1. Multiple comparisons - multiple pairwise tests - orthogonal contrasts - independent tests - labelling conventions

2. Card example number 1

3. Multiple tests Problem: Because we examine the same data in multiple comparisons, the result of the first comparison affects our expectation of the next comparison.

4. Multiple tests ANOVA shows at least one different, but which one(s)? significant Not significant significant T-tests of all pairwise combinations

5. Multiple tests T-test: <5% chance that this difference was a fluke… affects likelihood of finding a difference in this pair!

6. Multiple tests Solution: Make alpha your overall “experiment-wise” error rate affects likelihood (alpha) of finding a difference in this pair! T-test: <5% chance that this difference was a fluke…

7. Multiple tests Solution: Make alpha your overall “experiment-wise” error rate e.g. simple Bonferroni: Divide alpha by number of tests Alpha / 3 = 0.0167 Alpha / 3 = 0.0167 Alpha / 3 = 0.0167

8. Card example 2

9. Orthogonal contrasts Orthogonal = perpendicular = independent Contrast = comparison Example. We compare the growth of three types of plants: Legumes, graminoids, and asters. These 2 contrasts are orthogonal: 1. Legumes vs. non-legumes (graminoids, asters) 2. Graminoids vs. asters

10. Trick for determining if contrasts are orthogonal: 1. In the first contrast, label all treatments in one group with “+” and all treatments in the other group with “-” (doesn’t matter which way round). LegumesGraminoidsAsters + - -

11. Trick for determining if contrasts are orthogonal: 1. In the first contrast, label all treatments in one group with “+” and all treatments in the other group with “-” (doesn’t matter which way round). 2. In each group composed of t treatments, put the number 1/t as the coefficient. If treatment not in contrast, give it the value “0”. LegumesGraminoidsAsters +1 - 1/2 -1/2

12. Trick for determining if contrasts are orthogonal: 1. In the first contrast, label all treatments in one group with “+” and all treatments in the other group with “-” (doesn’t matter which way round). 2. In each group composed of t treatments, put the number 1/t as the coefficient. If treatment not in contrast, give it the value “0”. 3. Repeat for all other contrasts. LegumesGraminoidsAsters +1 - 1/2 -1/2 0+1 -1

13. Trick for determining if contrasts are orthogonal: 4. Multiply each column, then sum these products. LegumesGraminoidsAsters +1 - 1/2 -1/2 0+1 -1 0 - 1/2 +1/2 Sum of products = 0

14. Trick for determining if contrasts are orthogonal: 4. Multiply each column, then sum these products. 5. If this sum = 0 then the contrasts were orthogonal! LegumesGraminoidsAsters +1 - 1/2 -1/2 0+1 -1 0 - 1/2 +1/2 Sum of products = 0

15. What about these contrasts? 1. Monocots (graminoids) vs. dicots (legumes and asters). 2. Legumes vs. non-legumes

16. Important! You need to assess orthogonality in each pairwise combination of contrasts. So if 4 contrasts: Contrast 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, 3 and 4.

17. How do you program contrasts in JMP (etc.)? Treatment SS } Contrast 2 } Contrast 1

18. How do you program contrasts in JMP (etc.)? Normal treatments Legume11 Graminoid22 Aster32 SStreat 12267 Df treat21 MStreat60 MSerror10 Df error20 Legumes vs. non- legumes “There was a significant treatment effect (F…). About 53% of the variation between treatments was due to differences between legumes and non- legumes (F 1,20 = 6.7).” F 1,20 = (67)/1 = 6.7 10 From full model!

19. Even different statistical tests may not be independent ! Example. We examined effects of fertilizer on growth of dandelions in a pasture using an ANOVA. We then repeated the test for growth of grass in the same plots. Problem?

20. Multiple tests Not significant significant Not significant a a,b b Convention: Treatments with a common letter are not significantly different