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SEVERAL S AND MEDIANS: MORE ISSUES Business Statistics CONTENTS Post-hoc analysis ANOVA for 2 groups The equal variances assumption The Kruskal-Wallis test Old exam question Further study POST-HOC ANALYSIS After rejecting the null

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SEVERAL 𝜈S AND MEDIANS: MORE ISSUES

Business Statistics

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Post-hoc analysis ANOVA for 2 groups The equal variances assumption The Kruskal-Wallis test Old exam question Further study CONTENTS

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After rejecting the null hypothesis of equal means, we naturally want to know: ▪ which of the means differ (differs) significantly? ▪ is it (are they) lower or higher than the others? We are only allowed to go into this after 𝐼0 has been rejected ▪ therefore, we speak of a post-hoc analysis or post-hoc test For 𝑑 groups, there are

𝑑 𝑑−1 2

distinct pairs of means to be compared ▪ so-called multiple comparison tests POST-HOC ANALYSIS

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There are many such multiple comparison tests We focus on Tukey’s studentized range test (or HSD for “honestly significant difference” test) ▪ a multiple comparison test that is widely used ▪ named after statistician John Wilder Tukey (1915-2000) POST-HOC ANALYSIS

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POST-HOC ANALYSIS

This line, for instance, compares Club 1 to Club 3

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On the basis of significant differences, SPSS defines homogeneous subsets Rule: if two groups are in the same subset, they do not differ significantly POST-HOC ANALYSIS

The means of club 2 and club 3 cannot be discerned (statistically), and both differ significantly from the mean of club 1. And: club 1 is significantly better.

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Comparing 2 means ▪ Choice between:

▪ independent sample 𝑢-test ▪ ANOVA

▪ Example on Computer Anxiety Rating ANOVA FOR 2 GROUPS

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Result of 𝑢-test (which of the two?) Result of ANOVA ANOVA FOR 2 GROUPS

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Comparing two means (equality: 𝜈1 = 𝜈2) ▪ 𝑢-test

▪ null distribution: 𝑢~𝑢𝑜1+𝑜2−2 ▪ reject for small and large values ▪ equal variance required ▪ or the other test without this requirement ▪ normal populations required ▪ or symmetric populations and 𝑜1, 𝑜2 ≥ 15, or 𝑜1, 𝑜2 ≥ 30

▪ ANOVA for two groups (one factor with two levels)

▪ null distribution 𝐺~𝐺

1,𝑜1+𝑜2−2

▪ reject for large values ▪ equal variance required ▪ normal populations required

ANOVA FOR 2 GROUPS

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So, 𝑢-test is not superfluous now we have ANOVA You still need the independent samples 𝑢-test: ▪ more hypotheses possible (𝜈1 ≥ 𝜈2, 𝜈1 = 𝜈2 + 7, etc.) ▪ weaker requirement for population variances ▪ weaker requirement for population distributions ANOVA FOR 2 GROUPS

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Main assumption of ANOVA: equal variances Seen before in the independent samples 𝑢-test ▪ where the pooled variance was used to estimate 𝜏1

2 = 𝜏2 2

the assumption was tested with Levene’s test THE EQUAL VARIANCES ASSUMPTION

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Levene’s test is a homogeneity of variance test ▪ works for two variances (H0: 𝜏1

2 = 𝜏2 2)

▪ but also for several variances (H0: 𝜏1

2 = 𝜏2 2 = 𝜏3 2 = ⋯)

Example (golf clubs) ▪ 𝑞−value ≫ 0.1, so hypothesis of equal variances is not rejected ▪ validity of use of ANOVA is OK THE EQUAL VARIANCES ASSUMPTION

if not, escape to nonparametric ANOVA? see next ...

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▪ Recall that we used non-parametric methods when populations are not normally distributed

▪ Can we develop a non-parametric ANOVA?

▪ Yes: the Kruskal-Wallis test

▪ based on ranking of the observations on 𝑍 ▪ compares medians (𝐼0: 𝑁1 = 𝑁2 = 𝑁3 = ⋯) ▪ has lower power than ANOVA (is less sensitive) ▪ requires few assumptions

▪ Generalization of Wilcoxon-Mann-Whitney test, but for more than two groups THE KRUSKAL-WALLIS TEST

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Computational steps in Kruskal-Wallis test: ▪ Rank the observations 𝑧1, … , 𝑧𝑜, yielding 𝑠

1, … , 𝑠 𝑜

▪ 𝑜𝑘 size of group 𝑘; 𝑜 = σ𝑘=1

𝑑

𝑜𝑘

▪ Calculate the sum of ranks in every group

▪ 𝑈

𝑘 = σ𝑗=1 𝑜𝑘 𝑆𝑗𝑘 (for all groups 𝑘 = 1, … , 𝑑)

▪ Calculate test statistic

▪ 𝐼 =

12 𝑜 𝑜+1 σ𝑘=1 𝑑

𝑜𝑘 𝑈∙𝑘 − ന 𝑈

∙∙ 2 ; reject for large values

▪ Under 𝐼0: 𝐼 ∼ 𝜓𝑑−1

2

▪ test right-tailed (like ANOVA) ▪ for very small samples (groups<5), test not appropriate

THE KRUSKAL-WALLIS TEST

Required: populations of “similar” shape On formula sheet a slightly different form that works easier

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Example: comparing three golf clubs ▪ using SPSS THE KRUSKAL-WALLIS TEST

𝑞-value Kruskal-Wallis statistic (𝐼)

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Fill out the table EXERCISE 1

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21 May 2015, Q1n OLD EXAM QUESTION

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