Heiman G. Basic Statistics for the Behavioral Sciences
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368 CHAPTER 15 / Chi Square and Other Nonparametric Procedures
The Wilcoxon T Test for Two Related Samples
Perform the Wilcoxon test when you have related samples of ranked data. Recall
that related samples occur when you match samples or have repeated measures. For
example, say that we perform a study similar to the previous reaction time study, but
this time we measure the reaction times of the same participants to both the red and
black symbols. Table 15.7 gives the data we might obtain.
To compute the Wilcoxon
1. Determine the difference score for each pair of scores. Subtract the score in one
condition from the score in the other for each pair. It makes no difference which
score is subtracted from which, but subtract the scores the same way for all pairs.
2. Determine the of the nonzero difference scores. Ignore any differences equal to
zero and count the number of the other difference scores. In our study, one differ-
ence equals zero, so .
3. Assign ranks to the nonzero difference scores. Ignore the sign ( ) of each
difference. Assign the rank of 1 to the smallest difference, the rank of 2 to the
second-smallest difference, and so on. Record the ranked scores in a column.
4. Separate the ranks, using the sign of the difference scores. Create two columns
of ranks, labeled “ ” and “ .” The column contains the ranks assigned to
negative differences in step 3. The column contains the ranks assigned to
positive differences.
5. Compute the sums of ranks for the positive and negative difference scores.
Compute for the column labeled “ ” and for the column labeled “ .”
6. Determine the Wilcoxon . In the two-tailed test, the Wilcoxon equals
the smallest . In the example, the smallest equals 3, so . In the
one-tailed test, we predict whether most differences are positive or negative,
depending on our experimental hypotheses. Thus, we predict whether or
contains the smaller , and the one we predict is . (If we predicted that
red symbols would produce the largest reaction time scores, given the way
we subtracted, we would predict that for the column would be smaller,
so would be 42.)T
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TABLE 15.7
Data for the Wilcoxon T Test for Two Related Samples
Reaction Reaction Time
Time to Red to Black Ranked
Participant Symbols Symbols Difference Scores R R
1 540 760 220 6 6
2 580 710 130 4 4
3 600 1105 505 9 9
4 680 880 200 5 5
5 430 500 70 3 3
6 740 990 250 7 7
7 600 1050 450 8 8
8 690 640 50 2 2
9 605 595 10 1 1
10 520 520 0
©R 5 3©R 5 42N 5 9
12
Nonparametric Procedures for Ranked Data 369
7. Find the critical value of in Table 9 of Appendix C, entitled “Critical Values of
the Wilcoxon .” Find for the appropriate and , the number of nonzero
difference scores. In our study, and , so is 5.0.
8. Compare to . Again, watch out: is significant if it is equal to or less
than . (The critical value is the largest value that our smallest can be and
still reflect a significant difference.)
In the example, the is less than the , so we have a significant difference.
Therefore, we conclude that each sample represents a different distribution of ranks and
thus a different population of reaction time scores .
There is no way to compute for this procedure.
The Kruskal–Wallis H Test
The Kruskal–Wallis test is analogous to a one-way, between-subjects ANOVA for
ranks. It assumes that the study involves one factor involving at least three conditions,
and each is tested using independent samples, with at least five participants in each sam-
ple. The null hypothesis is that all conditions represent the same population of ranks.
As an example, say that we examine the independent variable of a golfer’s height and
the dependent variable of the distance he or she hits the ball. We test golfers classified
as either short, medium, or tall, measuring the distance each drives the ball in meters.
However, say that we cannot assume that the distance scores have homogeneous vari-
ance, so we cannot perform ANOVA. Instead, we perform the Kruskal–Wallis test.
The data are shown in Table 15.8.
To compute the Kruskal–Wallis test,
1. Assign ranks, using all scores in the experiment. Assign a rank of 1 to the lowest
score in the experiment, a 2 to the second-lowest score, and so on.
