# THE KRUSKAL WALLLIS TEST

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1 THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08

2 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2

3 NON PARAMETRIC TESTS: CHARACTERISTICS Distribution-free tests? Not exactly, they just less restrictive than parametric tests Based on ranked data By ranking the data we lose some information about the magnitude of difference between scores the non-parametric tests are less powerful than their parametric counterparts, i.e. a parametric test is more likely to detect a genuine effect in the data, if there is one, than a non-parametric test. 3

4 WHEN TO USE KRUSKAL-WALLIS We want to compare several independent groups but we don t meet some of the assumptions made in ANOVA: Data should come from a normal distribution Variances should be fairly similar (Levene s test) 4

5 EXAMPLE: EFFECT OF WEED ON CROP 4 groups: 0 weeds/meter 4 samples x group (N16) 1 weed/meter 3 weeds/meter 9 weeds/meter Weeds Corn Weeds Corn Weeds Corn Weeds Corn Corn crop by weeds (Ex , Moore & McCabe, 2005) 5

6 EFFECT OF WEED ON CROP: EXPLORING THE DATA Weeds n Mean Std.dev Summary statistics for Effect of Weed on Crop! For ANOVA: the largest standard deviation should NOT exceed twice the smallest. 6

7 EFFECT OF WEED ON CROP: EXPLORING THE DATA: Q-Q PLOTS Ex. 15.9, Moore & McCabe,

8 KRUSKAL-WALLIS: HYPOTHESISING H 0 : All four populations have the same median yield. H a : Not all four median yields are equal.! ANOVA F: H 0 : µ0 = µ1 = µ3 = µ9 H a : not all four means are equal. 8

9 KRUSKAL-WALLIS: HYPOTHESISING Non-parametric tests hypothesize about the median instead of the mean (as parametric tests do). mean a hypothetical value not necessarily present in the data (µ= Σ x i / n) median the middle score of a set of ordered observations. In the case of even number of observations, the median is the average of the two scores on each side of what should be in the middle The mean is more sensitive to outliers than the median. Ex: 1, 5, 2, 8, 38 µ= 10,7 [( )/5] median = 5 (1, 2, 5, 8, 38)! Median when the observations are even: Ex: 5, 2, 8, 38 = 6,5 (2, 5, 8, 38) (5+8)/2 = 6,5 9

10 THE KRUSKAL-WALLIS TEST: THE THEORY We take the responses from all groups and rank them; then we sum up the ranks for each group and we apply one way ANOVA to the ranks, not to the original observations. We order the scores that we have from lowest to highest, ignoring the group that the scores come from, and then we assign the lowest score a rank of 1, the next highest a rank of 2 and so on. 10

11 EFFECTS OF WEED ON CROP: KRUSKAL-WALLIS TEST: RANKING THE DATA Score 142,4 157,3 158,6 161,1 166,2 166,7 166,7 172,2 Rank Act. Rank ,5 12,5 14 Group ! Repeated values (tied ranks) are ranked as the average of the potential ranks for those scores, i.e. (12+13)/2=12,5 11

12 EFFECTS OF WEED ON CROP: KRUSKAL-WALLIS TEST: RANKS When the data are ranked we collect the scores back in their groups and add up the ranks for each group = R i (i determines the particular group) Weeds Ranks Sum of ranks , , ,5 33, , ,0 Ex , Moore & McCabe,

13 THE KRUSKAL-WALLIS TEST: THE THEORY! In ANOVA, we calculate the total variation (total sum of squares, SST) by adding up the variation among the groups (sum of squares for groups, SSG) with the variation within group (sum of squares for error, SSE): SST=SSG+SSE In Kruskal-Wallis: one way ANOVA to the ranks, not the original scores. If there are N observations in all, the ranks are always the whole numbers from 1 to N. The total sum of squares for the ranks is therefore a fixed number no matter what the data are no need to look at both SSG and SSE Kruskal-Wallis = SSG for the ranks 13

14 KRUSKAL-WALLIS TEST: H STATISTIC The test statistic H is calculated: H = 12 Σ R i 3(N + 1) N (N+1) n i 2 The Kruskal-Wallis test rejects the H o when H is large. 14

15 EFFECTS OF WEED ON CROP: KRUSKAL-WALLIS TEST: H STATISTIC Our Example: I = 4, N = 16, n i =4, R = 52.5, 33.5, 25.0, 25.0 H = 12 Σ R i 3(N + 1) N (N+1) n i 2 = 12 ( ) 3(17) (16)(17) = 12 ( ) = (1282,125) 51 = =

16 EFFECTS OF WEED ON CROP: KRUSKAL-WALLIS TEST: P VALUE H has approximately the chi-square distribution with k 1 degrees of freedom df = 3 (4-1) & H = < P >

17 THE STUDY: THE EFFECT OF SOYA ON CONCENTRATION 4 groups: O Soya Meals per week 1 Soya Meal per week 4 Soya Meals per week 9 Soya Meals per week 20 participants per group N=80 Tested after one year: RT when naming words 17

18 THE EFFECT OF SOYA ON CONCENTRATION: EXPLORATORY STATISTICS Soya n Mean Std.dev Summary Statistics for Soya on Concentration! Violation of the rule of thumb for using ANOVA: the largest standard deviation should NOT exceed twice the smallest. 18

19 THE EFFECT OF SOYA ON CONCENTRATION: TEST OF NORMALITY Tests of Normality RT (Ms) Number of Soya Meals Per Week No Soya Meals 1 Soya Meal Per Week 4 Soyal Meals Per Week 7 Soya Meals Per Week a. Lilliefors Significance Correction Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig.,181 20,085,805 20,001,207 20,024,826 20,002,267 20,001,743 20,000,204 20,028,912 20,071 Significance of data the distribution is significantly different from a normal distribution, i.e. it is non-normal. 19

