Regression designs
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Regression designs 0 1 2 3 4 5 6 7 8 9 Growth rate Y 1 10 Plant size X 1 X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 72
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9. 8. 7. Growth rate. 6. Y. 5. 4. 3. 2. 1. 0. Regression designs. 1. 10. Plant size. X 1. X Y 11.5 23.3 44.0 64.5 85.2 1072. 9. 9. 8. 8. 7. 7. Growth rate. Growth rate. 6. 6. Y. Y. 5. 5. 4. 4. 3. 3. 2. 2. 1. 1. 0. 0. 1. 10. Plant size. X 1. - PowerPoint PPT Presentation
Transcript of Regression designs
Regression designs
0123456789
Gro
wth
rat
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Y
1 10Plant size
X1
X Y1 1.52 3.34 4.06 4.58 5.210 72
Regression designs
0123456789
Gro
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rat
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Y
1 10Plant size
X1
0123456789
Gro
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Y
1 10Plant size
X1
X Y1 1.52 3.34 4.06 4.58 5.210 72
X Y1 0.81 1.71 3.010 5.210 7.010 8.5
Regression designs
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Gro
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Y
1 10Plant size
X1
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Gro
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1 10Plant size
X1
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Gro
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Y
0 1Plant size
X1
X Y1 1.52 3.34 4.06 4.58 5.210 7.2
X Y1 0.81 1.71 3.010 5.210 7.010 8.5
X Y0 0.80 1.70 3.01 5.21 7.01 8.5
Code 0=small, 1=large
0123456789
Gro
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0 1Plant size
X1
X Y0 0.80 1.70 3.01 5.21 7.01 8.5
Code 0=small, 1=large
Growth = m*Size + b
Questions on the general equation above:
1. What parameter predicts the growth of a small plant?
2. Write an equation to predict the growth of a large plant.
3. Based on the above, what does “m” represent?
0123456789
Gro
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0 1Plant size
X1
X Y0 0.80 1.70 3.01 5.21 7.01 8.5
Code 0=small, 1=large
Growth = m*Size + b
If small
Growth = m*0 + b
If large
Growth = m*1 + b
Large - small = m
Growth of smallDifference in growth
What about “covariates”…- looking at the effect of salmon on tree growth rates
Nitrogen
Compare tree growth around 2 streams, one with and one without salmon
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Salmon No Salmon
t(9) = 0.06, p = 0.64
In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous)
ANCOVA
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Plant height (cm)
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Salmon No Salmon
Fertilizer treatment (X1): code as 0 = No Salmon; 1 = Salmon
Plant height (X2): continuous
ANCOVA
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0 2 4 6
Plant height (cm)
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SalmonNo Salmon
ANCOVA
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Plant height (cm)
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X1 X2 Y0 1 1.10 2 4.0: : :1 1 3.11 2 5.2: : :1 5 11.3
X1*X200:12
5
?
?
Salmon
No Salmon
Fertilizer treatment (X1): code as 0 = No Salmon; 1 = Salmon
Plant height (X2): continuous
1. Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
ANCOVA
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Plant height (cm)
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SalmonNo Salmon
ANCOVA
Questions:
• Write out equation for No Salmon (X1= 0)
• Write out equation for Salmon (X1 = 1)
• What differs between two equations?
• If no interaction (i.e. m3 = 0) what differs between eqns?
1. Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
ANCOVA
If X1=0: Y=m1X1+ m2X2 +m3X1X2 +b
If X1=1: Y=m1 + m2X2 +m3X2 +b
Difference: m1 +m3X2
1. Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
Difference if
no interaction: m1 +m3X2
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0 2 4 6
Plant height (cm)
Gro
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0 2 4 6
Plant height (cm)
Gro
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Difference between categories….
Constant, doesn’t depend on covariate
Depends on covariate
= m1 (no interaction)= m1 + m3X2
(interaction)
1. Fit full model (categorical treatment, covariate, interaction)
2. Test for interaction (if significant- stop!)
ANCOVA
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Plant height (cm)
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If no interaction, the lines will be parallel
SalmonNo Salmon
1. Fit full model (categorical treatment, covariate, interaction)
2. Test for interaction (if significant- stop!)3. Test for differences in intercepts between
lines = m1
ANCOVA
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No interactionIntercepts differ
} m1
Multiple X variables:
Both categorical …………... ANOVA
One categorical, one continuous……………...ANCOVA
Both continuous …………....Multiple Regression
Regression’s deep dark secret:
Order matters!
Input: height p=0.001weight p=0.34age p=0.07
Input: height p=0.001age p=0.04weight p=0.88
Why? In the first order, even though weight wasn’t significant, it explained some of the variation before age was tested. Common when x-variables are correlated with each other.
