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560 14 Models for Tables

This hypothesis states that the population proportions of the three categories

are the same for the two groups Before and After. The alternative is a negation

of H

: (p

, p

) 6=(p

, p

The test statistic is

1·

−n

·1

)

2·

−n

·2

)

3·

−n

·3

)

2(n

)

, (14.6)

where

i j

=(n

i j

)/2.

The statistic

in (14.6) has a χ

-distribution with 2 degrees of freedom

and its large values are critical.

Rule: Reject H

at signiﬁcance level α if χ

> χ

2,1

−α

, where χ

2,1

−α

is a 1 − α quantile of a χ

-distribution with 2 degrees of freedom

(

chi2inv(1-alpha, 2)). The p-value of this test is 1-chi2cdf(chi2, 2) for

chi2 as in (14.6).

Example 14.14. Speckmann (1965) and Lawal (2003) provide data on the reli-

gious afﬁliation of husbands and wives in 264 marriages in Surinam.

Wife

Christian Muslim Hindu Total

Christian 17 1 7 25

Husband Muslim

1 66 6 73

Hindu

9 6 151 166

Total 27 73 164 264

Here Hindu counts combine Sanatan Dharm and Arya Samaj afﬁliations. Are

the population proportions of husbands who are Christian, Muslim, or Hindu

equal to the proportions of wives who belong to those three religious groups?

Note that the data are paired and the Stuart–Maxwell test is appropriate.

For the data in the table,

in (14.6) is 28/124 =0.2258. At 2 degrees of free-

dom, the p-value of the test is 0.8932 (as

1-chi2cdf(0.2258, 2). Thus, H

not rejected, the proportions of these three religious groups in the population

of Surinam husbands are not different than the proportions among the wives.

14.7 Exercises 561

14.7 Exercises

14.1. Amoebas and intestinal disease. When an epidemic of severe intestinal

disease occurred among workers in a plant in South Bend, Indiana, doctors

said that the illness resulted from infection by the amoeba Entamoeba his-

tolytica. There are actually two races of these amoebas, large and small, and

the large ones were believed to be causing the disease. Doctors suspected

that the presence of the small ones might help people resist infection by the

large ones. To check on this, public health ofﬁcials chose a random sample of

138 apparently healthy workers and determined if they were infected with

either the large or small amoebas. The table below [given by Cohen (1973)]

provides the resulting data. Is the presence of the large race independent

of the presence of the small one? Test at 5% signiﬁcance level.

Large race

Small race Present Absent Total

Present 12 23 35

Absent 35 68

103

Total 47 91 138

14.2. Drinking and Smoking. Alcohol and nicotine consumption during preg-

nancy are believed to be associated with certain characteristics in children.

Since drinking and smoking behaviors may be related, it is important to

understand the nature of this relationship when fully assessing their inﬂu-

ence on children. In a study by Streissguth et al. (1984), 452 mothers were

classiﬁed according to their alcohol intake prior to pregnancy recognition

and their nicotine intake during pregnancy. The data are summarized in

the following table.

Nicotine (mg/day)

Alcohol (ounces/day) None 1–15 16 or more

None 105 7 11

0.01–0.10 58 5 13

0.11–0.99 84 37 42

1.00 or more 57 16 17

(a) Calculate the column sums. In what way does the pattern of alcohol

consumption vary with nicotine consumption?

(b) Calculate the row sums. In what way does the pattern of nicotine con-

sumption vary with alcohol consumption?

and H

for assessing whether or not alcohol consumption

and nicotine consumption are independent.

(d) Compute the table of expected counts.

562 14 Models for Tables

(e) Find the χ

statistic. Report the degrees of freedom and the p-value.

(f) What do you conclude from the analysis of this table?

14.3. Alcohol and Marriage. A national survey was conducted to obtain infor-

mation on the alcohol consumption patterns of American adults by marital

status. A random sample of 1303 residents 18 years old and over yielded

the data below. Do the data suggest at a 5% signiﬁcance level that marital

status and alcohol consumption patterns are statistically dependent?

Abstain 1–60 Over 60 Total

Single 67 213 74 354

Widowed 85 633 129 847

Divorced 27 60 15 102

Total 179 906 218 1303

14.4. Family Size. A demographer surveys 1000 randomly chosen American

families and records their family sizes and family incomes:

Family Size

Family Income

2 3 4 5 6 7

Low 145 81 57 22 9 8

Middle

151 73 71 33 13 10

High

124 60 80 42 13 8

Do the data provide sufﬁcient evidence to conclude that family size and

family income are statistically dependent?

(a) State the H

and H

hypotheses.

(b) Perform the test. Use

α =0.05. Comment.

