analysis of covariance lecture modules using plots
- 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 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 Af 6 is a fixed factor called A with 6 levels Br 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: Er 10 Papers: Pf 2 Example 2: Populations: Pr 2 Herbivory: Hf 2 Example 3: Light: Lf 3 Nutrients: Nf 3 Blocks: Br 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 CO2 Double CO2
- 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 fertilizer Phosphorus 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 fertilizer Phosphorus fertilizer Strain A Strain B Strain C Strain D Strain E Strain F Grand mean Variance: Group
- 15. 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
- 16. 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
- 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. Af 6 x Br 5 tells us that this is a factorial design with factor A “crossed” with factor B Af 6 (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.
- 19. Group exercise (groups of 3) Experimental design handout Write out the factors and levels using formal notation
- 20. Example 1: Er 10 x Pf 2 Example 2: Pr 2 (Hf 2 ) Example 3: Br 3 x Lf 3 x Nf 3
- 21. Split plot design An experiment replicated within an experiment! Ambient CO2 Elevated CO2 4 Main plots, e.g. greenhouses
- 22. Split plot design An experiment replicated within an experiment! Main plot CO2 MS maintreat F Main plot error MS mainerror
- 23. Split plot design An experiment replicated within an experiment! Ambient CO2 Elevated CO2 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 CO2 MS subinteract F Subplot error MS suberror
- 26. Split plot design An experiment replicated within an experiment! Subplot nutrient MS subtreat F nutrient x CO2 MS subinteract F Subplot error MS suberror Main plot CO2 MS maintreat F Main plot error MS mainerror