Heiman G. Basic Statistics for the Behavioral Sciences

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348 CHAPTER 14 / The Two-Way Analysis of Variance

20. You conduct an experiment involving two levels of self-confidence ( is low and

is high) and examine participants’ anxiety scores after they speak to one of

four groups of differing sizes ( through represent speaking to a small,

medium, large, or extremely large group, respectively). You compute the follow-

ing sums of squares ( and ):

Source Sum of Squares df Mean Square F

Between

Factor A 8.42

㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Factor B 76.79 㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Interaction 23.71 㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Within 110.72 㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Total 219.64 㛬㛬㛬㛬㛬㛬

(a) Complete the ANOVA summary table. (b) With , what do you conclude

about each ? (c) Compute the appropriate values of HSD. (d) For the levels of

factor B, the means are , , , and What

should you conclude about the main effect of B? (e) How important is the size of the

audience in determining a person’s anxiety score? How important is the person’s

self-confidence?

21. You measure the dependent variable of participants’ relaxation level as a function

of whether they meditate before being tested, and whether they were shown a film

containing a low, medium, or high amount of fantasy. Perform all appropriate sta-

tistical analyses, and determine what you should conclude about this study.

Amount of Fantasy

Low Medium High

57 9

Meditation 65 8

2610

2910

5510

No 10 2 5

Meditation 10 5 6

94 5

10 3 7

10 2 6

INTEGRATION QUESTIONS

22. What does a 95% confidence interval for indicate? (Ch.11)

23. What does “significant” indicate about the results of any study? (Ch. 10)

24. (a) How do you recognize the independent variable in a study? (b) How do you

recognize the levels in a study? (c) How do you recognize the dependent variable

in a study? (Chs. 2, 12, 13, 14)

25. To select a statistical procedure for an experiment, what must you ask about how

participants are selected? (Chs. 12, 13, 14)



5 28.3.X

5 24.6X

5 20.02X

5 18.36

obt

 5 .05

N 5 32n 5 4

Integration Questions 349

26. (a) How do you recognize a two-sample t-test design? (b) What must be true

about the dependent variable? (c) Which versions of the t-test are available?

(Chs. 2, 10, 12)

27. (a) How do you recognize a correlational design? (b) To select a correlation coef-

ficient, what must you ask about the variables? (c) What is the parametric correla-

tion coefficient? (d) What is the nonparametric correlation coefficient? (Chs. 2, 7,

10, 11)

28. (a) How do you recognize a design that requires a one-way ANOVA? (b) What

must be true about the dependent variable? (c) Which versions of ANOVA are

available? (Chs. 10, 12 13)

29. (a) How do you recognize a design that fits a two-way ANOVA? (b) What must be

true about the dependent variable? (c) Which versions of ANOVA are available?

(Chs. 10, 12, 14)

30. For the following, identify the factor(s), the primary inferential procedure to

perform and the key findings we’d look for. If a correlational design, indicate the

predictor and criterion. If no parametric procedure can be used, indicate why.

(Chs. 7, 8, 10, 11, 12, 13, 14)

(a) We measure babies’ irritability when their mother is present and when she is

absent. (b) We test the driving ability of participants who are either high, medium,

or low in the personality trait of “thrill seeker.” For each type, we test some

participants who have had either 0, 1, or 2 oz. of alcohol. (c) Parents with

alcoholism may produce adult children who are more prone to alcoholism. We

compare the degree of alcoholism in participants with alcoholic parents to those

with nonalcoholic parents. (d) To study dark adaptation, participants were asked

to identify stimuli after sitting in a dim room for 1 minute, again after 15 minutes,

and again after 30 minutes. (e) We study whether people who smoke cigarettes are

more prone to be drug abusers. We identify participants who are smokers or non-

smokers, and for each, count the number who are high or low drug abusers. (f) To

test if creativity scores change with age, we test groups of 5-, 10-, or 15-year-olds.