14.5. Nightmares. Over the years numerous studies have sought to charac-

terize the nightmare sufferer. From these studies has emerged the stereo-

type of someone with high anxiety, low ego strength, feelings of inadequacy,

and poorer-than-average physical health. A study by Hersen (1971) ex-

plored whether gender is independent of having frequent nightmares. Us-

ing Hersen’s data summarized in the table below, test the hypothesis of

independence at level

α =0.05.

Men Women

Nightmares often (at least once a month) 55 60

Nightmares seldom (less than once in a month) 105 132

14.6. Independence of Segregation. According to a mathematical model of

inheritance, at a given meiosis, the probability of an allele at one locus

passing to the gamete is independent of an allele at any locus on another

chromosome passing to the gamete. This is usually referred as the law of

independent segregation of genes. Also, one allele from the pair at any locus

passes to the gamete with probability equal to 1/2.

14.7 Exercises 563

Roberts et al. (1939) conducted an extensive set of experiments for testing

independent segregation of the genes in mice and rats. One of results of a

mating of the form Aa Bb Dd

×aa bb dd is reported as:

ab aB Ab AB Total

d 427 440 509 460 1836

D 494 467 462 475 1898

Total 921 907 971 935 3734

Using χ

statistic test for the independence of segregation at α =5%.

Hint.

The expected number in each cell is 1/2 ×1/2 ×1/2 ×3734 =466.75.

14.7. Site of Corpus Luteum in Caesarean Births. Williams (1921) observed

that in Caesarean-section births, the corpus luteum was located in the right

ovary 23 times and 16 times in the left for male children. For female chil-

dren the numbers were 13 and 12, respectively. Test for the independence

of ovary side and gender of a child.

14.8. An Easy Grade? A student wants to take a statistics course with a pro-

fessor who is an easy grader. There are three professors scheduled to teach

the course sections next semester. The student manages to obtain a random

sample of grades given by the three professors this past year.

Observed Prof A Prof B Prof C Total

Grades A 10 12 28 50

Grades B 35 30 15 80

Grades C 15 30 25 70

Total 60 72 68 200

Using a signiﬁcance level of 1%, test the hypothesis that a student’s grade

is independent of the professor.

14.9. Importance of Bystanders. When a group of people is confronted with an

emergency, a diffusion-of-responsibility process can interfere with an indi-

vidual’s responsiveness. Darley and Latané (1968) asked subjects to partic-

ipate in a discussion carried over an intercom. Aside from the experimenter

to whom they were speaking, subjects thought that there were zero, one, or

four other people (bystanders) also listening over the intercom. Part way

through the discussion, the experimenter feigned serious illness and asked

for help. Darley and Latané noted how often the subject sought help for the

experimenter as a function of the number of supposed bystanders. The data

are summarized in the table:

Sought assistance No assistance

No bystanders 11 2

One bystander 16 10

Four bystanders 4 9

564 14 Models for Tables

What could Darley and Latané conclude from the results?

(a) State H

and H

(b) Perform the test at a 5% signiﬁcance level.

14.10. Baseball in 2003. As part of baseball’s attempt to make the All-Star Game

more meaningful, baseball Commissioner Bud Selig pushed through a pro-

posal to give the winning league home-ﬁeld advantage in the World Series.

The proposal, while controversial to many baseball purists, was just one of

many changes instituted by Selig since he became commissioner in 1998.

Despite Selig’s efforts, which are generally rated positively by baseball fans,

sports fans are most likely to say Major League Baseball has the most se-

rious problems of the major professional sports leagues. And nearly 4 in

10 sports fans say baseball is currently in a state of crisis or has major

problems. Nevertheless, the majority of sports fans say they are following

baseball as much or more than they did 3 years ago. Our interest is in how

the opinions of fans who are particular about baseball compare to those of

general sports fans regarding the Major League events. The question to 140

declared fans was: Are you following Major League Baseball more closely

than you did three years ago, less closely, or about the same as before?

More Less Same Total

Sports Fans 6 19 32 57

Baseball Fans 13 25 45 83

Total 19 44 77 140

Test the hypothesis that the type of fan is independent of a change in inter-

est level. Use

α =5%.

14.11. Psychosis in Adopted Children. Numerous studies have been done to

determine the etiology of schizophrenia. Such factors as biological, psy-

chocultural, and sociocultural inﬂuences have been suggested as possible

causes of this disorder. To test if schizophrenia has a hereditary compo-

nent, researchers compared adopted children whose biological mothers are

schizophrenic (“exposure”) to adopted children whose biological mothers are

normal (“nonexposure”).

Furthermore, the child-rearing abilities of adoptive families have been as-

sessed to determine if there is a relationship between those children who

become psychotic and the type of family into which they are adopted.

