14 - 1 © 2000 prentice-hall, inc. statistics nonparametric statistics chapter 14

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© 2000 Prentice-Hall, Inc.© 2000 Prentice-Hall, Inc.

StatisticsStatistics

Nonparametric StatisticsNonparametric StatisticsChapter 14Chapter 14

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Learning ObjectivesLearning Objectives

1.1. Distinguish Parametric & Nonparametric Distinguish Parametric & Nonparametric Test Procedures Test Procedures

2.2. Explain a Variety of Nonparametric Test Explain a Variety of Nonparametric Test ProceduresProcedures

3.3. Solve Hypothesis Testing Problems Solve Hypothesis Testing Problems Using Nonparametric TestsUsing Nonparametric Tests

4.4. Compute Spearman’s Rank CorrelationCompute Spearman’s Rank Correlation

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Hypothesis Testing Hypothesis Testing ProceduresProcedures

HypothesisTesting

Procedures

NonparametricParametric

Z Test

Kruskal-WallisH-Test

WilcoxonRank Sum

Test

t Test One-WayANOVA

HypothesisTesting

Procedures

NonparametricParametric

Z Test

Kruskal-WallisH-Test

WilcoxonRank Sum

Test

t Test One-WayANOVA

Many More Tests Exist!Many More Tests Exist!

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Parametric Test Parametric Test ProceduresProcedures

1.1. Involve Population ParametersInvolve Population Parameters Example: Population MeanExample: Population Mean

2.2. Require Interval Scale or Ratio ScaleRequire Interval Scale or Ratio Scale Whole Numbers or FractionsWhole Numbers or Fractions Example: Height in Inches (72, 60.5, 54.7)Example: Height in Inches (72, 60.5, 54.7)

3.3. Have Stringent AssumptionsHave Stringent Assumptions Example: Normal DistributionExample: Normal Distribution

4.4. Examples: Z Test, t Test, Examples: Z Test, t Test, 22 Test Test

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Nonparametric Test Nonparametric Test ProceduresProcedures

1.1. Do Not Involve Population ParametersDo Not Involve Population Parameters Example: Probability Distributions, IndependenceExample: Probability Distributions, Independence

2.2. Data Measured on Any ScaleData Measured on Any Scale Ratio or IntervalRatio or Interval OrdinalOrdinal

Example: Good-Better-BestExample: Good-Better-Best NominalNominal

Example: Male-FemaleExample: Male-Female

3.3. Example: Wilcoxon Rank Sum TestExample: Wilcoxon Rank Sum Test

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Advantages of Advantages of Nonparametric TestsNonparametric Tests

1.1. Used With All ScalesUsed With All Scales

2.2. Easier to ComputeEasier to Compute Developed Originally Before Developed Originally Before

Wide Computer UseWide Computer Use

3.3. Make Fewer AssumptionsMake Fewer Assumptions

4.4. Need Not Involve Need Not Involve Population ParametersPopulation Parameters

5.5. Results May Be as Exact as Results May Be as Exact as Parametric ProceduresParametric Procedures

© 1984-1994 T/Maker Co.

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Disadvantages of Disadvantages of Nonparametric TestsNonparametric Tests

1.1. May Waste Information May Waste Information If Data Permit Using If Data Permit Using

Parametric ProceduresParametric Procedures Example: Converting Data Example: Converting Data

From Ratio to Ordinal ScaleFrom Ratio to Ordinal Scale

2.2. Difficult to Compute by Difficult to Compute by Hand for Large SamplesHand for Large Samples

3.3. Tables Not Widely Tables Not Widely AvailableAvailable

© 1984-1994 T/Maker Co.

