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¼ the proportion of individuals in the population who

belong in category i of factor 1 and category j of factor 2

¼ Pða randomly selected individual falls in both category

i of factor 1 and category j of factor 2Þ

Then

i

¼Pða randomly selected individual falls in category i of factor 1)

j

¼Pða randomly selected individual falls in category j of factor 2)

Recall that two events A and B are independent if P (A \ B) ¼ P(A)·P(B). The

null hypothesis here says that an individual’s category with respect to factor 1 is

independent of the category with respect to factor 2. In symbols, this becomes

¼ p

i·

· p

·j

for every pair (i, j ).

The expected count in cell (i, j )isn · p

,sowhenH

is true, E(N

) ¼ n · p

i·

· p

·j

To obtain a chi-squared statistic, we must therefore estimate the p

i·

’s (i ¼ 1, ..., I )

and p

·j

’s ( j ¼ 1, ..., J ). The (maximum likelihood) estimates are

i

¼ sample proportion for category i of factor 1

and

j

¼ sample proportion for category j of factor 2

This gives estimated expec ted cell counts identical to those in the case of homo-

geneity.

¼ n 

i



j

¼ n 

i



j

i

 n

j

ðith row totalÞðjth column totalÞ

The test statistic is also identical to that used in testing for homogeneity, as is

the number of degrees of freedom. This is because the number of freely determined

cell counts is IJ – 1, sinc e only the total n is fixed in advance. There are I estimated

i·

’s, but only I – 1 are independently estimated since S p

i·

¼ 1, and similarly J –1

·j

’s are independently estimated, so I + J – two parameter s are independently

estimated. The rule of thumb now yields df ¼ IJ –1–(I + J –2)¼ IJ – I –

J +1¼ (I –1)·(J – 1).

748 CHAPTER 13 Goodness-of-Fit Tests and Categorical Data Analysis

Null hypothesis: H

: p

¼ p



 p

j

i ¼ 1; ...; I; j ¼ 1; ...; J

Alternative hypothesis: H

: H

is not true

Test statistic value:

all cells

ðobserved  estimated expectedÞ

estimated expected

i¼1

j¼1

ðn



Rejection region: w

 w

a;ðI1ÞðJ1Þ

Again, P-value information can be obtained as described in Section 13.1.

The test can safely be applied as long as

 5 for all cells.

Example 13.14 A study of the relationship between facility conditions at gasoline stations and

aggressiveness in the pricing of gasoline (“An Analysis of Price Aggressiveness in

Gasoline Marketing,” J. Market. Res., 1970: 36–42) reports the accompanying data

based on a sample of n ¼ 441 stations. At level .01, does the data suggest that

facility conditions and pricing policy are independent of one another? Observed

and estimated expected counts are given in Table 13.10.

Thus

ð24  17:02Þ

17:02

þþ

ð36  54:29Þ

54:29

¼ 22:47

and because w

:01;4

¼ 13:277, the hypothesis of independence is rejected.

We conclude that knowledge of a station’s pricing policy does give informa-

tion about the condition of facilities at the station. In particular, stations with an

aggressive pricing policy appear more likely to have substandard facilities than

stations with a neutral or nonaggressive policy.

■

Ordinal Factors and Logistic Regression

Sometimes a factor has ordinal categories, meaning that there is a natural ordering.

For example, there is a natural ordering to freshman, sophomore, junior, senior. In

such situations we can use a method that often has greater power to detect relation-

ships. Consider the case in which the first factor is ordinal and the other has two

categories. Denote by X the level of the first (ordinal) factor, the rows, which will

be the predictor in the model. Then Y designates the column, either one or two, and

