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. Two (or more) X variables:
Both categorical …………... ANOVA
One categorical,
one continuous……………...ANCOVA

7. ANCOVA
In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous)
Fertilized P
Fertilized N
Growth rate (g/day)
Plant height (cm)

8. ANCOVA Fertilizer treatment (X1): code as 0 = N; 1 =P
Plant height (X2): continuous
Fertilized P
Fertilized N
Growth rate (g/day)
Plant height (cm)

9. ANCOVA Fertilizer treatment (X1): code as 0 = N; 1 = P
Plant height (X2): continuous ?
X1 X2 Y
: : :
X1*X2 : 1 2 5
Growth rate (g/day) ?
Fertilized P
Fertilized N
Plant height (cm)

10. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
Fertilized P
Fertilized N
Growth rate (g/day)
Plant height (cm)

11. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Y=m1X1+ m2X2 +m3X1X2 +b
Questions:
Write out equation for N fertilizer (X1= 0)
Write out equation for P fertilizer (X1 = 1)
What differs between two equations?
If no interaction (i.e. m3 = 0) what differs between eqns?

12. 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

13. 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)

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

15. 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)

16. Multiple X variables:
Both categorical …………... ANOVA
One categorical,
one continuous……………...ANCOVA
Both continuous …………....?

17. Multiple regression
Herbivore damage
Higher nutrient trees
Lower nutrient trees
Tree age
Damage= m1*age + b

18. Herbivore damage
Tree age
Residuals of
herbivore damage
Tree nutrient concentration

19. Damage= m1*age + m2*nutrient + b
Herbivore damage
Tree age
Residuals of
herbivore damage
Tree nutrient concentration

20. No interaction (additive):
Interaction (non-additive): y y
Damage= m1*age + m2*nutrient + m3*age*nutrient +b

21. Non-linear regression? Just a special case of multiple regression!X2 X1
X X2 Y
Y = m1 x +m2 x2 +b
Y = m1 x1 +m2 x2 +b

22. 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.

23. Solutions?
1) Use a logical order. For example in ANCOVA it makes sense to test the interaction first
2) Stepwise regression: “tries out” various orders of removing variables.