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Frequently Used Frequently Used Nonparametric TestsNonparametric Tests

1.1. Sign Test Sign Test

2.2. Wilcoxon Rank Sum TestWilcoxon Rank Sum Test

3.3. Wilcoxon Signed Rank TestWilcoxon Signed Rank Test

4.4. Kruskal Wallis Kruskal Wallis HH-Test-Test

5.5. Friedman Friedman FFrr-Test-Test

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Sign Test Sign Test

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Sign Test Sign Test

1.1. Tests One Population Median, Tests One Population Median, (eta) (eta)

2.2. Corresponds to t-Test for 1 MeanCorresponds to t-Test for 1 Mean

3.3. Assumes Population Is ContinuousAssumes Population Is Continuous

4.4. Small Sample Test Statistic: # Sample Small Sample Test Statistic: # Sample Values Above (or Below) MedianValues Above (or Below) Median Alternative Hypothesis DeterminesAlternative Hypothesis Determines

5.5. Can Use Normal Approximation If Can Use Normal Approximation If nn 10 10

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Sign Test Uses P-Sign Test Uses P-Value Value

to Make Decisionto Make Decision

.031.109

.219.273

.219

.109.031.004 .004

0%

10%

20%

30%

0 1 2 3 4 5 6 7 8 X

P(X)

.031.109

.219.273

.219

.109.031.004 .004

0%

10%

20%

30%

0 1 2 3 4 5 6 7 8 X

P(X) Binomial: n = 8 p = 0.5

P-Value Is the Probability of Getting an Observation At P-Value Is the Probability of Getting an Observation At Least as Extreme as We Got. If 7 of 8 Observations Least as Extreme as We Got. If 7 of 8 Observations ‘Favor’ H‘Favor’ Haa, Then P-Value = , Then P-Value = PP((xx 7) = .031 + .004 = .035. 7) = .031 + .004 = .035.

If If = .05, Then Reject H = .05, Then Reject H00 Since P-Value Since P-Value ..

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Sign Test ExampleSign Test Example

You’re an analyst for Chef-You’re an analyst for Chef-Boy-R-Dee. You’ve asked Boy-R-Dee. You’ve asked 7 people to rate a new 7 people to rate a new ravioli on a 5-point Likert ravioli on a 5-point Likert scale (1 = terrible to scale (1 = terrible to 5 = excellent. The ratings 5 = excellent. The ratings are: are: 2 5 3 4 1 4 52 5 3 4 1 4 5. . At the At the .05.05 level, is there level, is there evidence that the evidence that the medianmedian rating is rating is at least 3at least 3??

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Sign Test SolutionSign Test Solution

HH00: :

HHaa: :

= =

Test Statistic:Test Statistic:

P-Value: P-Value:

Decision:Decision:

Conclusion:Conclusion:

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HH00: : = = 33

HHaa: : < 3 < 3

= =


P-Value: P-Value:

Decision:Decision:


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HH00: : = = 33

HHaa: : < 3 < 3

= = .05.05


P-Value: P-Value:

Decision:Decision:


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HH00: : = = 33

HHaa: : < 3 < 3

= = .05.05


P-Value: P-Value:

Decision:Decision:


S = 2 S = 2 (Ratings 1 & 2 Are (Ratings 1 & 2 Are Less Than Less Than = = 3:3:22, 5, 3, 4, , 5, 3, 4, 11, 4, 5), 4, 5)

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HH00: : = = 33

HHaa: : < 3 < 3

= = .05.05


P-Value: P-Value:

Decision:Decision:


PP((xx 2) = 1 - 2) = 1 - PP((xx 1) 1) = .937= .937

(Binomial Table, (Binomial Table, nn = 7, = 7, pp = 0.50) = 0.50)


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HH00: : = = 33

HHaa: : < 3 < 3

= = .05.05


P-Value: P-Value:

Decision:Decision:


Do Not Reject at Do Not Reject at = .05 = .05

PP((xx 2) = 1 - 2) = 1 - PP((xx 1) 1) = .937= .937



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HH00: : = = 33

HHaa: : < 3 < 3

= = .05.05


P-Value: P-Value:

Decision:Decision:



There Is No Evidence There Is No Evidence Median Is Less Than 3Median Is Less Than 3

PP((xx 2) = 1 - 2) = 1 - PP((xx 1) 1) = .937= .937



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Wilcoxon Rank Sum Wilcoxon Rank Sum Test Test

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1.1. Tests Two Independent Population Tests Two Independent Population Probability DistributionsProbability Distributions

2.2. Corresponds to t-Test for 2 Independent Corresponds to t-Test for 2 Independent MeansMeans

3.3. AssumptionsAssumptions Independent, Random SamplesIndependent, Random Samples Populations Are ContinuousPopulations Are Continuous

4.4. Can Use Normal Approximation If Can Use Normal Approximation If nnii 10 10

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ProcedureProcedure1.1. Assign Ranks, Assign Ranks, RRii, to the , to the nn11 + + nn22 Sample Sample

