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Statistical Analysis Using ANOVA: Independent and Repeated-Measures Comparisons

This paper applies ANOVA techniques to statistical analysis, comparing independent and repeated-measures data in research studies.

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    Statistical Analysis Using ANOVA: Independent and Repeated - Measures Comparisons 1. The following data summarize the results from an independent - measures study comparing three treatment conditions. Treatment I II III 0 2 4 N=18 0 3 2 G=36 0 1 4 ∑X 2 =114 3 3 3 0 2 4 0 1 4 M=0.5 M=2 M=3.5 T=3 T=12 T=21 SS=7.5 SS=4 SS=3.5 a. Use an ANOVA with α=.05 to determine whether there are any significan t differences among the three treatment means.

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    ( ) 0 1 2 3 1 total between within 2 2 2 total within inside each treatment betw : vs. : At least one treatment mean differs. .05 1 18 1 17 1 3 1 2 5 5 5 15 36 114 114 72 42 18 7.5 4 3.5 15 H H df N df k df G SS X N SS SS SS     = = = = − = − = = − = − = = + + = = − = − = − = = = + + =   een total within 42 15 27 Anova Table Source 27 13.50 Between 27 2 13.50 13.50 2 1.00 15 Within 15 15 1.00 15 Total 42 17 SS SS SS df MS F = − = − = = = = The F - critical value at 0.05 level with (2, 15) degrees of freedom is 3.68. Since observed F = 13.50 > 3.68, reject the null hypothesis. Therefore, at 0.05 level data provides sufficient evidence that there are significan t differences among the three treatment means . b. Calculate η 2 to measure the effect size for this study. 2 between total 27 0.643 . 42 SS SS  = = = 2. The following data summarize the results from an independent - measures study comparing three treatment conditions. Treatment I II II 4 1 0 N=12 6 4 2 G=36 3 5 0 ∑X 2 =164 7 2 2

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    M=5 M=3 M=1 T=20 T=12 T=4 SS=10 SS=10 SS=4 a. Calculate the sample variances for each of the three samples. 2 1 1 1 2 2 2 2 2 3 3 3 10 10 Sample variance of treatment I, 3.33 . 1 4 1 3 10 10 Sample variance of treatment II, 3.33 . 1 4 1 3 4 4 Sample variance of treatment III, 1.33 . 1 4 1 3 SS s n SS s n SS s n = = = = − − = = = = − − = = = = − − b. Use an ANOVA with α=.05 to determine whether there are any significant differences among the three treatment means. 0 1 2 3 1 : vs. : At least one treatment mean differs. .05 H H     = = = ( ) total between within 2 2 2 total within inside each treatment between total within 1 12 1 11 1 3 1 2 3 3 3 9 36 164 164 108 56 18 10 10 4 24 56 24 32 Anova Table Source Between df N df k df G SS X N SS SS SS SS SS SS df MS F = − = − = = − = − = = + + = = − = − = − = = = + + = = − = − =   32 16.00 32 2 16.00 6.00 2 2.67 24 Within 24 9 2.67 9 Total 56 11 = = = The F - critical value at 0.05 level with (2, 9 ) degrees of freedom is 4 . 2 6.

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