Post on 15-Dec-2015
Multiple comparisons
- multiple pairwise tests
- orthogonal contrasts
- independent tests
- labelling conventions
Card example number 1
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.
Multiple tests
ANOVA shows at least one different, but which one(s)?
significant
Not significant
significant
•T-tests of all pairwise combinations
Multiple tests
T-test: <5% chance that this difference was a fluke…
affects likelihood of finding a difference in this pair!
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…
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
Card example 2
Orthogonal contrastsOrthogonal = 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
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 + - -
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
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
Trick for determining if contrasts are orthogonal:
4. Multiply each column, then sum these products.
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!
Legumes Graminoids Asters +1 - 1/2 -1/2 0 +1 -1
0 - 1/2 +1/2
Sum of products = 0
What about these contrasts?
1. Monocots (graminoids) vs. dicots (legumes and asters).
2. Legumes vs. non-legumes
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.
How do you program contrasts in JMP (etc.)?
Treatment SS
}Contrast 2
}Contrast 1
How do you program contrasts in JMP (etc.)?
Normal treatments
Legume 1 1Legume 1 1Graminoid 2 2Graminoid 2 2Aster 3 2Aster 3 2
SStreat 122 67Df treat 2 1MStreat 60
MSerror 10Df error 20
Legumesvs. 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 (F1,20 = 6.7).”
F1,20 = (67)/1 = 6.7 10
From full model!
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?
Multiple tests
Not significantsignificant
Not significant
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bConvention:Treatments with a common letter are not significantly different
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