quadrats, anova. quadrat shape ? ? ? ? ? ? ? 1. edge effects best worst
DESCRIPTION
Quadrat shape 2. Variance bestTRANSCRIPT
Quadrats, ANOVA
Quadrat shape
?
? ?
?
?
?
?
1. Edge effects
bestworst
Quadrat shape
2. Variance
4 1
4
5
best
Quadrat size
1. Edge effects
??
?? ?
?
bestworst
3/5 on edge 3/8 on edge
Quadrat size
2. Variance
High variance Low variance
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)
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!
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)?
16 cm
22cm
ANOVA Example: formal notation
Example 1Ecologists: Er
10
Papers: Pf2
Example 2:Populations: Pr
2
Herbivory: Hf2
Example 3:Light: Lf
3
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 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
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
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
Strain A
Strain B
Strain C
O N P
Strain D
Strain E
Strain F
Fertilizer
No fertilizer Nitrogen fertilizer Phosphorus fertilizer
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
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
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)
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.
Group exercise (groups of 3)
Experimental design handout
Write out the factors and levels using formal notation
Example 1:Er
10 x Pf2
Example 2:Pr
2 (Hf2)
Example 3:Br
3 x Lf3 x Nf
3