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Nonparametric Test Distribution-Free Tests 1. 2. 3. 4. No assumptions of normality Focus on medians rather than means Not affected by outliers Des NOT really reduce power Mann-Whitney U Test 1. 2. 3. 4. 5. Test for two independent groups Assumptions for t-test cannot be made Data is neither ratio nor interval Data must be at least ordinal Tests if two groups have the same distribution n1 = # of cases in the smaller of the groups n2 = # of cases in the larger of the groups E = experimental group membership C = control group membership e.g., E scores = 11,15,9 and C scores = 6,8,13,10 Rank order all scores with group ID 6C,8C,9E,10C,11E,13C,15E Count # of Es that precede each C: U = 0 + 0 +1 + 2 = 3 Go top table for P under Null Hypothesis associated with the data; n1 = 3, n2 = 4, U = 3 U equal or greater than 3 p = .20 Thus, fail to reject at alpha .05. More than two groups: Kruskal-Wallis (H) (for ordinal data) One-Way ANOVA by ranks Null Hypothesis: k samples come from same population or identical populations with respect to central tendencies. Rank order all N scores from the smallest to the largest. e.g., Authoritarian Scores G1 G2 G3 96 82 115 128 124 149 83 132 166 61 135 147 101 109 Transformed 4 9 3 1 5 2 8 10 11 6 7 13 14 12 K-W (H) Ri2 12 H 3( N 1) N ( N 1) ni K = # of groups n = number of cases in a group N = total number of cases R = sum of ranks in a group 12 22 2 37 2 462 H [ 3(14 1) 14(14 1) 5 5 4 = 6.4 This is tested against Chi-Squared (k-1) df. Go to Chi-Squared Table , with 2 df and alpha .05 Friedman’s Test for k related samples One-way repeated measures Example Patients 1 2 3 4 5 6 7 8 9 10 Totals Psy(rank) 6(2) 4(1) 9(3) 5(2) 2(1) 6(1) 7(2) 4(1) 6(3) 72) 18 drug(rank) psy/drug(rank) 8(3) 5(1) 8(3) 6(2) 12 7(2) 4(1) 2F R 2j 3N ( k 1) Nk ( N 1) 4(1) 6(3) 7(3) 3(2) N = # of patients 8(3) 7(2) 9(3) 5(1) 8(3) 5(2) 5(2) 4(1) 12 2 2 2 ( 18 26 16 ) 3(10)(4) 8(3) 6(1) 10(3)(4) 26 16 = 5.6 This is test against Chi-Squared (k-1) df. Fail to reject at alpha .05.