2. Compute the sum of the ranks in each condition. Compute the in each column.
3. Compute the sum of squares between groups . Use the formula
For our example,
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Height
Short Medium Tall
㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭 㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭 㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭㛭
Score Rank Score Rank Score Rank
10 2 24 3 68 14
28 6 27 5 71 15
26 4 35 7 57 10
39 8 44 9 60 12
6 1 58 11 62 13
N 5 15n
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5 21
TABLE 15.8
Data for the
Kruskal–Wallis H Test
370 CHAPTER 15 / Chi Square and Other Nonparametric Procedures
4. Compute . Use the formula
where is the total of the study, and is computed as above.
In the example
5. Find the critical value of in the tables (Table 7 in Appendix C). Values of
have the same sampling distribution as . The degrees of freedom are
where is the number of levels in the factor.
In the example, is 3, so for and , is 5.99.
6. Compare the obtained value of to the critical value of . The is significant
if it is larger than the critical value. Above, the of 9.62 is larger than the
of 5.99, so it is significant. This means that at least two samples represent differ-
ent populations of ranks. Because the distance participants hit the ball underlies
each rank, we conclude that at least two of the populations of distances for short,
medium, and tall golfers are not the same .
7. Perform post hoc comparisons using the rank sums test. When is significant,
determine which specific conditions differ by performing the rank sums test on
every pair of conditions. This is analogous to Fisher’s protected t-test (discussed
in Chapter 13) and is used regardless of the in each group. For each pair, treat
the two conditions being compared as if they comprised the entire study: re-rank
the scores using only the two conditions being compared, and then perform the
previous rank sums test.
In the example, comparing short to medium golfers produces a of 1.36, compar-
ing short to tall golfers produces a of 2.62, and comparing medium to tall golfers
produces a of 2.40. With , from the -tables is . Therefore, the
scores of short and medium participants are not significantly different, but they both
differ significantly from those in the tall condition. We conclude that tall golfers pro-
duce one population of distances that is different from the population for short and
medium golfers.
8. If is significant, compute eta squared. Use the formula
where is computed in the Kruskal–Wallis test and is the total number of
participants. Above, and , so , or . Therefore,
the variable of a player’s height accounts for approximately 69% of the variance
in the distance scores.
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Nonparametric Procedures for Ranked Data 371
The Friedman Test
The Friedman test is analogous to a one-way, within-subjects ANOVA for ranks.
It assumes that the study involves one factor having at least three levels and that the
samples in each are related (because of either matching or repeated measures). With
only three levels, we must have at least ten participants in the study. With only four lev-
els, we must have at least five participants.
As an example, say that we consider the three teaching styles of Dr. Highman,
Dr. Shyman, and Dr. Whyman. A sample of students who have taken courses from all
three instructors is repeatedly measured. Table 15.9 shows the data.
To perform the Friedman test,
1. Assign ranks within the scores of each participant. If the scores are not already
ranks, assign the rank of 1 to the lowest score for participant 1, assign the rank of
2 to the second-lowest score for participant 1, and so on. Repeat the process for
each participant.
2. Compute the sum of the ranks in each condition. Find in each column.
3. Compute the sum of squares between groups . Use the formula
In the example,
4. Compute the Friedman statistic. Use the formula
where is the number of participants and is the number of levels of the factor.
In the example
2
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5 1.1021129822 120 5 129.8 2 120 5 9.80
2
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5 a
12
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2
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2
2
2
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x
2
x
2
Rankings for Three Instructors
Participant Dr. Highman Dr. Shyman Dr. Whyman
11 2 3
21 3 2
31 2 3
41 3 2
52 1 3
61 3 2
71 2 3
81 3 2
91 3 2
10 2 1 3
©R
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TABLE 15.9
Data for the Friedman
Test
372 CHAPTER 15 / Chi Square and Other Nonparametric Procedures
5. Find the critical value of in the -tables (Table 7 in Appendix C). The degrees
of freedom are
where is the number of levels in the factor.
For the example, , so for and , the critical value is 5.99.
6. Compare to the critical value of . If is larger than , the results are
significant. Our of 9.80 is larger than the of 5.99, so at least two of the
samples represent different populations .
7. When the is significant, perform post hoc comparisons using Nemenyi’s
Procedure. This procedure is analogous to Tukey’s HSD procedure. To perform
Nemenyi’s procedure,
a. Compute the critical difference. Use the formula
where is the number of levels of the factor, is the number of participants
(or rows in the study’s diagram), and is the critical value used to test the
Friedman .