We also identify them as Caucasian or non-Caucasian to determine if age-related

changes in creativity depend on race. (g) As in part (f) we measure the age scores

and creativity scores of all students in the school district, again separating

Caucasian and non-Caucasian. (Chs. 8, 12, 13, 14)

■ ■ ■ SUMMARY OF

FORMULAS

The general format for the summary table for a two-way, between-subjects ANOVA is

Summary Table of Two-Way ANOVA

Source Sum of Squares df Mean Square F

Between

Factor A SS

Factor B SS

Interaction SS

AB

Within SS

Total SS

tot

350 CHAPTER 14 / The Two-Way Analysis of Variance

1. To perform the two-way, between-subjects ANOVA

a. Compute the sum of squares,

b. Compute the degrees of freedom,

i. The between groups for factor A

equals , where is the number of

levels in factor A.

ii. The df between groups for factor B

equals , where is the number of lev-

els in factor B.

iii. The df between groups for the interaction

equals multiplied times .

iv. The df within groups equals

, where is the total of the

study and is the total number of cells in

the study.

c. Compute the mean square,

A3B

NNN 2 k

A3B

1df

A3B

2 1

1df

2 1

1df

2df

5 SS

tot

2 SS

A3B

5 SS

2 SS

5 © a

1Sum of scores in the row2

n of scores in the row

b2 a

1©X

tot

5 © a

1Sum of scores in the row2

n of scores in the row

b2 a

1©X

tot

2 a

1©X

tot

5 © a

1Sum of scores in the column2

n of scores in the column

tot

5 ©X

tot

2 a

1©X

tot

d. Compute ,

Critical values of are found in Table 5 of

Appendix C.

2. The formula for Tukey’s HSD post hoc

comparisons is

a. For a significant main effect: Find in Table

6 in Appendix C for k equal to the number of

levels in the factor. In the formula, n is the

number of scores used to compute each main

effect mean in the factor.

b. For a significant interaction effect: Determine

the adjusted k using the small table at the top

of Table 6 in Appendix C. Find in Table 6

using the adjusted k. In the formula for HSD,

n is the number of scores in each cell.

3. The formula for eta squared is



Sum of squares between groups for the factor

tot

HSD 5 1q

. b

A3B

obt

351

Chi Square and Other

Nonparametric Procedures

GETTING STARTED

To understand this chapter, recall the following:

■

From Chapter 2, the four types of measurement scales (nominal, ordinal,

interval, and ratio).

■

From Chapter 12, the types of designs that call for either the independent-

samples t-test or the related-samples t-test.

■

From Chapter 13, the one-way ANOVA, post hoc tests, and eta squared.

■

From Chapter 14, what a two-way interaction indicates.

Your goals in this chapter are to learn

■

When to use nonparametric statistics.

■

The logic and use of the one-way chi square.

■

The logic and use of the two-way chi square.

■

The nonparametric procedures corresponding to the independent-samples and

related-samples t-test and to the between-subjects and within-subjects ANOVA.

Previous chapters have discussed the category of inferential statistics called parametric

procedures. Now we’ll turn to the other category, called nonparametric procedures.

Nonparametric procedures are still inferential statistics for deciding whether the differ-

ences between samples accurately represent differences in the populations, so the logic

here is the same as in past procedures. What’s different is the type of dependent vari-

able involved. In this chapter, we will discuss (1) two common procedures used with

nominal scores called the one-way and two-way chi square and (2) review several less

common procedures used with ordinal scores.

WHY IS IT IMPORTANT TO KNOW ABOUT NONPARAMETRIC PROCEDURES?

Previous parametric procedures have required that dependent scores reflect an interval

or ratio scale, that the scores are normally distributed, and that the population variances

are homogeneous. It is better to design a study that allows you to use parametric proce-

dures because they are more powerful than nonparametric procedures. However, some-

times researchers don’t obtain data that fit parametric procedures. Some dependent

variables are nominal variables (for example, whether someone is male or female).

Sometimes we can measure a dependent variable only by assigning ordinal scores (for

example, judging this participant as showing the most of the variable, this one second-

most, and so on). And sometimes a variable involves an interval or ratio scale, but the

352 CHAPTER 15 / Chi Square and Other Nonparametric Procedures

populations are severely skewed and/or do not have homogeneous variance (for exam-

ple, previously we saw that yearly income forms a positively skewed distribution).

Parametric procedures will tolerate some violation of their assumptions. But if the

data severely violate the rules, then the result is to increase the probability of a Type I

error so that it is much larger than the alpha level we think we have.

Therefore, when data do not fit a parametric procedure, we turn to nonparametric

statistics. They do not assume a normal distribution or homogeneous variance, and the

scores may be nominal or ordinal. By using these procedures, we keep the probability of a

Type I error equal to the alpha level that we’ve selected. Therefore, it is important to know

about nonparametric procedures because you may use them in your own research, and you

will definitely encounter them when reading the research of others.

