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Tests
Jean-Yves Le Boudec
Contents
1. The Neyman Pearson framework
2. Likelihood Ratio Tests
3. ANOVA
4. Asymptotic Results
5. Other Tests
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Tests
Tests are used to give a binary answer to hypotheses of a statistical natureEx: is A better than B?
Ex: does this data come from a normal distribution ?
Ex: does factor n influence the result ?
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Example: Non Paired Data
Is red better than blue ?
For data set (a) answer is clear (by inspection of confidence interval) no test required
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Is this data normal ?
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5.1 The Neyman-Pearson Framework
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Example: Non Paired Data
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Is red better than blue ?
Critical Region, Size and Power
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Example : Paired Data
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Power
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Grey Zone
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p-value of a test
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Tests are just tests
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Test versus Confidence Intervals
If you can have a confidence interval, use it instead of a test
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2. Likelihood Ratio Test
A special case of Neyman-Pearson
A Systematic Method to define tests, of general applicability
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A Classical Test: Student Test
The model :
The hypotheses :
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Here it is the same as a Conf. Interval
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The “Simple Goodness of Fit” Test
Model
Hypotheses
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1. compute likelihood ratio statistic
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2. compute p-value
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Mendel’s Peas
P= 0.92 ± 0.05 => Accept H0
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3 ANOVA
Often used as “Magic Tool”
Important to understand the underlying assumptions
ModelData comes from iid normal sample with unknown means and same variance
Hypotheses
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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 M0
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The ANOVA Theorem
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Geometrical InterpretationAccept H0 if SS2 is small
The theorem tells us what “small” means in a statistical sense
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ANOVA Output: Network Monitoring
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The Fisher-F distribution
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Compare Test to Confidence Intervals
For non paired data, we cannot simply compute the differenceHowever CI is sufficient for parameter set 1
Tests disambiguate parameter sets 2 and 3
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Test the assumptions of the test…Need to test the assumptions
Normal In each group: qqplot…
Same variance
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4 Asymptotic Results
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2 x Likelihood ratio statistic
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The chi-square distribution
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Asymptotic Result
Applicable when central limit theorem holds
If applicable, radically simpleCompute 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
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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
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Apply the Generic Method
Compute likelihood ratio statistic
Compute p-value
Either use MC or the large n asymptotic
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Is it normal ?
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Mendel’s Peas
P= 0.92 ± 0.05 => Accept H0
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Test of Independence
Model
Hypotheses
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Apply the generic method
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5 Other TestsSimple Goodness of Fit
Model: iid data
Hypotheses: H0 common distrib has cdf F()H1 common distrib is anything
Kolmogorov-Smirnov: under H0, the distribution of
is independent of F()
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Anderson-Darling
An alternative to K-S, less sensitive to “outliers”
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Jarque Bera test of normality (Chapter 4)
Based on Kurtosis and SkewnessShould be 0 for normal distribution
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Robust TestsMedian Test
Model : iid sample
Hypotheses
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Median Test
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Wilcoxon Signed Rank Test
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Wilcoxon Rank Sum Test
Model: Xi and Yj independent samples, each is iid
Hypotheses: H0 both have same distributionH1 the distributions differ by a location shift
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Wilcoxon Rank Sum Test
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Turning Point
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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 ?
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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 ?
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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 ?
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