1. Regression designs Y X1 Plant size 1 2 3 4 5 6 7 8 9 Growth rate 1 101 2 3 4 5 6 7 8 9
Growth rate Y 1 10
Plant size X1
X Y
1 1.5
2 3.3
4 4.0
6 4.5
8 5.2
10 72

2. Regression designs Y Y X1 X1 Plant size Plant size 1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9
Growth rate Y 1 2 3 4 5 6 7 8 9
Growth rate Y 10
Plant size X1 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

3. Regression designs Y Y X1 Y X1 X1 Code 0=small, 1=large Plant size1 2 3 4 5 6 7 8 9
Growth rate Y 1 2 3 4 5 6 7 8 9
Growth rate Y 10
Plant size X1 1 2 3 4 5 6 7 8 9
Growth rate Y
Plant size X1 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
X Y
0 0.8
0 1.7
0 3.0
1 5.2
1 7.0
1 8.5

4. Growth = m*Size + b Y X1 Questions on the general equation above:
Code 0=small, 1=large
Growth = m*Size + b 1 2 3 4 5 6 7 8 9
Growth rate Y
Plant size X1
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?
X Y
0 0.8
0 1.7
0 3.0
1 5.2
1 7.0
1 8.5

5. Growth = m*Size + b If small Y Growth = m*0 + b If large X1
Difference in growth
Growth of small
Code 0=small, 1=large
Growth = m*Size + b 1 2 3 4 5 6 7 8 9
Growth rate Y
Plant size X1
If small
Growth = m*0 + b
If large
Growth = m*1 + b
X Y
0 0.8
0 1.7
0 3.0
1 5.2
1 7.0
1 8.5
Large - small = m

6. Nitrogen What about “covariates”…
- looking at the effect of salmon on tree growth rates
Nitrogen

7. Compare tree growth around 2 streams, one with and one without salmon
No Salmon
Growth rate (g/day)
t(9) = 0.06, p = 0.64

8. ANCOVA
In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous)
Salmon
No Salmon
Growth rate (g/day)
Plant height (cm)

9. ANCOVA Fertilizer treatment (X1): code as 0 = No Salmon; 1 = Salmon
Plant height (X2): continuous
Salmon
No Salmon
Growth rate (g/day)
Plant height (cm)

10. ANCOVA Fertilizer treatment (X1): code as 0 = No Salmon; 1 = Salmon
Plant height (X2): continuous ?
X1 X2 Y
: : :
X1*X2 : 1 2 5
Growth rate (g/day) ?
Salmon
No Salmon
Plant height (cm)

11. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
Salmon
No Salmon
Growth rate (g/day)
Plant height (cm)

12. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
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?

13. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
If X1=0: Y=m1X1+ m2X2 +m3X1X2 +b
If X1=1: Y=m m2X2 +m3X b
Difference: m1 +m3X2
Difference if
no interaction: m1 +m3X2

14. Difference between categories….
Constant, doesn’t
depend on covariate
Depends on covariate
= m1 + m3X2
(interaction)
= m1 (no interaction) 12 10 8
Growth rate (g/day)
Growth rate (g/day) 6 4 2 2 4 6
Plant height (cm)
Plant height (cm)

15. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Test for interaction (if significant- stop!)
If no interaction, the lines will be parallel
Salmon
No Salmon
Growth rate (g/day)
Plant height (cm)

16. ANCOVA
Fit full model (categorical treatment, covariate, interaction)
Test for interaction (if significant- stop!)
Test for differences in intercepts between lines = m1
} m1
Growth rate (g/day)
No interaction
Intercepts differ
Plant height (cm)

17. Multiple X variables:
Both categorical …………... ANOVA
One categorical,
one continuous……………...ANCOVA
Both continuous …………....Multiple Regression

18. Regression’s deep dark secret:
Order matters!
Input: height p=0.001
weight p=0.34
age p=0.07
age p=0.04
weight 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.