Multiple comparisons multiple pairwise tests orthogonal contrasts independent

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

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.

3 treatments: a, no active management b, selective logging c, replanting Look at the effect of each treatment on Orang-utan births

Multiple tests Births per km 2 ANOVA gives f(2, 27) = 5. 82, p < 0. 05. Therefore at least one different, but which one(s)? significant Not significant • T-tests of all pairwise combinations

Births per km 2 Multiple tests T-test: <5% chance that this difference was a fluke… affects likelihood of finding a difference in this pair!

Births per km 2 Multiple tests T-test: <5% chance that this difference was a fluke… Solution: Make alpha your overall “experiment-wise” error rate affects likelihood (alpha) of finding a difference in this pair!

Multiple tests Births per km 2 Solution: Make alpha your overall “experiment-wise” error rate e. g. simple Bonferroni: Divide alpha by number of tests Alpha / 3 = 0. 0167

Multiple tests Births per km 2 Tukey post-hoc testing does this for you Uses the qdistribution Does all the pairwise comparisons. 1 -2, p < 0. 05 1 -3, p > 0. 05 2 -3, p < 0. 05

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

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 +1 Graminoids - 1/2 Asters -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 +1 0 Graminoids - 1/2 +1 Asters -1/2 -1

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

How do you program contrasts in JMP (etc. )? Normal treatments Legumes vs. nonlegumes Legume Graminoid Aster 1 1 2 2 3 3 1 1 2 2 SStreat Df treat MStreat 122 2 60 67 1 MSerror Df error 10 20 “There was a significant treatment effect (F…). About 53% of the variation between treatments was due to differences between legumes and nonlegumes (F 1, 20 = 6. 7). ” F 1, 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 Births per km 2 b a, b a significant Not significant Convention: Treatments with a common letter are not significantly different