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

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Card example number 1

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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.

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

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

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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…

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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

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Card example 2

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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

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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 + - -

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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

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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

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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

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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

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

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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.

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

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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!

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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?

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

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