In the example, , , and , so
b. Compute the mean rank for each condition. For each condition, divide the
sum of ranks by the number of participants. In the example, the
mean ranks are 1.2, 2.3, and 2.5 for Highman, Shyman, and Whyman,
respectively.
c. Compute the differences between all pairs of mean ranks. Subtract each mean
rank from the other mean ranks. Any absolute difference between two means
that is greater than the critical difference indicates that the two conditions dif-
fer significantly. In the example, the differences between Dr. Highman and the
other two instructors are 1.10 and 1.30, respectively, and the difference
between Shyman and Whyman is . The critical difference is 1.09, so only
Dr. Highman’s ranking is significantly different from those of the other two
instructors. Thus, we conclude that Dr. Highman would be ranked superior to
the other two instructors by the population.
8. Describe a significant relationship, using eta squared. Use the formula
where is from the Friedman test, is the number of participants, and is
the number of levels of the factor. For the example,
Thus, the instructor variable accounts for 34% of the variability in rankings.
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Chapter Summary 373
Congratulations! You have read an entire statistics book, and that’s an accomplishment!
You should be proud of the sophisticated level of your knowledge because you are now
familiar with the vast majority of statistical procedures used in psychology and other
behavioral sciences. Even if you someday go to graduate school, you’ll find that there
is little in the way of basics for you to learn.
Using the SPSS Appendix As shown in Appendix B.9, SPSS will perform the one-
way or the two-way procedure. This includes reporting its significance level and, in
the two-way design, computing or . Appendix B.10 describes how to use SPSS to
compute the nonparametric procedures for ranked scores discussed in this chapter,
except for post hoc comparisons and .
CHAPTER SUMMARY
1. Nonparametric procedures are used when data do not meet the assumptions of
parametric procedures. Nonparametric procedures are less powerful than
parametric procedures.
2. Chi square x
2
is used with one or more nominal (categorical) variables, and the
data are the frequencies with which participants fall into each category.
3. The one-way compares the the frequency of category membership along one
variable. A significant indicates that the observed frequencies are unlikely to
represent the distribution of frequencies in the population described by .
4. The two-way tests whether category membership for one variable is independ-
ent of category membership for the other variable. A significant indicates that
the data represent dependent or correlated variables in the population.
5. With a significant two-way , describe the strength of the relationship with
(a) the phi coefficient if the design is a , or (b) the contingency
coefficient (C) if the design is not a . Squaring or gives the proportion
of variance accounted for, which indicates how much more accurately the
frequencies of category membership on one variable can be predicted by knowing
category membership on the other variable.
6. The two nonparametric versions of the independent-samples t-test for ranks are
the Mann–Whitney test, performed when both ns are less than 20, and the rank
sums test, performed when either is greater than 20.
7. The Wilcoxon test is the nonparametric, related-samples t-test for ranks.
8. The Kruskal–Wallis test is the nonparametric, one-way, between-subjects
ANOVA for ranks. The rank sums test is the post hoc test for identifying the
specific conditions that differ.
9. The Friedman test is the nonparametric, one-way, within-subjects ANOVA for
ranks. Nemenyi’s test is the post hoc test for identifying the specific conditions
that differ.
10. Eta squared describes the relationship found in experiments involving ranked
data.
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PUTTING IT
ALL TOGETHER
374 CHAPTER 15 / Chi Square and Other Nonparametric Procedures
KEY TERMS
-distribution 355
chi square procedure 352
contingency coefficient 362
expected frequency 354
Friedman test 371
goodness-of-fit test 353
Kruskal–Wallis test 369
Mann–Whitney test 365U
H
2
2
C
2
C
2
2
crit
2
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f
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f
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Nemenyi’s procedure 372
nonparametric statistics 352
observed frequency 353
one-way chi square 352
phi coefficient 361
rank sums test 366
test of independence 358
two-way chi square 357
Wilcoxon test 368T
REVIEW QUESTIONS
(Answers for odd-numbered questions are in Appendix D.)
1. What do all nonparametric inferential procedures have in common with all
parametric procedures?