20 THE EFFECT OF SOYA ON CONCENTRATION: HOMOGENEITY OF VARIANCE Test of Homogeneity of Variance RT (Ms) Based on Mean Based on Median Based on Median and with adjusted df Based on trimmed mean Levene Statistic df1 df2 Sig. 5, ,003 2, ,042 2, ,107,045 4, ,010 Significance of data the variances in different groups are significantly different data are not homogenous 20

21 21 KRUSKAL-WALLIS: SPSS

22 22 KRUSKAL-WALLIS: SPSS

23 THE EFFECT OF SOYA ON CONCENTRATION: SPSS: RANKS Ranks RT (Ms) Number of Soya Meals No Soya Meals 1 Soya Meal Per Week 4 Soyal Meals Per Week 7 Soya Meals Per Week Total N Mean Rank 20 46, , , ,

24 THE EFFECT OF SOYA ON CONCENTRATION: SPSS: TEST STATISTICS Test Statistics b,c Chi-Square df Asymp. Sig. Monte Carlo Sig. a. b. c. Sig. 99% Confidence Interval Lower Bound Upper Bound Based on sampled tables with starting seed Kruskal Wallis Test Grouping Variable: Number of Soya Meals Per Week RT (Ms) 8,659 3,034,031 a,027,036 Test significance p <.034 Confidence Interval does not cross the boundary of.05 a lot of confidence that the significant effect is genuine 24

25 THE EFFECT OF SOYA ON CONCENTRATION: THE CONCLUSION We know that there is difference but we don t know exactly where! 25

26 KRUSKAL-WALLIS: FINDING THE DIFFERENCE 25, , ,00 s) (M T R 10, ,00 0,00 No Soya Meals 1 Soya Meal Per Week 4 Soyal Meals Per Week Number of Soya Meals Per Week 7 Soya Meals Per Week 26

27 KRUSKAL-WALLIS: POST HOC TESTS: MANN-WHITNEY TEST Mann-Whitney tests = Wilcoxon Rank Sum Test a non-parametric test for comparing two independent groups based on ranking! Lots of Wilcoxon Rank Sum Tests inflation of the Type I error (the probability of falsely rejecting the H 0 ) Bonferroni correction -.05/ the number of test to be conducted The value of significance becomes too small, i.e.: 0 soya meals, 1 soya meal, 4 soya meals, 7 soya meals = 6 tests.05/6=

28 KRUSKAL-WALLIS: POST HOC TESTS: MANN-WHITNEY TEST Select a number of comparisons to make, i.e.: Test 1: 1 soya meal per week compared to 0 soya meals Test 2: 4 soya meals per week compared to 0 soya meals Test 3: 7 soya meals per week compared to 0 soya meals α level =.05/3=

29 KRUSKAL-WALLIS: POST HOC TESTS: MANN-WHITNEY TEST 1. 0 soya vs. 1 meal per week 2. 0 soya vs. 4 meals per week Test Statistics b Test Statistics b Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed) Exact Sig. [2*(1-tailed Sig.)] a. Not corrected for ties. RT (Ms) 191, ,000 -,243,808,820 a b. Grouping Variable: Number of Soya Meals Per Week 3. 0 soya vs. 7 meals per week Mann-Whitney U Wilcoxon W Z Test Statistics b Asymp. Sig. (2-tailed) Exact Sig. [2*(1-tailed Sig.)] a. Not corrected for ties. RT (Ms) 104, ,000-2,597,009,009 a b. Grouping Variable: Number of Soya Meals Per Week Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed) Exact Sig. [2*(1-tailed Sig.)] a. Not corrected for ties. RT (Ms) 188, ,000 -,325,745,758 a b. Grouping Variable: Number of Soya Meals Per Week! α level =.0167 not.05 29

30 KRUSKAL-WALLIS: TESTING FOR TRENDS: JONCKHEERE-TERPSTRA TEST If we expect that the groups we compare are ordered in a certain way. I.e. the more soya a person eats the more concentrated and faster they become (shorter RTs) 30

31 KRUSKAL-WALLIS: TESTING FOR TRENDS: JONCKHEERE- TERPSTRA TEST Jonckheere-Terpstra Test b Number of Levels in Number of Soya Meals Per Week N Observed J-T Statistic Mean J-T Statistic Std. Deviation of J-T Statistic Std. J-T Statistic Asymp. Sig. (2-tailed) Monte Carlo Sig. (2-tailed) Monte Carlo Sig. (1-tailed) Sig. 99% Confidence Interval Sig. 99% Confidence Interval Lower Bound Upper Bound Lower Bound Upper Bound a. Based on sampled tables with starting seed b. Grouping Variable: Number of Soya Meals Per Week RT (Ms) , , ,333-2,476,013,012 a,009,015,006 a,004,008 normal distribution z score calculated: -2,476 If > 1.65 significant result. - descending medians scores get smaller + ascending medians scores get bigger Medians get smaller the more soya meals we eat : RTs become faster more soya better concentration and 31 more speed!

32 CALCULATING AN EFFECT SIZE A standardized measure of the magnitude of the observed effect the measured effect is meaningful or important Cohen s d or Pearson s correlation coefficient r: 1 > r < 0 r =.10 small effect 1% of the total variance r =.30 medium effect 9 %of total variance r =.50 large effect 25 %of total varince Converting z score into the effect size estimate r = Z N 32

33 CALCULATING AN EFFECT SIZE 33

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