The families are classiﬁed as follows: healthy, moderately disturbed, and

severely disturbed.

The following data are from an experiment described in Carson and Butcher

(1992).

Type of Adoptive Family

Healthy Moderately Disturbed Severely Disturbed

Diagnosis Exp Nonexp Exp Nonexp Exp Nonexp

None 41 42 11 26 6 15

Psychotic 10 11 18 25 38 28

14.7 Exercises 565

(a) For moderately disturbed families ﬁnd the risk difference for a child’s

psychosis with the mother’s schizophrenia as a risk factor. Find a 95% con-

ﬁdence interval for the risk difference.

(b) For severely disturbed families ﬁnd the risk ratio and odds ratio for

psychosis with the mother’s schizophrenia as a risk factor. Find a 95% con-

ﬁdence interval for the risk ratio and odds ratio.

overall? Assess this using the Mantel–Haenszel test. Find the overall odds

ratio and corresponding 95% conﬁdence interval.

14.12. The Midtown Manhattan Study. The data set below has been analyzed

by many authors (Haberman, Goodman, Agresti, etc.) and comes from the

study by Srole et al. (1962). The data cross-classiﬁes 1660 young New York

residents with respect to two factors: Mental Health and parents’ Socioeco-

nomic Status (SES).

The Mental Health factor is classiﬁed by four categories: Well, Mild Symp-

tom Formation, Moderate Symptom Formation, and Impaired. The parents’

SES has six categories ranging from High (1) to Low (6).

Here is the table:

Well Mild Moderate Impaired

1 (High) 64 94 58 46

2 57 94 54 40

3 57 105 65 60

4 72 141 77 94

5 36 97 54 78

6 (Low) 21 71 54 71

(a) Test the hypothesis that Mental Health and parents’ SES are indepen-

dent factors at a 5% conﬁdence level. You can use

tablerxc.m code.

(b) The table below provides the expected frequencies. Explain what the

expected frequencies are. Explain how the number 104.0807 from the table

was obtained and show your work.

Well Mild Moderate Impaired

1 (High) 48.4542 95.0145 57.1349 61.3964

2 45.3102 88.8494 53.4277 57.4127

3 53.0777 104.0807 62.5867 67.2548

4 71.0169 139.2578 83.7398 89.9855

5 49.0090 96.1024 57.7892 62.0994

6 (Low) 40.1319 78.6952 47.3217 50.8512

14.13. Tonsillectomy and Hodgkin’s Disease. A study by Johnson and John-

son (1972) involved 85 patients with Hodgkin’s disease. Each of these had a

normal sibling (one who did not have the disease). In 26 of these pairs, both

individuals had had tonsillectomies (T); in 37 pairs, both individuals had

not had tonsillectomies (N); in 15 pairs, only the normal individual had had

566 14 Models for Tables

a tonsillectomy; in 7 pairs, only the one with Hodgkin’s disease had had a

tonsillectomy.

Normal/T Normal/N Total

Patient/T 26 15 41

Patient/N 7 37 44

Total 33 52 85

A goal of the study was to determine whether there was a link between

the disease and having had a tonsillectomy: Is the proportion of those who

had tonsillectomies the same among those with Hodgkin’s disease as among

those who do not have it?

14.14. School Spirit at Duke. Duke has always been known for its great

school spirit and support of its athletic teams, as evidenced by the famous

Cameron Crazies. One way that school enthusiasm is shown is by donning

Duke paraphernalia including shirts, hats, shorts, and sweatshirts. The

students’ project in an introductory statistics class was to explore possible

links between school spirit, measured by the number of students wearing

paraphernalia, and some other attributes. It was hypothesized that men

would wear Duke clothes more frequently than women. The data were col-

lected on the Bryan Center walkway starting at a random hour on ﬁve dif-

ferent days. Each day 100 men and 100 women were tallied.

Duke Paraphenalia No Duke Paraphenalia Total

Male 131 369 500

Female 52 448 500

Total 183 817 1000

14.15. Two Halloween Questions with Easy Answers. A study was designed

to test whether or not aggression is a function of anonymity. The study was

conducted as a ﬁeld experiment on Halloween (Fraser, 1974); 300 children

were observed unobtrusively as they made their rounds. Of these 300 chil-

dren, 173 wore masks that completely covered their faces, while 127 wore

no masks. It was found that 101 children in the masked group displayed ag-

gressive or antisocial behavior versus 36 children in the unmasked group.

(a) Are anonymity and aggression independent? Use

α =0.01.

