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Tests

Tests. Jean-Yves Le Boudec. Contents. The Neyman Pearson framework Likelihood Ratio Tests ANOVA Asymptotic Results Other Tests. Tests. Tests are used to give a binary answer to hypotheses of a statistical nature Ex: is A better than B?

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Tests

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  1. Tests Jean-Yves Le Boudec

  2. Contents • The Neyman Pearson framework • Likelihood Ratio Tests • ANOVA • Asymptotic Results • 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 • Is red better than blue ? 6

  8. Critical Region, Size and Power 7

  9. Example : Paired Data 8

  10. Power 9

  11. Grey Zone 10

  12. 11

  13. p-value of a test 12

  14. 13

  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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  22. 21

  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 H0 26

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

  29. 28

  30. 29

  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 M0 30

  32. The ANOVA Theorem 31

  33. Geometrical Interpretation • Accept H0 if SS2 is small • The theorem tells us what “small” means in a statistical sense 32

  34. 33

  35. ANOVA Output: Network Monitoring 34

  36. The Fisher-F distribution 35

  37. 36

  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

  40. 39

  41. 4 Asymptotic Results 2 x Likelihood ratio statistic 40

  42. 41

  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 subproblemslrs = = 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

  47. 46

  48. Is it normal ? 47

  49. 48

  50. 49

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