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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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**Tests**Jean-Yves Le Boudec**Contents**• The Neyman Pearson framework • Likelihood Ratio Tests • ANOVA • Asymptotic Results • 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**Example: Non Paired Data**• Is red better than blue ? 6**Power**9**Grey Zone**10**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**A Classical Test: Student Test**• The model : • The hypotheses : 19**The “Simple Goodness of Fit” Test**• Model • Hypotheses 23**Mendel’s Peas**• P= 0.92 ± 0.05 => Accept H0 26**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**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**Geometrical Interpretation**• Accept H0 if SS2 is small • The theorem tells us what “small” means in a statistical sense 32**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**4 Asymptotic Results**2 x Likelihood ratio statistic 40**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**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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