Table 13.10 Observed and estimated expected counts for Example 13.14

13.3 Two-Way Contingency Tables 749

Y will be the dependent variable in the model. It is convenient for purposes of

logistic regression to label column 1 as Y ¼ 0 (failure) and column 2 as Y ¼ 1

(success), corresponding to the usual notation for binomial trials. In terms of

logistic regression, p(x) is the probability of success given that X ¼ x:

pðxÞ¼PðY ¼ 1jX ¼ xÞ¼P ðj ¼ 2ji ¼ xÞ¼

þ p

Then the logistic model of Chapter 12 says that

þb

pðxÞ

1  pðxÞ

In terms of the odds of success in a row (estimated by the ratio of the two counts),

the model says that the odds change proportionally (by the fixed multiple e

) from

row to row. For example, suppose a test is given in grades 1, 2, 3, and 4 with

successes and failures as follows

Grade Failed Passed Estimated Odds

14545 1

23060 2

31872 4

41080 8

Here the model fits perfectly, with odds ratio e

¼ 2, so b

¼ ln(2) and b

¼ln(2).

In general, it should be clear that b

is the natural log of the odds ratio between

successive rows. If a table with I rows and 2 columns has roughly a common odds ratio

from row to row, then the logistic model should be a good fit if the rows are labeled

with consecutive integers.

We focus on the slope b

because the relationship between the two factors

hinges on this parameter. The hypothesis of no relationship is equivalent to H

¼ 0, which is usually tested against a two-tailed alternative.

Example 13.15 Is there a relationship between TV watching and physical fitness? For an answer

we refer to the article “Television Viewing and Physical Fitness in Adults”

(Res. Quart. Exercise Sport, 1990: 315–320). Subjects were asked about their

television-viewing habits and were classified as physically fit if they scored in

the excellent or very good category on a step test. Table 13.11 shows the results in

the form of a 4  2 table. The TV column gives the hours per day

Table 13.11 TV versus fitness results

750 CHAPTER 13 Goodness-of-Fit Tests and Categorical Data Analysis

The rows need to be given specific numeric values for computational pur-

poses, and it is convenient to make these just 1, 2, 3, 4, because consecutive integers

correspond to the assumption of a common odds ratio from row to row. The

columns may need to be labeled as 0 and 1 for input to a program. The logist ic

regression results from MINITAB are shown in Figure 13.5, where the estimated

coefficient

for TV is given as –.29 and the odds ratio is given as .75 ¼ e

–.29

.This

means that, for each increase in TV watchi ng category, the odds of being fit decline

to about 3/4 of the previous value. There is a loss of 25% for each increment in TV.

The outpu t shows two tests for b

,az based on the ratio of the coefficient to

its estimated standard err or and G, which is based on a likelihood ratio test and

gives the chi-squared approximation for the difference of log likelihoods. The two

tests usually give very similar results, with G being approximately the square of z.

In this case they agree that the P-value is around .02, which means that we should

reject at the .05 level the hypothesis that b

¼ 0, and we can conclude that there is a

relationship between TV watching and fitness. Of course, the existence of a

relationship does not im ply anything about one causing the other. By the way, a

chi-squared test yields w

¼ 6.161 with 3 df, P ¼ .104, so with this test we would

not conclude that there is a relationship, even at the 10% level. There is an

advantage in using logistic regression for this kind of data.

Suppose there are two ordinal factors, each with more than two levels. This

too can be handled with logistic regression, but it requires a procedure called

ordinal logistic regression that allows an ordinal dependent variable. When one

factor is ordinal and the other is not, the analysis can be done with multinomial

(also called nominal or polytomous) logistic regression, which allows a non-ordinal

dependent variable.

Models and methods for analyzing data in which each individual is cate-

gorized with respect to three or more factors (multidimensional contingency tables)

are discusse d in several of the references in the chapter bibliography.

Figure 13.5 Logistic regression for TV versus ﬁtness ■

13.3 Two-Way Contingency Tables 751

Exercises Section 13.3 (23–35)

23. Reconsider the Cubs data of Exercise 56

in Chapter 10. Form a 2  2 table for the data

and use a w

statistic to test the hypothesis of

equal population proportions. The w

statistic

should be the square of the z statistic in Exer-

cise 56 of Chapter 10. How are the P-values

related?

24. The accompanying data refers to leaf marks

found on white clover samples selected from

both long-grass areas and short-grass areas

(“The Biology of the Leaf Mark Polymorphism

in Trifolium repens L.,” Heredity, 1976:

306–325). Use a w

test to decide whether the

true proportions of different marks are identical

for the two types of regions.

