Human Stupidity: Historical: Opinions II

When making decisions in the real world, the situation can easily become even far more biased than our natural tendencies shown in those artificial studies. Not only we tend to keep our initial opinions much longer than we should, we also directly decide the sources of information we will use. And that almost always means looking for opinions of those we already agree with, while disregarding people who opposes our own views. Of course, this will simply make us more sure about what we thought, even when that should not be the case. While doing that, we just learn the reasons why our opinion might be right, but we rarely come to know the reasons why it might actually be wrong. Test yourself: Can you make a convincing argument about some political or religious idea you oppose? You don't have to believe the argument is enough to change your mind, but it should be considered a solid argument ("It is the mark of an educated mind to be able to entertain a thought without accepting it'', attributed to Aristotle).

Of course, anyone would like to think that their beliefs are reasonable, rational, and well justified. After all, if they weren't, we wouldn't have them, right? But evidence, unfortunately, is not on our side. In a very interesting example, Jervis  observed an effect he called irrational consistency (Baron uses the term belief overkill). This consists of the fact that when people hold a specific belief, for example in a policy, they usually have many independent ideas they believe in and all of them support the said policy. And those who oppose the policy tend to defend the opposite set of ideas. However, if those ideas are independent, any rational being could defend some and oppose others, while a consideration about the total effect would lead to the final point of view on the policy. That people are too consistent is a clear sign reason is not playing the role it should in this problem.

Jervis mentions as an example the case of people who supported or opposed a ban on nuclear tests. Among the issues behind a decision to support or ban, he presents three issues: if the tests would cause serious medical danger; if the tests would lead to major weapon improvements; and if they would be a source of international tension. It is important to notice that it is completely reasonable to believe that the tests would not cause serious medical danger but would cause international tension. These evaluations are independent and any of the four possible combinations of beliefs make just as sense as the other three. That means that, if people were reasoning in a competent and independent way, no correlation between those beliefs should be observed. And yet those who were in favor of the ban held all the beliefs that the tests would cause healthy problems, would lead to more dangerous weapons, and would increase international tension. And, as it should be obvious by now, those who opposed the ban, disagreed in all the subjects with those who were in favor. Apparently, people felt somehow led to have a consistent set of beliefs, even when there was no reason at all for that consistency.

As a matter of fact, when our beliefs seem to conflict with each other, a phenomenon called cognitive dissonance, we have a tendency to change some of those beliefs to avoid the conflict. This was observed in a series of experiments conducted by Festinger . The typical experiment included performing some task and be paid either a very small amount for it ($1.00) or a more reasonable amount ($20.00, in 1962). When the subjects were asked about their feelings about the task, those who had been paid very little had a better evaluation of it than those who had received more. The explanation proposed by Festinger is that people wouldn't perform that task for just one dollar. But they had done it, what created the cognitive dissonance that the subjects solved by evaluating the task as more entertaining. After all, doing an entertaining task for basically no money makes more sense than doing a boring task.

Human Stupidity: Historical: Opinions I

Before moving on to other issues, there are still a couple of other examples of our probabilistic thinking that I'd like to discuss. As we have seen in the AIDS problem, the correct way to solve whether the patient was sick or not was first to consider the initial probability for the disease and then change it to a new, posteriori value as we learn the result of the exam. This method is the basis of the Bayesian view in Statistics and there is actual evidence that we reason in a way that resembles Bayesian methods even as early as 12-month old .

But we do not do that in a perfect way, of course. In the AIDS problem, we saw that people generally simple disregard the initial, prior information contained in the initial chance of 1 in 1,000. We just use the information about the exam, as if we had no initial good guess about the chance of the patient being sick. And, as a matter of fact, if we had started with equal chances, 50% instead of 0.1%, the final chance for the patient to be sick would, indeed, in that case, be 98%. But we knew better and, by ignoring the prior information, we could cause a lot of unnecessary damage.

This effect is known as base rate neglect. In 1973, Kahneman and Tversky presented a problem to several people where they would have to guess, based on a text description, if the described person was a lawyer or an engineer. It was clearly stated that the described person was part of a group with 30 engineers and 70 lawyers. This should mean that it was more likely the person would be a lawyer than an engineer. However, this piece of information was completely disregarded and only the text used to make that inference. When the text was non-informative, with no clues pointing to engineer or lawyer, people would state there was a 50-50% chance, instead of the correct 30-70%.

At this point, it should be expected this is not the only mistake we do on how we change our opinions. As a matter of fact, the base rate neglect is not exactly an effect on how we change opinions. It actually happens in subjects that we have no initial opinions about. In those cases, even if there is evidence to be used as an initial opinion, that initial evidence is disregarded and only the new information is used. In many cases, however, people have an opinion before new information is provided. In this case, they should use that information as prior and update it following Bayes theorem.

