Experimental Design & Analysis
Factorial ANOVA
Copyright, Gerry Quinn & Mick Keough, 1998
Please do not copy or distribute this file
without the authors’ permission
Two factor factorial ANOVA
• Two factors (2 independent 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
Quinn (1988) - fecundity of
limpets
• Factor A is season with 2 levels:
– spring, summer
• Factor B is density with 4 levels:
– 8, 15, 30, 45 per 225cm2
• n = 3 fences in each combination:
– each combination is termed a cell (8 cells)
• Dependent variable:
– fecundity (no. egg masses per limpet)
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
• Dependent variable:
– growth of Douglas fir trees seedlings
Data layout
Factor A
1
j
1
........
i
2
j
2
Factor B
1
Reps
y111
y112
yij1
yij2
y11k
yijk
Cell means y11
2
2
1
j
yij
Linear model
• yijk = m + ai + bj + (ab)ij + eijk
where
• m is overall mean
• ai is effect of factor A
• bi is effect of factor B
• (ab)ii is effect of interaction between A
and B
• eijk is unexplained variation
Worked example
Season
Spring
Density 8 15 30 45
Reps
Summer
8 15 30 45
n = 3 in each of 8 groups (cells)
p = 2 seasons, q = 4 densities
Worked example
Cell means:
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
Density
marginal 2.13 1.68
means
1.19 0.89
1.47
Grand mean
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):
– m1, m2...mi
• Factor B marginal means (pooling A):
– m1, m2...mj
• HO: no difference between marginal
means of factor A, pooling levels of B
– HO: m1 = m2 = mi = m
• HO: no main effect of factor A, pooling
levels of B (a1 = a2 = … = ai = 0)
• Example:
– No difference between season marginal
means
– No effect of season, pooling densities
Cell means
Density
Spring
8
15
30
45 Season
means
2.42 2.18 1.57 1.20
1.84 m spring
Summer 1.83 1.18 0.81 0.59
1.10 m summer
Density
means 2.13 1.68 1.19 0.89
1.47 m overall
m8
m 15 m 30 m 45
• HO: no difference between marginal
means of factor B, pooling levels of A
– HO: m1 = m2 = mj = m
• HO: no main effect of factor B, pooling
levels of A (b1 = b2 = … = bi = 0)
• Example:
– No difference between density marginal
means
– No effect of density, pooling seasons
Cell means
Density
8
15
30
45 Season
means
2.42 2.18 1.57 1.20
1.84
m spring
Summer 1.83 1.18 0.81 0.59
1.10
m summer
Density
means 2.13 1.68 1.19 0.89
1.47
m overall
Spring
m 8 = m 15 = m 30 = m 45
Interaction
• An interaction between 2 factors:
– effect of factor A is dependent on level of
factor B and vice-versa
• HO: no interaction between factor A and
factor B:
– effects of factor A and factor B are
independent of each other
– no joint effects of A & B acting together
(abij = 0)
– mij - mi - mj + m = 0
Interaction example from Underwood
(1981)
Season: Summer Autumn Winter Spring
Area:
1
2 1
2 1 2 1
2
• n = 3 plankton tows for each season/area
combination
• DV is no. plankton in each tow
• cell means (each season/area combination)
• main effect means:
•season means pooling areas
•area means pooling seasons
MEAN NUMBER PER HAUL
Interaction plot
Main effects
Cell means
Season means Area means
800
600
400
200
A1
800
600
400
200
A2
A2
A1
800
600
400
200
A2
A1
S A W S
SEASON
S A W S
SEASON
1
2
AREA
Residual variation
• Variation between replicates within each
cell
• Pooled across cells if homogeneity of
variance assumption holds
 ( yijk  y )
2
Partitioning total variation
SSTotal
SSA
SSA
SSB
SSAB
SSResidual
+ SSB
+
SSAB + SSResidual
variation between A marginal means
variation between B marginal means
variation due to interaction between A
and B
variation between replicates within
each cell
Factorial ANOVA table
Source
SS
df
MS
Factor A
SSA
p-1
Factor B
SSB
q-1
Interaction
AXB
SSAB
(p-1)(q-1)
SS A
p-1
SS B
q-1
SS AB
(p-1)(q-1)
Residual
SSResidual
pq(n-1) SS Residual
pq(n-1)
Worked example
Source
SS
Season
3.25
1
3.25
Density
5.28
3
1.76
Interaction
0.17
3
0.06
Residual
2.92
16
0.18
11.62
23
Total
df
MS
Expected mean squares
Both factors fixed:
• MSA
s2 + nqai2/p-1
• MSB
s2 + npbi2/q-1
• MSA X B
s2 + n(ab)ij2/(p-1)(q-1)
• MSResidual
s2
HO: no interaction
• If no interaction:
– HO: interaction (abij) = 0 true
• F-ratio:
– MSAB / MSResidual  1
• Compare F-ratio with F-distribution with