25. The following data resulted from an experiment

to study the effects of leaf removal on the ability

of fruit of a certain type to mature (“Fruit Set,

Herbivory, Fruit Reproduction, and the Fruiting

Strategy of Catalpa speciosa,” Ecology, 1980:

57–64). Does the data suggest that the chance

of a fruit maturing is affected by the number of

leaves removed? State and test the appropriate

hypotheses at level .01.

Treatment

Number

of Fruits

Matured

Number

of Fruits

Aborted

Control 141 206

Two leaves removed 28 69

Four leaves removed 25 73

Six leaves removed 24 78

Eight leaves removed 20 82

26. The article “Human Lateralization from Head to

Foot: Sex-Related Factors” (Science, 1978:

1291–1292) reports for both a sample of right-

handed men and a sample of right-handed

women the number of individuals whose feet

were the same size, had a bigger left than right

foot (a difference of half a shoe size or more), or

had a bigger right than left foot.

Does the data indicate that gender has a strong

effect on the development of foot asymmetry?

State the appropriate null and alternative hypoth-

eses, compute the value of w

, and obtain infor-

mation about the P-value.

27. The article “Susceptibility of Mice to Audio-

genic Seizure Is Increased by Handling Their

Dams During Gestation” (Science, 1976:

427–428) reports on research into the effect of

different injection treatments on the frequencies

of audiogenic seizures.

Does the data suggest that the true percentages

in the different response categories depend on

the nature of the injection treatment? State and

test the appropriate hypotheses using a ¼ .005.

28. The accompanying data on sex combinations of

two recombinants resulting from six different

male genotypes appears in the article “A New

Method for Distinguishing Between Meiotic and

Premeiotic Recombinational Events in Drosoph-

ila melanogaster”(Genetics, 1979: 543–554).

Does the data support the hypothesis that the

frequency distribution among the three sex com-

binations is homogeneous with respect to the dif-

ferent genotypes? Define the parameters of

interest, state the appropriate H

and H

,and

perform the analysis.

752

CHAPTER 13 Goodness-of-Fit Tests and Categorical Data Analysis

Sex Combination

M/M M/F F/F

Male 135 80 39

241 84 45

333 87 31

Genotype 4 8 26 8

5 5 11 6

630 65 20

29. Each individual in a random sample of high school

and college students was cross-classified with

respect to both political views and marijuana

usage, resulting in the data displayed in the accom-

panying two-way table (“Attitudes About Mari-

juana and Political Views,” Psych. Rep., 1973:

1,051–1,054). Does the data support the hypothesis

that political views and marijuana usage level are

independent within the population? Test the appro-

priate hypotheses using level of significance .01.

30. Show that the chi-squared statistic for the test of

independence can be written in the form

i¼1

j¼1

 n

Why is this formula more efficient computation-

ally than the defining formula for w

31. Suppose that in Exercise 29 each student had

been categorized with respect to political views,

marijuana usage, and religious preference, with

the categories of this latter factor being Protes-

tant, Catholic, and other. The data could be dis-

played in three different two-way tables, one

corresponding to each category of the third factor.

With p

ijk

¼ P(political category i, marijuana cat-

egory j, and religious category k), the null hypoth-

esis of independence of all three factors states that

ijk

¼ p

i··

·j·

··k

Let n

ijk

denote the observed

frequency in cell (i, j , k). Show how to estimate

the expected cell counts assuming that H

is true

(

ijk

¼ n

ijk

, so the

ijk

’s must be determined).

Then use the general rule of thumb to determine

the number of degrees of freedom for the chi-

squared statistic.

32. Suppose that in a particular state consisting of four

distinct regions, a random sample of n

voters is

obtained from the kth region for k ¼ 1, 2, 3, 4.

