Thinking, Fast and Slow (21 page)

• The correlation between the size of objects measured with precision in English or in metric units is 1. Any factor that influences one measure also influences the other; 100% of determinants are shared.
  
• The correlation between self-reported height and weight among adult American males is .41. If you included women and children, the correlation would be much higher, because individuals’ gender and age influence both their height ann wd their weight, boosting the relative weight of shared factors.
  
• The correlation between SAT scores and college GPA is approximately .60. However, the correlation between aptitude tests and success in graduate school is much lower, largely because measured aptitude varies little in this selected group. If everyone has similar aptitude, differences in this measure are unlikely to play a large role in measures of success.
  
• The correlation between income and education level in the United States is approximately .40.
  
• The correlation between family income and the last four digits of their phone number is 0.
  

It took Francis Galton several years to figure out that correlation and regression are not two concepts—they are different perspectives on the same concept. The general rule is straightforward but has surprising consequences: whenever the correlation between two scores is imperfect, there will be regression to the mean. To illustrate Galton’s insight, take a proposition that most people find quite interesting:

You can get a good conversation started at a party by asking for an explanation, and your friends will readily oblige. Even people who have had some exposure to statistics will spontaneously interpret the statement in causal terms. Some may think of highly intelligent women wanting to avoid the competition of equally intelligent men, or being forced to compromise in their choice of spouse because intelligent men do not want to compete with intelligent women. More far-fetched explanations will come up at a good party. Now consider this statement:

This statement is obviously true and not interesting at all. Who would expect the correlation to be perfect? There is nothing to explain. But the statement you found interesting and the statement you found trivial are algebraically equivalent. If the correlation between the intelligence of spouses is less than perfect (and if men and women on average do not differ in intelligence), then it is a mathematical inevitability that highly intelligent women will be married to husbands who are on average less intelligent than they are (and vice versa, of course). The observed regression to the mean cannot be more interesting or more explainable than the imperfect correlation.

You probably sympathize with Galton’s struggle with the concept of regression. Indeed, the statistician David Freedman used to say that if the topic of regression comes up in a criminal or civil trial, the side that must explain regression to the jury will lose the case. Why is it so hard? The main reason for the difficulty is a recurrent theme of this book: our mind is strongly biased toward causal explanations and does not deal well with “mere statistics.” When our attention is called to an event, associative memory will look for its cause—more precisely, activation will automatically spread to any cause that is already stored in memory. Causal explanations will be evoked when regression is detected, but they will be wrong because the truth is that regression to the mean has an explanation but does not have a cause. The event that attracts our attention in the golfing tournament is the frequent deterioration of the performance of the golfers who werecte successful on day 1. The best explanation of it is that those golfers were unusually lucky that day, but this explanation lacks the causal force that our minds prefer. Indeed, we pay people quite well to provide interesting explanations of regression effects. A business commentator who correctly announces that “the business did better this year because it had done poorly last year” is likely to have a short tenure on the air.

Our difficulties with the concept of regression originate with both System 1 and System 2. Without special instruction, and in quite a few cases even after some statistical instruction, the relationship between correlation and regression remains obscure. System 2 finds it difficult to understand and learn. This is due in part to the insistent demand for causal interpretations, which is a feature of System 1.

I made up this newspaper headline, but the fact it reports is true: if you treated a group of depressed children for some time with an energy drink, they would show a clinically significant improvement. It is also the case that depressed children who spend some time standing on their head or hug a cat for twenty minutes a day will also show improvement. Most readers of such headlines will automatically infer that the energy drink or the cat hugging caused an improvement, but this conclusion is completely unjustified. Depressed children are an extreme group, they are more depressed than most other children—and extreme groups regress to the mean over time. The correlation between depression scores on successive occasions of testing is less than perfect, so there will be regression to the mean: depressed children will get somewhat better over time even if they hug no cats and drink no Red Bull. In order to conclude that an energy drink—or any other treatment—is effective, you must compare a group of patients who receive this treatment to a “control group” that receives no treatment (or, better, receives a placebo). The control group is expected to improve by regression alone, and the aim of the experiment is to determine whether the treated patients improve more than regression can explain.

Incorrect causal interpretations of regression effects are not restricted to readers of the popular press. The statistician Howard Wainer has drawn up a long list of eminent researchers who have made the same mistake—confusing mere correlation with causation. Regression effects are a common source of trouble in research, and experienced scientists develop a healthy fear of the trap of unwarranted causal inference.

One of my favorite examples of the errors of intuitive prediction is adapted from Max Bazerman’s excellent text
Judgment in Managerial Decision Making
:

Having read this chapter, you know that the obvious solution of adding 10% to the sales of each store is wrong. You want your forecasts to be regressive, which requires adding more than 10% to the low-performing branches and adding less (or even subtracting) to others. But if you ask other people, you are likely to encounter puzzlement: Why do you bother them with a
n obvious question? As Galton painfully discovered, the concept of regression is far from obvious.

