Introduction to factorial ANOVA 2015

1

Introduction to Factorial ANOVA

Read from the bottom up!!!!

Two factor factorial ANOVA

• Two factors (2 predictor variables)– Factor A (with p groups or levels)

– Factor B (with q groups or levels)

• Crossed design:– every level of one factor crossed with every

level of second factor

– all combinations (i.e. cells) of factor A and factor B

2

Quinn (1988) - fecundity of limpets

• Factor A - season with 2 levels:– spring, summer

• Factor B - density with 4 levels:– 8, 15, 30, 45 per 225cm2

• n = 3 fences in each combination:– each combination is termed a cell (8 cells)

• Response variable:– fecundity (no. egg masses per limpet)

Fecundity of limpets

3

Stehman & Meredith (1995) -growth of fir tree seedlings

• Factor A is nitrogen with 2 levels– present, absent

• Factor B is phosphorous with 4 levels– 0, 100, 300, 500 kg.ha-1

• 8 cells, n replicate seedlings in each cell

• Response variable:– growth of Douglas fir trees

seedlings

Data layout

Factor A 1 2 ........ i

Factor B 1 2 j 1 2 j 1 2 j

Reps y111 yij1

y112 yij2

y11k yijk

Cell means y11 yij

Note levels in factor B are the same for all levels in Factor A (not nested)

4

Linear model

yijk = + i + j + ()ij + ijk

where overall meani effect of factor Aj effect of factor B

()ij effect of interaction between A & B

ijk unexplained variation (error term)

Worked example - Limpets

Season Spring Summer

Density 8 15 30 45 8 15 30 45

Reps n = 3 in each of 8 groups (cells)

p = 2 seasons, q = 4 densities

5

Worked example

Density 8 15 30 45 Season Marginal means

Spring 2.42 2.18 1.57 1.20 1.84

Summer 1.83 1.18 0.81 0.59 1.10

Densitymarginal 2.13 1.68 1.19 0.89 1.47means Grand mean

Cell means

Null hypotheses• Main effect:

– effect of one factor, pooling over levels of other factor

– effect of one factor, independent of other factor

• Factor A marginal means (pooling B):– 1, 2...i

• Factor B marginal means (pooling A):– 1, 2...j

6

Main effect A• H0: no difference between marginal means of

factor A, pooling levels of B– H0: 1 = 2 = i =

• H0: no main effect of factor A, pooling levels of B (1 = 2 = … = i = 0)

• Example:– No difference between season marginal means

– No effect of season, pooling densities

Density 8 15 30 45 Seasonmeans

Spring 2.42 2.18 1.57 1.20 1.84 spring

Summer 1.83 1.18 0.81 0.59 1.10 summer

Densitymeans 2.13 1.68 1.19 0.89 1.47 overall

8 15 30 45

Cell means

7

Main effect B

• H0: no difference between marginal means of factor B, pooling levels of A– H0: 1 = 2 = j =

• H0: no main effect of factor B, pooling levels of A (1 = 2 = … = j = 0)

• Example:– No difference between density marginal means

– No effect of density, pooling seasons

Density 8 15 30 45 Seasonmeans

Spring 2.42 2.18 1.57 1.20 1.84 spring

Summer 1.83 1.18 0.81 0.59 1.10 summer

Densitymeans 2.13 1.68 1.19 0.89 1.47 overall

8 = 15 = 30 = 45

Cell means

8

Interaction

• An interaction between 2 factors:– effect of factor A dependent on level of factor B and vice-

versa

• H0: no interaction between factor A & factor B:– effects of factor A & factor B are independent of each

other

– no joint effects of A & B acting together (ij = 0)

– ij - i - j + = 0, which is equal to

ij – (i + j) + = 0

Factors A and B are additive

ij – (i + j) + = 0

µ+α+β+ αβ –( µ+α+µ+β)+ µ = 0

µ+α+β+ αβ – µ-α-µ-β+ µ = 0

µ+α+β+ αβ – µ-α-µ-β+ µ = 0

αβ = 0

9

No interaction

• Root:shoot ratio of invasive grass:– C. maculosa

• Two factors:– AM fungus

(present/absent)

– Type of phosphorous (organic/inorganic)

• Factorial design– replicate pots

0

0.1

0.2

0.3

0.4

0.5

+ AM fungi - AM fungiR

oot:

shoo

t ra

tio

Inorganic POrganic P

Interaction

• Root:shoot ratio of native grass:– F. idahoensis

• Two factors:– AM fungus

(present/absent)

