Cognitive Biases I

Optional Reading

Today’s lecture is based primarily on:

“How We Know What Isn’t So,” Chapter 1.

By Thomas Gilovich, a psychologist

Patterns

Pattern Recognition

Seeing patterns in your data is a good thing, and humans are natural pattern finders.

Watson & Crick discovered the structure of DNA by recognizing the “fuzzy X” pattern it left when bombarded with X-rays.

Pattern Recognition

But sometimes we see patterns when there’s really nothing to see.

Consider this famous photograph from 1976 by the Viking I spacecraft. Look! A face on the surface of mars!

Pattern Recognition

But sometimes we see patterns when there’s really nothing to see.

But this was what we were seeing…

Jesus Tree

Jesus Toast

Jesus Sock

Why do we see extra patterns?

Our brains are very good at finding patterns when they exist, and this is important.

But part of our success comes by seeing patterns everywhere, even when they don’t exist, including in random data.

The Clustering Illusion

I flipped a coin (really!) 20 times in a row. ‘X’ is Queen Elizabeth II and ‘O’ is the lion with the crown. Here is what I got:

XXXXOOXOOXXOOXOOXOOO

The Clustering Illusion

I flipped a coin (really!) 20 times in a row. ‘X’ is Queen Elizabeth II and ‘O’ is the lion with the crown. Here is what I got:

XXXXOOXOOXXOOXOOXOOO

That doesn’t look random. But it is. The coin lands the same as the previous toss 10 times and different from the previous toss 9 times.

The Clustering Illusion

I flipped a coin (really!) 20 times in a row. ‘X’ is Queen Elizabeth II and ‘O’ is the lion with the crown. Here is what I got:

XXXXOOXOOXXOOXOOXOOO

That doesn’t look random. But it is. The coin lands the same as the previous toss 10 times and different from the previous toss 9 times.

The Clustering Illusion

Ask anyone who watches basketball whether this is true:

“If a player makes a shot, they’re more likely to make the next; if they miss a shot, they’re less likely to make the next.”

The Clustering Illusion

Most people will say ‘yes, of course’. But it’s not true, they’re subject to the clustering illusion.

Gilovich, Vallone & Tversky (1985) analyzed records of made and missed shots, and they found:

The Clustering Illusion

• Players who made a shot, on average, scored on the very next shot 51% of the time.

• Players who missed a shot, on average, scored on the very next shot 54% of the time.

• Players who made two shots in a row, scored on the very next shot 50% of the time.

• Players who missed two shots in a row, scored on the very next shot 53% of the time.

Clustering + Explanation

Often, when people see a pattern (even one that isn’t real) they try to explain it.

Once they have an explanation, it can become very difficult to convince them that the pattern is fake. “I know players shoot better after making a shot– this gives them confidence to make another.”

Participation

Examples of people seeing patterns that aren’t there are good for participation!

I went to a cave near Guilin and every rock was named after a different animal.

Rabbit, Frog, Monkey, Man?

REGRESSION TO THE MEAN

Variables

• Height• Profit• Spiciness• Time• Speed• Intelligence• Hours worked

Positive Correlation

Negative Correlation

Measuring Correlations

Perfect Correlation

We say that two variables are perfectly correlated when knowing the value of one variable allows you to know the value of the other variable with certainty. R = 1 or R = -1.

For example, the area of a triangle whose base is 5 (one variable) is perfectly correlated with the height of the triangle (another variable).

Imperfect Correlation

Two variables are imperfectly correlated when the value of one gives you some evidence about the value of the other.

For example height of parents (one variable) is imperfectly correlated with height of children (another variable). Tall parents have tall children, on average, and short parents have short children, on average.

Regression to the Mean

Whenever two variables are imperfectly correlated, extreme values of one variable tend to be paired with less extreme values of the other.

Tall parents have tall children, but the children tend to be less tall than the parents. Students who do very well on Exam 1 tend to do well on Exam 2, but not as well as they did on Exam 1.

Average of Parents Height

# of Parents # of Children Shorter by 0.67 sd

# of Children Taller by 0.67 sd

Bottom 2% 7 0 14

2%-9% 45 2 32

9%-25% 80 1 9

Middle 50% 219 6 10

75%-91% 50 16 0

91%-98% 14 29 0

Top 2% 4 100 0

Regression to the Mean

This is true of any two imperfectly correlated variables.

Companies that do very well one year on average do well the next year, but not quite as well as the previous. Students who do well in high school on average do well in college, but not as well as in high school.

Regression to the Mean

“Regression” just means going back, and “mean” means average. “Regression to the mean” is just a fancy way of saying going back to average.