2. (a) Which variable in an experiment determines whether to use parametric or
nonparametric procedures? (b) In terms of the dependent variable, what are the
two categories into which all nonparametric procedures can be grouped?
3. (a) Which two scales of measurement always require nonparametric procedures?
(b) What two things can be “wrong” with interval/ratio scores that lead to
nonparametric procedures (c) What must you do to the interval/ratio scores first?
4. (a) Why, if possible, should we design a study that meets the assumptions of a
parametric procedure? (b) Why shouldn’t you use parametric procedures for data
that clearly violate their assumptions?
5. (a) When do you use the one-way chi square? (b) When do you use the two-way
chi square?
6. (a) What is the symbol for observed frequency? What does it mean? (b) What is
the symbol for expected frequency? What does it mean?
7. (a) What does a significant one-way chi square indicate? (b) What does a signifi-
cant two-way chi square indicate?
8. What is the logic of and in all procedures for ranked data?
9. (a) What is the phi coefficient, and when is it used? (b) What does the squared phi
coefficient indicate? (c) What is the contingency coefficient, and when is it used?
(d) What does the squared contingency coefficient indicate?
10. What is the nonparametric version of each of the following? (a) A one-way, between-
subjects ANOVA (b) An independent-samples t-test (c) A related-samples
t-test (d) An independent-samples t-test (e) A one-way, within-subjects
ANOVA (f) Fisher’s protected t-test (g) Tukey’s HSD test
APPLICATION QUESTIONS
11. In the population, political party affiliation is 30% Republican, 55% Democratic,
and 15% other. To determine whether this distribution is also found among the
elderly, in a sample of 100 senior citizens, we find 18 Republicans, 64 Democrats,
and 18 other. (a) What procedure should we perform? (b) What are and ? H
a
H
0
1n 7 202
1n 6 202
H
a
H
0
(c) What is for each group? (d) Compute . (e) With , what do you
conclude about party affiliation in the population of senior citizens?
12. A survey finds that, given the choice, 34 females prefer males much taller than them-
selves, and 55 females prefer males only slightly taller than themselves. (a) What
procedure should we perform? (b) What are and ? (c) With , what do
you conclude about the preference of females in the population? (d) Describe how
you would graph these results.
13. Foofy counts the students who like Professor Demented and those who like Profes-
sor Randomsampler. She then performs a one-way to determine if there is a sig-
nificant difference between the frequency with which students like each professor.
(a) Why is this approach incorrect? (Hint: Check the assumptions of .) (b) How
should she analyze the data?
14. The following data reflect the frequency with which people voted in the last
election and were satisfied with the officials elected:
Satisfied
Yes N o
Yes 48 35
Voted
No 33 52
(a) What procedure should we perform? (b) What are and ? (c) What is in
each cell? (d) Compute . (e) With , what do you conclude about the
correlation between these variables? (f) How consistent is this relationship?
15. A study determines the frequency of the different political party affiliations for
male and female senior citizens. The following data are obtained:
Affiliation
Republican Democrat Other
Male 18 43 14
Gender
Female 39 23 18
(a) What procedure should we perform? (b) What are and ? (c) What is in
each cell? (d) Compute . (e) With , what do you conclude about gen-
der and party affiliation in the population of senior citizens? (f) How consistent is
this relationship?
16. Select the noparametric procedure to use when we study: (a) The effect of a pain
reliever on rankings of the emotional content of words describing pain. One group
is tested before and after taking the drug. (b) The effect of eight colors of
spaghetti sauce on its tastiness. A different sample tastes each color of sauce, and
tastiness scores are ranked. (c) The (skewed) reaction time scores after one group
of participants consumed 1, 3, and then 5 alcoholic drinks. (d) Whether two levels
of family income influence the percentage of income spent on clothing last year.
Percentages are then ranked.
17. After testing 40 participants, a significant of 13.31 was obtained. With
and , how would this result be reported in a publication?df 5 2 5 .05
2
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Application Questions 375
376 CHAPTER 15 / Chi Square and Other Nonparametric Procedures
18. We compare the attitude scores of people tested in the morning to their scores
when tested in the afternoon. We obtain the following interval data but it does not
have homogeneous variance. With , is there a significant difference in
scores as a function of testing times?