– Invasive competitor C. maculosa

• Factorial design– replicate pots

0

1

2

3

F. idahoensis C. maculosa

Competitor

Roo

t:sh

oot

rati

o

+ AM- AM

10

If interaction absent

• Factors A & B affect Y independently of each other

• Examine/test main effects and marginal means

Interaction examination

0

0.1

0.2

0.3

0.4

0.5

+ AM fungi - AM fungi

Roo

t:sh

oot

rati

o

Inorganic POrganic P

Organic Inorganic Row means

+AM Fungi

0.36 0.29 0.325

-AM Fungi

0.33 0.23 0.28

Column means

0.345 0.26 Overall mean =

0.3025

ij – (i + j) + = 0

0.36-(0.325+0.345)+0.3025 = -0.0075

11

If interaction present

• Factors A & B interact in their effect on Y

• Factors A & B do NOT affect Yindependently of each other

• Difficult to examine/test main effect and marginal means

• Must examine results to determine if main effects are interpretable

Interaction examination

F. idahoensis C. maculosa Row means

+AM Fungi

1.2 2.3 1.75

-AM Fungi

1.6 1.4 1.5

Column means

1.4 1.85 Overall mean =

1.625

ij – (i + j) + = 0

1.4-(1.85+1.5)+1.625 = -0.325

0

1

2

3

F. idahoensis C. maculosa

Competitor

Roo

t:sh

oot

rati

o

+ AM- AM

12

Residual variation

• Variation between replicates within each cell

• Pooled across cells if homogeneity of variance assumption holds

2)( ijijk yy

Partitioning total variation

SSTotal

SSA + SSB + SSAB + SSResidual

SSA variation between A marginal meansSSB variation between B marginal meansSSAB variation due to interaction between A

and BSSResidual variation between replicates within

each cell

13

Source SS df MS

Factor A SSA p-1 SS Ap-1

Factor B SSB q-1 SS Bq-1

Interaction SSAB (p-1)(q-1) SS ABA X B (p-1)(q-1)

Residual SSResidual pq(n-1) SS Residualpq(n-1)

Factorial ANOVA table

Expected mean squares

Both factors fixed:

MSA

MSB

MSA X B

MSResidual2

22

22

22

)1)(1(

)(

1

1

qp

n

q

np

p

nq

ij

i

i

14

H0: no interaction

• If no interaction:– H0: interaction (ij)

= 0 true

• F-ratio:– MSAB / MSResidual 1

2

22

22

22

)1)(1(

)(

1

1

qp

n

q

np

p

nq

ij

i

i

H0: no main effect

• If no main effect of factor A:– H0: 1 = 2 = i = (i = 0) is

true

• F-ratio:– MSA / MSResidual 1

• If no main effect of factor B:– H0: 1 = 2 = j = (j = 0) is

true

• F-ratio:– MSB / MSResidual 1 2

22

22

22

)1)(1(

)(

1

1

qp

n

q

np

p

nq

ij

i

i

15

Worked example

Density 8 15 30 45 Season Marginal means

Spring 2.42 2.18 1.57 1.20 1.84

Summer 1.83 1.18 0.81 0.59 1.10

Densitymarginal 2.13 1.68 1.19 0.89 1.47means Grand mean

Cell means

Testing of H0’s

• Test H0 of no interaction first:– no significant interaction between density

and season (P = 0.824)• If not significant, test main effects:

– significant effects of season (P < 0.001) and density (P < 0.001)

• Planned and unplanned comparisons:– applied to interaction and to main effects– try to limit unplanned comparisons

16

Interpreting interactions

1. Hopefully you will have a hypothesis or set of hypotheses for the interaction term

2. Plot cell means (This is usually the most informative thing to do)

3. Test hypotheses concerning the interaction (using Specify command)

4. If appropriate examine hypotheses concerned with main effects (using contrast or specify)

Interaction plot

0

1

2

3

0 20 40 60

Density

No

. eg

g m

asse

s p

er li

mp

et

SpringSummer

• Effect of density same for both seasons• Difference between seasons same for all densities• Parallel lines in cell means (interaction) plot

17

Worked example II

• Low shore Siphonaria– larger limpets

• Two factors– Season (spring and summer)

– Density (6, 12, 24 limpets per 225cm2)

• Response variable:– no. egg masses per limpet

• n = 3 enclosures per season/density combination

Worked example II

Season 1 17.15 119.85 < 0.001

Density 2 2.00 13.98 0.001

Interaction 2 0.85 5.91 0.016

Residual 12 0.14

Total 17

Source df MS F P

18

Interaction plot

0

1

2

3

4

0 10 20 30

Density

No

. eg

g m

asse

s p

er li

mp

etSpring

Summer

• Effect of density different for each season• Difference between seasons varies for each density• Non-parallel lines in cell means (interaction) plot• Is the effect of Season interpretable??

