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# note07 - Chapter 3 K-Sample Methods 0 Comparing more than...

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Chapter 3: K -Sample Methods 0. Comparing more than two groups (treatments) overall comparison: whether or not differences exist among groups multiple comparison: which groups differ significantly from the others Assumption: the experimental units are assigned to the k treatments in a completely random design or the observations have been randomly selected from k populations Hypothesis: H 0 : F 1 ( x ) = F 2 ( x ) = · · · = F k ( x ) H a : F i ( x ) F j ( x ) or F i ( x ) F j ( x ) for at least one pair ( i, j ) with strict inequality holding for at least one x Shift Alternative Hypothesis: H 0 : F 1 ( x ) = F 2 ( x ) = · · · = F k ( x ) = F ( x ) H a : F i ( x ) = F ( x μ i ) for at least one i of i = 1 , . . . , k 1. K -Sample Permutation F -Tests One-Way Data Layout X ij = μ i + ǫ ij where ǫ ij iid F ( ǫ ) N = n 1 + n 2 + · · · + n k Treatments Observations Sample Sizes Means Variances 1 X 11 , X 12 , . . . , X 1 n 1 n 1 ¯ X 1 S 2 1 2 X 21 , X 22 , . . . , X 2 n 2 n 2 ¯ X 2 S 2 2 . . . . . . . . . . . . . . . k X k 1 , X k 2 , . . . , X kn k n k ¯ X k S 2 k 1

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One-Way ANOVA test Assumptions: for i = 1 , . . . , k j = 1 , . . . , n i X ij iid N ( μ i , σ 2 ) X i ’s are independent Hypothesis: H 0 : μ 1 = μ 2 = · · · = μ k H a : μ i negationslash = μ j for at least one pair ( i, j ) F test: ¯ X = k i =1 n i j =1 X ij /N Source DF SS MS F Treatment k 1 SST r = k i =1 n i ( ¯ X i ¯ X ) 2 MST = SST k - 1 F = MST MSE Error N k SSE = k i =1 ( n i 1) S 2 i MSE = SSE N - k Total N 1 SS total = k i =1 n i j =1 ( X ij ¯ X ) 2 Under H 0 , F F k - 1 ,N - k reject H 0
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