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© Copyright McGraw-Hill 200013-1
CHAPTER 13
Nonparametric Statistics
© Copyright McGraw-Hill 200013-2
Objectives
State the advantages and disadvantages of nonparametric methods.
Test hypotheses using the sign test.
Test hypotheses using the Wilcoxon rank sum test.
Test hypotheses using the signed-rank test.
© Copyright McGraw-Hill 200013-3
Objectives (cont’d.)
Test hypotheses using the Kruskal-Wallis test.
Compute the Spearman Rank correlation coefficient.
Test hypotheses using the runs test.
© Copyright McGraw-Hill 200013-4
Introduction
Nonparametric statistics or distribution-free statistics are used when the population from which the samples are selected is not normally distributed.
© Copyright McGraw-Hill 200013-5
Advantages of Nonparametric Methods
They can be used to test population parameters when the variable is not normally distributed.
They can be used when the data are nominal or ordinal.
They can be used to test hypotheses that do not involve population parameters.
© Copyright McGraw-Hill 200013-6
Advantages of Nonparametric Methods
In most cases, the computations are easier than those for the parametric counterparts.
They are easier to understand.
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Disadvantages of Nonparametric Methods
They are less sensitive than their parametric counterparts when the assumptions of the parametric methods are met. Therefore, larger differences are needed before the null hypothesis can be rejected.
They tend to use less information than the parametric tests. For example, the sign test requires the researcher to determine only whether the data values are above or below the median, not how much above or below the median each value is.
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Disadvantages of Nonparametric Methods
They are less efficient than their parametric counterparts when the assumptions of the parametric methods are met; that is, larger sample sizes are needed to overcome the loss of information. For example, the nonparametric sign test is about 60% as efficient as its parametric counterpart, the z test. Thus, a sample size of 100 is needed for use of the sign test, compared with a sample size of 60 for use of the z test to obtain the same results.
© Copyright McGraw-Hill 200013-9
Ranking the Data
Many nonparametric tests involve the ranking of data — that is, the positioning of a data value in a data array according to some rating scale.
Speaker A B C DRating 4 3 5 1
Speaker D B A CRating 1 3 4 5Ranking 1 2 3 4
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Nonparametric Methods
Sign test
Wilcoxon rank sum test
Wilcoxon signed-rank test
Kruskal-Wallis test
Spearman rank coefficient
Runs test
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Single-Sample Sign Test
The sign test is the simplest of the nonparametric tests and is used to test the value of a median for a specific sample.
When using the sign test, the researcher hypothesizes the specific value for the median of a population; then he or she selects a sample of data and compares each value with the conjectured median.
© Copyright McGraw-Hill 200013-12
Single-Sample Sign Test (cont’d.)
If the data value is above the conjectured median, it is assigned a “+” sign.
If it is below the conjectured median, it is assigned a “–” sign.
If it is exactly the same as the conjectured median, it is assigned a “0”.
© Copyright McGraw-Hill 200013-13
Single-Sample Sign Test (cont’d.)
If the null hypothesis is true, the number of + and – signs should be approximately equal.
If the null hypothesis is not true, there will be a disproportionate number of + or – signs.
The test value is the smaller number of + or – signs.
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z Test Value in the Sign Test when n 26
where
X = smaller number of + or – signs
n = sample size
zX n
n 05 2
2. /
/
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Wilcoxon Rank Sum Test
The Wilcoxon rank sum test is used for independent samples.
Both sample sizes must be 10.
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Formula for Wilcoxon Rank Sum Test
where
R = sum of the ranks for the smaller sample size (n1)
n1 = smaller of the sample sizes, n1 10
n2 = larger of the sample sizes , n2 10
zR R
R
Rn n n
1 1 2 1
2
b g
R
n n n n
1 2 1 2 1
12
b g
© Copyright McGraw-Hill 200013-17
The Wilcoxon Signed-Rank Test
When the samples are dependent, as they would be in a before-and-after test using the same subjects, the Wilcoxon signed-rank test can be used in place of the t test for dependent samples.
This test does not require the condition of normality.
When n 30, the normal distribution can be used to approximate the Wilcoxon distribution.
© Copyright McGraw-Hill 200013-18
The Wilcoxon Signed-Rank Test (cont’d.)
The formula for the Wilcoxon signed-rank test is:
where
n = number of pairs where the difference is not 0
ws = smaller sum in absolute value of the signed
ranks
zw
n n
n n n
s
14
1 2 124
© Copyright McGraw-Hill 200013-19
The Kruskal-Wallis Test
The Kruskal-Wallis test, also called the H test, is used to compare three or more means.
Data values are grouped and then are ranked.
© Copyright McGraw-Hill 200013-20
Formula for the Kruskal-Wallis Test
Where
R1 = sum of the ranks of
sample 1
n1 = size of sample 1
R2 = sum of the ranks of
sample 2
n2 = size of sample 2
Rk = sum of the ranks of
sample k
nk = size of sample k
N = n1 + n2 + … + nk
k = number of samples
HN N
Rn
Rn
R
nNk
k
FHG
IKJ 12
13 11
2
1
22
2
2
© Copyright McGraw-Hill 200013-21
The Spearman Rank Correlation Coefficient
where
d = difference in the ranks
n = number of data pairs
rd
n ns
16
1
2
2c h
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Run Test
A run is a succession of identical letters preceded or followed by a different letter or no letter at all, such as the beginning or end of the succession.
© Copyright McGraw-Hill 200013-23
Summary
In many research situations, the assumptions for the use of parametric statistics cannot be met.
Some statistical studies do not involve parameters such as means, variances, and proportions.
For both situations, statisticians have developed nonparametric statistical methods, also called distribution-free methods.
© Copyright McGraw-Hill 200013-24
Summary (cont’d.)
There are several advantages to the use of nonparametric methods — the most important one is that no knowledge of the population distribution is required.
The major disadvantage is that they are less efficient than their parametric counterparts when the assumptions for the parametric methods are met.
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Summary (cont’d.)
NonparametricTest
ParametricTest
Condition
Single-sample sign z or t 1 sample
Paired-sample sign z or t 2 dependent samples
Wilcoxon rank sum z or t 2 independent samples
Wilcoxon signed-rank t 2 dependent samples
Kruskal Wallis ANOVA 3 independent samples
Spearman rankcorrelation coefficient
Pearson’scorrelationcoefficient
Relationships betweenvariables
Runs Test none Randomness
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Conclusions
Nonparametric or distribution-free tests are used when situations are not normally distributed. A sportswriter may wish to know whether
there is a relationship between the rankings of two Olympic swimming judges.
A sociologist may wish to determine whether men and women enroll at random for a specific rehabilitation program.
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