ANCOVA
Regression with more than one line
Andrew Jackson
a.jackson@tcd.ie
Focus on gene / environment effects
• Prof. Donal Manahan’s seminar “Evolution and
development: an ecological perspective”
3/11/2011
• Growth rates are affected by extrinsic
environmental conditions
• Growth rates are affected by intrinsic
physiological factors which may be governed
by genetic factors
Experiments
• Rear larval oysters at different temperatures
• Record simple growth rate as mm/week
• Repeat the experiment with different
genotypes
The effect of temperature
30
25
20
15
10
growth rate (mm/week)
Intercept = 5, slope = 1.3
4
6
8
10
12
temperature
14
16
18
30
25
20
15
10
growth rate (mm/week)
The effect of genotype
1
2
genotype
30
25
20
15
10
growth rate (mm/week)
The effect of both together
4
6
8
10
12
temperature
14
16
18
30
• Temperature affects both
genotypes equally
• There is a fixed effect of
genotype
25
– Constant for all
temperatures
15
20
• The red genotype grows
faster than the black one
• coefficients
10
growth rate (mm/week)
How do these lines differ?
4
6
8
10
12
temperature
14
16
18
– Slopes = 1.3
– red intercept = 8
– black intercept=5
40
30
20
10
growth rate (mm/week)
A different genotype
4
6
8
10
12
temperature
14
16
18
– Green slope = 1.8, intercept = 8
– Black slope = 1.3, intercept = 5
40
30
20
10
• There is still an effect of
temperature
• But, now it is different for each
genotype
• The effect of genotype is no
longer fixed for all
temperatures
• There is an interaction
between temperature
(environment) and genotype
• Coefficients
growth rate (mm/week)
How do these lines differ?
4
6
8
10
12
temperature
14
16
18
A slightly different question
And why its important to consider
the linear covariate when comparing
between groups
How do we compare two lines statistically?
• Known as:
• Analysis of Covariance:
ANCOVA
• Also a GLM with fixed
factors and linear
covariates
An alternative dataset
• Experiment to study
effect of herbivores on
primary productivity in
ecosystems
• Series of in situ
exclusion experiments
• Measured:
– Seed mass (g)
– Grazed / Ungrazed
– Root diameter at start of
experiment
The Data
Covariate
Response
Fixed Factor
Root
Fruit
Grazing
6.225
59.77
Ungrazed
6.487
60.98
Ungrazed
4.919
14.73
Ungrazed
5.13
19.28
Ungrazed
5.417
34.25
Ungrazed
5.359
35.53
Ungrazed
8.643
78.28
Grazed
7.916
41.48
Grazed
9.351
98.47
Grazed
7.066
40.15
Grazed
8.158
52.26
Grazed
7.382
46.64
Grazed
8.515
71.01
Grazed
8.53
83.03
Grazed
Questions to ask
• How does grazing affect seed production?
• Why was root diameter recorded?
– How might this have changed the picture if it were
omitted?
• What do we need to
test statistically to
address our hypothesis?
Testing parallel lines
• Pick one line to be the
reference (e.g. Grazed)
• What is the equation
for the Grazed line?
– Seed = b0 + b1Root
• What is the equation
for Ungrazed line?
– Seed = b0 + bug + b1Root
Testing parallel Lines in R
•
•
Call:
glm(formula = Fruit ~ Root + Grazing, data = mydata)
•
•
•
Deviance Residuals:
Min
1Q
-17.1920
-2.8224
•
•
•
•
•
•
•
Coefficients:
•
AIC: 271.13
•
Number of Fisher Scoring iterations: 2
Median
0.3223
3Q
3.9144
Max
17.3290
Estimate Std. Error t value
(Intercept)
-127.829
9.664 -13.23
Root
23.560
1.149
20.51
GrazingUngrazed
36.103
3.357
10.75
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’
Pr(>|t|)
1.35e-15 ***
< 2e-16 ***
6.11e-13 ***
0.05 ‘.’ 0.1 ‘ ’ 1
Testing non-parallel lines
• Pick one line to be the
reference (e.g. Grazed)
• What is the equation
for the Grazed line?
– Seed = b0 + b1Root
• What is the equation
for Ungrazed line?
– Seed = b0 + bug +
(b1+b2)Root
– Seed = b0 + bug +
b1Root + b2Root_UG
Testing parallel Lines in R
•
•
•
•
•
Call:
glm(formula = Fruit ~ Root * Grazing, data = mydata)
Deviance Residuals:
Min
1Q
Median
3Q
Max
-17.3177
-2.8320
0.1247
3.8511
17.1313
•
•
•
•
•
•
•
•
Coefficients:
•
AIC: 273.01
•
Number of Fisher Scoring iterations: 2
Estimate Std. Error t value Pr(>|t|)
(Intercept)
-125.173
12.811 -9.771 1.15e-11 ***
Root
23.240
1.531 15.182 < 2e-16 ***
GrazingUngrazed
30.806
16.842
1.829
0.0757 .
Root:GrazingUngrazed
0.756
2.354
0.321
0.7500
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1