Complex interaction

• Behavioural response of larval newts in lab

• Factor A– Chemical cues from

adult newts

• Factor B– Earthworm prey

• Factorial design– replicate aquaria

0

10

20

30

40

50

Newtpresent

Newtabsent

% la

rval

new

ts in

op

en

+ worms- worms

19

Multiple comparisons

• Use Tukey’s test, Bonferroni t-tests etc.:– compare all cell means in interaction

• Usually lots of means:– lots of non-independent comparisons

• Often ambiguous results

• Not very informative, not very powerful

Simple main effects

• Tests across levels of one factor for each level of second factor separately.– Is there effect of density for spring?– Is there effect of density for summer?Alternatively– Is there effect of season for density = 6?– etc.

• Equivalent to series of one factor ANOVAs• Use dfResidual and MSResidual from original

factorial ANOVA

20

Season 1 17.15 119.85 < 0.001

Density 2 2.00 13.98 0.001

Interaction 2 0.85 5.91 0.016

Residual 12 0.14

Total 17

Source df MS F P

Worked example II: Low shore Siphonaria


Density 2 2.00 13.98 0.001

Season 1 17.15 119.85 <0.001

Density x Season 2 0.85 5.91 0.016Simple main effectsDensity in spring 2 0.17 1.21 0.331Density in summer 2 2.67 18.69 <0.001

Residual 12 0.14

Source df MS F P

21

0

1

2

3

4

0 10 20 30

Density

No

. eg

g m

ass

es p

er li

mp

etSpring

Summer


Simple main effects df MS F PDensity in spring 2 0.17 1.21 0.331Density in summer 2 2.67 18.69 <0.001

Mixed ModelsTraditional ANOVA approach

• At least one of the Factors is Random

• At least one of the Factors is Fixed

• Not too difficult if only one Factor is Random

• Very complex if more than one Factor is Random (realm or psuedo – F stats)

22

Palm seedlings in Peru (Losos 1995)

Mixed model

• Survivorship of palm seedlings in Peru

• Factor A - fixed– 4 successional zones

– early-seral, mid-seral, late-seral, Heliconia

• Factor B - random– 2 randomly located transects

• 5 replicate plots of seedlings within each zone-transect combination (cell)

23

Mixed model

• Age at metamorphosis of copepod larvae

• Factor A - fixed– 4 food treatments

– high food, high to low, etc.

• Factor B - random– 15 randomly chosen sibships

• 4 replicate dishes of larvae within each food-sibship combinations (cell)

Expected mean squares

Factor A fixed, B random:

• MSA

• MSB

• MSA X B

• MSResidual2

22

22

222

1

n

np

p

nqn i

24

Tests in mixed model

• H0: no effect of random interaction A*B:– F-ratio: MSAB / MSResidual

• H0: no effect of random factor B:– F-ratio: MSB / MSResidual

• H0: no effect of fixed factor A:– F-ratio: MSA / MSAB

2

22

22

222

1

n

np

p

nqn i

Palm seedlings

Source df MS F P Denom.

Zone 3 0.060 0.31 0.819 Z x T

Transect 1 0.045 3.10 0.041 Residual

Zone x transect 3 0.191 13.33 <0.001 Residual

Residual 30 0.014

25

Assumptions of factorial ANOVA

Assumptions apply to yijk within each cell

• Normality– boxplots etc.

• Homogeneity of residual variance– residual plots, variance vs mean plots etc.

• Independence

Assumptions not met?

• Robust if equal n

• Transformations important

• No suitable non-parametric (rank-based) test

26

More complex ANOVAs

• Three or more factors

• Three factor ANOVA:– 3 main effects

– 3 two-way interactions

– 1 three-way interaction

• Test three-way interaction first, then two ways, then main effects

Canola (Brassica) germination

• Factor A - seed type– 3 levels (different genetic lines & controls)

• Factor B - light– 3 levels (full, dark, shade)

• Factor C - nutrients– 2 levels (low, high)

• Residual:– 5 petri dishes (with seeds) in each of 18 cells

• Response variable:– average time to germinate

27

3 factor exampleSource df F P

Seed type (S) 2 17.23 <0.001Light (L) 2 3.04 0.063Nutrient (N) 1 0.16 0.690S x L 4 5.95 0.001S x N 2 0.05 0.954N x L 2 0.21 0.815S x N x L 4 0.43 0.787Residual 30

Complex mixed models

Source df F-test denominator

Season 1 Season x TransZone 3 Zone x TransTransect (random) 1 ResidualSeason X Zone 3 Season x Zone x TransSeason X Trans 1 ResidualZone x Trans 3 ResidualSeason X Zone x Trans 3 Residual

Residual 55

28

General scheme for testing ANOVA models with fixed factors

Source df dfdenomonator F

A dfA = p-1 dfResidual MSA/MSResidual

B dfB = q-1 dfResidual MSB/MSResidual

AB dfAB = (p-1)(q-1) dfResidual MSAB/MSResidual

Residual pq(n-1)


