Nonparametric statistics ppt @ bec doms

Chapter Goals ,[object Object],[object Object],[object Object],[object Object],[object Object]

Nonparametric Statistics ,[object Object],[object Object],[object Object],[object Object]

Wilcoxon Signed Rank Test ,[object Object],[object Object],[object Object],[object Object]

The W Test Statistic ,[object Object],[object Object],[object Object],[object Object],[object Object]

The W Test Statistic ,[object Object],[object Object],[object Object],[object Object],[object Object],(continued)

The W Test Statistic ,[object Object],[object Object],[object Object],[object Object],(continued)

Example ,[object Object],[object Object],[object Object],Class size = x i Difference d i = x i – 40 | d i | 23 45 34 78 34 66 61 95 -17 5 -6 38 -6 26 21 55 17 5 6 38 6 26 21 55

Example ,[object Object],tied (continued) | d i | Rank 5 6 6 17 21 26 38 55 1 2.5 2.5 4 5 6 7 8

Example ,[object Object],[object Object],(continued) These three are below the claimed median, the others are above Class size = x i Difference d i = x i – 40 | d i | Rank R+ R- 23 45 34 78 34 66 61 95 -17 5 -6 38 -6 26 21 55 17 5 6 38 6 26 21 55 4 1 2.5 7 2.5 6 5 8 1 7 6 5 8 4 2.5 2.5  = 27  = 9

Completing the Test ,[object Object],[object Object],Test at the  = .05 level: This is a two-tailed test and n = 8, so find W L and W U in appendix P: W L = 3 and W U = 33 The calculated test statistic is W =  R+ = 27

Completing the Test ,[object Object],[object Object],W L = 3 and W U = 33 W L < W < W U so do not reject H 0 (there is not sufficient evidence to conclude that the median class size is different than 40) (continued) W L = 3 do not reject H 0 reject H 0 W =  R+ = 27 W U = 33 reject H 0

If the Sample Size is Large ,[object Object],[object Object],where W = sum of the R+ ranks d = number of non-zero d i values

Nonparametric Tests for Two Population Centers ,[object Object],[object Object],[object Object],Wilcoxon Matched-Pairs Signed Rank Test Mann-Whitney U-test Large Samples Small Samples Large Samples Small Samples

Mann-Whitney U-Test Used to compare two samples from two populations Assumptions: The two samples are independent and random The value measured is a continuous variable The measurement scale used is at least ordinal If they differ, the distributions of the two populations will differ only with respect to the central location

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Mann-Whitney U-Test (continued)

[object Object],Mann-Whitney U-Test (continued)

Mann-Whitney U-Test (continued) Mann-Whitney U-Statistics where: n 1 and n 2 are the two sample sizes  R 1 and  R 2 = sum of ranks for samples 1 and 2

Mann-Whitney U-Test (continued) Claim: Median class size for Math is larger than the median class size for English A random sample of 9 Math and 9 English classes is selected (samples do not have to be of equal size) Rank the combined values and then split them back into the separate samples

[object Object],(continued) Mann-Whitney U-Test Class size (Math, M) Class size (English, E) 23 45 34 78 34 66 62 95 81 30 47 18 34 44 61 54 28 40

Mann-Whitney U-Test Ranking for combined samples tied (continued) Size Rank 18 1 23 2 28 3 30 4 34 6 34 6 34 6 40 8 44 9 Size Rank 45 10 47 11 54 12 61 13 62 14 66 15 78 16 81 17 95 18

[object Object],Mann-Whitney U-Test (continued) Class size (Math, M) Rank Class size (English, E) Rank 23 45 34 78 34 66 62 95 81 2 10 6 16 6 15 14 18 17 30 47 18 34 44 61 54 28 40 4 11 1 6 9 13 12 3 8  = 104  = 67

