SIMPSON'S PARADOX:

Designed by Engin A. Sungur

THE HOT HAND IN BASKETBALL

SCENARIO

School children routinely learn to identify optical illusions. It is arguably as important that the general public learn to identify statistical illusions. A person unfamiliar with a counterintuitive result, such as Simpson's paradox, may give the wrong interpretation to the pattern in their data. Simpson's paradox refers to the reversal of the direction of a comparison or an association when data from several groups are combined to form a single group.

In this activity you will look at the hot hand theory in basketball. Suppose that a basketball player plans to attempt 20 shots, with each shot resulting in a hit or a miss. (A statistician might assume tentatively that the assumption of Bernoulli trials are appropriate for this experiment). Suppose that the experiment is performed and the player obtains the following data

HMHMM MHHHM HHHMM HMHHH

Are the three occurrences of three successive hits convincing evidence of the player having a "hot hand"?

QUESTION

How can two individuals end up with two different conclusions when they have the same source and use "scientific" methods?

OBJECTIVES

The goal is to learn how to interpret two and higher dimensional tables and how to deal with counterintuitive results. At the end of this activity you will also be able to find and interpret conditional empirical probabilities.

ACTIVITY

1. WARM-UP ACTIVITIES

Do you think that when shooting free throws, a player has a better chance of making his second shot after making his first shot than after missing his first shot?

A "yes" response can be interpreted as indicating belief in the existence of the hot hand phenomenon, and a "no" response as indicating disbelief. Tversky and Gilovich (1989) asked this question to a sample of 100 "avid basketball fans" from Cornell and Stanford. Sixty-eight of the fans responded "yes" and the other 32 "no".

2. MAIN ACTIVITY

Consider two of the Boston Celtics players, Larry Bird and Rick Robey.

a. By referring to the following tables fill in the blanks:

LARRY BIRD

Second:
First:
Hit
Miss
Total
Hit
251
34
285
Miss
48
5
53
Total
299
39
338


RICK ROBEY

Second:
First:
Hit
Miss
Total
Hit
54
37
91
Miss
49
31
80
Total
103
68
171

During the 1980-1981 and 1981-1982 seasons, Larry Bird shot a pair of free throws on

occasions. times he missed both shots, he made both shots, times he made only the first shot, and times he made only the second shot.

On the other hand, Rick Robey shot a pair of free throws on occasions. times he missed both shots, he made both shots, times he made only the first shot, and times he made only the second shot.

For the Larry Bird, the proportion of the first shot hits that are followed by a hit was , and the proportion of first shot misses that are followed by a hit was . For Robey, these numbers are and , respectively.

Based on your answers, is there an evidence that the hot hand theory is valid?

b. It is possible, of course, to ignore the identity of the player attempting the shots and examine the data in the collapsed table given below.

COLLAPSED TABLE

Second:
First:
Hit
Miss
Total
Hit
305
71
376
Miss
97
36
133
Total
402
107
509

Now, fill in the blanks

Larry Bird and Rick Robey shot a pair of free throws on occasions. times they missed both shots, they made both shots, times they made only the first shot, and times they made only the second shot. For the collapsed table, the proportion of the first shot hits that are followed by a hit was , and the proportion of first shot misses that are followed by a hit was .

Based on your answers, is there an evidence that the hot hand theory is valid?

c. The data from Bird and Robey shows that, contrary to the hot hand theory each player shot slightly better after a miss than after a hit. On the other hand for the collapsed table proportion of first shot hits that are followed by a hit is greater than the proportion of first shot misses that are followed by a hit. This is called Simpson's Paradox. Could you explain why even though both Bird and Robey shot better after a miss than after a hit, the collapsed values show the reverse pattern?

d. What is the main difference between Bird and Robey?

What is the proportion of hits and misses for the Bird?

What is the proportion of hits and misses for the Robey?

EXTENSIONS

3. Suppose that 100 recent graduates are surveyed and their income at first job (high, low), gender (female, male), and major discipline (liberal arts, science or engineering) are recorded. Fill in the following tables so that in liberal arts percentage of low income is 60% for both females and males, and in science or engineering percentage of high income is 80% for both females and males.

LIBERAL ARTS

Low IncomeHigh Income Total
Female 30
Male
Total 50

SCIENCE OR ENGINEERING

Low IncomeHigh Income Total
Female 10
Male
Total 50

Now collapse these tables over major discipline and produce a table which includes gender and income.

COLLAPSED TABLE

Low IncomeHigh Income Total
Female
Male
Total 100

By using the collapsed table find the proportion of females in the low income group , also find the proportion of males in the low income group .

Interpret your results.

4. Death Penalty and Race: The following is a 2x2x2 contingency table-two rows, two columns, and two layers-from an article that studied effects of racial characteristics on whether individuals convicted of homicide receive the death penalty. The 674 subjects classified in the table were the defendants in indictments involving cases with multiple murders in Florida between 1976 and 1987. The variables in the table are Y="death penalty verdict", having two categories (yes, no), and X="race of defendant" and Z="race of victims", each having categories (white, black).

Death Penalty
Victim's Race
Defendant's Race
Yes
No
Total
White
White
53
414
467
Black
11
37
48
Black
White
0
16
16
Black
4
139
143
Total
White
53
430
483
Black
15
176
191

Source: M.L. Radelet and G.L. Pierce, Florida Law Rev. 43:1-34 (1991)

What type of effect the defendant's race have on the death penalty verdict?