Sunday, September 16, 2012
Odds Are ... Behind Door Number 3
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.
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??