1. Tests Jean-Yves Le Boudec

2. Contents 1.The Neyman Pearson framework 2.Likelihood Ratio Tests 3.ANOVA 4.Asymptotic Results 5.Other Tests 1

3. Tests Tests are used to give a binary answer to hypotheses of a statistical nature Ex: is A better than B? Ex: does this data come from a normal distribution ? Ex: does factor n influence the result ? 2

4. Example: Non Paired Data Is red better than blue ? For data set (a) answer is clear (by inspection of confidence interval) no test required 3

5. Is this data normal ? 4

6. 5.1 The Neyman-Pearson Framework 5

7. Example: Non Paired Data 6  Is red better than blue ?

8. Critical Region, Size and Power 7

9. Example : Paired Data 8

10. Power 9

11. 10 Grey Zone

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13. p-value of a test 12

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15. Tests are just tests 14

16. Test versus Confidence Intervals If you can have a confidence interval, use it instead of a test 15

17. 2. Likelihood Ratio Test A special case of Neyman-Pearson A Systematic Method to define tests, of general applicability 16

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20. A Classical Test: Student Test The model : The hypotheses : 19

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23. Here it is the same as a Conf. Interval 22

24. The “Simple Goodness of Fit” Test Model Hypotheses 23

25. 1. compute likelihood ratio statistic 24

26. 2. compute p-value 25

27. Mendel’s Peas P= 0.92 ± 0.05 => Accept H 0 26

28. 3 ANOVA Often used as “Magic Tool” Important to understand the underlying assumptions Model Data comes from iid normal sample with unknown means and same variance Hypotheses 27

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31. The ANOVA Theorem We build a likelihood ratio statistic test The assumption that data is normal and variance is the same allows an explicit computation it becomes a least square problem = a geometrical problem we need to compute orthogonal projections on M and M 0 30

32. The ANOVA Theorem 31

33. Geometrical Interpretation Accept H 0 if SS2 is small The theorem tells us what “small” means in a statistical sense 32

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35. ANOVA Output: Network Monitoring 34

36. The Fisher-F distribution 35

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38. Compare Test to Confidence Intervals For non paired data, we cannot simply compute the difference However CI is sufficient for parameter set 1 Tests disambiguate parameter sets 2 and 3 37

39. Test the assumptions of the test… Need to test the assumptions Normal In each group: qqplot… Same variance 38

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41. 4 Asymptotic Results 40 2 x Likelihood ratio statistic

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43. The chi-square distribution 42

44. Asymptotic Result Applicable when central limit theorem holds If applicable, radically simple Compute likelihood ratio statistic Inspect and find the order p (nb of dimensions that H1 adds to H0) This is equivalent to 2 optimization subproblems lrs = = max likelihood under H1 - max likelihood under H0 The p-value is 43

45. Composite Goodness of Fit Test We want to test the hypothesis that an iid sample has a distribution that comes from a given parametric family 44

46. Apply the Generic Method Compute likelihood ratio statistic Compute p-value Either use MC or the large n asymptotic 45

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48. Is it normal ? 47

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51. Mendel’s Peas P= 0.92 ± 0.05 => Accept H 0 50

52. Test of Independence Model Hypotheses 51

53. Apply the generic method 52

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55. 5 Other Tests Simple Goodness of Fit Model: iid data Hypotheses: H 0 common distrib has cdf F() H 1 common distrib is anything Kolmogorov-Smirnov: under H 0, the distribution of is independent of F() 54

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57. Anderson-Darling An alternative to K-S, less sensitive to “outliers” 56

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60. Jarque Bera test of normality (Chapter 4) Based on Kurtosis and Skewness Should be 0 for normal distribution 59

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62. Robust Tests Median Test Model : iid sample Hypotheses 61

63. Median Test 62

64. Wilcoxon Signed Rank Test 63

65. Wilcoxon Rank Sum Test Model: X i and Y j independent samples, each is iid Hypotheses: H 0 both have same distribution H 1 the distributions differ by a location shift 64

66. Wilcoxon Rank Sum Test 65

67. Turning Point 66

68. Questions What is the critical region of a test ? What is a type 1 error ? Type 2 ? The size of a test ? What is the p-value of a test ? 67

69. Questions What are the hypotheses for ANOVA ? How do you compute a p-value by Monte Carlo simulation ? A Monte Carlo simulation returns p = 0; what can we conclude ? What is a likelihood ratio statistic test ? What can we say about its p-value ? 68

70. We have data X_1,…,X_m and Y_1, …,Y_m. Explain how we can compute the p-value of a test that compares the variance of the two samples ? We have a collection of random variables X[i,j] that corresponds to the result of the ith simulation when the machine uses configuration j. How can you test whether the configuration plays a role or not ? 69