ObservationsObservations If Unequal Sample Sizes, Let If Unequal Sample Sizes, Let nn11 Refer to Smaller- Refer to Smaller-

Sized SampleSized Sample Smallest Value = 1Smallest Value = 1 Average TiesAverage Ties

2.2. Sum the Ranks, Sum the Ranks, TTii, for Each Sample, for Each Sample

3.3. Test Statistic Is Test Statistic Is TTA A (Smallest Sample)(Smallest Sample)

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ExampleExampleYou’re a production planner. You want to You’re a production planner. You want to see if the operating rates for 2 factories is see if the operating rates for 2 factories is the same. For factory 1, the rates (% of the same. For factory 1, the rates (% of capacity) are capacity) are 7171, , 8282, , 7777, , 9292, , 8888. For . For factory 2, the rates are factory 2, the rates are 8585, , 8282, , 9494 & & 9797. Do the factory rates . Do the factory rates have the same have the same probability probability distributionsdistributions at the at the .10.10 level? level?

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SolutionSolutionHH00::

HHaa::

==

nn11 = = nn22 = =

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:


RanksRanks

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SolutionSolutionHH00: : Identical Distrib.Identical Distrib.

HHaa: : Shifted Left or RightShifted Left or Right

==

nn11 = = nn22 = =



Decision:Decision:


RanksRanks

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= = .10.10

nn11 = = 4 4 nn22 = = 5 5



Decision:Decision:


RanksRanks

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Wilcoxon Rank Sum Wilcoxon Rank Sum Table (Portion)Table (Portion)

n1

3 4 5 ..

TL TU TL TU TL TU ..

3 6 15 7 17 7 20 ..n2 4 7 17 12 24 13 27 ..

5 7 20 13 27 19 36 ..: : : : : : : :

n1

3 4 5 ..

TL TU TL TU TL TU ..

3 6 15 7 17 7 20 ..n2 4 7 17 12 24 13 27 ..

5 7 20 13 27 19 36 ..: : : : : : : :

= .05 one-tailed; = .05 one-tailed; = .10 two-tailed = .10 two-tailed

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= = .10.10

nn11 = = 4 4 nn22 = = 5 5



Decision:Decision:

Conclusion:Conclusion:RejectReject RejectRejectDo Not Do Not

RejectReject

1313 2727 RanksRanks

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Wilcoxon Rank Sum Wilcoxon Rank Sum Test Computation Test Computation

TableTable

Factory 1 Factory 2Rate Rank Rate Rank

Rank Sum

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TableTable


71 8582 8277 9492 9788 ... ...

Rank Sum

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TableTable


71 1 8582 8277 9492 9788 ... ...

Rank Sum

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TableTable


71 1 8582 8277 2 9492 9788 ... ...

Rank Sum

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TableTable


71 1 8582 3 82 477 2 9492 9788 ... ...

Rank Sum

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TableTable


71 1 8582 3 3.5 82 4 3.577 2 9492 9788 ... ...

Rank Sum

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TableTable


71 1 85 582 3 3.5 82 4 3.577 2 9492 9788 ... ...

Rank Sum

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TableTable


71 1 85 582 3 3.5 82 4 3.577 2 9492 9788 6 ... ...

Rank Sum

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TableTable


71 1 85 582 3 3.5 82 4 3.577 2 9492 7 9788 6 ... ...

Rank Sum

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TableTable


71 1 85 582 3 3.5 82 4 3.577 2 94 892 7 9788 6 ... ...

Rank Sum

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TableTable


71 1 85 582 3 3.5 82 4 3.577 2 94 892 7 97 988 6 ... ...

Rank Sum

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TableTable


71 1 85 582 3 3.5 82 4 3.577 2 94 892 7 97 988 6 ... ...