While this is a qualitatively correct description of what we seem to do, it is not exact when we try to see it in numeric terms. Phillips and Edwards  observed that, while people do change their opinions in the correct direction, making a statement more or less probable when given data, the amount of the change is smaller than it would be expected from a simple use of the Bayes theorem. They have named this bias as conservatism, as people tend to conserve their initial opinions more than they should. And it is worth mentioning that they observed this in a completely neutral situation, where people were asked to evaluate from which bag a set of red and blue chips had come. They informed the subjects that there were two possible bags, one with 700 red chips and 300 blue ones, while the other bag had 300 red chips and 700 blue ones. If, after taking 12 chips from one of the bags, it was observed that 8 were red and 4 were blue, the question is how likely it is that those chips came from the bag with a majority of red chips. You can ask yourself, as a reader, what probability value would you state. Phillip and Edwards observed people tended to answer a value around 70%. However, the correct value, if you assume both bags were just as likely initially is a much larger change from the initial 50%, the correct final probability is actually approximately 97%.

While this tendency to change opinions too little might look at first a simple effect of analysing an unknown problem, that is not the case. Even in the world of corporate finance, evidence was observed that investors tend to under-react to information. Baron has an interesting section on this problem, that he calls the persistence of irrational beliefs where he cites some of the literature in the area. This includes studies that show that the order the data is presented affects the final opinion of individuals, even when that order was irrelevant and contained no new information.

One interesting study on this primacy effect, where first observations carry more weight than later ones, even when they should not, was conducted by Peterson and DuCharme. Like the Phillips and Edwards study, they had the question was to find out from which bag a set of poker was more likely to have been drawn for. Urn C had 3 red, 2 blue, 2 yellow, 1 green and 2 white chips, while urn D contained 2 red, 3 blue, 1 yellow, 2 green and 2 white ships. One urn was shown to the subjects and chips were taken from that urn and return to it, one at a time. After each draw, the subjects were asked to evaluate the probability that the urn they were drawing from was urn C. However, the draws were not random but arranged so that the first 30 draws favored the idea that was urn C, while the 30 following draws favored D, in an exact mirror way to the first 30. That is, the total evidence in favor of each urn canceled after the 60 draws and the final opinion should be equal chances for each urn. But, since the individuals started believing C was more probable, they observed a very clear tendency to keep that initial evaluation. It typically took series of 50 draws favoring D in order to counter the initial 30 draws supporting C.

Human Stupidity: Historical: Probability Thinking IV

One particular troublesome example of how disastrous probabilistic mistakes can be and one I use every time for my students is the classical example of testing for the existence of a rare but serious disease. Most of the time, texts refer to the disease as AIDS, but which one is not relevant. Let's just assume that there is a treatment to the disease that has serious side effects and any physician would prefer not to administer it unless really necessary. Luckily, most of the population has not contracted the disease; we actually know only one person in 1,000 has it. We also have a test for it that is reasonably reliable. Whenever someone is sick, it provides a positive result for detecting the virus 98% of the time. Whenever someone does not have the virus it gives a negative result 98% of the times as well (the two 98% chances are not necessarily equal and are often not). In other words, in both cases, it gives an erroneous result only 2% of the times. Assume the test is applied at a person you have no other information about and the result comes positive. This suggests that person might have the disease, but it is also possible that the test has failed. Given all the information above, if you were evaluating this patient, how likely would you say it is that this person is actually sick?

The importance of getting the result correctly can not be overstated here. If it is very likely that the person is sick, treatment should start immediately. If it is very unlikely, it might make sense to prepare additional tests, but, since the treatment has serious side effects, it should not be applied. And, if it is the case that we are not sure at all, for more central probabilities, close to 50%, an assessment of the risk involved in each decision must be made with the proper caution. But that all depends on getting the right evaluation. Ask yourself, just by reading the problem, how likely you think it is that the patient is sick. If you are like the majority of humankind, you will reason that, since the test is correct 98% of time, the probability that the patient would be sick should be around 98% as well. So, you would start the treatment immediately.

But the truth is not so simple. That reasoning simply ignores one extremely important information that you had. And that is the initial chance the patient were sick. I did tell you that was, before the test, 1 in 1,000. Reason a little. If it were the opposite, with 99,9% of the population sick, a positive result should mean extra evidence in favor of the disease and the chance should be even larger than 99,9%, not as small as 98%. By the same reasoning, if you knew for sure, at first, this patient was healthy, you would continue to know it and simply conclude this was one of the 2% cases where the test goes wrong. So, that 1 in 1,000 has important information and you ignored it. And, as a matter of fact, the chance that this patient is sick is not 98%, it is actually smaller than 5%. If the side effects of the treatment are severe, a person that was had a 95% chance of being healthy would have to suffer it, without enough evidence to support the need for it.

