EPI 809 / Spring 2008 Chapter 9 Nonparametric Statistics.

EPI 809 / Spring 2008EPI 809 / Spring 2008

Chapter 9Chapter 9

Nonparametric StatisticsNonparametric Statistics


Learning ObjectivesLearning Objectives

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

2.2. Explain commonly used Explain commonly used Nonparametric Test ProceduresNonparametric Test Procedures

3.3. Perform Hypothesis Tests Using Perform Hypothesis Tests Using Nonparametric ProceduresNonparametric Procedures


Hypothesis Testing ProceduresHypothesis Testing Procedures

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!


Parametric Test ProceduresParametric Test Procedures

1.1. Involve Population Parameters (Mean)Involve Population Parameters (Mean)

2.2. Have Stringent Assumptions Have Stringent Assumptions

(Normality)(Normality)

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

F testF test


Nonparametric Test Nonparametric Test ProceduresProcedures

1.1. Do Not Involve Population Do Not Involve Population ParametersParameters

Example: Probability Distributions, Example: Probability Distributions, IndependenceIndependence

2.2. Data Measured on Any Scale (Data Measured on Any Scale (Ratio Ratio or or Interval, Ordinal or Nominal)Interval, Ordinal or Nominal)

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


Advantages of Advantages of Nonparametric TestsNonparametric Tests

1.1. Used With All ScalesUsed With All Scales

2.2. Easier to ComputeEasier to Compute

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 Results May Be as Exact

as Parametric Proceduresas Parametric Procedures

© 1984-1994 T/Maker Co.


Disadvantages of Disadvantages of Nonparametric TestsNonparametric Tests

1.1. May Waste Information May Waste Information Parametric model more efficient Parametric model more efficient

if data Permitif data Permit

2.2. Difficult to Compute byDifficult to Compute by

hand for Large Sampleshand for Large Samples

3.3. Tables Not Widely AvailableTables Not Widely Available

© 1984-1994 T/Maker Co.


Popular Nonparametric TestsPopular Nonparametric 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


Sign Test Sign Test


Sign Test Sign Test

1.1. Tests One Population Median, Tests One Population Median,

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 Values Small Sample Test Statistic: # Sample Values Above (or Below) MedianAbove (or Below) Median

5. Can Use Normal Approximation If 5. Can Use Normal Approximation If nn 10 10


Sign Test ConceptsSign Test Concepts

Make null hypothesis about true medianMake null hypothesis about true median

Let S = number of values greater than medianLet S = number of values greater than median

Each sampled item is independentEach sampled item is independent

If null hypothesis is true, S should have binomial If null hypothesis is true, S should have binomial distribution with success probability .5distribution with success probability .5


Sign Test ExampleSign Test Example

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


Sign Test SolutionSign Test Solution

HH00: : HHaa: : = = Test Statistic:Test Statistic:

P-Value: P-Value:

Decision:Decision:

Conclusion:Conclusion:



HH00: : = = 33 HHaa: : < 3 < 3 = = Test Statistic:Test Statistic:

P-Value: P-Value:

Decision:Decision:




HH00: : = = 33 HHaa: : < 3 < 3 = = .05.05 Test Statistic:Test Statistic:

P-Value: P-Value:

Decision:Decision:





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)Is observing 2 or more a small prob event?




P-Value: P-Value:

Decision:Decision:


PP(S (S 2) = 1 - 2) = 1 - PP(S (S 1) 1) = .9297= .9297

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





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) = . 9297= . 9297






P-Value: P-Value:

Decision:Decision:



There is No evidence for There is No evidence for Median < 3Median < 3

PP((xx 2) = 1 - 2) = 1 - PP((xx 1) 1) == = . 9297= . 9297


S = 2 S = 2 (Ratings 1 & 2 are (Ratings 1 & 2 are < < = = 3:3:22, 5, 3, 4, , 5, 3, 4, 11, 4, 5), 4, 5)Is observing 2 or more a small prob event?


Wilcoxon Rank Sum Wilcoxon Rank Sum TestTest


Wilcoxon Rank Sum Test Wilcoxon Rank Sum Test

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.AssumptionsAssumptionsIndependent, Random SamplesIndependent, Random Samples

Populations Are ContinuousPopulations Are Continuous

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


Wilcoxon Rank Sum Test Wilcoxon Rank Sum Test ProcedureProcedure

1.1. 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-Sized Sample Refer to Smaller-Sized Sample

Smallest Value = 1Smallest Value = 1

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

Test Statistic Is Test Statistic Is TTA A (Smallest Sample)(Smallest Sample)Null Null hypothesis: both samples come from the same underlying hypothesis: both samples come from the same underlying

distributiondistribution

Distribution of T is not quite as simple as binomial, but it can be Distribution of T is not quite as simple as binomial, but it can be computedcomputed


Wilcoxon Rank Sum Test Wilcoxon Rank Sum Test ExampleExample

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


Wilcoxon Rank Sum Test Wilcoxon Rank Sum Test SolutionSolution

HH00:: HHaa:: == nn11 = = nn22 = = Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:


RanksRanks



HH00: : Identical Distrib.Identical Distrib. HHaa: : Shifted Left or Shifted Left or

RightRight == nn11 = = nn22 = = Critical Value(s):Critical Value(s):


Decision:Decision:


RanksRanks




RightRight = = .10.10 nn11 = = 4 4 nn22 = = 5 5 Critical Value(s):Critical Value(s):


Decision:Decision:


RanksRanks


Wilcoxon Rank Sum Wilcoxon Rank Sum Table 12 (Rosner) (Portion)Table 12 (Rosner) (Portion)

n1 4 5 6 ..

TL TU TL TU TL TU ..

4 10 26 16 34 23 43 .. n2 5 11 29 17 38 24 48 .. 6 12 32 18 42 26 52 .. : : : : : : : :

n1 4 5 6 ..

TL TU TL TU TL TU ..

4 10 26 16 34 23 43 .. n2 5 11 29 17 38 24 48 .. 6 12 32 18 42 26 52 .. : : : : : : : :

= .05 two-tailed= .05 two-tailed






Decision:Decision:

Conclusion:Conclusion:RejectReject RejectRejectDo Not Do Not

RejectReject

1212 2828 RanksRanks


Wilcoxon Rank Sum Test Wilcoxon Rank Sum Test Computation TableComputation Table

Factory 1 Factory 2Rate Rank Rate Rank

Rank Sum




71 8582 8277 9492 9788 ... ...

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum




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

Rank Sum 19.5 25.5






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)






Decision:Decision:


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

RejectReject








Decision:Decision:



There is No evidence for There is No evidence for unequal distribunequal distrib

RejectReject RejectRejectDo Not Do Not RejectReject



EPI 809 / Spring 2008 Chapter 9 Nonparametric Statistics.

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