fixed, random, mixed-model anovas factorial vs. nested designs formal design notation split-plot...

•Fixed, random, mixed-model ANOVAs

• Factorial vs. nested designs

• Formal design notation

• Split-plot designs

ANOVA lecture

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

Factor – a variable of interest e.g. temperature

Level – a particular value / state of a factore.g. hot, cold

In this example, temperature is a factor with two levels.

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

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!

Formal notation

Af6 is a fixed factor called A with 6 levels

Br5 is a random factor called B with 5 levels

Group exercise (groups of 3)

Experimental design handout

Write out the factors and levels using formal notation

ANOVA Example: formal notation

Example 1Ecologists: Er

10

Papers: Pf2

Example 2:Populations: Pr

4

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

4 (Hf2)

 Example 3:Br

3 x Lf3 x Nf

3

Split plot design

An experiment replicated within an experiment!

Ambient CO2Elevated CO2

4 Main plots, e.g. greenhouses

Split plot design

An experiment replicated within an experiment!

Main plotCO2 MS maintreat F

Main plot error MS mainerror

Split plot design

An experiment replicated within an experiment!

Ambient CO2Elevated CO2

4 Main plots, e.g. greenhouses

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

Split plot design

An experiment replicated within an experiment!

Subplotnutrient MS subtreat F nutrient x CO2 MS subinteract F

Subplot error MS suberror

Split plot design

An experiment replicated within an experiment!

Subplotnutrient MS subtreat F nutrient x CO2 MS subinteract F

Subplot error MS suberror

Main plotCO2 MS maintreat F

Main plot error MS mainerror