Nonparametric Statistics Chapter 9 Nonparametric Statistics 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 As a result of this class, you will be able to ... EPI 809 / Spring 2008

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

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

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

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. 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 3. Tables Not Widely Available © 1984-1994 T/Maker Co. EPI 809 / Spring 2008

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

Sign Test EPI 809 / Spring 2008 9 47

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

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

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? Assume that the population is normally distributed. Allow students about 10 minutes to solve this. EPI 809 / Spring 2008

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

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

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

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

Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05 Test Statistic: P-Value: Decision: Conclusion: P(S  2) = 1 - P(S  1) = .9297 (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

Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05 Test Statistic: P-Value: Decision: Conclusion: P(x  2) = 1 - P(x  1) = . 9297 (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

Sign Test Solution H0:  = 3 P-Value: Ha:  < 3 Decision:  = .05 Test Statistic: P-Value: Decision: Conclusion: P(x  2) = 1 - P(x  1) = = . 9297 (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

Wilcoxon Rank Sum Test EPI 809 / Spring 2008 9 47

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample) Do Not Reject Reject Reject 12 28  Ranks EPI 809 / Spring 2008

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 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample) Do Not Reject at  = .10 Do Not Reject Reject Reject 12 28  Ranks EPI 809 / Spring 2008

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 = 5 + 3.5 + 8+ 9 = 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