Quadrats, ANOVA

1. Quadrats, ANOVA

2. ? ? ? ? ? ? Quadrat shape 1. Edge effects ? best worst

3. Quadrat shape 2. Variance 4 5 4 1 best

4. 3/5 on edge 3/8 on edge best worst Quadrat size 1. Edge effects ? ? ? ? ? ?

5. Low variance High variance Quadrat size 2. Variance

6. Quadrat size So should we always use as large a quadrat as possible? Tradeoff with cost (bigger quadrats take l o n g e r to sample)

7. Size & shape affect! • Quadrat lab • Use a cost (“time is money”): benefit (low variance) approach to determine the optimal quadrat design for 10 tree species. • Hendrick’s method • Wiegert’s method • Cost: • total time = time to locate quadrat + time to census quadrat • Benefit: • Variance

8. Quadrat lab What is better quadrat shape? Square or rectangle? What is better quadrat size? 4, 9 ,16, 25 cm2 ? Does your answer differ with tree species (distribution differs)? 22cm 16 cm

9. ANOVA Example: formal notation Example 1 Ecologists: Er10 Papers: Pf2 Example 2: Populations: Pr2 Herbivory: Hf2 Example 3: Light: Lf3 Nutrients: Nf3 Blocks: Br3

10. 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!

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

12. 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! Double CO2 Ambient CO2 Pea plant Bean plant Corn plant

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

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

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

16. 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

17. 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

18. 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

19. 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.

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

21. Example 1: Er10 x Pf2 Example 2: Pr2 (Hf2) Example 3: Br3 x Lf3 x Nf3