Rank Sum 19.5 25.5

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= = .10.10

nn11 = = 4 4 nn22 = = 5 5



Decision:Decision:

Conclusion:Conclusion:RejectReject RejectRejectDo Not Do Not

RejectReject


TT22 = 5 + 3.5 + 8+ 9 = 25.5 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)(Smallest Sample)

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= = .10.10

nn11 = = 4 4 nn22 = = 5 5



Decision:Decision:


Do Not Reject at Do Not Reject at = .10 = .10RejectReject RejectRejectDo Not Do Not

RejectReject



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= = .10.10

nn11 = = 4 4 nn22 = = 5 5



Decision:Decision:



There Is No Evidence There Is No Evidence Distrib. Are Not EqualDistrib. Are Not Equal

RejectReject RejectRejectDo Not Do Not RejectReject



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Wilcoxon Signed Rank Wilcoxon Signed Rank Test Test

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Wilcoxon Wilcoxon Signed Rank Test Signed Rank Test

1.1. Tests Probability Distributions of Tests Probability Distributions of 2 Related Populations2 Related Populations

2.2. Corresponds to t-test for Dependent Corresponds to t-test for Dependent (Paired) Means(Paired) Means

3.3. AssumptionsAssumptions Random SamplesRandom Samples Both Populations Are ContinuousBoth Populations Are Continuous

4.4. Can Use Normal Approximation If Can Use Normal Approximation If nn 25 25

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Signed Rank Test Signed Rank Test ProcedureProcedure

1.1. Obtain Difference Scores, Obtain Difference Scores, DDii = = XX11ii - - XX22ii

2.2. Take Absolute Value of Differences, Take Absolute Value of Differences, DDii

3.3. Delete Differences With 0 ValueDelete Differences With 0 Value

4.4. Assign Ranks, Assign Ranks, RRii, Where Smallest = 1, Where Smallest = 1

5.5. Assign Ranks Same Signs as Assign Ranks Same Signs as DDii

6.6. Sum ‘+’ Ranks (TSum ‘+’ Ranks (T++) & ‘-’ Ranks (T) & ‘-’ Ranks (T--)) Test Statistic Is TTest Statistic Is T-- (One-Tailed Test) (One-Tailed Test)

Test Statistic Is Smaller of TTest Statistic Is Smaller of T-- or T or T++ (2-Tail) (2-Tail)

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Signed Rank Test Signed Rank Test Computation TableComputation Table

X1i X2i Di = X1i - X2i |Di| Ri Sign Sign Ri

X11 X21 D1 = X11 - X21 |D1| R1 ± ± R1

X12 X22 D2 = X12 - X22 |D2| R2 ± ± R2

X13 X23 D3 = X13 - X23 |D3| R3 ± ± R3

: : : : : : :

X1n X2n Dn = X1n - X2n |Dn| Rn ± ± Rn

Total T+ & T-

X1i X2i Di = X1i - X2i |Di| Ri Sign Sign Ri

X11 X21 D1 = X11 - X21 |D1| R1 ± ± R1

X12 X22 D2 = X12 - X22 |D2| R2 ± ± R2

X13 X23 D3 = X13 - X23 |D3| R3 ± ± R3

: : : : : : :

X1n X2n Dn = X1n - X2n |Dn| Rn ± ± Rn

Total T+ & T-

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Signed Rank TestSigned Rank TestExampleExample

You work in the finance department. Is the You work in the finance department. Is the newnew financial package financial package fasterfaster ( (.05.05 level)? You collect the level)? You collect the following data entry times:following data entry times:

UserUser CurrentCurrent NewNewDonnaDonna 9.989.98 9.889.88SantoshaSantosha 9.889.88 9.869.86SamSam 9.909.90 9.839.83TamikaTamika 9.999.99 9.809.80BrianBrian 9.949.94 9.879.87JorgeJorge 9.849.84 9.849.84

© 1984-1994 T/Maker Co.

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Signed Rank Test Signed Rank Test SolutionSolution

HH00::

HHaa::

= =

n’ =n’ =



Decision:Decision:


TT00

RejectReject Do Not Do Not RejectReject

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HH00: : Identical Distrib.Identical Distrib.

