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note03 - Chapter 2 Two-Sample Methods 1 Two-Sample...

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Chapter 2: Two-Sample Methods 1. Two-Sample Permutation Test Two-sample t-test Assumptions X 1 , . . . , X m iid N ( μ X , σ 2 X ) and Y 1 , . . . , Y n iid N ( μ Y , σ 2 Y ) X 1 , . . . , X m and Y 1 , . . . , Y n are independent σ 2 X = σ 2 Y Test H 0 : μ X = μ Y vs H a : μ X > μ Y Pooled sample variance: Test statistic: Reject H 0 if – Can we guarantee that the assumptions of the t-test are al- ways met? Two-sample permutation test Assumptions X 1 , . . . , X m iid F 1 and Y 1 , . . . , Y n iid F 2 ( F 1 and F 2 are continuous cumulative distribution) X 1 , . . . , X m and Y 1 , . . . , Y n are independent Assume F 1 ( x ) = F 2 ( x − △ ) 1
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Hypotheses: H 0 : = 0 H a : > 0 H 0 : = 0 H a : < 0 H 0 : = 0 H a : △ negationslash = 0 H 0 asserts that the X variable and Y variable have the same proba- bility distribution, but the common distribution is not specified Test steps D the difference between two groups Compute D obs Permute m + n observations: m obs for group 1 and n obs for group 2 Compute D 1 , . . . , D N
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