A dfA = p-1 dfResidual MSA/MSResidual


C dfC = r-1 dfResidual MSC/MSResidual


AC dfAC = (p-1)(r-1) dfResidual MSAC/MSResidual

BC dfBC = (q-1)(r-1) dfResidual MSBC/MSResidual

ABC dfABC = (p-1)(q-1)(r-1) dfResidual MSABC/MSResidual

Residual pqr(n-1)

A, B

A, B, C

And so on

General scheme for testing Mixed ANOVA models (fixed and random factors)


A dfA = p-1 dfAB MSA/MSAB



Residual pq(n-1)


A dfA = p-1 dfAC MSA/MSAC

B dfB = q-1 dfBC MSB/MSBC

C dfC = r-1 dfResidual MSC/MSResidual

AB dfAB = (p-1)(q-1) dfABC MSAB/MSABC

AC dfAC = (p-1)(r-1) dfResidual MSAC/MSResidual

BC dfBC = (q-1)(r-1) dfResidual MSBC/MSResidual

ABC dfABC = (p-1)(q-1)(r-1) dfResidual MSABC/MSResidual

Residual pqr(n-1)

A = fixedB = random

A = fixedB = fixedC = random

And so on

29

Effect of Blocking on Power of Test

Relationship between supplemental watering and oak seedling germination

Ho: No difference in seedling number in watered and control plots

How to set this up!!!

Options

• Fully randomized• Randomized Block - with no replication

– Account for underlying but unknown spatial variation

• Randomized block - with replication– Account for underlying but unknown spatial variation– Tradeoff between number of Blocks and replicates

within Block

• Constraints - we can only logistically handle 24 replicate plots

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Completely RandomWatered

Control

Randomized Block - no replicationWatered

Control

31

Randomized Block - MaximizeBlocks minimize replication within Blocks

Watered

Control

Randomized Block - MinimizeBlocks Maximize replication within Blocks

Watered

Control

32

Mea

n an

d S

EM

Control WaterTTT

9.0

11.6

14.2

16.8

19.4

22.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

9.0

11.6

14.2

16.8

19.4

22.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Completely randomized design24 replicated plots (12 per TTT)

Randomized Block design2 Blocks 6 reps of each TTT per Block


Randomized Block design12 Blocks (no TTT reps in each Block)

Control WaterTTT

9.0

11.6

14.2

16.8

19.4

22.0

SE

ED

ING

S

Control WaterTTT

9.0

11.6

14.2

16.8

19.4

22.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

9.0

11.6

14.2

16.8

19.4

22.0

SE

ED

ING

S

Control WaterTTT

9.0

11.6

14.2

16.8

19.4

22.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Control WaterTTT

10.0

12.2

14.4

16.6

18.8

21.0

SE

ED

ING

S

Completely randomized design24 replicated plots (12 per TTT)



Randomized Block design12 Blocks (no TTT reps in each Block)

Compare Models

Design Source df1 df2 MS F P Test Notes

CR TTT$ 1 22 121.5 4.2 0.052 MSTTT/MSResidual No BlocksCR Residual 22 28.7

RB - no rep TTT$ 1 11 121.5 92.2 0.000001 MSTTT/MSResidual 12 BlocksRB - no rep Block 11 56.2 No testRB - no rep Residual 11 1.3

RB - 2 reps TTT$ 1 5 121.5 43.4 0.001 MSTTT/MSTTT*Block 6 BlocksRB - 2 reps Block 5 11 120.9 103.6 0.0000001 MSBlock/MSResidual

RB - 2 reps TTT$*Blk 5 11 2.8 2.4 0.10 MSTTT*Block/MSResidual

RB - 2 reps Residual 11 1.2

RB - 6 reps TTT$ 1 1 121.56 11.4 0.18 MSTTT/MSTTT*Block 2 BlocksRB - 6 reps Block 1 20 450.7 52.7 0.000001 MSBlock/MSResidual

RB - 6 reps TTT$*Blk 1 20 10.7 1.2 0.28 MSTTT*Block/MSResidual

RB - 6 reps Residual 20 8.6

33

0.0000010.0000020.0000030.0000050.000007

0.000010.000020.000030.000040.00006

0.00010.00020.00030.00040.0006

0.0010.0020.0030.0050.007

0.010.020.030.050.07

0.1

1 5 11 22

DF Denominator

Tradeoff between blocking and degrees of freedom

CR

RB, 2, 6

RB, 6, 2

RB, 12, 1

P =0.05Randomized block, 2 blocks 6 reps of each treatment per block

Randomized block, 6 blocks 2 reps of each treatment per block

Randomized block, 12 blocks 1 reps of each treatment per block

Completely randomized design

RB, 2, 6

RB, 6, 2

RB, 12, 1

CR

Documents

Introduction to factorial ANOVA 2015