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•Fixed, random, mixed-model ANOVAs
• Factorial vs. nested designs
• Formal design notation
• Split-plot designs
ANOVA lecture
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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
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Factor – a variable of interest e.g. temperature
Level – a particular value / state of a factore.g. hot, cold
In this example, temperature is a factor with two levels.
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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
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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!
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Formal notation
Af6 is a fixed factor called A with 6 levels
Br5 is a random factor called B with 5 levels
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Group exercise (groups of 3)
Experimental design handout
Write out the factors and levels using formal notation
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ANOVA Example: formal notation
Example 1Ecologists: Er
10
Papers: Pf2
Example 2:Populations: Pr
4
Herbivory: Hf2
Example 3:Light: Lf
3
Nutrients: Nf3
Blocks: Br3
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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!
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Two-way factorial ANOVAHow 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 AMS Error
MS BMS Error
MS A x BMS Error
MS AMS A x B
MS BMS A x B
MS A x BMS Error
MS A x BMS Error
MS BMS Error
MS AMS A x B
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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 CO2 Double CO2
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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
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Strain A
Strain B
Strain C
O N P
Strain D
Strain E
Strain F
Fertilizer
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No fertilizer Nitrogen fertilizer Phosphorus fertilizer
Strain A Strain B Strain C Strain D Strain E Strain F
Grand mean
Variance: Group
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No fertilizer Nitrogen fertilizer Phosphorus fertilizer
Strain A Strain B Strain C Strain D Strain E Strain F
Variance: Subgroup within a group
Grand mean
Variance: Group
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No fertilizer Nitrogen fertilizer Phosphorus 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
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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 groupsMS Among all subgroups
MS GroupsMS Subgroups within groups
B-1
Subgroups within groups
B(A-1)
ABn-1
df F
Among all subgroups
AB(n-1)
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Formal notation cont.
Af6 x Br
5 tells us that this is a factorial design with factor A “crossed” with factor B
Af6 (Br
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.
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Group exercise (groups of 3)
Experimental design handout
Write out the factors and levels using formal notation
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Example 1:Er
10 x Pf2
Example 2:Pr
4 (Hf2)
Example 3:Br
3 x Lf3 x Nf
3
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Split plot design
An experiment replicated within an experiment!
Ambient CO2Elevated CO2
4 Main plots, e.g. greenhouses
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Split plot design
An experiment replicated within an experiment!
Main plotCO2 MS maintreat F
Main plot error MS mainerror
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Split plot design
An experiment replicated within an experiment!
Ambient CO2Elevated CO2
4 Main plots, e.g. greenhouses
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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
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Split plot design
An experiment replicated within an experiment!
Subplotnutrient MS subtreat F nutrient x CO2 MS subinteract F
Subplot error MS suberror
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Split plot design
An experiment replicated within an experiment!
Subplotnutrient MS subtreat F nutrient x CO2 MS subinteract F
Subplot error MS suberror
Main plotCO2 MS maintreat F
Main plot error MS mainerror