REMEMBER Use nonparametric statistics when dependent scores form very

nonnormal distributions, when the population variance is not homogeneous,

or when scores are measured using ordinal or nominal scales.

CHI SQUARE PROCEDURES

Chi square procedures are used when participants are measured using a nominal vari-

able. With nominal variables, we do not measure an amount, but rather we categorize

participants. Thus, we have nominal variables when counting how many individuals

answer yes, no, or maybe to a question; how many claim to vote Republican, Democra-

tic, or Socialist; how many say that they were or were not abused as children; and so

on. In each case, we count the number, or frequency, of participants in each category.

The next step is to determine what the data represent. For example, we might find

that out of 100 people, 40 say yes to a question and 60 say no. These numbers indicate

how the frequencies are distributed across the categories of yes/no. As usual, we want

to draw inferences about the population: Can we infer that if we asked the entire popu-

lation this question, 40% would say yes and 60% would say no? Or would the frequen-

cies be distributed in a different manner? To make inferences about the frequencies in

the population, we perform chi square (pronounced “kigh square”). The chi square

procedure is the nonparametric inferential procedure for testing whether the frequen-

cies in each category in sample data represent specified frequencies in the population.

The symbol for the chi square statistic is .

REMEMBER Use the chi square procedure when you count the number

of participants falling into different categories.

Theoretically, there is no limit to the number of categories—levels—you may have

in a variable and no limit to the number of variables you may have. Therefore, we

describe a chi square design in the same way we described ANOVAs: When a study has

only one variable, perform the one-way chi square; when a study has two variables,

perform the two-way chi square; and so on.

ONE-WAY CHI SQUARE

The one-way chi square is used when data consist of the frequencies with which

participants belong to the different categories of one variable. Here we examine the

relationship between the different categories and the frequency with which participants

1



One-Way Chi Square 353

fall into each. We ask, “As the categories change, do the frequencies in the categories

also change?”

Here is an example. Being right-handed or left-handed is related to brain organiza-

tion, and many of history’s great geniuses were left-handed. Therefore, using an IQ

test, we select a sample of 50 geniuses. Then we ask them whether they are left-

or right-handed (ambidextrous is not an option). The total numbers of left- and right-

handers are the frequencies in the two categories. The results are shown here:

Handedness

Left-Handers Right-Handers

 10 f

 40

k  2

N  total f

 50

Each column contains the frequency in that category. We call this the observed fre-

quency, symbolized by . The sum of the from all categories equals , the total

number of participants. Notice that stands for the number of categories, or levels, and

here .

Above, 10 of the 50 geniuses (20%) are left-handers, and 40 of them (80%) are right-

handers. Therefore, we might argue that the same distribution of 20% left-handers and

80% right-handers would occur in the population of geniuses. But, there is the usual

problem: sampling error. Maybe, by luck, the people in our sample are unrepresenta-

tive, so in the population of geniuses, we would not find this distribution of right- and

left-handers. Maybe our results poorly represent some other distribution.

What is that “other distribution” of frequencies that the sample poorly represents?

To answer this, we create a model of the distribution of the frequencies we expect to

find in the population when is true. The model describes the distribution of

frequencies in the population if there is not the predicted relationship. It is because

we test this model that the one-way chi square procedure is also called a goodness-

of-fit test. Essentially, we test how “good” the “fit” is between our data and the

model. Thus, the goodness-of-fit test is another way of asking whether sample data

are likely to represent the distribution of frequencies in the population as described

by .

Hypotheses and Assumptions of the One-Way Chi Square

The one-way tests only two-tailed hypotheses. Usually, researchers test the that

there is no difference among the frequencies in the categories in the population, mean-

ing that there is no relationship in the population. For the handedness study, for the

moment we’ll ignore that there are more right-handers than left-handers in the world.

Therefore, if there is no relationship in the population, then our is that the frequen-

cies of left- and right-handed geniuses are equal in the population. There is no conven-

tional way to write this in symbols, so simply write : all frequencies in the

population are equal. This implies that, if the observed frequencies in the sample are

not equal, it’s because of sampling error.

The alternative hypothesis always implies that the study did demonstrate the pre-

dicted relationship, so we have : not all frequencies in the population are equal. For

our handedness study, implies that the observed frequencies represent different fre-

quencies of left- and right-handers in the population of geniuses.



k 5 2

354 CHAPTER 15 / Chi Square and Other Nonparametric Procedures

The one-way has five assumptions:

1. Participants are categorized along one variable having two or more categories, and

we count the frequency in each category.

