THE KRUSKAL–WALLLIS TEST

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THEKRUSKAL–WALLLISTESTTEODORA H. MEHOTCHEVAWednesday, 23rd April 08Seminar in Methodology & Statistics

THE KRUSKAL-WALLIS TEST:The non-parametric alternative to ANOVA:testing for difference between several independentgroups2Seminar in Methodology & Statistics

NON PARAMETRIC TESTS:CHARACTERISTICSDistribution-free tests? Not exactly, they just less restrictive than parametric tests Basedon ranked data Byranking the data we lose some information about the magnitudeof 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 , ifthere is one, than a non-parametric test.3Seminar in Methodology & Statistics

WHEN TO USE KRUSKAL-WALLISWe want to compare several independent groups butwe don’t meet some of the assumptions made inANOVA: Datashould come from a normal distribution Variancesshould be fairly similar (Levene’s test)4Seminar in Methodology & Statistics

EXAMPLE:EFFECT OF WEED ON CROP4 groups: 0 weeds/meter1 weed/meter3 weeds/meter9 weeds/meter4 samples x group 73153.19162.70176.91161.13165.09162.45Corn crop by weeds (Ex. 15.13, Moore & McCabe, 2005)Seminar in Methodology & Statistics

EFFECT OF WEED ON CROP:EXPLORING THE 34161.02510.49394157.57510.118Summary statistics for Effect of Weed on Crop! For ANOVA: the largest standard deviation should NOT exceed twicethe smallest.6Seminar in Methodology & Statistics

EFFECT OF WEED ON CROP:EXPLORING THE DATA: Q-Q PLOTS7Ex. 15.9, Moore & McCabe, 2005Seminar in Methodology & Statistics

KRUSKAL-WALLIS: HYPOTHESISINGH0: All four populations have the same median yield.Ha: Not all four median yields are equal.! ANOVA F:H0: µ0 µ1 µ3 µ9Ha: not all four means are equal.8Seminar in Methodology & Statistics

KRUSKAL-WALLIS: HYPOTHESISING Non-parametric tests hypothesize about the median instead of the mean (asparametric tests do).mean – a hypothetical value not necessarily present in the data (µ Σ xi / n)median – the middle score of a set of ordered observations. In the case of evennumber of observations, the median is the average of the two scoreson each side of what should be in the middleThe mean is more sensitive to outliers than the median.Ex: 1, 5, 2, 8, 38µ 10,7 [(1 5 2 8 38)/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 6,5Seminar in Methodology & Statistics9

THE KRUSKAL-WALLIS TEST:THE THEORY We take the responses from all groups and rank them; thenwe sum up the ranks for each group and we apply one wayANOVA 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 weassign the lowest score a rank of 1, the next highest a rankof 2 and so on.10Seminar in Methodology & Statistics

EFFECTS OF WEED ON CROP:KRUSKAL-WALLIS TEST: RANKING THEDATAScore142,4157,3158,6161,1 166,2166,7166,7172,2 Rank1234 11121314 Act. Rank1234 1112,512,514 Group9331 1010 ! Repeated values (tied ranks) are ranked as the average of the potential ranks for thosescores, i.e.(12 13)/2 12,511Seminar in Methodology & Statistics

EFFECTS OF WEED ON CROP:KRUSKAL-WALLIS TEST: RANKSWhen the data are ranked we collect the scores back in their groups and addup the ranks for each group Ri (i determines the particular group)WeedsRanksSum of ,0Ex. 15.14, Moore & McCabe, 200512Seminar in Methodology & Statistics

THE KRUSKAL-WALLIS TEST: THE THEORY!In ANOVA , we calculate the total variation (total sum of squares, SST) byadding 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 SSEIn 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 from1 to N. The total sum of squares for the ranks is therefore a fixed number nomatter what the data are no need to look at both SSG and SSE Kruskal-Wallis SSG for the ranksSeminar in Methodology & Statistics13

KRUSKAL-WALLIS TEST: H STATISTICThe test statistic H is calculated:H 12N (N 1)ΣRi2ni3(N 1) The Kruskal-Wallis test rejects the Ho when H is large.14Seminar in Methodology & Statistics

