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EPI 809 / Spring 2008 Chapter 9 Nonparametric Statistics.

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Presentation on theme: "EPI 809 / Spring 2008 Chapter 9 Nonparametric Statistics."— Presentation transcript:

1 EPI 809 / Spring 2008 Chapter 9 Nonparametric Statistics

2 EPI 809 / Spring 2008 Learning Objectives 1.Distinguish Parametric & Nonparametric Test Procedures 2.Explain commonly used Nonparametric Test Procedures 3. Perform Hypothesis Tests Using Nonparametric Procedures

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

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

5 EPI 809 / Spring 2008 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

6 EPI 809 / Spring 2008 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 © 1984-1994 T/Maker Co.

7 EPI 809 / Spring 2008 Disadvantages of Nonparametric Tests 1.May Waste Information Parametric model more efficient if data Permit 2.Difficult to Compute by hand for Large Samples hand for Large Samples 3.Tables Not Widely Available © 1984-1994 T/Maker Co.

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

9 EPI 809 / Spring 2008 Sign Test

10 EPI 809 / Spring 2008 Sign Test 1. Tests One Population Median,  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

11 EPI 809 / Spring 2008 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

12 EPI 809 / Spring 2008 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?

13 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  H a :   =  Test Statistic: P-Value:Decision:Conclusion:

14 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  = 3  H a :  < 3   =  Test Statistic: P-Value:Decision:Conclusion:

15 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  = 3  H a :  < 3   =.05  Test Statistic: P-Value:Decision:Conclusion:

16 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  = 3  H a :  < 3   =.05  Test Statistic: P-Value:Decision:Conclusion: 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?

17 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  = 3  H a :  < 3   =.05  Test Statistic: P-Value:Decision:Conclusion: P(S  2) = 1 - P(S  1) =.9297 (Binomial Table, n = 7, p = 0.50) 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?

18 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  = 3  H a :  < 3   =.05  Test Statistic: P-Value:Decision:Conclusion: Do Not Reject at  =.05 P(x  2) = 1 - P(x  1) =. 9297 (Binomial Table, n = 7, p = 0.50) 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?

19 EPI 809 / Spring 2008 Sign Test Solution  H 0 :  = 3  H a :  < 3   =.05  Test Statistic: P-Value:Decision:Conclusion: Do Not Reject at  =.05 There is No evidence for Median < 3 P(x  2) = 1 - P(x  1) = =. 9297 (Binomial Table, n = 7, p = 0.50) S = 2 (Ratings 1 & 2 are <  = 3: 2, 5, 3, 4, 1, 4, 5) Is observing 2 or more a small prob event?

20 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test

21 EPI 809 / Spring 2008 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 n i  10

22 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Procedure 1.Assign Ranks, R i, to the n 1 + n 2 Sample Observations If Unequal Sample Sizes, Let n 1 Refer to Smaller-Sized Sample Smallest Value = 1 2.Sum the Ranks, T i, for Each Sample Test Statistic Is T A (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

23 EPI 809 / Spring 2008 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?

24 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 :  H a :   =  n 1 =n 2 =  Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

25 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 : Identical Distrib.  H a : Shifted Left or Right   =  n 1 =n 2 =  Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

26 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 : Identical Distrib.  H a : Shifted Left or Right   =.10  n 1 = 4 n 2 = 5  Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

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

28 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 : Identical Distrib.  H a : Shifted Left or Right   =.10  n 1 = 4 n 2 = 5  Critical Value(s): Test Statistic: Decision:Conclusion: RejectReject Do Not Reject 1228  Ranks

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

30 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 7185 82 7794 9297 88... Rank Sum

31 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 82 7794 9297 88... Rank Sum

32 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 82 77294 9297 88... Rank Sum

33 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 823 4 77294 9297 88... Rank Sum

34 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 823 3.5824 3.5 77294 9297 88... Rank Sum

35 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 9297 88... Rank Sum

36 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 9297 886... Rank Sum

37 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 92797 886... Rank Sum

38 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 92797 886... Rank Sum

39 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 927979 886... Rank Sum

40 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 927979 886... Rank Sum19.525.5

41 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 : Identical Distrib.  H a : Shifted Left or Right   =.10  n 1 = 4 n 2 = 5  Critical Value(s): Test Statistic: Decision:Conclusion: RejectReject Do Not Reject 1228  Ranks T 2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

42 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 : Identical Distrib.  H a : Shifted Left or Right   =.10  n 1 = 4 n 2 = 5  Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.10 RejectReject Do Not Reject 1228  Ranks T 2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

43 EPI 809 / Spring 2008 Wilcoxon Rank Sum Test Solution  H 0 : Identical Distrib.  H a : Shifted Left or Right   =.10  n 1 = 4 n 2 = 5  Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.10 There is No evidence for unequal distrib RejectReject Do Not Reject 1228  Ranks T 2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

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