2. Each participant can be in only one category (that is, you cannot have repeated

measures).

3. Category membership is independent: The fact that an individual is in a category

does not influence the probability that another participant will be in any category.

4. We include the responses of all participants in the study (that is, you would not

count only the number of right-handers, or in a different study, you would count

both those who do and those who do not agree with a statement).

5. For theoretical reasons, each “expected frequency” discussed below must be at

least 5.

Computing the One-Way Chi Square

The first step in computing is to translate into the expected frequency for each

category. The expected frequency is the frequency we expect in a category if the sam-

ple data perfectly represent the distribution in the population described by the null

hypothesis. The symbol for an expected frequency is . Our is that the frequencies

of left- and right-handedness are equal. If the sample perfectly represents this, then out

of our 50 participants, 25 should be right-handed and 25 should be left-handed. Thus,

the expected frequency in each category is .

Notice that, whenever we are testing the of no difference among the categories,

the will be the same for all categories, and it will always equal . Thus, above,

our . Also notice that sometimes may contain a decimal. For ex-

ample, if we included a third category, ambidextrous, then , and each would

be 16.67.

After computing each , the next step is to compute , which we call .

obt



k 5 3

5 50>2 5 25

N>kf

5 25



This says to find the difference between and in each category, square that differ-

ence, and then divide it by the for that category. After doing this for all categories,

sum the quantities, and the answer is .

For the handedness study, we have these frequencies:

Handedness

Left-Handers Right-Handers

 10 f

 40

 25 f

 25



obt

The formula for chi square is

obt

5 © a

2 f

One-Way Chi Square 355

Filling in the formula gives

In the numerator of each fraction, first subtract the from the , giving

Squaring each numerator gives

After dividing,

So our

Interpreting the One-Way Chi Square

If the sample perfectly represents the situation where is true, then each “should”

equal its corresponding . Then the difference between and should equal zero, and

so should equal zero. If not, says this is due to sampling error. However, the

larger the differences between and (and the larger the ), the harder it is for us to

accept that this is simply due to sampling error. At the same time, larger differences

between and are produced because of a larger observed frequency in one category

and a smaller one in another, so the more it looks like we are really representing a rela-

tionship. Therefore, the larger the , the less likely it is that is true and the more

likely it is that is true.

To determine if our is large enough, we examine the -distribution. Like previ-

ous sampling distributions, it is as if we have infinitely selected samples from the situa-

tion where is true. The -distribution is the sampling distribution containing all

possible values of when is true. Thus, for the handedness study, the -distribu-

tion is the distribution of all possible values of when there are two categories and the

frequencies in the two categories in the population are equal. You can envision the -

distribution as shown in Figure 15.1.

Even though the -distribution is not at all nor-

mal, it is used in the same way as previous sam-

pling distributions. Most often the data perfectly

represent the situation so that each equals its

, and then is zero. However, sometimes by

chance, the observed frequencies differ from the

expected frequencies, producing a greater than

zero. The larger the , the larger are the differ-

ences between the observed and the expected and

then the less likely they are to occur when

is true.

Because can only get larger, we again

have two-tailed hypotheses but one region of

rejection. To determine if is significant, we

obt



obt



␹



obt



obt



obt



obt



obt

5 18.0



obt

5 9 1 9



obt

5 a

225

b1 a

225



obt

5 a

12152

b1 a

1152



obt

5 © a

2 f

b5 a

110 2 252

b1 a

140 2 252

α = .05

greater than 0

crit

Region of rejection

FIGURE 15.1

Sampling distribution of

when H

is true

compare it to the critical value, symbolized by . As with previous statistics, the

-distribution changes shape as the degrees of freedom change, so we must

first determine the degrees of freedom. Here, , where is the number of

categories.

REMEMBER In a one-way chi square, .

Find the critical value of in Table 7 in Appendix C, entitled “Critical Values of

Chi Square.” For the handedness study, , so , and with , the

. Our of 18.0 is larger than this , so the results are significant:

This indicates that the differences between our observed and expected frequen-

cies are very unlikely to occur if our data represent no relationship in the population.

Therefore we reject that this is what the data represent. We then accept the that

the sample represents frequencies in the population that are not equal. In fact, we

would expect to find about 20% left-handers and 80% right-handers in the popula-

tion of geniuses. We conclude that we have evidence of a relationship between hand-

edness and genius. Then, as usual, we interpret the relationship in terms of the

behaviors and variables involved.

