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

2. Goals
Describe your ANOVA design to a statistician (who can then help you analyse it).
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 CO2 vs. double CO2 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
Af6 is a fixed factor called A with 6 levels
Br5 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
Ecologists: Er10
Papers: Pf2
Example 2:
Populations: Pr4
Herbivory: Hf2
Example 3:
Light: Lf3
Nutrients: Nf3
Blocks: Br3

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
MS A
MS Error
MS A
MS A x B
MS A
MS A x B
Factor B
MS B
MS A x B
MS B
MS Error
MS B
MS Error
A x B
MS A x B
MS Error
MS A x B
MS Error
MS A x B
MS Error

11. If you can fill in a table with unique replicates, it’s factorial!
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!
Ambient CO2
Double CO2
Pea plant
Bean plant
Corn plant

12. In this example, strain type is “nested within” fertilizer.
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 fertilizer Phosphorus fertilizer
Strain A Strain B Strain C Strain D Strain E Strain F

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

14. Grand mean Variance: Group
No fertilizer Nitrogen fertilizer Phosphorus fertilizer
Strain A Strain B Strain C Strain D Strain E Strain F

15. Variance: Subgroup within a group
Grand mean
Variance: Group
No fertilizer Nitrogen fertilizer Phosphorus fertilizer
Variance: Subgroup within a group
Strain A Strain B Strain C Strain D Strain E Strain F

16. Variance: Subgroup within a group
Grand mean
Variance: Group
No fertilizer Nitrogen fertilizer Phosphorus fertilizer
Variance: Subgroup within a group
Strain A Strain B Strain C Strain D Strain E Strain F
Variance: Among all subgroups

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

18. Formal notation cont.
Af6 x Br5 tells us that this is a factorial design with factor A “crossed” with factor B
Af6 (Br5) 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:
Er10 x Pf2
Example 2:
Pr4 (Hf2)
Example 3:
Br3 x Lf3 x Nf3

21. An experiment replicated within an experiment!
Split plot design
An experiment replicated within an experiment!
4 Main plots, e.g. greenhouses
Ambient CO2
Elevated CO2

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

23. An experiment replicated within an experiment!
Split plot design
An experiment replicated within an experiment!
4 Main plots, e.g. greenhouses
Ambient CO2
Elevated CO2

24. An experiment replicated within an experiment!
Split plot design
An experiment replicated within an experiment!
Subplots with six different nutrient concentrations

25. Split plot design
An experiment replicated within an experiment!
Subplot
nutrient MS subtreat F
nutrient x CO2 MS subinteract F
Subplot error MS suberror

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