Nonparametric Statistics Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000...

Nonparametric Statistics

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Nonparametric Statistics

• Many of the nonparametric statistical tests answer the same sorts of questions as the parametric tests. With nonparametric tests the assumptions can be relaxed considerably. Consequently, nonparametric methods are used for situations that violate the assumptions of parametric procedures.


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Test for Randomness;The Runs Test


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number of arrangement An!

n1 !n2 !


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The Runs Test (Small Samples)

• Sequence 1: HHHHH TTTTT

Run 1 Run 2

• Sequence 3: T HHH T H T H TT

Run 1 Run 7

• R = number of RunsIntroduction to Business Statistics, 5e

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The Runs Test (Small Samples)

Ho: The sequence was generated in a random mannerHa: The sequence was not generated in a random manner

P(Rk1 ) 2

.025

P(Rk2 ) 2

.025


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Runs Test (Large Samples)

R 1 2n1n2

n1 n2 R

2n1n2 (2n1n2 n1 n2 )

(n1 n2 )2 (n1 n2 1)

ZR R

R

Ho: The sequence was generated in a random mannerHa: The sequence was not generated in a random manner

Reject Ho if |Z| > Z/2Introduction to Business Statistics, 5e

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Nonparametric Test Central Tendency: Two Populations


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Mann-Whitney U(Independent Samples)

Ho: The two populations have identical probability distributionsHa: The two populations differ in location

T1 = sum of the ranks of the observations from the first sample T2 = sum of the ranks of the observations from the second sample


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Mann-Whitney USmall Samples

U1 n1n2 n1 (n1 1)

2 T1

U2 n1n2 n2 (n2 1)

2 T2


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Procedure:1. Assume that n1 < n2 (reverse the samples if necessary)

2. Determine U1 and U2

3. Use the value from Table A.10 to test Ho vs. Ha

Two-Sided TestHa: the two populations differ in location

Reject Ho if Table A.10 value for U is < /2, where U isthe minimum of U1 and U2.


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Procedure:1. Assume that n1 < n2 (reverse the samples if necessary)

2. Determine U1 and U2

3. Use the value from Table A.10 to test Ho vs. Ha

One-Sided Test

Ha: Population 1 is shifted to Ha: Population 1 is shifted to the right of population2 the left of population 2 Reject Ho if Table A.10 Reject Ho if Table A.10 value for U is less than value for U is less than where U = U1. Where U = U2


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Mann-Whitney ULarge Samples

ZU2 U 2

U 2

U 2n1n2

2 U 2

n1n2 (n1 n2 1)

12


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Ho: The two populations have identical probabilitydistribution

Determine U2

Two-Sided Test

Ha: The two populations differ in location

Reject Ho if |Z| > Z /2


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Ho: The two populations have identical probabilitydistribution

Determine U2

One-Sided Test

Ha: Population 1 is shifted to Ha: Population 1 is shifted to the right of population 2 the left of population 2

Reject Ho if Z > Z Reject Ho if Z < - Z


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Wilcoxon Signed Rank Test for Paired Samples

When small samples from suspected nonnormal populations are used, a nonparametric technique is required. The Wilcoxon test is used for such situations


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• Determine the difference for each sample.

• Arrange the absolute value of these differences in order, assigning a rank to each.

• T+ = sum of ranks having a positive value and T- = sum of ranks having a negative value


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Wilcoxon Signed Rank Test for Paired Samples (small n)

Ho: The population differences are centered at 0

Two-Sided Test

Ha: the population differences are not centered at 0

Using the two-sided value from Table A.11, Reject Ho if T table value,

where T = the minimum of T+ and T-.Introduction to Business Statistics, 5e

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Wilcoxon Signed Rank Test for Paired Samples (small n)


One-Sided TestHa:the population differences Ha:the population differences are centered at a value >0 are centered at a value < 0

Using the one-sided value from Using the one-sided value from Table A.11, Table A.11, Reject Ho if T- table value. Reject Ho if T- table value.


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Wilcoxon Signed Rank Test for Paired Samples (Large n)

T n(n1)

4T

n(n1)(2n1)

24

ZT T

T


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Two-Sided Test

Ha: the population differences are not centered at 0

Reject Ho if |Z| > Z/2


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One-Sided Test

Ha:the population differences Ha:the population differences are centered at a value >0 are centered at a value < 0

Reject Ho if Z > Z Reject Ho if Z < -Z Introduction to Business Statistics, 5e

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Kruskal-Wallis Test

The nonparametric counterpart to the one-way ANOVA test.


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Kruskal-Wallis Test

Ho: the k populations have identical probability distributionsHa: at least two of the populations differ in location

KW12

n(n1)

Ti2

nii1

k 3(n1)

Reject Ho if KW > 2.dfIntroduction to Business Statistics, 5e

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p-value for KW


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Kruskal-Wallis Test


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The Friedman Test

The nonparametric counterpart to the randomized block ANOVA test.

FR 12

bk(k1)Ti

2

i1

k 3b(k1)

Reject Ho if FR > 2.dfIntroduction to Business Statistics, 5e

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The Friedman Test

Ho: the k populations have identical probability distributionsHa: at least two of the populations differ in location

FR 12

bk(k1)Ti

2

i1

k 3b(k1)


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Spearman’s Rank Correlation

Spearman’s is the nonparametric counterpart to the Pearson Correlation

rs R(x)R(y) [R(x)][ R(y)] / n

R2 (x) [R(x)]2 / n R(y)] [R(y)]2 / n


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rs R(x)R(y) [R(x)][ R(y)] / n

R2 (x) [R(x)]2 / n R(y)] [R(y)]2 / n

If there are no ties then a shortcut equation may be used.0

rs 16d2

n(n2 1)Introduction to Business Statistics, 5e

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Ho: No association between X and Y existsHa: An association between X and Y exists

Two-Sided Test

Determine rs

Use Table A.12 for /2

Reject Ho if |rs| > table valueIntroduction to Business Statistics, 5e

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Ho: No association between X and Y exists

One-Sided Test

Ha: A positive association Ha: A negative association between X and Y exists between X and Y exists

Determine rs Determine rs

Use Table A.12 for Use Table A.12 for Reject Ho if rs > table value Reject Ho if rs < table value


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Nonparametric Statistics Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000...

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