Quadrats, ANOVA

Quadrat shape

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? ?

?

?

?

?

1. Edge effects

bestworst

Quadrat shape

2. Variance

4 1

4

5

best

Quadrat size

1. Edge effects

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?? ?

?

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