Speaking of Regression to Mediocrity

Life presents us with many occasions to forecast. Economists forecast inflation and unemployment, financial analysts forecast earnings, military experts predict casualties, venture capitalists assess profitability, publishers and producers predict audiences, contractors estimate the time required to complete projects, chefs anticipate the demand for the dishes on their menu, engineers estimate the amount of concrete needed for a building, fireground commanders assess the number of trucks that will be needed to put out a fire. In our private lives, we forecast our spouse’s reaction to a proposed move or our own future adjustment to a new job.

Some predictive judgments, such as those made by engineers, rely largely on look-up tables, precise calculations, and explicit analyses of outcomes observed on similar occasions. Others involve intuition and System 1, in two main varieties. Some intuitions draw primarily on skill and expertise acquired by repeated experience. The rapid and automatic judgments and choices of chess masters, fireground commanders, and physicians that Gary Klein has described in
Sources of Power
and elsewhere illustrate these skilled intuitions, in which a solution to the current problem comes to mind quickly because familiar cues are recognized.

Other intuitions, which are sometimes subjectively indistinguishable from the first, arise from the operation of heuristics that often substitute an easy question for the harder one that was asked. Intuitive judgments can be made with high confidence even when they are based on nonregressive assessments of weak evidence. Of course, many judgments, especially in the professional domain, are influenced by a combination of analysis and intuition.

Nonregressive Intuitions

Let us return to a person we have already met:

People who are familiar with the American educational scene quickly come up with a number, which is often in the vicinity of 3.7 or 3.8. How does this occur? Several operations of System 1 are involved.

• A causal link between the evidence (Julie’s reading) and the target of the prediction (her GPA) is sought. The link can be indirect. In this instance, early reading and a high GDP are both indications of academic talent. Some connection is necessary. You (your System 2) would probably reject as irrelevant a report of Julie winning a fly fishing competitiowhired D=n or excelling at weight lifting in high school. The process is effectively dichotomous. We are capable of rejecting information as irrelevant or false, but adjusting for smaller weaknesses in the evidence is not something that System 1 can do. As a result, intuitive predictions are almost completely insensitive to the actual predictive quality of the evidence. When a link is found, as in the case of Julie’s early reading, WY SIATI applies: your associative memory quickly and automatically constructs the best possible story from the information available.
  
• Next, the evidence is evaluated in relation to a relevant norm. How precocious is a child who reads fluently at age four? What relative rank or percentile score corresponds to this achievement? The group to which the child is compared (we call it a reference group) is not fully specified, but this is also the rule in normal speech: if someone graduating from college is described as “quite clever” you rarely need to ask, “When you say ‘quite clever,’ which reference group do you have in mind?”
  
• The next step involves substitution and intensity matching. The evaluation of the flimsy evidence of cognitive ability in childhood is substituted as an answer to the question about her college GPA. Julie will be assigned the same percentile score for her GPA and for her achievements as an early reader.
  
• The question specified that the answer must be on the GPA scale, which requires another intensity-matching operation, from a general impression of Julie’s academic achievements to the GPA that matches the evidence for her talent. The final step is a translation, from an impression of Julie’s relative academic standing to the GPA that corresponds to it.
  

Intensity matching yields predictions that are as extreme as the evidence on which they are based, leading people to give the same answer to two quite different questions:

By now you should easily recognize that all these operations are features of System 1. I listed them here as an orderly sequence of steps, but of course the spread of activation in associative memory does not work this way. You should imagine a process of spreading activation that is initially prompted by the evidence and the question, feeds back upon itself, and eventually settles on the most coherent solution possible.

Amos and I once asked participants in an experiment to judge descriptions of eight college freshmen, allegedly written by a counselor on the basis of interviews of the entering class. Each description consisted of five adjectives, as in the following example:

We asked some participants to answer two questions:

The questions require you to evaluate the evidence by comparing the description to your norm for descriptions of students by counselors. The very existence of such a norm is remarkable. Although you surely do not know how you acquired it, you have a fairly clear sense of how much enthusiasm the description conveys: the counselor believes that this student is good, but not spectacularly good. There is room for stronger adjectives than
intelligent
(
brilliant
,
creative
),
well-read
(
scholarly, erudite, impressively knowledgeable
), and
hardworking
(
passionate
,
perfectionist
). The verdict: very likely to be in the top 15% but unlikely to be in the top 3%. There is impressive consensus in such judgments, at least within a culture.

The other participants in our experiment were asked different questions:

You need another look to detect the subtle difference between the two sets of questions. The difference should be obvious, but it is not. Unlike the first questions, which required you only to evaluate the evidence, the second set involves a great deal of uncertainty. The question refers to actual performance at the end of the freshman year. What happened during the year since the interview was performed? How accurately can you predict the student’s actual achievements in the first year at college from five adjectives? Would the counselor herself be perfectly accurate if she predicted GPA from an interview?

