ANOVA: Why analyzing variance to compare means?

22 s12s12 s22s22 s32s32 22 22

s j 2 = s 1 2, s 2 2, s 3 2 (s s s 3 2 )  3   2 MS w   2  SS w  df w

22 s12s12 s22s22 s32s32 

Sampling Distribution of means    2 MS b   2 SS b df b

MS w   2 MS b   df = 2, 30

SourceSSdfMSF Betweenk-1 WithinN-k Total ANOVA Table

= 3.33 = 2.00 = = RogerianBehavior ModificationControl ( ) 2 ( ) 2 ( ) 2 (4-3.33) 2 (3-2.00) 2 (8-6.33) 2 = 3.89 (3-3.33) 2 (2-2.00) 2 (6-6.33) 2 (3-3.33) 2 (1-2.00) 2 (5-6.33) 2 RogerianBehavior ModificationControl ( - ) 2 g X

Source of variation SSdfMS (Variance) F test Between groups 29.56k-1 = = Within groups 7.32N-k = = = = 12.11

Post Hoc Comparison When you want to compare two means, you calculate a t and then compare your computed (or observed) t with the critical t. When you do multiple comparisons, the two means you want to compare are from a bunch of means. The steps between each pair of the bunch of means you have should be taken into consideration in that, if the null is true, it is more likely to commit Type I error when comparing two means which are farther away from each other than two means which are close to each other. M2 = 5;M4 = 6;M1 = 9;M3 = 15 A larger critical t makes it harder to reject the Null: μ2 – μ3 = 0 A smaller critical t makes it easier to reject the Null: μ2 – μ4 = 0 Most Liberal: Tukey’s Least Significant Difference Test (LSD) Moderate: Tukey’s Honestly Significant Difference Test (HSD) Most Conservative: Scheffe’s Multiple Comparison Test

SourceSSdfMSF Between A B AxB K – 1 J – 1 (K-1)(J-1) WithinN – K – J Total Two-Way ANOVA Table