Each voter is then classified according to which

candidate (1, 2, or 3) he or she prefers and accord-

ing to voter registration (1 ¼ Dem., 2 ¼ Rep.,

3 ¼ Indep.). Let p

ijk

denote the proportion of

voters in region k who belong in candidate cate-

gory i and registration category j. The null hypoth-

esis of homogeneous regions is H

ij1

¼ p

ij2

¼ p

ij3

¼ p

ij4

for all i, j (i.e., the propor-

tion within each candidate/registration combina-

tion is the same for all four regions). Assuming

that H

is true, determine

ijk

and

ijk

as functions

of the observed n

ijk

’s, and use the general rule of

thumb to obtain the number of degrees of freedom

for the chi-squared test.

33. Consider the accompanying 2  3 table displaying

the sample proportions that fell in the various com-

binations of categories (e.g., 13% of those in the

sample were in the first category of both factors).

a. Suppose the sample consisted of n ¼ 100 peo-

ple. Use the chi-squared test for independence

with significance level .10.

b. Repeat part (a) assuming that the sample size

was n ¼ 1000.

c. What is the smallest sample size n for which

these observed proportions would result in

rejection of the independence hypothesis?

34. Use logistic regression to test the relationship

between leaf removal and fruit growth in Exer-

cise 25. Compare the P-value with what was

found in Exercise 25. (Remember that w

¼ z

Explain why you expected the logistic regression

to give a smaller P-value.

35. A random sample of 100 faculty at a university

gives the results shown below for professorial

rank versus gender.

a. Test for a relationship at the 5% level using a

chi-squared statistic.

b. Test for a relationship at the 5% level using

logistic regression.

13.3 Two-Way Contingency Tables 753

c. Compare the P-values in parts (a) and (b). Is

this in accord with your expectations? Explain.

d. Interpret your results. Assuming that today’s

assistant professors are tomorrow’s associate

professors and professors, do you see implica-

tions for the future?

Rank Male Female

Professor 25 9

Assoc Prof 20 8

Asst Prof 18 20

Supplementary Exercises (36–47)

36. The article “Birth Order and Political Success”

(Psych. Rep., 1971: 1,239–1,242) reports that

among 31 randomly selected candidates for

political office who came from families with

four children, 12 were firstborn, 11 were mid-

dleborn, and 8 were lastborn. Use this data to

test the null hypothesis that a political candidate

from such a family is equally likely to be in any

one of the four ordinal positions.

37. The results of an experiment to assess the effect

of crude oil on fish parasites are described in the

article “Effects of Crude Oils on the Gastrointes-

tinal Parasites of Two Species of Marine Fish”

(J. Wildlife Diseases, 1983: 253–258). Three

treatments (corresponding to populations in the

procedure described) were compared: (1) no con-

tamination, (2) contamination by 1–year-old

weathered oil, and (3) contamination by new

oil. For each treatment condition, a sample of

fish was taken, and then each fish was classified

as either parasitized or not parasitized. Data com-

patible with that in the article is given. Does the

data indicate that the three treatments differ with

respect to the true proportion of parasitized and

nonparasitized fish? Test using a ¼ .01.

Treatment Parasitized Nonparasitized

Control 30 3

Old oil 16 8

New oil 16 16

38. Qualifications of male and female head and

assistant college athletic coaches were compared

in the article “Sex Bias and the Validity of

Believed Differences Between Male and Female

Interscholastic Athletic Coaches” (Res. Q. Exer-

cise Sport, 1990: 259–267). Each person in ran-

dom samples of 2225 male coaches and 1141

female coaches was classified according to num-

ber of years of coaching experience to obtain the

accompanying two-way table. Is there enough

evidence to conclude that the proportions falling

into the experience categories are different for

men and women? Use a ¼ .01.

Years of Experience

Gender 1–3 4–6 7–9 10–12 13+

Male 202 369 482 361 811

Female 230 251 238 164 258

39. The authors of the article “Predicting Profes-

sional Sports Game Outcomes from Intermediate

Game Scores” (Chance, 1992: 18–22) used a chi-

squared test to determine whether there was any

merit to the idea that basketball games are not

settled until the last quarter, whereas baseball

games are over by the seventh inning. They

also considered football and hockey. Data was

collected for 189 basketball games, 92 baseball

games, 80 hockey games, and 93 football games.

