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# Multiple comparisons

Multiple comparisons. - multiple pairwise tests - orthogonal contrasts - independent tests - labelling conventions. Card example number 1. Multiple tests. Problem:

## Multiple comparisons

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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)? • T-tests of all pairwise combinations significant significant Not significant

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

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

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

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). Legumes Graminoids Asters + - -

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”. Legumes Graminoids Asters +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. Legumes Graminoids Asters +1 - 1/2 -1/2 0 +1 -1

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

14. Legumes Graminoids Asters +1 - 1/2 -1/2 0 +1 -1 0 - 1/2 +1/2 Sum of products = 0 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!

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 1 } Contrast 2

18. F1,20 = (67)/1 = 6.7 10 From full model! How do you program contrasts in JMP (etc.)? Legumes vs. non-legumes Normal treatments “There was a significant treatment effect (F…). About 53% of the variation between treatments was due to differences between legumes and non-legumes (F1,20 = 6.7).” Legume 1 1 Legume 1 1 Graminoid 2 2 Graminoid 2 2 Aster 3 2 Aster 3 2 SStreat 122 67 Df treat 2 1 MStreat 60 MSerror 10 Df error 20

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 b Convention: Treatments with a common letter are not significantly different a,b a significant Not significant Not significant

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