What is going on? It is actually not so hard to understand when we look at the whole picture. What we know is that the test gave a positive result. Two things might have happened to cause that result. The patient could actually be sick and that would have happened initially with one chance in 1,000. Or the test could have failed, providing a positive answer for a healthy person. This case had a chance of 20 in 1,000. Clearly, the failure of the test is much more likely than the hypothesis that the patient is sick, 20 times more likely, actually. The chance of the patient being sick actually increases from the initial 1 in 1,000 to a posterior probability a little less than 50 in 1,000. That is a huge increase and this large change is due to the fact the test is reasonably reliable. But, by ignoring one very important piece of information in the problem, completely wrong decisions about our health can happen (and almost certainly do happen) every day (the correct way to calculate the final probability is using Bayes Theorem. Its use will be explained later here).

As a matter of fact, the simple misuse of basic probabilistic and statistical concepts in health problems is so widespread and so serious that efforts already exist to better educate physicians and patients. Gerd Gigerenzer  has been championing this urgent need for better education, with good reason. He has also collected a number of stories on how that misuse can cause bad health decisions and bad policies, leading governments to spend money on non-existant problems. One of those stories I particularly like I heard from him in a conference. The problem started when  Rudi Giuliani said in a campaign advertisement in 2007 that: ``I had prostate cancer, 5, 6 years ago. My chance of surviving prostate cancer -- and thank God I was cured of it -- in the United States? 82%. My chance of surviving prostate cancer in England? Only 44% under socialized medicine''(original story in the Washington Post is here).

This certainly looks as if there were a very serious difference in the quality of the treatments in USA and England. But the actual mortality rates in both countries are basically the same. And yet, Giuliani was not lying, just using numbers without all the competence required to analyse them. Those figures he cited were 5-year survival rates. What was actually happening was that in the USA men are under a lot of pressure to screen for prostate cancer and many actually do participate in the prostate-specific antigen (PSA) screening. That does not happen in England. Even forgetting the important and not clear question of whether screening actually saves lives or not, there is one other extremely important effect here. The percentages do not refer to the groups treated equally and, as such, they just are NOT comparable. Gigerenzer has the perfect example to explain why.

Imagine two men, one American and one English, who will die of prostate cancer at age 70, regardless of treatment. In both cases, the cancer could be detected by screening as early as they were 60. But only the American does the screening and he discovers the disease at age 60. Even if the treatment fails and he dies at 70, his 5-year survival rate is still 100%, he does get to 65 without dying. The English man, on the other hand, only finds out he is sick when the disease is advanced and there are clear symptoms, at age 67. He also dies at the same age, but he does not pass the 5-year survival period. Not because of any difference in the health systems, both men contracted the disease at the same age and died the same age as well. The only difference is that one knew it much earlier. And, since American men are screened, they do know about it earlier and the almost double death rate in England is nothing more than a trick of measuring things differently.

Human Stupidity: Historical: Probability Thinking III

If we just made mistakes when interpreting probability values but were able to analyse data correctly, the mistakes I just described wouldn't be so serious. After all, probability is a recent discipline we invented to deal with uncertain outcomes. If our brains were able to interpret data correctly but not stated probability values, that would just mean we should be careful in how we present evidence and results. Indeed, Gigerenzer and Hoffrage  have observed that, under some circumstances, people make less mistakes when presented with frequency information rather than probability values.

Unfortunately, although different ways to present a problem can have a positive impact in our capacity to analyse it, we also have a tendency to misinterpret data. When asked to estimate the probability of a result in a two-stage lottery with equal probabilities of winning or losing at each stage, Cohen et al.  observed people tend to overestimate their chances of winning. Instead of the correct chance of 25%, they actually observed an average estimate of 45%. Evaluating chances wrongly can easily cause anyone to make wrong decisions.

In similar lines to the errors in mathematical analysis that Kahan et al observed, people also see things in data that are not there, while failing to notice effects that are real, according to the data they observe. Chapman and Chapman (see also this) presented pairs of words on a large screen to the several people in order to test how we perceive correlations. The pairs were presented such that each first word was shown an equal number of times together with each of the different word from the second set. However, the subjects of the experiment described the pairs that made more sense, such as lion and tiger, or eggs and bacon, had, according to their perception showed up more often than the pairs where there was no logical relation between the words. This illusory correlation effect, where a non-existent correlation is perceived was confirmed by later studies.

The opposite case was also observed, that is, in problems where people expect to find no correlation between variables, they fail to notice real correlations in the data. Or, when the correlation is very strong, they tend to consider it smaller than it really is. This was observed initially by Hamilton and Rose , who called the effect invisible correlation. One interesting thing about both illusory and invisible correlation biases is that our prior beliefs seem to play a central role in how we interpret the data we obtain. While prior information should indeed affect our final opinions after observing data, it should not change what that data says.

The list of probabilistic biases that have been observed is huge, however it is not the objective of this text to get anywhere close to an extensive treatment of the problem. My goal here is just to convince people that our so often praised human rationality is actually far worse than we would like to admit. Or, at least, to convince that they should check the existing literature and see for themselves. Still, it is worth showing a few more cases before going on to other subjects.