1. Nonparametric Statistics
Chapter 9
Nonparametric Statistics
EPI 809 / Spring 2008

2. Learning Objectives
1. Distinguish Parametric & Nonparametric Test Procedures
2. Explain commonly used Nonparametric Test Procedures
3. Perform Hypothesis Tests Using Nonparametric Procedures
As a result of this class, you will be able to ...
EPI 809 / Spring 2008

3. Hypothesis Testing Procedures
Many More Tests Exist!
EPI 809 / Spring 2008 12

4. Parametric Test Procedures
1. Involve Population Parameters (Mean)
2. Have Stringent Assumptions
(Normality)
3. Examples: Z Test, t Test, 2 Test,
F test
EPI 809 / Spring 2008

5. Nonparametric Test Procedures
1. Do Not Involve Population Parameters
Example: Probability Distributions, Independence
2. Data Measured on Any Scale (Ratio or Interval, Ordinal or Nominal)
3. Example: Wilcoxon Rank Sum Test
EPI 809 / Spring 2008

6. Advantages of Nonparametric Tests
1. Used With All Scales
2. Easier to Compute
3. Make Fewer Assumptions
4. Need Not Involve Population Parameters
5. Results May Be as Exact
as Parametric Procedures
© T/Maker Co.
EPI 809 / Spring 2008

7. 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
© T/Maker Co.
EPI 809 / Spring 2008

8. Popular Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
EPI 809 / Spring 2008

9. Sign Test
EPI 809 / Spring 2008 9 47

10. Sign Test Tests One Population Median, 
2. Corresponds to t-Test for 1 Mean
3. Assumes Population Is Continuous
Small Sample Test Statistic: # Sample Values Above (or Below) Median
5. Can Use Normal Approximation If n  10
EPI 809 / Spring 2008

11. 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
EPI 809 / Spring 2008

12. 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:
At the .05 level, is there evidence that the median rating is at least 3?
Assume that the population is normally distributed.
Allow students about 10 minutes to solve this.
EPI 809 / Spring 2008

13. Sign Test Solution H0: P-Value: Ha: Decision:  = Test Statistic:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
EPI 809 / Spring 2008

14. Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  =
Test Statistic:
P-Value:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
EPI 809 / Spring 2008

15. Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05
Test Statistic:
P-Value:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
EPI 809 / Spring 2008

16. Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05
Test Statistic:
P-Value:
Decision:
Conclusion:
Note: More than 5 have been sold (6.4), but not enough to be significant.
S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)
Is observing 2 or more a small prob event?
EPI 809 / Spring 2008

17. Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05
Test Statistic:
P-Value:
Decision:
Conclusion:
P(S  2) = 1 - P(S  1) = (Binomial Table, n = 7, p = 0.50)
Note: More than 5 have been sold (6.4), but not enough to be significant.
S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)
Is observing 2 or more a small prob event?
EPI 809 / Spring 2008

18. Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05
Test Statistic:
P-Value:
Decision:
Conclusion:
P(x  2) = 1 - P(x  1) = (Binomial Table, n = 7, p = 0.50)
Note: More than 5 have been sold (6.4), but not enough to be significant.
S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)
Do Not Reject at  = .05
Is observing 2 or more a small prob event?
EPI 809 / Spring 2008

19. Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05
Test Statistic:
P-Value:
Decision:
Conclusion:
P(x  2) = 1 - P(x  1) = = (Binomial Table, n = 7, p = 0.50)
Note: More than 5 have been sold (6.4), but not enough to be significant.
S = 2 (Ratings 1 & 2 are <  = 3: 2, 5, 3, 4, 1, 4, 5)
Do Not Reject at  = .05
There is No evidence for Median < 3
Is observing 2 or more a small prob event?
EPI 809 / Spring 2008

20. Wilcoxon Rank Sum Test
EPI 809 / Spring 2008 9 47

21. Wilcoxon Rank Sum Test
1. Tests Two Independent Population Probability Distributions
2. Corresponds to t-Test for 2 Independent Means
3. Assumptions
Independent, Random Samples
Populations Are Continuous
4. Can Use Normal Approximation If ni  10
EPI 809 / Spring 2008

22. 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 Sample
Smallest Value = 1
2. Sum the Ranks, Ti, for Each Sample
Test 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
EPI 809 / Spring 2008

23. 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?
EPI 809 / Spring 2008 51

24. Wilcoxon Rank Sum Test Solution
H0:
Ha:
 =
n1 = n2 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
 Ranks
EPI 809 / Spring 2008

25. Wilcoxon Rank Sum Test Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 =
n1 = n2 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
 Ranks
EPI 809 / Spring 2008

26. Wilcoxon Rank Sum Test Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
 Ranks
EPI 809 / Spring 2008

27. Wilcoxon Rank Sum Table 12 (Rosner) (Portion)
 = .05 two-tailed
EPI 809 / Spring 2008

28. Wilcoxon Rank Sum Test Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do Not Reject
Reject
Reject 12 28
 Ranks
EPI 809 / Spring 2008

29. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank
Rank Sum
EPI 809 / Spring 2008

30. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 85 82 82 77 94 92 97 88
...
...
Rank Sum
EPI 809 / Spring 2008

31. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 82 82 77 94 92 97 88
...
...
Rank Sum
EPI 809 / Spring 2008

32. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 82 82 77 2 94 92 97 88
...
...
Rank Sum
EPI 809 / Spring 2008

33. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 82 3 82 4 77 2 94 92 97 88
...
...
Rank Sum
EPI 809 / Spring 2008

34. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 82 3
3.5 82 4
3.5 77 2 94 92 97 88
...
...
Rank Sum
EPI 809 / Spring 2008

35. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 5 82 3
3.5 82 4
3.5 77 2 94 92 97 88
...
...
Rank Sum
EPI 809 / Spring 2008

36. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 5 82 3
3.5 82 4
3.5 77 2 94 92 97 88 6
...
...
Rank Sum
EPI 809 / Spring 2008

37. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 5 82 3
3.5 82 4
3.5 77 2 94 92 7 97 88 6
...
...
Rank Sum
EPI 809 / Spring 2008

38. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 5 82 3
3.5 82 4
3.5 77 2 94 8 92 7 97 88 6
...
...
Rank Sum
EPI 809 / Spring 2008

39. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 5 82 3
3.5 82 4
3.5 77 2 94 8 92 7 97 9 88 6
...
...
Rank Sum
EPI 809 / Spring 2008

40. Wilcoxon Rank Sum Test Computation Table
Factory 1
Factory 2
Rate
Rank
Rate
Rank 71 1 85 5 82 3
3.5 82 4
3.5 77 2 94 8 92 7 97 9 88 6
...
...
Rank Sum
19.5
25.5
EPI 809 / Spring 2008

41. Wilcoxon Rank Sum Test Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
T2 = = 25.5 (Smallest Sample)
Do Not Reject
Reject
Reject 12 28
 Ranks
EPI 809 / Spring 2008

42. Wilcoxon Rank Sum Test Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
T2 = = 25.5 (Smallest Sample)
Do Not Reject at  = .10
Do Not Reject
Reject
Reject 12 28
 Ranks
EPI 809 / Spring 2008

43. Wilcoxon Rank Sum Test Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
T2 = = 25.5 (Smallest Sample)
Do Not Reject at  = .10
Do Not Reject
Reject
Reject
There is No evidence for unequal distrib 12 28
 Ranks
EPI 809 / Spring 2008