If had not been significant, we would have no evidence—one way or the other—

regarding how handedness is distributed among geniuses.

Note: If a study involves three categories, a significant chi square is not followed by

post hoc comparisons. We simply assume that the observed frequency in each category

represents frequencies that would be found in the population. Also, there is no measure

of effect size for this design.

Testing Other Hypotheses with the One-Way Chi Square

The procedure can also be used to test an other than that there is no difference

among the categories. For example, only about 10% of the general population is actu-

ally left-handed, so we should test whether handedness in geniuses is distributed dif-

ferently than this. Our is that geniuses are like the general population, being 10%

left-handed and 90% right-handed. Our is that our data represent a population of

geniuses that does not have this distribution (or for simplicity, we can write as

“not ”).

Each is again based on our . Say that we test our previous 50 geniuses. Our

says that left-handed geniuses should occur 10% of the time: 10% of 50 is 5, so

. Right-handed geniuses should occur 90% of the time: 90% of 50 is 45, so

. We found for left-handers and for right-handers. We compute

using the previous formula, comparing the to for left-handers and the to

for right-handers.

With and , the critical value of is again 3.84. Because the of

5.56 is larger than , we reject and conclude that the observed frequencies are

significantly different from what we would expect if handedness in the population of

geniuses was distributed as it is in the general population. Instead, we estimate that in

the population of geniuses, 20% are left-handers and 80% are right-handers.



crit



obt



k 5 2 5 .05



obt

5 © a

2 f

b5 a

110 2 52

b1 a

140 2 452

b5 5.56



obt

5 40f

5 10f

5 45

5 5



obt



obt



crit



obt



crit

5 3.84

 5 .05df 5 1k 5 2



df 5 k 2 1

kdf 5 k 2 1



crit

356 CHAPTER 15 / Chi Square and Other Nonparametric Procedures

■

The one-way is used when counting the

frequency of category membership on one variable.

MORE EXAMPLES

Below are the number of acts of graffiti that occur on

walls painted white, painted blue, or covered with

chalkboard. is that there are no differences in the

population. With , for

each category.

White Blue Chalk

 8 f

 5 f

 17

 10 f

 10

With and , ,

the wall coverings produce a significant difference in



crit

5 5.99 5 .05df 5 k 2 1 5 2



obt

5 7.80



obt

110 2 82

110 2 52

110 2 172



obt

5 © a

2 f

5 N>k 5 30>3 5 10N 5 30



the frequency of graffiti acts. In the population, we

expect 27% of graffiti on white walls, 17% on blue

walls, and 57% on chalkboard walls.

For Practice

1. The one-way chi square is used when we count

the ____ with which participants fall into

different ____.

2. We find in category A and in cate-

gory B. is that the frequencies are equal. The

for A is ____, and the for B is ____.

3. Compute .

4. The is ____, so at , is ____.

5. The is ____, so in the population we expect

membership is around ____% in A and around

____% in B.

Answers

1. frequency; categories

2. Each

4. 1; 3.84

5. significant; 35%; 65%



obt

130 2 212

130 2 392

5 5.40

5 60>2 5 30



obt



crit

 5 .05df



obt

5 39f

5 21

A QUICK REVIEW

The Two-Way Chi Square 357

THE TWO-WAY CHI SQUARE

The two-way chi square procedure is used when you count the frequency of category

membership along two variables. (The assumptions of the two-way chi square are the

same as for the one-way chi square.) Here is an example. At one time, psychologists

claimed that someone with a Type A personality tends to be a very pressured, hostile

individual who never seems to have enough time. The Type B personality tends not to

be so time pressured and is more relaxed and mellow. A controversy developed over

whether Type A people are less healthy, especially when it comes to the “big one”—

having heart attacks. Therefore, say that we select a sample of

80 people and determine how many are Type A and how

many Type B. We then count the frequency of heart attacks in

each type. We must also count how many in each type have

not had heart attacks (see our assumption 4). Therefore, we

have two categorical variables: personality type (A or B) and

health (heart attack or no heart attack). Table 15.1 shows the

layout of this study. Notice, this is a matrix, so it is

called a design. Depending on the number of categories

in each variable, a study might be a , a , and so on.3 3 42 3 3

2 3 2

Personality Type

Type A Type B

Heart

Attack

Health

No Heart

Attack

TABLE 15.1

A Two-Way Chi Square

Design Comparing

Participants’ Personality

Type and Health