The games analyzed were sampled randomly

from all games played during the 1990 season

for baseball and football and for the 1990–1991

season for basketball and hockey. For each game,

the late-game leader was determined, and then it

was noted whether the late-game leader actually

ended up winning the game. The resulting data is

summarized in the accompanying table.

Sport

Late-Game

Leader Wins

Late-Game

Leader Loses

Basketball 150 39

Baseball 86 6

Hockey 65 15

Football 72 21

The authors state, “Late-game leader is defined

as the team that is ahead after three quarters in

basketball and football, two periods in hockey,

and seven innings in baseball. The chi-square

754

CHAPTER 13 Goodness-of-Fit Tests and Categorical Data Analysis

value on three degrees of freedom is 10.52

(P < .015).”

a. State the relevant hypotheses and reach a

conclusion using a ¼ .05.

b. Do you think that your conclusion in part (a)

can be attributed to a single sport being an

anomaly?

40. The accompanying two-way frequency table

appears in the article “Marijuana Use in

College” (Youth and Society, 1979: 323–334).

Each of 445 college students was classified

according to both frequency of marijuana

use and parental use of alcohol and psychoactive

drugs. Does the data suggest that parental usage

and student usage are independent in

the population from which the sample was

drawn? Use the P-value method to reach a con-

clusion.

41. In a study of 2989 cancer deaths, the location of

death (home, acute-care hospital, or chronic-care

facility) and age at death were recorded, resulting

in the given two-way frequency table (“Where

Cancer Patients Die,” Public Health Rep., 1983:

173). Using a .01 significance level, test the null

hypothesis that age at death and location of death

are independent.

Location

Age Home Acute-Care Chronic-Care

15–54 94 418 23

55–64 116 524 34

65–74 156 581 109

Over 74 138 558 238

42. In a study to investigate the extent to which

individuals are aware of industrial odors in a

certain region (“Annoyance and Health Reac-

tions to Odor from Refineries and Other Indus-

tries in Carson, California,” Environ. Res., 1978:

119–132), a sample of individuals was obtained

from each of three different areas near industrial

facilities. Each individual was asked whether he

or she noticed odors (1) every day, (2) at least

once/week, (3) at least once/month, (4) less often

than once/month, or (5) not at all, resulting in the

output from SPSS on the next page. State and test

the appropriate hypotheses.

43. Many shoppers have expressed unhappiness

because grocery stores have stopped putting

prices on individual grocery items. The article

“The Impact of Item Price Removal on Grocery

Shopping Behavior” (J. Market., 1980: 73–93)

reports on a study in which each shopper in a

sample was classified by age and by whether he

or she felt the need for item pricing. Based on the

accompanying data, does the need for item pric-

ing appear to be independent of age?

Age

< 30 30–39 40–49 50–59  60

Number

in Sample

150 141 82 63 49

Number

Who Want

127 118 77 61 41

Item Pricing

44. Let p

denote the proportion of successes in a

particular population. The test statistic value in

Chapter 9 for testing H

: p

¼ p

was z ¼

 p

Þ=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

,wherep

¼ 1–p

Show that for the case k ¼ 2, the chi-squared sta-

tistic value of Section 13.1 satisfies w

¼ z

.[Hint:

First show that (n

– np

)

¼ (n

– np

)

45. The NCAA basketball tournament begins with 64

teams that are apportioned into four regional tour-

naments, each involving 16 teams. The 16 teams

in each region are then ranked (seeded) from 1 to

16. During the 12-year period from 1991 to 2002,

the top-ranked team won its regional tournament

22 times, the second-ranked team won 10 times,

the third-ranked team won 5 times, and the

remaining 11 regional tournaments were won

by teams ranked lower than 3. Let P

denote

the probability that the team ranked i in its region

is victorious in its game against the team ranked

j. Once the P

’s are available, it is possible to

compute the probability that any particular seed

wins its regional tournament (a complicated

calculation because the number of outcomes

13.3 Supplementary Exercises 755

in the sample space is quite large). The paper

“Probability Models for the NCAA Regional

Basketball Tournaments”(Amer. Statist., 1991:

35–38) proposed several different models for

the P

’s.