HHaa: : Current Shifted Current Shifted Right Right

= =

n’ =n’ =



Decision:Decision:


TT00


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Signed Rank Test Signed Rank Test Computation TableComputation Table

X1i X2i Di |Di| Ri Sign Sign Ri

9.98 9.88 +0.10 0.10 4 + +4

9.88 9.86 +0.02 0.02 1 + +1

9.90 9.83 +0.07 0.07 2 2.5 + +2.5

9.99 9.80 +0.19 0.19 5 + +5

9.94 9.87 +0.07 0.07 3 2.5 + +2.5

9.84 9.84 0.00 0.00 ... ... Discard

Total T+ = 15, T- = 0

X1i X2i Di |Di| Ri Sign Sign Ri

9.98 9.88 +0.10 0.10 4 + +4

9.88 9.86 +0.02 0.02 1 + +1

9.90 9.83 +0.07 0.07 2 2.5 + +2.5

9.99 9.80 +0.19 0.19 5 + +5

9.94 9.87 +0.07 0.07 3 2.5 + +2.5

9.84 9.84 0.00 0.00 ... ... Discard

Total T+ = 15, T- = 0

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= = .05.05

n’ = n’ = 5 (not 6; 1 elim.)5 (not 6; 1 elim.)



Decision:Decision:

Conclusion:Conclusion:RejectReject Do Not Do Not RejectReject

TT00

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Wilcoxon Signed Wilcoxon Signed Rank Rank

Table (Portion)Table (Portion)

One-Tailed Two-Tailed n = 5 n = 6 n = 7 .. = .05 = .10 1 2 4 .. = .025 = .05 1 2 .. = .01 = .02 0 .. = .005 = .01 ..

n = 11 n = 12 n = 13

: : : :

One-Tailed Two-Tailed n = 5 n = 6 n = 7 .. = .05 = .10 1 2 4 .. = .025 = .05 1 2 .. = .01 = .02 0 .. = .005 = .01 ..

n = 11 n = 12 n = 13

: : : :

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= = .05.05




Decision:Decision:


11 TT00

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= = .05.05




Decision:Decision:


11 TT00

Since One-Tailed Since One-Tailed Test & Current Test & Current Shifted Right, Use TShifted Right, Use T--: :

TT-- = 0 = 0

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= = .05.05




Decision:Decision:


Reject at Reject at = .05 = .05


11 TT00


TT-- = 0 = 0

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= = .05.05




Decision:Decision:



There Is Evidence New There Is Evidence New Package Is FasterPackage Is Faster


11 TT00


TT-- = 0 = 0

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Kruskal-Wallis H-TestKruskal-Wallis H-Test

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14 - 14 - 6363


Kruskal-Wallis H-Kruskal-Wallis H-TestTest

1.1. Tests the Equality of More Than 2 (Tests the Equality of More Than 2 (pp) ) Population Probability DistributionsPopulation Probability Distributions

2.2. Corresponds to ANOVA for More Than 2 Corresponds to ANOVA for More Than 2 MeansMeans

3.3. Used to Analyze Completely Used to Analyze Completely Randomized Experimental Designs Randomized Experimental Designs

4.4. Uses Uses 22 Distribution with Distribution with pp - 1 df - 1 df If At Least 1 Sample Size If At Least 1 Sample Size nnjj > 5 > 5

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Kruskal-Wallis H-Kruskal-Wallis H-Test AssumptionsTest Assumptions

1.1. Independent, Random Samples Independent, Random Samples

2.2. At Least 5 Observations Per SampleAt Least 5 Observations Per Sample

3.3. Continuous Population Probability Continuous Population Probability DistributionsDistributions

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Kruskal-Wallis H-Kruskal-Wallis H-Test ProcedureTest Procedure

1.1. Assign Ranks, Assign Ranks, RRi i , to the , to the nn Combined Combined

ObservationsObservations Smallest Value = 1; Largest Value = Smallest Value = 1; Largest Value = nn Average TiesAverage Ties

2.2. Sum Ranks for Each GroupSum Ranks for Each Group

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Kruskal-Wallis H-Kruskal-Wallis H-Test ProcedureTest Procedure

1.1. Assign Ranks, Assign Ranks, RRi i , to the , to the nn Combined Combined

ObservationsObservations Smallest Value = 1; Largest Value = Smallest Value = 1; Largest Value = nn Average TiesAverage Ties

2.2. Sum Ranks for Each GroupSum Ranks for Each Group

3.3. Compute Test StatisticCompute Test Statistic

Hn n

R

nnj

jj

p

FHG

IKJ

12

13 1

2

1a f a fHn n

R

nnj

jj

p

FHG

IKJ

12

13 1

2

1a f a fSquared total of Squared total of each groupeach group

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Kruskal-Wallis H-Kruskal-Wallis H-Test ExampleTest Example

As production manager, As production manager, you want to see if 3 filling you want to see if 3 filling machines have different machines have different filling times. You assign filling times. You assign 15 similarly trained & 15 similarly trained & experienced workers, experienced workers, 5 per machine, to the 5 per machine, to the machines. At the machines. At the .05.05 level, is level, is there a difference in the there a difference in the distributiondistribution of filling times?of filling times?