(a-1)(b-1) and ab(n-1) df
• Determine P-value
HO: no main effects
• If no main effect of factor A:
– HO: m1 = m2 = mi = m (ai = 0) is true
• F-ratio:
– MSA / MSResidual  1
• If no main effect of factor B:
– HO: m1 = m2 = mj = m (bj = 0) is true
• F-ratio:
– MSB / MSResidual  1
Worked example
Source
df
MS
F
P
Season
1
3.25
17.84
0.001
Density
3
1.76
9.67
0.001
Interaction
3
0.06
0.30
0.824
Residual
16
0.18
Total
23
Testing of HO’s
• Test HO 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
Interpreting interactions
• Plotting cell means
• Multiple comparison across interaction
term
• Simple main effects
• Treatment-contrast tests
• Contrast-contrast tests
No. egg masses per limpet
Interaction plot
3
2
Spring
Summer
1
0
0
20
40
60
Density
• Effect of density same for both seasons
• Difference between seasons same for all densities
• Parallel lines in cell means (interaction) plot
Worked example II
• Low shore Siphonaria
– larger limpets
• Two factors
– Season (spring and summer)
– Density (6, 12, 24 limpets per 225cm2)
• DV = no. egg masses per limpet
• n = 3 enclosures per season/density
combination
Worked example II
Source
df
MS
F
P
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
No. egg masses per limpet
Interaction plot
4
3
Spring
Summer
2
1
0
0
20
10
30
Density
• Effect of density different for each season
• Difference between seasons varies for each density
• Non-parallel lines in cell means (interaction) plot
Multiple comparison
• 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 an effect of density for winter?
– Is there an effect of density for summer?
Alternatively
– Is there an effect of season for density = 6?
– etc.
• Equivalent to series of one factor ANOVAs
• Use dfResidual and MSResidual from original 2
factor ANOVA
Treatment-contrast interaction
• Do contrasts or trends in one factor
interact with levels of other factor?
• Does the density contrast [6 vs the
average of 12 & 24] interact with
season?
Worked example II
Source
df
MS
F
P
Density
2
2.00
13.98
0.001
2
2
0.17
2.67
1.21
18.69
Simple main effects
Density in winter
Density in summer
Season
1
Density x Season
2
0.85
5.91
1
1.53
10.66
12
0.14
Treatment-contrast inter.
6 vs (avg 12 & 24) x season
Residual
0.331
<0.001
17.15 119.85 <0.001
0.016
0.007
Mixed model
Factor A fixed, B random:
• MSA
s2 + nsab2 + nqai2/p-1
• MSB
s2 + npsb2
• MSA X B
s2 + nsab2
• MSResidual
s2
Tests in mixed model
• HO: no effect of random interaction A*B:
– F-ratio: MSAB / MSResidual
• HO: no effect of random factor B:
– F-ratio: MSB / MSResidual
• HO: no effect of fixed factor A:
– F-ratio: MSA / MSAB
Assumptions of factorial ANOVA
Assumptions apply to DV within each cell
• Normality
– boxplots etc.
• Homogeneity of residual variance
– residual plots, variance vs mean plots etc.
• Independence
Factorial ANOVAs in the
literature
Copyright, Gerry Quinn & Mick Keough, 1998
Please do not copy or distribute this file
without the authors’ permission
Barbeau et al. (1994)
• J. Exp. Mar. Biol. Ecol. 180:103-136
• Experiment on consumption rate of
crabs feeding on scallops
• Two factors:
– Crab size (3 levels - small, medium, large)
– Scallop size (3 levels - small, medium,
large)
• Dependent variable:
– consumption rate (number of scallops per
predator per day) of crabs (Cancer
irroratus)
• Sample size:
– n = 2 - 4 replicate aquaria in each of 9 cells
Source
Crab size
Scallop size
Interaction
Residual
df
MS
F
2
2
4
18
123.0
220.3
68.0
42.6
2.89
5.18
1.60
P
0.082
0.017
0.218
Unplanned multiple comparison on main effect of
scallop size (pooling crab sizes):
S = M <
L
McIntosh et al. (1994)
• Ecology 75 : 2078-2090
• Effects of predatory fish on drifting
behaviour of insects (mayflies) in streams
• Factors:
– Predator cues (3 levels - no fish, galaxids,
trout)
– Time (2 levels - day, night)
• Dependent variable:
– number of mayflies drifitng
• Sample size:
– n=3 stream channels in each cell
Source
df
Predator cues 2
Time
1
Interaction
2
Residual
18
F
P
1.45
131.88
1.05
0.261
<0.001
0.369
Mayflies
20
15
Day
Night
10
5
0
no fish
galaxids
trout
Assumptions not met?
• Robust if equal n
• Transformations important
• No suitable non-parametric (rankbased) test