Mann-Whitney U-Test H 0 : Median M ≤ Median E H A : Median M > Median E Claim: Median class size for Math is larger than the median class size for English Note: U 1 + U 2 = n 1 n 2 (continued) Math: English:

[object Object],[object Object],[object Object],Mann-Whitney U-Test (continued)

[object Object],[object Object],[object Object],[object Object],[object Object],Mann-Whitney U-Test (continued)

[object Object],[object Object],Mann-Whitney U-Test Since U  U  , do not reject H 0 (continued) U  = 7 U = 19 do not reject H 0 reject H 0

Mann-Whitney U-Test for Large Samples ,[object Object],[object Object],[object Object]

Mann-Whitney U-Test for Large Samples ,[object Object],(continued) Where n 1 and n 2 are sample sizes from populations 1 and 2

Mann-Whitney U-Test for Large Samples ,[object Object],(continued)

Large Sample Example ,[object Object],[object Object],[object Object],[object Object],[object Object],H 0 : Median 1  Median 2 H A : Median 1 < Median 2

[object Object],Large Sample Example Since the alternative hypothesis indicates that population 2 has a higher median, use U 2 as the test statistic Compute the U statistics: (continued)

Large Sample Example Since z = -2.80 < -1.645, we reject H 0 Reject H 0  = .05 Do not reject H 0 0 (continued)

Wilcoxon Matched-Pairs Signed Rank Test ,[object Object],[object Object],[object Object]

The Wilcoxon T Test Statistic ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Wilcoxon T Test Statistic ,[object Object],[object Object],[object Object],[object Object],(continued)

Small Sample Example ,[object Object],Claim: Median value is smaller after than before Value (before) Value (after) 38 45 34 58 30 46 42 55 41 30 47 18 34 34 31 24 38 40

Small Sample Example ,[object Object],(continued) Value (before) Value (after) Difference d Rank of d Ranks with smaller expected sum 36 45 34 58 30 46 42 55 41 30 47 18 54 38 31 24 62 40 6 -2 16 4 -8 15 18 -7 1 4 -2 8 3 -6 7 9 -5 1 2 6 5  = T = 13

[object Object],[object Object],[object Object],[object Object],Small Sample Example Since T  T  , do not reject H 0 T  = 6 T = 13 do not reject H 0 reject H 0 (continued)

Wilcoxon Matched Pairs Test for Large Samples ,[object Object],[object Object],[object Object]

[object Object],Wilcoxon Matched Pairs Test for Large Samples (continued) where n is the number of paired values

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Kruskal-Wallis One-Way ANOVA

Kruskal-Wallis Test Procedure ,[object Object],[object Object],[object Object],[object Object]

Kruskal-Wallis Test Procedure ,[object Object],[object Object],where: N = Sum of sample sizes in all samples k = Number of samples R i = Sum of ranks in the i th sample n i = Size of the i th sample (continued)

[object Object],[object Object],[object Object],[object Object],[object Object],Kruskal-Wallis Test Procedure (continued)

[object Object],Kruskal-Wallis Example Class size (Math, M) Class size (English, E) Class size (History, H) 23 45 54 78 66 55 60 72 45 70 30 40 18 34 44

[object Object],Kruskal-Wallis Example Class size (Math, M) Ranking Class size (English, E) Ranking Class size (History, H) Ranking 23 41 54 78 66 2 6 9 15 12 55 60 72 45 70 10 11 14 8 13 30 40 18 34 44 3 5 1 4 7  = 44  = 56  = 20

[object Object],Kruskal-Wallis Example (continued)

[object Object],[object Object],Kruskal-Wallis Example (continued) Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and  = .05: There is not sufficient evidence to reject that the population medians are all equal

Kruskal-Wallis Correction ,[object Object],[object Object],[object Object],where: g = Number of different groups of ties t i = Number of tied observations in the i th tied group of scores N = Total number of observations

H Statistic Corrected for Tied Rankings ,[object Object]

Nonparametric statistics ppt @ bec doms

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