Chap. 9: Nonparametric Statistics - SRM University 809 / Spring 2008 Advantages of Nonparametric Tests 1. Used With All Scales 2. Easier to Compute 3. Make Fewer Assumptions 4. Need

EPI 809 / Spring 2008

Chapter 9

Nonparametric Statistics


Learning Objectives

1. Distinguish Parametric & Nonparametric Test Procedures

2. Explain commonly used Nonparametric Test Procedures

3. Perform Hypothesis Tests Using Nonparametric Procedures


Hypothesis Testing Procedures

Many More Tests Exist!


Parametric Test Procedures

1. Involve Population Parameters (Mean)

2. Have Stringent Assumptions (Normality)

3. Examples: Z Test, t Test, χ2 Test, F test


Nonparametric Test Procedures

1. Do Not Involve Population ParametersExample: Probability Distributions, Independence

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

3. Example: Wilcoxon Rank Sum Test


Advantages of Nonparametric Tests

1. Used With All Scales2. Easier to Compute3. Make Fewer Assumptions4. Need Not Involve

Population Parameters5. Results May Be as Exact

as Parametric Procedures

© 1984-1994 T/Maker Co.


Disadvantages of Nonparametric Tests

1.May Waste Information Parametric model more efficient if data Permit

2.Difficult to Compute by

hand for Large Samples

3.Tables Not Widely Available

© 1984-1994 T/Maker Co.


Popular Nonparametric Tests

1. Sign Test

2. Wilcoxon Rank Sum Test

3. Wilcoxon Signed Rank Test


Sign Test


Sign Test 1. Tests One Population Median, η

2. Corresponds to t-Test for 1 Mean

3. Assumes Population Is Continuous

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

5. Can Use Normal Approximation If n ≥ 10


Sign Test Concepts

Make null hypothesis about true median

Let S = number of values greater than median

Each sampled item is independent

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


Sign Test Example

You’re an analyst for Chef-Boy-R-Dee. You’ve asked 7 people to rate a new ravioli on a 5-point scale (1 = terrible,…, 5 = excellent) The ratings are: 2 5 3 4 1 4 5.

At the .05 level, is there evidence that the median rating is at least 3?


Sign Test Solution

H0: Ha: α = Test Statistic:

P-Value:

Decision:

Conclusion:


Sign Test Solution

H0: η = 3Ha: η < 3α = Test Statistic:

P-Value:

Decision:

Conclusion:


Sign Test Solution

H0: η = 3Ha: η < 3α = .05Test Statistic:

P-Value:

Decision:

Conclusion:


Sign Test Solution


P-Value:

Decision:

Conclusion:

S = 2 (Ratings 1 & 2 Are Less Than η = 3:2, 5, 3, 4, 1, 4, 5)Is observing 2 or more a small prob event?


Sign Test Solution


P-Value:

Decision:

Conclusion:

P(S ≥ 2) = 1 - P(S ≤ 1) = .9297

(Binomial Table, n = 7, p = 0.50)



Sign Test Solution


P-Value:

Decision:

Conclusion:Do Not Reject at α = .05

P(x ≥ 2) = 1 - P(x ≤ 1) = . 9297




Sign Test Solution


P-Value:

Decision:


There is No evidence for Median < 3

P(x ≥ 2) = 1 - P(x ≤ 1) == . 9297


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


Wilcoxon Rank Sum Test


Wilcoxon Rank Sum Test

1.Tests Two Independent Population Probability Distributions

2.Corresponds to t-Test for 2 Independent Means

3.AssumptionsIndependent, Random SamplesPopulations Are Continuous

4.Can Use Normal Approximation If ni ≥ 10


Wilcoxon Rank Sum Test Procedure

1. Assign Ranks, Ri, to the n1 + n2 Sample Observations

If Unequal Sample Sizes, Let n1 Refer to Smaller-Sized SampleSmallest Value = 1

2. Sum the Ranks, Ti, for Each SampleTest Statistic Is TA (Smallest Sample)

Null hypothesis: both samples come from the same underlying distribution

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


Wilcoxon Rank Sum Test Example

You’re a production planner. You want to see if the operating rates for 2 factories is the same. For factory 1, the rates (% of capacity) are 71, 82, 77, 92, 88. For factory 2, the rates are 85, 82, 94 & 97. Do the factory rates have the same probability distributions at the .10 level?


Wilcoxon Rank Sum Test Solution

H0:Ha:α =n1 = n2 = Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Σ Ranks



H0: Identical Distrib.Ha: Shifted Left or Rightα =n1 = n2 = Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Σ Ranks



H0: Identical Distrib.Ha: Shifted Left or Rightα = .10n1 = 4 n2 = 5 Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Σ Ranks


Wilcoxon Rank Sum Table 12 (Rosner) (Portion)α = .05 two-tailed




Test Statistic:

Decision:

Conclusion:Reject RejectDo Not Reject

12 28 Σ Ranks


Wilcoxon Rank Sum Test Computation Table

Factory 1 Factory 2Rate Rank Rate Rank

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 8582 8277 9492 9788 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 8582 8277 9492 9788 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 8582 8277 2 9492 9788 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 8582 3 82 477 2 9492 9788 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 8582 3 3.5 82 4 3.577 2 9492 9788 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 85 582 3 3.5 82 4 3.577 2 9492 9788 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 85 582 3 3.5 82 4 3.577 2 9492 9788 6 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 85 582 3 3.5 82 4 3.577 2 9492 7 9788 6 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 85 582 3 3.5 82 4 3.577 2 94 892 7 9788 6 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 85 582 3 3.5 82 4 3.577 2 94 892 7 97 988 6 ... ...

Rank Sum



Factory 1 Factory 2Rate Rank Rate Rank71 1 85 582 3 3.5 82 4 3.577 2 94 892 7 97 988 6 ... ...

Rank Sum 19.5 25.5




Test Statistic:

Decision:

Conclusion:Reject RejectDo Not Reject

12 28 Σ Ranks

T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)




Test Statistic:

Decision:


Reject RejectDo Not Reject

12 28 Σ Ranks





Test Statistic:

Decision:


There is No evidence for unequal distrib

Reject RejectDo Not Reject

12 28 Σ Ranks


Documents

Chap. 9: Nonparametric Statistics - SRM University 809 / Spring 2008 Advantages of Nonparametric Tests 1. Used With All Scales 2. Easier to Compute 3. Make Fewer Assumptions 4. Need