the probability of having Cancer if you get a
positive test result. Using Bayesian analysis, that is equal to the P
(C) * P (+ | C) = 0.01 (the probability of Cancer) times 0.9 (the
probability that you will get a positive test result if you do have
Cancer) = 0.009. It is also possible that you got a positive test and
don’t have Cancer. The probability of that is 0.198 calculated from P
(¬C) * P (+ | ¬C) similar to the above. These results are then combined
following Bayes Rule.
I will skip the rest of the math (do
I hear a sigh of relief?) and just give the final answer of what
percentage of people who get a positive test result actually have
Cancer?
It is approximately 0.043 or 4.3%. Now isn’t that
odd that a test that seems 90% accurate only gives a little over four
percent results? The reason is that, in our problem (and with prostate
Cancer) it is a lot more likely you don’t have Cancer without taking any
tests. The more rare the outcome we’re looking for in the general
population, the more accurate the test must be to be meaningful.
So,
even thought the P.S.A. test is quite accurate, the rarity of the
disease will give us a lot of false positive results and a lot of
biopsies and possible treatment where it wasn’t needed.
So
just how common is Prostate Cancer. (Note the actual numbers I used
above are not necessarily the values for Prostate Cancer or the P.S.A.
test, although they aren’t far off.) Well I can tell you that in men
approximately 65 years old — that’s me, when I was diagnosed I learned
that one of my Navy buddies from the 60’s had the surgery just a month
before me. I later learned another Navy buddy had it a year ago. And an
old high school friend just had the surgery last month. So that is three
I know of in my circle of friends which is probably just a couple of
hundred men. Prostate Cancer is the most common Cancer in men and the
mathematical assumptions above of 1% of the population have it is likely
too small. Prostate Cancer is not that rare once you limit the
population to men over a certain age.
But back to my original point that it just doesn’t seem like statistics follows common sense, although it actually does if you look carefully enough at the result.
You have to be very careful what assumptions you make and what causes
you associate with which effects. It scares me that this is so tricky
because I know the general public has terrible math skills. So when
newspapers start quoting statistics I think of what Mark Twain said,
“There are lies, damn lies, and statistics.”
Let me
conclude with an interesting little problem that statistics professors
love to discuss. Remember the old “Let’s Make a Deal” show with Monty
Hall. Remember at the end of the show, a contestant was given a chance
to choose three doors, door “1,” door “2,” or door “3”. Now behind two
of the doors were booby prizes sometimes called “goats.” But behind one
door was a new car.
So the contestant would pick one door.
Now, what are the odds that the contestant picked the good prize, the
car? Well, there are three choices and the contestant picks one, so the
odds are one in three or 0.33.
After choosing a door, but
before opening it, Monty would go to one of the other doors and open it
up to reveal it had a “goat.” Monty knew which door had the car and,
since the contestant chooses one door, there was at least one more door
with a booby prize, even if the contestant also had a “goat.”
Monty would then offer the contestant the opportunity to change his or her choice. "Do you want to stay with the door you've chosen, or go with the other one?" he'd ask.
So now the question is, what is the probability now that
the contestant has the car once it is revealed that one of the other
doors does not have the auto behind it?
It is still 0.33.
The odds didn’t change just because one door was opened. The contestant
had 0.33 or 33% chance of being right. That doesn’t change when Monty
opens another door that does not have the car.
So, now the
really tough question. Should the contestant stick with his or her
original pick or change to the remaining door. Or does it matter. The
answer is simple. The contestant should change doors. The odds that the
car is behind the other door is 0.67! Of course, there are now two
doors, and so the total probability must be 1.0. The first door selected
has the probability of 0.33, so the other door must have 0.67.
It
was always the best move to change doors. Didn’t guarantee you would
win the car, but the odds were twice as good. Does that make sense? If
you think about it hard enough, it will. If not, give me a call and I’ll
explain it further. Note that many people argue with this result. I
remember once, years ago, on an IBM discussion forum on a text only
green screen on the IBM mainframe network called VNET, where there was
great argument about the probability of the gender of children if you
knew the sex of one child or if you knew the sex of the first child.
These are similar problems to the Monty Hall exercise. Probability can
be very tricky and sometimes counter-intuitive.
If you google the “Monty Hall Problem” you will find lots of interesting discussion on this problem — and SOME WRONG ANSWERS!!
When
a professor wrote a text book on probability and wanted to include the
“Monty Hall” example, he went to Monty and asked for permission to use
his name. Monty was gracious and agreed and then asked the professor
what the odds were. Monty never would believe the professor because
thought, “there are two doors, the odds are 50 — 50.” Monty didn’t
realize his little game had a mathematical best answer.
So, now, what do you think? Is the recommendation from the United States Preventive Services Task Force
to reduce diagnostics tests such as mammograms or P.S.A. tests good
advice, or is it just a way to save money? Will you take what's behind
door number 3??
Sunday, September 16, 2012
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