Tests JeanYves Le Boudec Contents 1 The Neyman

Tests Jean-Yves Le Boudec

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

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

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

Is this data normal ? 4

5. 1 The Neyman-Pearson Framework 5

Example: Non Paired Data q Is red better than blue ? 6

Critical Region, Size and Power 7

Example : Paired Data 8

Power 9

Grey Zone 10

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p-value of a test 12

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Tests are just tests 14

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

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

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

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Here it is the same as a Conf. Interval 22

The “Simple Goodness of Fit” Test Model Hypotheses 23

1. compute likelihood ratio statistic 24

2. compute p-value 25

Mendel’s Peas P= 0. 92 ± 0. 05 => Accept H 0 26

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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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

The ANOVA Theorem 31

Geometrical Interpretation Accept H 0 if SS 2 is small The theorem tells us what “small” means in a statistical sense 32

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ANOVA Output: Network Monitoring 34

The Fisher-F distribution 35

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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

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

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4 Asymptotic Results 2 x Likelihood ratio statistic 40

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The chi-square distribution 42

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 H 1 adds to H 0) This is equivalent to 2 optimization subproblems lrs = = max likelihood under H 1 - max likelihood under H 0 The p-value is 43

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

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

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Is it normal ? 47

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Mendel’s Peas P= 0. 92 ± 0. 05 => Accept H 0 50

Test of Independence Model Hypotheses 51

Apply the generic method 52

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

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

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Robust Tests Median Test Model : iid sample Hypotheses 61

Median Test 62

Wilcoxon Signed Rank Test 63

Wilcoxon Rank Sum Test Model: Xi and Yj independent samples, each is iid Hypotheses: H 0 both have same distribution H 1 the distributions differ by a location shift 64

Wilcoxon Rank Sum Test 65

Turning Point 66

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

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

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