(b) If p

is the (population) proportion for aggressive behavior for subjects

wearing a mask and p

is the proportion for subjects not wearing a mask,

ﬁnd a 95% conﬁdence interval for the odds ratio:

/(1 − p

)

/(1 − p

)

14.16. Runners and Heart Attack. The inﬂuence of running on preventing

heart attacks has been studied by a local runners club. The following two-

way table classiﬁes 350 people as runners or nonrunners, and whether or

14.7 Exercises 567

not they have had a heart attack. The factors are runner status and history

of heart attack.

Heart attack No heart attack Total

Runner 12 112 124

Nonrunner 36 190 226

Total 48 302 350

(a) Test for the independence of factors. Use α =0.05.

(b) Explain in words (in terms of this problem) what constitutes errors of

the ﬁrst and second kind in the above testing.

14.17. Perceptions of Dangers of Smoking. A poll was conducted to determine

if perceptions of the hazards of smoking were dependent on whether or not

the person smoked. One hundred (100) smokers and 100 nonsmokers were

randomly selected and surveyed. The results are given below.

Smoking is: Very Somewhat Not

Dangerous Dangerous Dangerous Dangerous

Smokers 21 (35.5) 29 (30) 29 ( ) 21 ( )

Nonsmokers 50 (35.5) 31 ( ) 11 ( ) 8 ( )

Test the hypothesis that smoking status does not affect perception of the

dangers of smoking at

α = 0.05 (three theoretical frequencies are given in

parentheses).

14.18. Red Dye No. 2. Fienberg (1980) discusses an experiment in which the food

additive Red Dye No. 2 was fed to two groups of rats at various dosages.

Some rats died during the experiment, which lasted 131 weeks, and the

remainder were sacriﬁced at the end of the 131st week. All rats were exam-

ined for tumors.

Age of Death 0–131 weeks Terminal sacriﬁce

Dosage Low High Low High

Tumor present 4 7 0 7

Tumor absent 26 16 14 14

Test the dependence among the three factors.

14.19. Cooper Hawks. The data in the table below originates from a study de-

signed to assess whether tape-recorded alarm calls enhanced the detectabil-

ity of Cooper’s hawks (Accipiter cooperii) (Rosenﬁeld et al., 1988). Observa-

tions were made on each sampling transect both with and without recorded

calls. Thus, the observations are paired rather than independent.

Detected Not detected Total # Transects

Tapes used 9 9 18

Tapes not used 4 14 18

Total 13 23 36

568 14 Models for Tables

According to Engeman and Swanson (2003), Rosenﬁeld et al. (1988) have

assumed that the observations under the two conditions from each transect

could be considered independent. Their analysis for the nestling stage data

uses Fisher’s exact test and results in a one-tailed p-value of 0.08. They con-

cluded that during the nestling stage, broadcast recordings “can markedly

increase the chance of detecting Cooper’s hawks near their nests.” However,

an appropriate analysis would have been McNemar’s test for paired data.

Engeman and Swanson (2003) continue: “. . . let us presume that for some

reason we could assume that observations with and without recorded calls

were independent rather than paired. Then the data set, by all criteria of

which we are aware, still is of sufﬁcient size to apply Pearson’s

, rather

than Fisher’s exact test as the authors have done.”

We note that rather than size, the design of an experiment is critical for a

suggestion to use Fisher’s exact test. The marginals should be ﬁxed before

the experiment; this is not the case here.

Find the p-value for the correct McNemar test.

14.20. Hepatic Arterial Infusion. Allen-Mersh et al. (1994), as well as Spiegel-

halter et al. (2004), reported the results of a trial in which patients under-

going therapy for liver metastasis were randomized to receive it either sys-

tematically (as is standardly done) or via hepatic arterial infusion (HAI). Of

51 randomized to HAI, 44 died, and of 49 randomized to systematic therapy,

46 died. Estimate the log odds ratio and ﬁnd a 95% conﬁdence interval.

14.21. Vaccine Efﬁcacy Study. Consider the data from a vaccine efﬁcacy study

(Chan, 1998). In a randomized clinical trial of 30 subjects, 15 were inoc-

ulated with a recombinant DNA inﬂuenza vaccine and the remaining 15

were inoculated with a placebo. Twelve of the 15 subjects in the placebo

group (80%) eventually became infected with inﬂuenza, whereas for the

vaccine group, only 7 of the 15 subjects (47%) became infected. Suppose p

is the probability of infection for the vaccine group and p

is the probability

of infection for the placebo group.

What is the one-sided p-value for testing the null hypothesis H

: p

= p

obtained by Fisher’s exact test?

MATLAB FILES USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch14.Tables/

concoef.m, fisherexact.m, hindu.m, kappa1.m, mantelhaenszel.m,

mcnemart.m, partialrxcxp.m, PAS.m, ponvexam.m, stuartmaxwell.m,

table2x2.m, tablerxc.m, tablerxcxp.m, unmatch.m

Chapter References 569

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