Regression plus part 2

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Regression (plus) part 2

• Categorical variables in regressions
• Analysis of covariance (ANCOVA)
• Multiple tests
• P.S. remember to do readings on statistical power
  for the next lab

2
Regression designs
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10
Plant size
X1
X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 72
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Regression designs
1
10
Plant size
X1
X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 7.2
X Y 1 0.8 1 1.7 1 3.0 10 5.2 10 7.0 10 8.5
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Regression designs
Code 0small, 1large
1
10
Plant size
X1
X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 72
X Y 1 0.8 1 1.7 1 3.0 10 5.2 10 7.0 10 8.5
X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5
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Code 0small, 1large
Growth mSize b
Questions 1. What is mean growth of small
plants (in general)? 2. What is mean growth of
large plants? 3. What does m represent? b?
Hint use above answers
X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5
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Code 0small, 1large
Growth mSize b
If small Growth m0 b
If large Growth m1 b
X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5
Difference in growth?
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ANCOVA

• In an Analysis of Covariance, we look at the
  effect of a treatment (categorical) while
  accounting for a covariate (continuous)

Fertilized NP
Fertilized N
Growth rate (g/day)
Plant height (cm)
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ANCOVA

• Fertilizer treatment (X1) code as 0 N 1 NP
• Plant height (X2) continuous

Fertilized NP
Fertilized N
Growth rate (g/day)
Plant height (cm)
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ANCOVA

• Fertilizer treatment (X1) code as 0 N 1 NP
• Plant height (X2) continuous

X1 X2 Y 0 1 1.1 0 2 4.0 1 1 3.1 1 2 5.2
1 5 11.3
X1X2 0 0 1 2 5
Growth rate (g/day)
Plant height (cm)
10
ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Ym1X1 m2X2 m3X1X2 b

Fertilized NP
Fertilized N
Growth rate (g/day)
Plant height (cm)
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ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Ym1X1 m2X2 m3X1X2 b

• Questions
• Write out equation for N only fertilizer (X1 0)
• Write out equation for NP fertilizer (X1 1)
• What differs between two equations?
• If no interaction (i.e. m3 0) what differs
  between eqns?

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ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Ym1X1 m2X2 m3X1X2 b

If X11 Ym1 m2X2 m3X2 b
Difference m1 m3X2
If no interaction m1
13
ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Test for interaction (if significant- stop!)

If no interaction, the lines will be parallel
Fertilized NP
Fertilized N
Growth rate (g/day)
Plant height (cm)
14
ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Test for interaction (if significant- stop!)
• Test for differences in intercept

m1
Fertilized NP
Fertilized N
Growth rate (g/day)
No interaction Intercepts differ
Plant height (cm)
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Bromeliad food web
c. William H. Bond
Detritus
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Experimental design
Top trophic level
X
Bromeliad complexity
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Expt. 1


Predation x complexity or complexity2 Plt0.05
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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)?
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Multiple tests
Solution Make alpha your overall
experiment-wise error rate
T-test lt5 chance that this difference was a
fluke
affects likelihood (alpha) of finding a
difference in this pair!
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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
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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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