1. Quadrat sampling
Quadrat shape
Quadrat size
Lab
Regression and ANCOVA
Review
Categorical variables
ANCOVA if time

2. Quadrat shape
1. Edge effects ? ? ?
worst
best

3. Quadrat shape
2. Variance 4 5 4 1
best

4. Quadrat size 1. Edge effects best worst 3/5 on edge 3/8 on edge ? ? ?

5. Quadrat size
1. Edge effects
Density
Quadrat size

6. Quadrat size
2. Variance
High variance
Low variance

7. (bigger quadrats take l o n g e r to sample)
Quadrat size
So should we always use
as large a quadrat as possible?
Tradeoff with cost
(bigger quadrats take l o n g e r to sample)

8. Quadrat lab What is better quadrat shape? Square or rectangle?
What is better quadrat size? 4, 9 ,16, 25 cm2 ?
Does your answer differ with tree species (distribution differs)?
22cm
16 cm

9. Quadrat lab
Use a cost (“time is money”): benefit (low variance) approach to determine the optimal quadrat design for 10 tree species.
Hendrick’s method
Wiegert’s method
Cost:
total time = time to locate quadrat + time to census quadrat
Benefit:
Variance
Size & shape affect!

10. Quadrat lab Quadrats can also be used to determine spatial pattern!
We will analyze our data for spatial pattern (only) in the computer lab next week (3-5 pm).

11. Quadrat lab: points for thought
1. You need to establish if any species shows a density gradient. How will you do this?
2. You will have a bit of time to do something extra; what would be useful? Group work fine here.
3. Rules:
- if quadrat doesn’t fit on map ?
- if leaves are on edge of quadrat ?

12. Regression
Problem: to draw a straight line through the points that best explains the variance

13. Regression
Problem: to draw a straight line through the points that best explains the variance

14. Regression
Problem: to draw a straight line through the points that best explains the variance

15. Regression Test with F, just like ANOVA:
Variance explained by x-variable / df
Variance still unexplained / df
Variance
explained
(change in
line lengths2)
Variance
unexplained
(residual
line lengths2)

16. Regression Test with F, just like ANOVA:
Variance explained by x-variable / df
Variance still unexplained / df
In regression, each x-variable will normally have 1 df

17. Regression Test with F, just like ANOVA:
Variance explained by x-variable / df
Variance still unexplained / df
Essentially a cost: benefit analysis –
Is the benefit in variance explained worth the cost in using up degrees of freedom?

18. Regression example Total variance for 32 data points is 300 units.
An x-variable is then regressed against the data, accounting for 150 units of variance.
What is the R2?
What is the F ratio?

19. Regression example Total variance for 32 data points is 300 units.
An x-variable is then regressed against the data, accounting for 150 units of variance.
What is the R2?
What is the F ratio?
R2 = 150/300 = 0.5
F 1,30 = 150/1 = 30
150/30
Why is df error = 30?

20. 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

21. 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

22. 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 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

23. 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 “b” represent?
X Y
0 0.8
0 1.7
0 3.0
1 5.2
1 7.0
1 8.5

24. 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

25. 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)

26. 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)

27. 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)

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

29. 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?

30. 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: m m3X2
Difference if
no interaction: m m3X2

31. 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)

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

33. ANCOVA
Fit full model (categorical treatment, covariate, interaction)
Test for interaction (if significant- stop!)
Test for differences in intercept
} m1
Fertilized N+P
Fertilized N
Growth rate (g/day)
No interaction
Intercepts differ
Plant height (cm)