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 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 6 Is red better than blue ?
Critical Region, Size and Power 7
Example : Paired Data 8
Power 9
10 Grey Zone
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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 SS2 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 40 2 x Likelihood ratio statistic
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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 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
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: 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
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