EFFECTS OF WEED ON CROP:KRUSKAL-WALLIS TEST: H STATISTICOur Example: I 4, N 16, ni 4, R 52.5, 33.5, 25.0, 25.0H 12N (N 1)((16)(17) 41252.522ΣRini3(N 1)33.5225244252)43(17)12 (689.0625 280.5625 156.25 156.25) – 51272 0.0441(1282,125) – 51 56.5643– 51 5.56Seminar in Methodology & Statistics15

EFFECTS OF WEED ON CROP:KRUSKAL-WALLIS TEST: P VALUEhas approximately the chi-square distribution with k 1degrees of freedom H df 3 (4-1) & H 5.56 0.10 P 0.1516Seminar in Methodology & Statistics

THE STUDY:THE EFFECT OF SOYA ON CONCENTRATION 4groups:O Soya Meals per week1 Soya Meal per week4 Soya Meals per week9 Soya Meals per week 20participants per group N 80 Testedafter one year: RT when naming words17Seminar in Methodology & Statistics

THE EFFECT OF SOYA ONCONCENTRATION:EXPLORATORY 524.672634204.11014.409919201.65301.10865Summary Statistics for Soya on Concentration! Violation of the rule of thumb for using ANOVA:the largest standard deviation should NOT exceed twice the smallest.18Seminar in Methodology & Statistics

THE EFFECT OF SOYA ON CONCENTRATION:TEST OF NORMALITYTests of NormalityRT (Ms)Number of Soya MealsPer WeekNo Soya Meals1 Soya Meal Per Week4 Soyal Meals Per Week7 Soya Meals Per 912Shapiro-Wilkdf20202020Sig.,001,002,000,071a. Lilliefors Significance CorrectionSignificance of data the distribution is significantly differentfrom a normal distribution, i.e. it is non-normal.19Seminar in Methodology & Statistics

THE EFFECT OF SOYA ON CONCENTRATION:HOMOGENEITY OF VARIANCETest of Homogeneity of VarianceRT (Ms)Based on MeanBased on MedianBased on Median andwith adjusted dfBased on trimmed 0422,860358,107,0454,070376,010Significance of data the variances in different groupsare significantly different data are not homogenous20Seminar in Methodology & Statistics

KRUSKAL-WALLIS: SPSSSeminar in Methodology & Statistics21

KRUSKAL-WALLIS: SPSSSeminar in Methodology & Statistics22

THE EFFECT OF SOYA ONCONCENTRATION: SPSS: RANKSRanksRT (Ms)Number of Soya MealsNo Soya Meals1 Soya Meal Per Week4 Soyal Meals Per Week7 Soya Meals Per WeekTotalN2020202080Mean Rank46,3544,1544,1527,3523Seminar in Methodology & Statistics

THE EFFECT OF SOYA ON CONCENTRATION:SPSS: TEST STATISTICSTest StatisticsChi-SquaredfAsymp. Sig.Monte CarloSig.Sig.99% ConfidenceIntervalb,cLower BoundUpper BoundRT (Ms)8,6593,034,031a,027,036a. Based on 10000 sampled tables with starting seed2000000.b. Kruskal Wallis Testc. Grouping Variable: Number of Soya Meals Per Week Test significance p .034 Confidence Interval .028-.037 – does not cross the boundary of .05 a lot of confidence that the significant effect is genuineSeminar in Methodology & Statistics24

THE EFFECT OF SOYA ON CONCENTRATION:THE CONCLUSIONWe know that there is difference but we don’t know exactlywhere!25Seminar in Methodology & Statistics

KRUSKAL-WALLIS:FINDING THE 00No Soya Meals1 Soya Meal PerWeek4 Soyal Meals Per 7 Soya Meals PerWeekWeekNumber of Soya Meals Per WeekSeminar in Methodology & Statistics26

