Download ANOVA: Analysis of Variance and more Schemes and Mind Maps Economic statistics in PDF only on Docsity! ANOVA: Analysis of Variance An example ANOVA problem 25 individuals split into three between-subject conditions: A, B and C • A: 5,6,6,7,7,8,9,10 [8 participants, mean: 7.25] • B: 7,7,8,9,9,10,10,11 [8 participants, mean: 8.875] • P: 7,9,9,10,10,10,11,12,13 [9 participants, mean: 10.11] Are the differences between the conditions significant? How ANOVA works ANOVA measures two sources of variation in the data and compares their relative sizes • variation BETWEEN groups • for each data value look at the difference between its group mean and the overall mean 𝑥)* − ?̅? - • variation WITHIN groups • for each data value we look at the difference between that value and the mean of its group 𝑥). − 𝑥)* - F-score • The ANOVA F-statistic is a ratio of the Between Group Variaton divided by the Within Group Variation: 𝐹 = 1234225 6)37)5 • A large F is evidence against H0, since it indicates that there is more difference between groups than within groups. ANOVA Output for Our Example Analysis of Variance summary Source DF SS MS F P Treatment 2 34.74 17.37 6.45 0.006 [between groups] Error 22 59.26 2.69 [within groups] Total 24 94.00 ANOVA Output for Our Example MSG = SSG / DFG MSE = SSE / DFE F = MSG / MSE P-value comes from F(DFG,DFE) Analysis of Variance summary Source DF SS MS F P Treatment 2 34.74 17.37 6.45 0.006 [between groups] Error 22 59.26 2.69 [within groups] Total 24 94.00 34.74/2 = 17.37 Post-hoc analysis • ANOVA indicates that the groups do not all appear to have the same means… what next? How do we know what the differences really are? • If we only had two groups, then we’re done, we know the difference between them is significant. • If we have three or more groups, then a post hoc test is needed to determine which groups are significantly different from each other A: 5,6,6,7,7,8,9,10 [8 participants, mean: 7.25] B: 7,7,8,9,9,10,10,11 [8 participants, mean: 8.875] P: 7,9,9,10,10,10,11,12,13 [9 participants, mean: 10.11] Post-hoc analysis • Multiple post hoc analysis methods exist • We most commonly see the Tukey test • Results for our example dataset: HSD[.05]=2.02; HSD[.01]=2.61 M1 vs M2 nonsignificant M1 vs M3 P<.01 M2 vs M3 nonsignificant HSD = the absolute (unsigned) difference between any two sample means required for significance at the designated level.