a. One model postulated P

¼ .5 – l(i – j) with

l ¼

(from which P

16;1

, P

16;2

etc.). Based on this, P(seed #1 wins) ¼ .27477,

P(seed #2 wins) ¼ .20834, and P(seed #3

wins) ¼ .15429. Does this model appear to

provide a good fit to the data?

b. A more sophisticated model has P

¼ .5 +

.2813625(z

– z

), where the z’s are measures

of relative strengths related to standard normal

percentiles [percentiles for successive highly

seeded teams are closer together than is the

case for teams seeded lower, and .2813625

ensures that the range of probabilities is the

same as for the model in part (a)]. The resulting

probabilities of seeds 1, 2, or 3 winning their

regional tournaments are .45883, .18813, and

.11032, respectively. Assess the fit of this

model.

46. Have you ever wondered whether soccer players

suffer adverse effects from hitting “headers”?

The authors of the article “No Evidence of

Impaired Neurocognitive Performance in Colle-

giate Soccer Players” (Amer. J. Sports Med.

2002: 157–162) investigated this issue from

several perspectives.

a. The paper reported that 45 of the 91 soccer

players in their sample had suffered at least one

concussion, 28 of 96 nonsoccer athletes had suf-

fered at least one concussion, and only 8 of 53

student controls had suffered at least one con-

cussion. Analyze this data and draw appropriate

conclusions.

b. For the soccer players, the sample correlation

coefficient calculated from the values of

x ¼ soccer exposure (total number of

competitive seasons played prior to enrollment

in the study) and y ¼ score on an immediate

memory recall test was r ¼–.220. Interpret this

result.

c. Here is summary information on scores on a

controlled oral word-association test for the

soccer and nonsoccer athletes:

¼ 26;



¼ 37:50; s

¼ 9:13;

¼ 56;



¼ 39:63; s

¼ 10:19

Analyze this data and draw appropriate conclu-

sions.

Crosstabulation: AREA By CATEGORY

756 CHAPTER 13 Goodness-of-Fit Tests and Categorical Data Analysis

d. Considering the number of prior nonsoccer

concussions, the values of mean  SD for the

three groups were soccer players, .30  .67;

nonsoccer athletes, .49  .87; and student con-

trols, .19  .48. Analyze this data and draw

appropriate conclusions.

47. Do the successive digits in the decimal expansion

of p behave as though they were selected from a

random number table (or came from a computer’s

random number generator)?

a. Let p

denote the long-run proportion of digits

in the expansion that equal 0, and define p

, ...,

analogously. What hypotheses about these

proportions should be tested, and what is df for

the chi-squared test?

b. H

of part (a) would not be rejected for the

nonrandom sequence 012 ... 901 ... 901 ... .

Consider nonoverlapping groups of two digits,

and let p

denote the long-run proportion of

groups for which the first digit is i and the

second digit is j. What hypotheses about these

proportions should be tested, and what is df for

the chi-squared test?

c. Consider nonoverlapping groups of 5 digits.

Could a chi-squared test of appropriate hypoth-

eses about the p

ijklm

’s be based on the first

100,000 digits? Explain.

d. The paper “Are the Digits of p an Independent

and Identically Distributed Sequence?” (Amer.

Statist., 2000: 12–16) considered the first

1,254,540 digits of p, and reported the follow-

ing P-values for group sizes of 1, ..., 5 digits:

.572, .078, .529, .691, .298. What would you

conclude?

Bibliography

Agresti, Alan, An Introduction to Categorical Data

Analysis (2nd ed.), Wiley, New York, 2007. An

excellent treatment of various aspects of categori-

cal data analysis by one of the most prominent

researchers in this area.

Everitt, B. S., The Analysis of Contingency Tables (2nd

ed.), Halsted Press, New York, 1992. A compact

but informative survey of methods for analyzing

categorical data, exposited with a minimum of

mathematics.

Mosteller, Frederick, and Richard Rourke, Sturdy Sta-

tistics, Addison-Wesley, Reading, MA, 1973. Con-

tains several very readable chapters on the varied

uses of chi-square.

Bibliography 757