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Nonparametric Statistical Methods

Nonparametric Statistical Methods. Presented by Guo Cheng, Ning Liu , Faiza Khan, Zhenyu Zhang, Du Huang, Christopher Porcaro, Hongtao Zhao, Wei Huang. Introduction. Definition.

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Nonparametric Statistical Methods

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  1. Nonparametric Statistical Methods Presented by Guo Cheng, Ning Liu , Faiza Khan, Zhenyu Zhang, Du Huang, Christopher Porcaro, Hongtao Zhao, Wei Huang

  2. Introduction

  3. Definition Nonparametric methods 1: rank-based methods are used when we have no idea about the population distribution from which the data is sampled. Used for small sample sizes. Used when the data are measured on an ordinal scale and only their ranks are meaningful.

  4. Outline • 1. Sign Test • 2. Wilcoxon Signed Rank Test • 3. Inferences for Two Independent Samples • 4. Inferences for Several Independent Samples • 5. Friedman Test • 6. Spearman’s Rank Correlation • 7. Kendall’s Rank Correlation Coefficient

  5. 1 .Sign Test

  6. Parameter of interest: Median Median is used as a parameter because it is a better measure of data as compared to the mean for skewed distributions.

  7. Hypothesis test H0: µ = µ0 vs Ha: µ > µ0 where µ0 is a specified value and µ is unknown median

  8. Testing Procedure • Step 1: Given a random sample x1, x2, …, xn from a population with unknown median µ, count the number of xi’s that exceed µ0. • Denote them by s+. • s-= n - s+ • Step 2: Reject H0 if s+ is large or s- is small.

  9. How to reject H0? • To determine how large s+ must be in order to reject H0, we need to find out the distribution of the corresponding random variable S+. • Xi: random variable corresponding to the observed values xi • S-: random variable corresponding to s-

  10. Distribution of S+ and S-

  11. Calculating P-value

  12. Rejection criteria

  13. Large sample z-test

  14. Confidence Interval

  15. Example

  16. SAS code DATA themostat; INPUT temp; datalines; 202.2 203.4 … ; PROCUNIVARIATEDATA=themostat loccountmu0=200; VAR temp; RUN;

  17. SAS Output Basic Statistical Measures Location Variability Mean 201.7700 Std Deviation 2.41019 Median 201.7500 Variance 5.80900 Mode . Range 8.30000 Interquartile Range 2.90000 Tests for Location: Mu0=200 Test -Statistic- -----p Value------ Student's t t 2.322323 Pr > |t| 0.0453 Sign M 3 Pr >= |M| 0.1094 Signed Rank S 19.5 Pr >= |S| 0.048

  18. 2. Wilcoxon signed rank test

  19. Inventor Frank Wilcoxon (2 September 1892 in County Cork, Ireland – 18 November 1965, Tallahassee, Florida, USA) was a chemist and statistician, known for development of several statistical tests.

  20. What is it used for? • Two related samples • Matched samples • Repeated measurements on a single sample

  21. Hypothesis

  22. Testing procedure

  23. Example

  24. SAS codes DATA thermo; INPUT temp; datalines; 202.2 203.4 … ; PROCUNIVARIATEDATA=thermo loccountmu0=200; TITLE"Wilcoxon signed rank test the thermostat"; VAR temp; RUN;

  25. 8 SAS outputs (selected results) Basic Statistical Measures Location Variability Mean 201.7700 Std Deviation 2.41019 Median 201.7500 Variance 5.80900 Mode . Range 8.30000 Interquartile Range 2.90000 Tests for Location: Mu0=200 Test -Statistic- -----p Value------ Student's t t 2.322323 Pr > |t| 0.0453 Sign M 3 Pr >= |M| 0.1094 Signed Rank S 19.5 Pr >= |S| 0.048

  26. Large sample approximation

  27. Derive E(x) & Var(x)

  28. Rejection region:

  29. 3. Inferences for Two Independent Samples

  30. Hypothesis

  31. Definition

  32. Definition

  33. Wilcoxon sum rank test

  34. Mann-Whitney-U test

  35. Between two tests

  36. Advantages

  37. For large samples

  38. For large samples

  39. Treatment of ties

  40. Example • To test if the grades of two classes which have the same teacher are the same, we randomly pick 7 students from Class A and 9 from Class B, their scores are as follows • A: 8.50 9.48 8.65 8.16 8.83 7.76 8.63 • B: 8.27 8.20 8.25 8.14 9.00 8.10 7.20 8.32 7.70

  41. Example

  42. Example

  43. Example

  44. SAS code Data exam; Input group $ score @@; Datalines; A 8.50 A 9.48 A 8.65 A 8.16 A 8.83 A 7.76 A 8.63 B 8.27 B 8.20 B 8.25 B 8.14 B 9.00 B 8.10 B 7.20 B 8.32 B 7.70 ;

  45. SAS code Proc npar1way data=exam wilcoxon; Var score; Class group; Exact wilcoxon; Run;

  46. Output

  47. Output

  48. Output

  49. 4. Inferences for Several Independent Samples

  50. Introduction • We know that if our data is normally distributed and that the population standard deviations are equal, we can test for a difference among several populations by using the One-way ANOVA F test.

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