Mach1Mach1 Mach2Mach2 Mach3Mach325.4025.40 23.4023.40 20.0020.0026.3126.31 21.8021.80 22.2022.2024.1024.10 23.5023.50 19.7519.7523.7423.74 22.7522.75 20.6020.6025.1025.10 21.6021.60 20.4020.40

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2200

Kruskal-Wallis H-Kruskal-Wallis H-Test Test

SolutionSolutionHH00::

HHaa::

==

dfdf = =



Decision:Decision:


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2200



HHaa: : At Least 2 DifferAt Least 2 Differ

==

dfdf = =



Decision:Decision:


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2200




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


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2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:

Conclusion:Conclusion: = .05= .05

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SolutionSolution

Raw DataRaw Data


RanksRanks

Mach1Mach1 Mach2Mach2 Mach3Mach3

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SolutionSolution

Raw DataRaw Data


RanksRanks


11

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SolutionSolution

Raw DataRaw Data


RanksRanks


11

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SolutionSolution

Raw DataRaw Data


RanksRanks


11

33

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SolutionSolution

Raw DataRaw Data


RanksRanks

Mach1Mach1 Mach2Mach2 Mach3Mach31414 99 221515 66 771212 1010 111111 88 441313 55 33

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SolutionSolution

Raw DataRaw Data


RanksRanks

Mach1Mach1 Mach2Mach2 Mach3Mach31414 99 221515 66 771212 1010 111111 88 441313 55 336565 3838 1717TotalTotal

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SolutionSolution

Hn n

R

nnj

jj

p

FHG

IKJ

FHG

IKJ

FHG

IKJ

FH IK

12

13 1

1215 16

65

5

38

5

17

53 16

12240

1916 48

1158

2

1

2 2 2

a f a f

afafa f a f af af

a f.

.

Hn n

R

nnj

jj

p

FHG

IKJ

FHG

IKJ

FHG

IKJ

FH IK

12

13 1

1215 16

65

5

38

5

17

53 16

12240

1916 48

1158

2

1

2 2 2

a f a f

afafa f a f af af

a f.

.

14 - 14 - 7979


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


H = 11.58H = 11.58

14 - 14 - 8080


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


Reject at Reject at = .05 = .05 = .05= .05

H = 11.58H = 11.58

14 - 14 - 8181


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:



There Is Evidence Pop. There Is Evidence Pop. Distrib. Are DifferentDistrib. Are Different

= .05= .05

H = 11.58H = 11.58

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Friedman FFriedman Frr-Test-Test

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14 - 14 - 8484


Friedman FFriedman Frr-Test-Test

1.1. Tests the Equality of 2 or More (Tests the Equality of 2 or More (pp) ) Population Probability Distributions Population Probability Distributions When Blocking Variable UsedWhen Blocking Variable Used

2.2. Corresponds to Randomized Block F-TestCorresponds to Randomized Block F-Test

3.3. Used to Analyze Randomized Block Used to Analyze Randomized Block Designs Designs

4.4. Uses Uses 22 Distribution with Distribution with pp - 1 df - 1 df If Number of Blocks or Treatments > 5If Number of Blocks or Treatments > 5

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Friedman FFriedman Frr-Test -Test AssumptionsAssumptions

1.1. Independent, Random Samples Independent, Random Samples

2.2. Measurements Can Be Ranked Within Measurements Can Be Ranked Within BlocksBlocks

3.3. Continuous Population Probability Continuous Population Probability DistributionsDistributions

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Friedman FFriedman Frr-Test -Test ProcedureProcedure

1.1. Assign Ranks, Assign Ranks, RRi i , to the Observations , to the Observations

Within Each BlockWithin Each Block Smallest Value = 1; Largest Value = Smallest Value = 1; Largest Value = nnjj

