SEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics

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1 SEVERAL μs AND MEDIANS: MORE ISSUES Business Statistics

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

3 POST-HOC ANALYSIS 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 H 0 has been rejected therefore, we speak of a post-hoc analysis or post-hoc test For c groups, there are c c 1 compared so-called multiple comparison tests 2 distinct pairs of means to be

4 POST-HOC ANALYSIS 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 ( )

5 POST-HOC ANALYSIS This line, for instance, compares Club 1 to Club 3

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

7 ANOVA FOR 2 GROUPS Comparing 2 means Choice between: independent sample t-test ANOVA Example on Computer Anxiety Rating

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

9 ANOVA FOR 2 GROUPS Comparing two means (equality: μ 1 = μ 2 ) t-test null distribution: t~t n1 +n 2 2 reject for small and large values equal variance required or the other test without this requirement normal populations required or symmetric populations and n 1, n 2 15, or n 1, n 2 30 ANOVA for two groups (one factor with two levels) null distribution F~F 1,n1 +n 2 2 reject for large values equal variance required normal populations required

10 ANOVA FOR 2 GROUPS So, t-test is not superfluous now we have ANOVA You still need the independent samples t-test: more hypotheses possible (μ 1 μ 2, μ 1 = μ 2 + 7, etc.) weaker requirement for population variances weaker requirement for population distributions

11 THE EQUAL VARIANCES ASSUMPTION Main assumption of ANOVA: equal variances Seen before in the independent samples t-test where the pooled variance was used to estimate σ 1 2 = σ 2 2 the assumption was tested with Levene s test

12 THE EQUAL VARIANCES ASSUMPTION Levene s test is a homogeneity of variance test works for two variances (H 0 : σ 1 2 = σ 2 2 ) but also for several variances (H 0 : σ 1 2 = σ 2 2 = σ 3 2 = ) Example (golf clubs) p value 0.1, so hypothesis of equal variances is not rejected validity of use of ANOVA is OK if not, escape to nonparametric ANOVA? see next...

13 THE KRUSKAL-WALLIS TEST 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 Y compares medians (H 0 : M 1 = M 2 = M 3 = ) has lower power than ANOVA (is less sensitive) requires few assumptions Generalization of Wilcoxon-Mann-Whitney test, but for more than two groups

14 THE KRUSKAL-WALLIS TEST Computational steps in Kruskal-Wallis test: Rank the observations y 1,, y n, yielding r 1,, r n n j size of group j; n = σ j=1 Calculate the sum of ranks in every group n j T j = σ i=1 R ij (for all groups j = 1,, c) Calculate test statistic c H = 12 σ c 2 n n+1 j=1 n j T j നT ; reject for large values 2 Under H 0 : H χ c 1 test right-tailed (like ANOVA) for very small samples (groups<5), test not appropriate n j On formula sheet a slightly different form that works easier Required: populations of similar shape

15 THE KRUSKAL-WALLIS TEST Example: comparing three golf clubs using SPSS Kruskal-Wallis statistic (H) p-value

16 EXERCISE 1 Fill out the table

17 OLD EXAM QUESTION 21 May 2015, Q1n

18 FURTHER STUDY Doane & Seward 5/E , 16.5 Tutorial exercises week 4 Homogeneous subsets, Kruskal-Wallis test