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CHAPTER 13

Nonparametric Statistics

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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.

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Objectives (cont’d.)

Test hypotheses using the Kruskal-Wallis test.

Compute the Spearman Rank correlation coefficient.

Test hypotheses using the runs test.

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Introduction

Nonparametric statistics or distribution-free statistics are used when the population from which the samples are selected is not normally distributed.

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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.

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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.

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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.

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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”.

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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

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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.

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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

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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.

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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

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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.

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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.

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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.