analysis of covariance lecture modules using plots

anova

1. •Fixed, random, mixed-model ANOVAs
• Factorial vs. nested designs
• Formal design notation
• Split-plot designs
ANOVA lecture

2. Goals
1.Describe your ANOVA design to a
statistician (who can then help you
analyse it).
2.Recognize three common types of
ANOVA designs:
•Factorial: fixed, randomized block
•Nested
•Split-plot
3. For your reference: formulas for F
tests for each

3. Factor – a variable of interest
e.g. temperature
Level – a particular value / state of a factor
e.g. hot, cold
In this example, temperature is a factor with two levels.

4. Fixed factor
Either
(1) The investigator chooses the levels of the factor for
some purpose.
Eg. Ambient CO
2 vs. double CO
2
OR
(2) The levels used represent all possible levels.
Eg. Biological sex: Male, female

5. Random factor
The levels of the factor are chosen randomly from a
universe of possible levels.
Eg. We want to look at whether butterfly collectors
differ in their diversity estimates for 4 plots. We select
5 collectors “randomly” from a village.
Eg. We use three breeding lines of fruit flies as blocks
in a genetics experiment. Blocks are typically random
effects!

6. Formal notation
A
f
6 is a fixed factor called A with 6 levels
B
r
5 is a random factor called B with 5 levels

7. Group exercise (groups of 3)
Experimental design handout
 Write out the factors and levels using
formal notation

8. ANOVA Example: formal notation
Example 1
Ecologists: E
r
10
Papers: P
f
2
Example 2:
Populations: P
r
2
Herbivory: H
f
2
Example 3:
Light: L
f
3
Nutrients: N
f
3
Blocks: B
r
3

9. Fixed-effects ANOVA (Model I)
•All factors are fixed
Random-effects ANOVA (Model II)
•All factors are random
Mixed-model ANOVA (Model III)
•Contains both fixed and random effects, e.g.
randomized block!

10. Two-way factorial ANOVA
How to calculate “F”
Fixed effect
(factors A & B
fixed)
Random effect
(factors A & B
random)
Mixed model
(A fixed, B
random)
Factor A
Factor B
A x B
MS A
MS Error
MS B
MS Error
MS A x B
MS Error
MS A
MS A x B
MS B
MS A x B
MS A x B
MS Error
MS A x B
MS Error
MS B
MS Error
MS A
MS A x B

11. Factorial design:
All levels of one factor crossed by all levels
of another factor, i.e. all possible
combinations are represented.
If you can fill in a table with unique
replicates, it’s factorial!
Pea plant
Bean plant
Corn plant
Ambient CO
2
Double CO
2

12. Nested design
In this example, strain type is “nested within”
fertilizer.
Fertilizer is often called “group”, strain “subgroup”
The nested factor is always random
No fertilizer Nitrogen fertilizerPhosphorus fertilizer
Strain A Strain B Strain C Strain D Strain E Strain F

13. Strain A
Strain B
Strain C
O N P
Strain D
Strain E
Strain F
Fertilizer

14. No fertilizer Nitrogen fertilizerPhosphorus
fertilizer

Strain A Strain B Strain C Strain D Strain E Strain F
Grand mean
Variance: Group

15. No fertilizer Nitrogen fertilizerPhosphorus
fertilizer

Strain A Strain B Strain C Strain D Strain E Strain F
Variance: Subgroup within a group
Grand mean
Variance: Group

16. No fertilizer Nitrogen fertilizerPhosphorus
fertilizer

Strain A Strain B Strain C Strain D Strain E Strain F
Variance: Subgroup within a group
Variance: Among all subgroups
Grand mean
Variance: Group

17. Nested ANOVA: “A” Subgroups nested within “B”
Groups, with n replicates
In our example, A=2, B=3 and n=2
Total
Groups
MS Subgroups within groups
MS Among all subgroups
MS Groups
MS Subgroups within groups
B-1
Subgroups
within groups
B(A-1)
ABn-1
df F
Among all
subgroups
AB(n-1)

18. Formal notation cont.
A
f
6 x B
r
5 tells us that this is a factorial design
with factor A “crossed” with factor B
A
f
6
(B
r
5
) tells us that this is a nested design
with factor A “nested within” with factor B. In
other words, A is subgroup, B is group.

19. Group exercise (groups of 3)
Experimental design handout
 Write out the factors and levels using
formal notation

20. Example 1:
E
r
10
x P
f
2
 
Example 2:
P
r
2
(H
f
2
)
 
Example 3:
B
r
3
x L
f
3
x N
f
3

21. Split plot design
An experiment replicated within an
experiment!
Ambient CO
2
Elevated CO
2
4 Main plots, e.g. greenhouses

22. Split plot design
An experiment replicated within an
experiment!
Main plot
CO2 MS maintreat F
Main plot errorMS mainerror

23. Split plot design
An experiment replicated within an
experiment!
Ambient CO
2
Elevated CO
2
4 Main plots, e.g. greenhouses

24. Split plot design
An experiment replicated within an
experiment!
3 5 6 2 4 1
5 3 6 4 2 1
6 3 2 5 1 4
1 5 6 3 2 4
Subplots with six different
nutrient concentrations

25. Split plot design
An experiment replicated within an
experiment!
Subplot
nutrient MS subtreat F
nutrient x CO
2
MS subinteractF
Subplot error MS suberror

26. Split plot design
An experiment replicated within an
experiment!
Subplot
nutrient MS subtreat F
nutrient x CO
2
MS subinteractF
Subplot error MS suberror
Main plot
CO2 MS maintreat F
Main plot error MS mainerror