Three-Factor Experiments
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Transcript Three-Factor Experiments
Three-Factor Experiments
Design selection, treatment composition, and
randomization remain the same
Each additional factor adds a layer of complexity to the
analysis
In a 3-factor experiment we estimate and test:
– 3 main effects
– 3 two-factor interactions (first order)
– 1 three-factor interaction (second order)
Same Song - Second Verse...
Construct tables of means
Complete an ANOVA table
Perform significance tests
Compute appropriate means and standard errors
Interpret the results
In General...
We have ‘a’ levels of A, ‘b’ levels of B, and ‘c’ levels of C
– Total number of plots per replication will be a x b x c
SSTot =
(Yijkl Y)
2
– Excel Spreadsheet:
= DEVSQ(Range of all observations)
– With a calculator:
= s2*(n-1) = 2*n, where n = rabc
Table of Block Means
Block
I
R1
II
R2
III
R3
Mean
Y
Table for Main Effects
Level
1
2
...
f
Factor A
A1
A2
...
Aa
Factor B
B1
B2
...
Bb
Factor C
C1
C2
...
Cc
Where A1 represents the mean of all of the treatments
involving Factor A at level 1, averaged across all of the
other factors.
Table of Means for Treatments
Treatment Means over Reps
A1B1C1
A1B1C2
A1B1C...
A1B1Cc
T111
T112
T11..
T11c
A1B2C1
A1B2C2
A1B2C...
A1B2Cc
....
AaBbCc
T121
T122
T12..
T12c
Tabc
Compute a mean over
replications for each
treatment.
Total number of treatments
= a x b x c.
First-Order Interactions
Factor B
Factor A
1
2
...
b
1
2
...
a
T11.
T21.
...
Ta1.
T12.
T22.
...
Ta2.
...
...
...
...
T1b.
T2b.
...
Tab.
Compute a table such as this for each first order interaction:
AxB
AxC
BxC
ANOVA Table (fixed model)
Source
df
Total
rabc-1
Block
r-1
SS
SSTot=
ijkl Yijkl Y
SSR=
SSA=
SSB=
rac Y
b-1
j
C
c-1
SSC=
. j..
2
Y
rbci Yi... Y
B
F
MSR=
SSR/(r-1)
FR
MSR/MSE
MSA=
FA
SSA/(a-1)
MSB=
MSA/MSE
FB
SSB/(b-1)
MSC=
MSB/MSE
FC
SSC/(c-1)
MSC/MSE
2
abcl Y...l Y
Main Effects
A
a-1
MS
rab k Y..k. Y
2
2
2
ANOVA Table Continued...
Source
df
SS
MS
F
SSA SSB
MSAB
FAB
Y SSB SSC
MSAC
FAC
MSBC
FBC
MSABC
FABC
First Order Interactions
AB
(a-1)(b-1)
SSAB=
rcij Yij.. Y
AC
(a-1)(c-1)
SSAC=
SSBC=
ra Y
2
2
rbik Yi.k. Y SSA SSC
BC
(b-1)(c-1)
jk
3-Factor Interaction
ABC (a-1)(b-1)(c-1)
. jk.
SSABC=
2
2
r ijk Yijk. Y SSA SSB SSC
SSAB SSAC SSBC
Error
(r-1)(abc-1)
SSE=
SSTot-SSR-SSA-SSB-SSC
-SSAB-SSAC-SSBC-SSABC
MSE
Standard Errors
Factor
Std Err of Mean
Std Err of Difference
A
MSE/rbc
2MSE/rbc
B
MSE/rac
2MSE/rac
C
MSE/rab
2MSE/rab
AB
MSE/rc
2MSE/rc
AC
MSE/rb
2MSE/rb
BC
MSE/ra
2MSE/ra
ABC
MSE/r
2MSE/r
Interpretation
Depends on the outcome of the F tests for main effects
and interactions
If the 3-factor (AxBxC) interaction is significant
– None of the factors are acting independently
– Summarize with 3-way table of means for each treatment
combination
If 1st order interactions are significant (and not the 3-factor
interaction)
– Neither of the main effects are independent
– Summarize with 2-way table of means for significant interactions
If Main Effects are significant (and not any of the
interactions)
– Summarize significant main effects with a 1-way table of factor
means
Example
Study the effect of three production factors:
– Variety (2)
– Phosphorus Fertilization (3)
• None, 25 kg/ha, 50 kg/ha
– Weed Control (2)
• None, Herbicide
Using RBD design in three blocks
ANOVA
Source
df
SS
MS
F
Total
Block
35
2
1936.75
270.17
135.08
5.93**
Variety (V)
Phosphorus (P)
Herbicide (W)
1
2
1
306.25
32.00
12.25
306.25
16.00
12.25
13.44**
.70
.54
VxP
VxW
PxW
2
1
2
18.67
283.36
468.67
9.33
283.36
234.33
.41
12.44**
10.29**
VxPxW
2
44.22
22.11
.97
22
501.16
22.78
Error
Means and Standard Errors
Herbicide
Variety
None
Some
V1
56.89
52.44
V2
57.11
63.89
Mean seed yield (kg/plot)
from two varieties of chickpeas with and without
herbicide
*Standard error = 1.59
Mean seed yield
(kg/plot) of chickpeas at three levels
of phosphorus
fertilization with and
without herbicide
Phosphorus
Herbicide
None 25 kg/ha 50 kg/ha
None
60.00
57.83
53.17
Some
52.50
58.67
63.33
*Standard error = 1.95
Interpretation
The effect of herbicide depended on variety
– The addition of herbicide reduced the yield for variety 1
– The yield of variety 2 was increased by the use of
herbicide
Response to added phosphorus depended on
whether or not herbicide was used
– If no herbicide, seed yield was reduced when
phosphorus was added
– However, seed yield increased when phosphorus was
added in addition to herbicide
A Picture is Worth a Thousand Words
Herbicide x Variety Interactions
V2
64
64
62
62
60
60
Yield
Yield
Phosphorus x Herbicide Interactions
58
56
58
56
54
V1
52
None
Herbicide
Some
54
52
0
25
50
Phosphorus in kg/ha
Without Herbicide
With Herbicide