Morning Afternoon
14 36
18 31
20 19
28 48
310
34 49
20 20
24 29
19. We measure the maturity level of students who have completed statistics and
students who have not. Maturity scores tend to be skewed. For the following
interval scores,
Nonstatistics Statistics
43 51
52 58
65 72
23 81
31 92
36 64
(a) Do the groups differ significantly ? (b) What do you conclude about
maturity scores in the population?
20. A therapist evaluates a sample in a new treatment program after 1 month, after
2 months, and again after 3 months. Such data do not have homogeneous variance.
(a) What procedure should be used? Why? (b) What must the therapist do to the
data first? (c) If the results are significant, what procedure should be performed?
(d) Ultimately, what will the therapist be able to identify?
21. An investigator evaluated the effectiveness of a therapy on three types of patients.
She collected the following improvement ratings, but these data form skewed
distributions.
Depressed Manic Schizophrenic
16 7 13
11 9 6
12 6 10
20 4 15
21 8 9
(a) Which procedure should be used? Why? (b) What should the investigator
do to the data first? (c) If the results are significant, what should she do next?
(d) Ultimately, what will we learn from this study?
1 5 .052
5 .05
Integration Questions 377
22. A report indicates that the Friedman test was significant. (a) What does this test
indicate about the design of the study? (b) What does it indicate about the raw
scores? (c) What two procedures do you also expect to be reported? (d) What will
you learn about the relationship?
23. A report indicates that the Wilcoxon test was significant. (a) What does this test
indicate about the design of the study? (b) What does it indicate about the raw
scores? (c) What will you learn about the relationship here?
24. You show participants a picture of a person either smiling, frowning, or smirking.
For each, they indicate whether the person was happy or sad. (a) What are the fac-
tor(s) and level(s) in this design, and (b) What is the dependent variable? (c) How
will you analyze the results? (d) What potential flaw is built into the study in
terms of the statistical errors we may make? (c) When would this flaw be a
concern? (d) How can you eliminate the flaw?
INTEGRATION QUESTIONS
25. Thinking back on the previous few chapters, what three aspects of your
independent variable(s) and one aspect of your dependent variable determine
the specific inferential procedure to perform in a particular experiment?
(Ch’s. 10, 11, 12, 13, 14, 15)
26. For the following, what inferential statistical procedures should be performed
and what is the key information for answering the research question? Unless
described otherwise, assume scores are parametric. (a) To test the influence of
dietary fat levels on visual accuracy, rats are placed on one of four different
diets. Three weeks later, all are tested for visual accuracy. (b) We measure the
preferences of three groups of students for their classmates after either 1, 5, or
10 weeks into the semester. To be friendly, everyone’s scores are skewed
toward being positive. (c) To examine if body image is related to personality
traits, we measure the self-confidence and degree of obesity in a sample of
adults. (d) We find that among all Americans without marital problems, people
nod their head 32 times during a conversation with a spouse. In a sample of
people having problems, the mean number of nods is only 22, with .
(e) We select 30 left-handed and 30 right-handed men and measure their man-
ual dexterity scores. Ten of each type is tested under either low, medium, or
high illumination levels. (f) Last semester a teacher gave 25 As, 35 Bs, 20 Cs,
10 Ds, and 10 Fs. According to college policy, each grade should occur 20% of
the time. Is the teacher different from the college’s model? (g) We compare the
introversion of two groups of autistic children, one after receiving a new
improved therapy and the other after traditional therapy. (h) To determine if
handedness is related to aphasia, in a sample we count the number of left- and
right-handed individuals and in each group count those who do and do not
exhibit aphasia. (i) We create two groups of 15 participants, who either slept 4
or 8 hours the night before. Then we rank their driving ability. (j) To determine
how colors influence consumer behavior, we compare the mean attractiveness
scores from a group of participants who viewed a product’s label when printed
in red, in green, and in blue. (k) Does chewing gum while taking an exam lead
to better performance? We obtain exam grades for students who do or do not
chew gum during an exam, when the exam is either multiple choice or essay.
(Chs. 7, 8, 11, 12, 13, 14, 15)
s
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