Average TiesAverage Ties

2.2. Sum Ranks Within Each BlockSum Ranks Within Each Block

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Friedman FFriedman Frr-Test -Test ProcedureProcedure


Within Each BlockWithin Each Block Smallest Value = 1; Largest Value = Smallest Value = 1; Largest Value = nnjj

Average TiesAverage Ties

2.2. Sum Ranks Within Each BlockSum Ranks Within Each Block

3.3. Compute Test StatisticCompute Test Statistic

Fb p p

R b pr jj

p

12

13 12

1a f a fFb p p

R b pr jj

p

12

13 12

1a f a fSquared total of Squared total of each blockeach block

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Friedman FFriedman Frr-Test -Test ExampleExample

You’re a research assistant for the NIH. You’re You’re a research assistant for the NIH. You’re investigating the effects of plants on human stress. investigating the effects of plants on human stress. You record finger You record finger temperatures under 3 temperatures under 3 conditions: presence conditions: presence of a live plant, plant of a live plant, plant photo, nothing. At photo, nothing. At the the .05.05 level, does level, does finger temperature finger temperature depend on experimental condition?depend on experimental condition?

Subj. Live Photo None1 91.4 93.5 96.62 94.9 96.6 90.53 97.0 95.8 95.44 93.7 96.2 96.75 96.0 96.6 93.5

14 - 14 - 8989


2200

Friedman FFriedman Frr-Test -Test SolutionSolution

HH00::

HHaa::

= =

dfdf = =



Decision:Decision:


14 - 14 - 9090


2200




= =

dfdf = =



Decision:Decision:


14 - 14 - 9191


2200




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


14 - 14 - 9292


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


14 - 14 - 9393



Raw DataRaw Data

PlantPlant PhotoPhoto NoneNone91.491.4 93.593.5 96.696.694.994.9 96.696.6 90.590.597.097.0 95.895.8 95.495.493.793.7 96.296.2 96.796.796.096.0 96.696.6 93.593.5

RanksRanks

PlantPlant PhotoPhoto NoneNone

14 - 14 - 9494



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11

14 - 14 - 9595



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22

14 - 14 - 9696



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22 33

14 - 14 - 9797



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22 33

11

14 - 14 - 9898



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22 3322 11

14 - 14 - 9999



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22 3322 33 11

14 - 14 - 100100



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22 3322 33 1133 22 1111 22 3322 33 11

14 - 14 - 101101



Raw DataRaw Data


RanksRanks

PlantPlant PhotoPhoto NoneNone11 22 3322 33 1133 22 1111 22 3322 33 1199 1212 99TotalTotal

14 - 14 - 102102



Fb p p

R b pr jj

p

FHIK

12

13 1

125 3 4

9 12 9 3 5 4

1260

306 60

12

2

1

2 2 2

a f a f

afafaf af af af afaf

a f.

Fb p p

R b pr jj

p

FHIK

12

13 1

125 3 4

9 12 9 3 5 4

1260

306 60

12

2

1

2 2 2

a f a f

afafaf af af af afaf

a f.

14 - 14 - 103103


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


FFrr = 1.2 = 1.2

14 - 14 - 104104


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:


Do Not Reject at Do Not Reject at = .05 = .05 = .05= .05

FFrr = 1.2 = 1.2

14 - 14 - 105105


2200 5.9915.991




= = .05.05

dfdf = = pp - 1 = 3 - 1 = 2 - 1 = 3 - 1 = 2



Decision:Decision:



There Is No Evidence There Is No Evidence Distrib. Are DifferentDistrib. Are Different

= .05= .05

FFrr = 1.2 = 1.2

14 - 14 - 106106


Spearman’s Rank Spearman’s Rank Correlation CoefficientCorrelation Coefficient

14 - 14 - 107107


Spearman’s Rank Spearman’s Rank Correlation Correlation CoefficientCoefficient

1.1. Measures Correlation Between RanksMeasures Correlation Between Ranks

2.2. Corresponds to Pearson Product Corresponds to Pearson Product Moment Correlation CoefficientMoment Correlation Coefficient

3.3. Values Range from -1 to +1Values Range from -1 to +1

14 - 14 - 108108


Spearman’s Rank Spearman’s Rank Correlation Correlation CoefficientCoefficient

1.1. Measures Correlation Between RanksMeasures Correlation Between Ranks

2.2. Corresponds to Pearson Product Corresponds to Pearson Product Moment Correlation CoefficientMoment Correlation Coefficient

