cd-rom chapter 15 introduction to nonparametric statistics

CD-ROM Chapter 15CD-ROM Chapter 15

Introduction to Introduction to Nonparametric Nonparametric

StatisticsStatistics

Chapter 15 - Chapter 15 - Chapter Chapter OutcomesOutcomesAfter studying the material in this chapter, you should be able to:Recognize when and how to use the runs test and testing for randomness.Know when and how to perform a Mann-Whitney U test.Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it in a decision-making context.Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA.

Nonparametric StatisticsNonparametric Statistics

Nonparametric statistical Nonparametric statistical proceduresprocedures are those statistical methods that do not concern themselves with population distributions and/or parameters.

The Runs TestThe Runs Test

The runs testruns test is a statistical procedure used to determine whether the pattern of occurrences of two types of observations is determined by a random process.

The Runs TestThe Runs Test

A runrun is a succession of occurrences of a certain type preceded and followed by occurrences of the alternate type or by no occurrences at all.

The Runs TestThe Runs Test(Table 15-1)(Table 15-1)

Sequence Number Code Sequence Number Code1 0.34561 - 11 0.67201 +2 0.42789 - 12 0.23790 -3 0.36925 - 13 0.24509 -4 0.89563 + 14 0.01467 -5 0.25679 - 15 0.78345 +6 0.92001 + 16 0.69112 +7 0.58345 + 17 0.46023 -8 0.23114 - 18 0.38633 -9 0.12672 - 19 0.60914 +

10 0.88569 + 20 0.95234 +

The Runs TestThe Runs Test(Small Sample Example)(Small Sample Example)

H0: Computer-generated numbers are random between 0.0 and 1.0.

HA: Computer-generated numbers are not random .

--- + - ++ -- ++ --- ++ -- ++Runs: 1 2 3 4 5 6 7 8 9 10

There are r = 10 runsFrom runs table (Appendix K) with n1 = 9 and n2 = 11, the

critical value of r is 6

The Runs TestThe Runs Test(Small Sample Example)(Small Sample Example)

Test Statistic:

r = 10 runs

Critical Values from Runs Table:

Possible

Runs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Reject HReject H00Reject HReject H00Do not reject HDo not reject H00

Decision:

Since r = 10, we do not reject the null hypothesis.

Large Sample Runs TestLarge Sample Runs Test

MEAN AND STANDARD DEVIATION FOR MEAN AND STANDARD DEVIATION FOR rr

where:n1 = Number of occurrences of first type

n2 = Number of occurrences of second type

12

21

21

nn

nnr

)1()(

)2)(2(

212

21

212121

nnnn

nnnnnnr

Large Sample Runs TestLarge Sample Runs Test

TEST STATISTIC FOR LARGE TEST STATISTIC FOR LARGE SAMPLE RUNS TESTSAMPLE RUNS TEST

r

rrz

Large Sample Runs TestLarge Sample Runs Test(Example 15-2)(Example 15-2)

OOOUOOUOUUOOUUOOOOUUOUUOOO

UUUOOOOUUOOUUUOUUOOUUUUU

OOOUOUUOOOUOOOOUUUOUUOOOU

OOUUOUOOUUUOUUOOOOUUUOOO

Table 15-2

n1 = 53 “O’s” n2 = 47 “U’s”

r = 45 runs

96.1025. z0

Large Sample Runs Test Large Sample Runs Test (Example 15-2)(Example 15-2)

Rejection Region /2 = 0.025

Since z= -1.174 > -1.96 and < 1.96, we do not reject H0,

96.1025. z

Rejection Region /2 = 0.025

H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level.H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce level.

= 0.05

174.195659.4

82.5045

r

rrz

Mann-Whitney U TestMann-Whitney U Test

The Mann Whitney U test can be used to compare two samples from two populations if the following assumptions are satisfied:

• The two samples are independent and random.

• The value measured is a continuous variable.

• The measurement scale used is at least ordinal.

• If they differ, the distributions of the two populations will differ only with respect to the central location.

Mann-Whitney U TestMann-Whitney U Test

U-STATISTICSU-STATISTICS

where:n1 and n2 are the two sample sizes

R1 and R2 = Sum of ranks for samples

1 and 2

111

211 2

)1(R

nnnnU

222

212 2

)1(R

nnnnU

Mann-Whitney U TestMann-Whitney U Test- Large Samples -- Large Samples -

MEAN AND STANDARD DEVIATION FOR MEAN AND STANDARD DEVIATION FOR THE THE UU-STATISTIC-STATISTIC

where:n1 and n2 = Sample sizes from

populations 1 and 2

221nn

12

)1)()(( 2121

nnnn

Mann-Whitney U TestMann-Whitney U Test- Large Samples -- Large Samples -

MANN-WHITNEY U-TEST STATISTICMANN-WHITNEY U-TEST STATISTIC

12)1)()((

2

2121

21

nnnn

nnU

z

0~~21

Mann-Whitney U TestMann-Whitney U Test(Example 15-4)(Example 15-4)

Since z= -1.027 > -1.645, we do not reject H0,

645.1z

Rejection Region = 0.05

05.0

0~~:

0~~:

21

210

AH

H

027.1

12)1404144)(404)(144(

088,29412,27

12)1)()((

2

2121

21

nnnn

nnU

z

Wilcoxon Matched-Pairs Wilcoxon Matched-Pairs TestTest

The Wilcoxon matched pairs signed rank test can be used in those cases where the following assumptions are satisfied:

• The differences are measured on a continuous variable.

• The measurement scale used is at least interval.

• The distribution of the population differences is symmetric about their median.


WILCOXON MEAN AND STANDARD WILCOXON MEAN AND STANDARD DEVIATIONDEVIATION

where:n = Number of paired values

4

)1(

nn

24

)12)(1(

nnn


WILCOXON TEST STATISTICWILCOXON TEST STATISTIC

24)12)(1(

4)1(

nnn

nnT

z

Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance

Kruskal-Wallis one-way analysis of variance can be used in one-way analysis of variance if the variables satisfy the following:

• They have a continuous distribution.• The data are at least ordinal.• The samples are independent.• The samples come from populations

whose only possible difference is that at least one may have a different central location than the others.


H-STATISTICH-STATISTIC

where:N = Sum of sample sizes in all samplesk = Number of samplesRi = Sum of ranks in the ith sample

ni = Size of the ith sample

1),1(3)1(

12

1

2

kdfwithNn

R

NNH

k

i i

i


CORRECTION FOR TIED RANKINGSCORRECTION FOR TIED RANKINGS

where:g = Number of different groups of tiesti = Number of tied observations in the

ith tied group of scoresN = Total number of observations

NN

ttg

iii

31

3 )(1


H-STATISTIC CORRECTED FOR TIED H-STATISTIC CORRECTED FOR TIED RANKINGSRANKINGS

NN

tt

NnR

NNH g

iii

k

i i

i

31

3

1

2

)(1

)1(3)1(

12

Key TermsKey Terms

• Kruskal-Wallis One-Way Analysis of Variance

• Mann-Whitney U Test

• Nonparametric Statistical Procedure

• Run

• Runs Test

• Wilcoxon Test

cd-rom chapter 15 introduction to nonparametric statistics

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