Imperfect Correlation

If I flip a fair coin and ten times in a row it lands heads each time, it will not “regress to the mean” and land tails more than heads to balance things out.Since it’s a fair coin, it is most likely to land heads half the times and tails the other half. Future tosses of a coin are independent of past ones. Fair coins land, on average, 50% heads and 50% tails.

Imperfect Correlation

Regression to the mean happens when we have two imperfectly correlated variables X and Y, and X takes on a very extreme value. Then we expect Y to take on a less extreme value.

But coin flips are not imperfectly correlated. They are not correlated at all. Past coin flips do not influence future coin flips.

Group Effect

It’s important to note that regression to the mean is not an individual effect. If you score low on the first exam you can’t assume you will do better next time.

What you can assume is that if you take the group of all who did poorly and then test them again, the average score will be closer to the mean.

Not Causal

It’s also important to note that regression to the mean is not a causal relation between variables.

If I give an exam, there will be some random variation– some measure of “luck.” Imagine that I’m going to give an exam with 5 questions on it. I give all the students 10 questions to study, and for each exam, I randomly select 5 of those 10 questions for the exam.

What the Students Study

Now imagine that each student studies only 5 of the possible questions. So each student knows the same amount– half of the material. Intuitively, each student “deserves” the same grade.

Giving the Exam

But now imagine that I give an exam, randomly selecting 5 questions. On average, students will know the answers to 2.5 of the questions, since on average they know half of the material.But some students will be “lucky”– I will pick only the questions they studied, and they will get 5/5. Other students will be unlucky: I will pick only the questions they didn’t study. Everyone else will be in the middle.

Regression to the Mean

But if I give a second exam, we will see regression to the mean. The lucky students on the first exam probably won’t be the lucky ones on the second. The same goes for the unlucky students.So the lucky students on exam #1 will get worse grades on exam #2 (on average), and the unlucky students on #1 will get better grades on #2 (on average). Their grades regressed to the mean (2.5).

Real Life

This isn’t exactly how real life works, but the idea is the same. Every exam is a random sampling of the questions the professor could have asked, and each student knows some of the material. On any given exam, sometimes you will be lucky, sometimes the exam will give your “true” score, and sometimes you will be unlucky.

Regression Fallacy

The regression fallacy involves attributing a causal explanation to what is nothing more than regression to the mean.

The Sports Illustrated Jinx

Some people believe in the “Sports Illustrated jinx”: when you appear on the cover of Sports Illustrated, you do very poorly in your sport.

The Sports Illustrated Jinx

For example, while this was the cover of SI in February, Jeremy Lin shot 1-for-11 in a game where the New York Knicks lost to the Miami Heat 102-88.

The Sports Illustrated Jinx

And when this was the cover of SI, Lin’s team snapped a 7 game winning streak when they lost at home in New York to the New Orleans Hornets.

Regression to the Mean

But there is no SI jinx. This is just regression to the mean.

You get on the cover of SI when you are the best athlete in all sports (in America) during the previous week. Your performance this week is imperfectly correlated with your performance last week. You’re unlikely to be the best athlete in all sports two weeks in a row!

Traffic Survey

Suppose that the government conducts a survey of traffic intersections to find out which had the most accidents in the past month. At every intersection where there was a large number of accidents, the government installs cameras. Next month they do a survey again and notice than on average there are fewer accidents at the locations where cameras are installed.

Government Claims

The government claims: Claim: Installing the cameras reduced the number of accidents.

Regression Fallacy

The government might be right. But this also might be the regression fallacy.

The variables “traffic accidents this month” and “traffic accidents next month” are imperfectly correlated. Intersections with lots of accidents this month will likely have lots next month; intersections with few this month will likely have few next month.

Regression to the Mean

If you have two imperfectly correlated variables, and one of them (“traffic accidents this month”) takes on an extreme value, the other will have a more moderate value. So the intersections that were the worst this month will be, on average, bad-but-less-bad next month. The will, on average, improve– simply through regression to the mean.

The Regression Fallacy

The regression fallacy is attributing a cause (the cameras the government installed) to an effect (the decrease in accidents at the intersections that had the most accidents last month) that is really just regression to the mean.

Note

None of this means the cameras didn’t work!

To truly test whether cameras work, you must install them at a random sampling of intersections– good and bad ones. If accidents then go down on average, you can be confident the cameras worked. We’ll talk more about random samples later in the course.

Reward and Punishment

We can see how people might become convinced that punishment works better than reward.

You punish someone when they do something exceptionally bad. Even if the punishment does nothing, we expect their behavior to regress back to normal. So it will look like punishment works.

Reward and Punishment

We can see how people might become convinced that punishment works better than reward.

You reward someone when they do something exceptionally good. Even if the reward does nothing, we expect their behavior to regress back to normal. So it will look like rewards make them behave worse.

Participation

Parenting examples or student/ teacher examples are good here!