3.3. Values Range from -1 to +1Values Range from -1 to +1

4.4. Equation (Shortcut)Equation (Shortcut)

rd

n ns

1

6

1

2

2d ird

n ns

1

6

1

2

2d i

14 - 14 - 109109


Spearman’s Rank Spearman’s Rank Correlation Correlation ProcedureProcedure


of Each Variable Separatelyof Each Variable Separately

2.2. Calculate Differences, Calculate Differences, ddi i , Between , Between

Each Pair of RanksEach Pair of Ranks

3.3. Square Differences, Square Differences, ddi i 22, Between Ranks, Between Ranks

4.4. Sum Squared Differences for Each VariableSum Squared Differences for Each Variable

5.5. Use Shortcut Approximation FormulaUse Shortcut Approximation Formula

14 - 14 - 110110


Spearman’s Rank Spearman’s Rank Correlation ExampleCorrelation Example

You’re a research assistant for the FBI. You’re You’re a research assistant for the FBI. You’re investigating the relationship between a person’s investigating the relationship between a person’s attempts at deception attempts at deception & % changes in their & % changes in their pupil size. You ask pupil size. You ask subjects a series of subjects a series of questions, some of questions, some of which they must which they must answer dishonestly. answer dishonestly. At the At the .05.05 level, what is the level, what is the correlation coefficientcorrelation coefficient??

Subj. DeceptionPupil

1 87 102 63 63 95 114 50 75 43 0

14 - 14 - 111111


Spearman’s Rank Spearman’s Rank Correlation TableCorrelation Table

Subj. Decep. R1i Pupil R2i di di2

Total


Total

14 - 14 - 112112




1 87 10

2 63 6

3 95 11

4 50 7

5 43 0

Total


1 87 10

2 63 6

3 95 11

4 50 7

5 43 0

Total

14 - 14 - 113113




1 87 4 10

2 63 3 6

3 95 5 11

4 50 2 7

5 43 1 0

Total


1 87 4 10

2 63 3 6

3 95 5 11

4 50 2 7

5 43 1 0

Total

14 - 14 - 114114




1 87 4 10 4

2 63 3 6 2

3 95 5 11 5

4 50 2 7 3

5 43 1 0 1

Total


1 87 4 10 4

2 63 3 6 2

3 95 5 11 5

4 50 2 7 3

5 43 1 0 1

Total

14 - 14 - 115115




1 87 4 10 4 0

2 63 3 6 2 1

3 95 5 11 5 0

4 50 2 7 3 -1

5 43 1 0 1 0

Total


1 87 4 10 4 0

2 63 3 6 2 1

3 95 5 11 5 0

4 50 2 7 3 -1

5 43 1 0 1 0

Total

14 - 14 - 116116




1 87 4 10 4 0 0

2 63 3 6 2 1 1

3 95 5 11 5 0 0

4 50 2 7 3 -1 1

5 43 1 0 1 0 0

Total 2


1 87 4 10 4 0 0

2 63 3 6 2 1 1

3 95 5 11 5 0 0

4 50 2 7 3 -1 1

5 43 1 0 1 0 0

Total 2

14 - 14 - 117117


Spearman’s Rank Spearman’s Rank Correlation SolutionCorrelation Solution

r

d

n ns

ii

n

1

6

1

16 2

5 5 1

1 010

0 90

2

12

2

d iafd i.

.

r

d

n ns

ii

n

1

6

1

16 2

5 5 1

1 010

0 90

2

12

2

d iafd i.

.

14 - 14 - 118118


ConclusionConclusion

1.1. Distinguished Parametric & Distinguished Parametric & Nonparametric Test Procedures Nonparametric Test Procedures

2.2. Explained a Variety of Nonparametric Explained a Variety of Nonparametric Test ProceduresTest Procedures

3.3. Solved Hypothesis Testing Problems Solved Hypothesis Testing Problems Using Nonparametric TestsUsing Nonparametric Tests

4.4. Computed Spearman’s Rank CorrelationComputed Spearman’s Rank Correlation

14 - 1 © 2000 prentice-hall, inc. statistics nonparametric statistics chapter 14

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