1. Quadrats, ANOVA

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

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

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

5. Quadrat size 2. Variance High variance Low 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. 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 Size & shape affect!

8. Quadrat lab What is better quadrat shape? Square or rectangle? What is better quadrat size? 4, 9,16, 25 cm 2 ? Does your answer differ with tree species (distribution differs)? 16 cm 22cm

9. ANOVA Example: formal notation Example 1 Ecologists: E r 10 Papers: P f 2 Example 2: Populations: P r 2 Herbivory: H f 2 Example 3: Light: L f 3 Nutrients: N f 3 Blocks: B r 3

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” 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 Error MS A x B MS Error MS B MS Error MS A MS A x B

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! Pea plant Bean plant Corn plant Ambient CO 2 Double CO 2

13. 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 fertilizerNitrogen fertilizerPhosphorus fertilizer Strain A Strain B Strain C Strain D Strain E Strain F

14. Strain A Strain B Strain C ONP Strain D Strain E Strain F Fertilizer

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

16. No fertilizerNitrogen fertilizerPhosphorus fertilizer Strain A Strain B Strain C Strain D Strain E Strain F Variance: Subgroup within a group Grand mean Variance: Group

17. No fertilizerNitrogen fertilizerPhosphorus 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

18. 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)

19. Formal notation cont. A f 6 x B r 5 tells us that this is a factorial design with factor A “crossed” with factor B A f 6 (B r 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.

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

21. Example 1: E r 10 x P f 2 Example 2: P r 2 (H f 2 ) Example 3: B r 3 x L f 3 x N f 3