1. One Way ANOVA SS total SS between SS within Logic of Two Way ANOVA

2. Two Way ANOVA SS total SS between SS within Logic of Two Way ANOVA SS factorA SS factorB SS interaction Main Effect 1Main Effect 2Interaction

3. Logic of Two Way ANOVA This is no surpriseThis is news

4. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 1: Get the cell means, row means, column means, and grand mean

5. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 1: Get the SS total = 56

6. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 2: Get the SS within = 14

7. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 3: Get the SS time = 18

8. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 4: Get the SS temp = 12

9. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 5: Get the SS interaction = 12 SS total SS between SS within SS time SS temp SS interaction = 12= 18 = 14 = 56

10. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 Step 6: Finish the table

11. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 SourceSSdfs2s2 F Between SS time 18 SS temp 12 SS timXtemp 12 Within 14 Total56 Step 6: Finish the table

12. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 SourceSSdfs2s2 F Between SS time 181 SS temp 122 SS timXtemp 122 Within 1412 Total5617 Step 6: Finish the table

13. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 SourceSSdfs2s2 F Between SS time 181 SS temp 1226 SS timXtemp 1226 Within 14121.17 Total5617 Step 6: Finish the table

14. Steps for Two Way ANOVA Morning Afternoon 70 o 50 o 90 o 1 2 34 5 63 1 2 1 4 10 1 2 0 0 0 SourceSSdfs2s2 F Between SS time 181 15.38 SS temp 1226 5.12 SS timXtemp 1226 5.12 Within 14121.17 Total5617 Step 6: Finish the table p <.01 p <.05

15. Steps for Two Way ANOVA Since there are only two levels of time, you know they are significantly different from one another Since there are three levels of temp, you will need to use post-hoc multiple comparisons (e.g. Tukey’s HSD) to determine which levels differ from one another.

16. Steps for Two Way ANOVA To understand the interaction more fully, you might want to look at the “simple effects” You could compute the effect of temp just in the morning, for example. You would simply compute SS temp at AM

17. Assumptions for the Two Factor ANOVA 1.Observations within each sample are independent. 2.Populations are normally distributed. 3.Populations from which the samples are selected must have equal variances (homogeneity of variance). 4.Equal cell Ns (if calculating by hand)