The objective of this study was to compare the percentile judgments that the participants made when evaluating the evidence in one case, and when predicting the ultimate outcome in another. The results are easy to summarize: the judgments were identical. Although the two sets of questions differ (one is about the description, the other about the student’s future academic performance), the participants treated them as if they were the same. As was the case with Julie, the prediction of the future is not distinguished from an evaluation of current evidence—prediction matches evaluation. This is perhaps the best evidence we have for the role of substitution. People are asked for a prediction but they substitute an evaluation of the evidence, without noticing that the question they answer is not the one they were asked. This process is guaranteed to generate predictions that are systematically biased; they completely ignore regression to the mean.

During my military service in the Israeli Defense Forces, I spent some time attached to a unit that selected candidates for officer training on the basis of a series of interviews and field tests. The designated criterion for successful prediction was a cadet’s final grade in officer school. The validity of the ratings was known to be rather poor (I will tell more about it in a later chapter). The unit still existed years later, when I was a professor and collaborating with Amos in the study of intuitive judgment. I had good contacts with the people at the unit and asked them for a favor. In addition to the usual grading system they used to evaluate the candidates, I asked for their best guess of the grade that each of the future cadets would obtain in officer school. They collected a few hundred such forecasts. The officers who had produced the prediof рctions were all familiar with the letter grading system that the school applied to its cadets and the approximate proportions of A’s, B’s, etc., among them. The results were striking: the relative frequency of A’s and B’s in the predictions was almost identical to the frequencies in the final grades of the school.

These findings provide a compelling example of both substitution and intensity matching. The officers who provided the predictions completely failed to discriminate between two tasks:

• their usual mission, which was to evaluate the performance of candidates during their stay at the unit
  
• the task I had asked them to perform, which was an actual prediction of a future grade
  

They had simply translated their own grades onto the scale used in officer school, applying intensity matching. Once again, the failure to address the (considerable) uncertainty of their predictions had led them to predictions that were completely nonregressive.

A Correction for Intuitive Predictions

Back to Julie, our precocious reader. The correct way to predict her GPA was introduced in the preceding chapter. As I did there for golf on successive days and for weight and piano playing, I write a schematic formula for the factors that determine reading age and college grades:

The shared factors involve genetically determined aptitude, the degree to which the family supports academic interests, and anything else that would cause the same people to be precocious readers as children and academically successful as young adults. Of course there are many factors that would affect one of these outcomes and not the other. Julie could have been pushed to read early by overly ambitious parents, she may have had an unhappy love affair that depressed her college grades, she could have had a skiing accident during adolescence that left her slightly impaired, and so on.

Recall that the correlation between two measures—in the present case reading age and GPA—is equal to the proportion of shared factors among their determinants. What is your best guess about that proportion? My most optimistic guess is about 30%. Assuming this estimate, we have all we need to produce an unbiased prediction. Here are the directions for how to get there in four simple steps:

1. Start with an estimate of average GPA.
   
2. Determine the GPA that matches your impression of the evidence.
   
3. Estimate the correlation between your evidence and GPA.
   
4. If the correlation is .30, move 30% of the distance from the average to the matching GPA.
   

Step 1 gets you the baseline, the GPA you would have predicted if you were told nothing about Julie beyond the fact that she is a graduating senior. In the absence of information, you would have predicted the average. (This is similar to assigning the base-rate probability of business administration grahavрduates when you are told nothing about Tom W.) Step 2 is your intuitive prediction, which matches your evaluation of the evidence. Step 3 moves you from the baseline toward your intuition, but the distance you are allowed to move depends on your estimate of the correlation. You end up, at step 4, with a prediction that is influenced by your intuition but is far more moderate.

This approach to prediction is general. You can apply it whenever you need to predict a quantitative variable, such as GPA, profit from an investment, or the growth of a company. The approach builds on your intuition, but it moderates it, regresses it toward the mean. When you have good reasons to trust the accuracy of your intuitive prediction—a strong correlation between the evidence and the prediction—the adjustment will be small.

Intuitive predictions need to be corrected because they are not regressive and therefore are biased. Suppose that I predict for each golfer in a tournament that his score on day 2 will be the same as his score on day 1. This prediction does not allow for regression to the mean: the golfers who fared well on day 1 will on average do less well on day 2, and those who did poorly will mostly improve. When they are eventually compared to actual outcomes, nonregressive predictions will be found to be biased. They are on average overly optimistic for those who did best on the first day and overly pessimistic for those who had a bad start. The predictions are as extreme as the evidence. Similarly, if you use childhood achievements to predict grades in college without regressing your predictions toward the mean, you will more often than not be disappointed by the academic outcomes of early readers and happily surprised by the grades of those who learned to read relatively late. The corrected intuitive predictions eliminate these biases, so that predictions (both high and low) are about equally likely to overestimate and to underestimate the true value. You still make errors when your predictions are unbiased, but the errors are smaller and do not favor either high or low outcomes.