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

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

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.

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

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!

Formal notation Af6 is a fixed factor called A with 6 levels Br5 is a random factor called B with 5 levels

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

ANOVA Example: formal notation Ecologists: Er10 Papers: Pf2 Example 2: Populations: Pr4 Herbivory: Hf2 Example 3: Light: Lf3 Nutrients: Nf3 Blocks: Br3

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!

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

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

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

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

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

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

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

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.

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

Example 1: Er10 x Pf2   Example 2: Pr4 (Hf2) Example 3: Br3 x Lf3 x Nf3

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

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

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

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

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

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