KRUSKAL-WALLIS:POST HOC TESTS: MANN-WHITNEY TESTMann-Whitney tests Wilcoxon Rank Sum Testa non-parametric test for comparing two independent groups based on ranking! Lots of Wilcoxon Rank Sum Tests inflation of the Type I error (theprobability of falsely rejecting the H0) 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 .0083Seminar in Methodology & Statistics27

KRUSKAL-WALLIS:POST HOC TESTS: MANN-WHITNEY TESTSelect a number of comparisons to make, i.e.:Test 1: 1 soya meal per week compared to 0 soya mealsTest 2: 4 soya meals per week compared to 0 soya mealsTest 3: 7 soya meals per week compared to 0 soya meals α level .05/3 .016728Seminar in Methodology & Statistics

KRUSKAL-WALLIS:POST HOC TESTS: MANN-WHITNEY TEST1. 0 soya vs. 1 meal per weekTest StatisticsbTest StatisticsbRT (Ms)191,000401,000-,243,808a,820Mann-Whitney UWilcoxon WZAsymp. Sig. (2-tailed)Exact Sig. [2*(1-tailedSig.)]RT (Ms)188,000398,000-,325,745a,758a. Not corrected for ties.a. Not corrected for ties.b. Grouping Variable: Number of Soya Meals Per Weekb. Grouping Variable: Number of Soya Meals Per Week3. 0 soya vs. 7 meals per weekTest StatisticsbMann-Whitney UWilcoxon WZAsymp. Sig. (2-tailed)Exact Sig. [2*(1-tailedSig.)]RT (Ms)104,000314,000-2,597,009! α level .0167 not .05Seminar in Methodology & StatisticsMann-Whitney UWilcoxon WZAsymp. Sig. (2-tailed)Exact Sig. [2*(1-tailedSig.)]2. 0 soya vs. 4 meals per weeka,009a. Not corrected for ties.b. Grouping Variable: Number of Soya Meals Per Week29

KRUSKAL-WALLIS:TESTING FOR TRENDS:JONCKHEERE-TERPSTRA TESTIf 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 fasterthey become (shorter RTs)30Seminar in Methodology & Statistics

KRUSKAL-WALLIS:TESTING FOR TRENDS: JONCKHEERETERPSTRA TESTJonckheere-Terpstra TestbRT (Ms)Number of Levels in Number of Soya Meals Per Week4NObserved J-T StatisticMean J-T StatisticStd. Deviation of J-T StatisticStd. J-T StatisticAsymp. Sig. (2-tailed)Monte Carlo Sig.(2-tailed)Monte Carlo Sig.(1-tailed)80912,0001200,000116,333Sig.99% ConfidenceIntervalLower BoundUpper BoundSig.99% ConfidenceIntervalLower BoundUpper Bounda. Based on 10000 sampled tables with starting seed 2000000.b. Grouping Variable: Number of Soya Meals Per Week-2,476,013,012a,009,015,006a,004,008normal distributionz score calculated: -2,476If 1.65 significant result .“-” descending mediansscores get smaller“ ” ascending mediansscores get bigger Medians get smaller the moresoya meals we eat : RTs become faster moresoyabetterconcentration and31more speed!Seminar in Methodology & Statistics

CALCULATING AN EFFECT SIZEA standardized measure of the magnitude of the observed effect the measured effect is meaningful or importantCohen’s d or Pearson’s correlation coefficient r:1 r 0r .10 small effect 1% of the total variancer .30 medium effect 9 %of total variancer .50 large effect 25 %of total varinceConverting z score into the effect size estimater Z NSeminar in Methodology & Statistics32

CALCULATING AN EFFECT SIZE33Seminar in Methodology & Statistics

KRUSKAL-WALLIS: SPSS Seminar in Methodology & Sta t istics 21. KRUSKAL-WALLIS: SPSS Seminar in Methodology & Sta t istics 22. THE EFFECT OF SOYA ON CONCENTRATION: SPSS: RANKS Ranks 20 46,35 20 44,15 20 44,15 20 27,35 80 Number of Soya Meals No Soya Meals 1 Soya Meal Per Week 4 Soyal Meals Per Week 7 Soya Meals Per WeekFile Size: 1MB

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