M09_Factorial.pptx
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Agronomy Trials
Usually interested in the factors of production:
– When to plant?
– What seeding rate?
– Fertilizer? What kind?
– Irrigation? When? How much?
– When should we harvest?
Interactions of Treatment Factors
Could consider one factor at a time
– Hold all other factors constant
– This is ok if the factors act independently
But often factors are not independent of one
another
Examples:
– Plant growth habit and plant density
– Crop maturity group and response to fertilizer or
planting date
– Breed of animal and levels of a nutritional supplement
– Others?
Interactions
Consider 3 varieties at four rates of nitrogen
V1
V2
V3
Yield
V1
V1
V2
V2
V3
V3
20 40 60 80
No interaction
Relative yield
of varieties is
the same at all
fertilizer levels
20 40 60 80
20 40 60 80
Noncrossover
Interactions
Crossover
Interactions
Magnitude of
differences among
varieties depends
on fertilizer level
Ranks of varieties
depend on fertilizer
level
Interactions – numerical example
Effect of two levels of phosphorous and
potassium on crop yield
No interaction
P0
P1
Mean
K0
10
18
14
K1
14
22
18
Mean
12
20
(22-14)-(18-10) = 0
Positive interaction
P0
P1
Mean
K0
10
18
14
K1
12
26
19
Mean
11
22
(26-12)-(18-10) = +6
Negative interaction
P0
P1
Mean
K0
10
18
14
K1
16
14
15
Mean
13
16
(14-16)-(18-10) = -10
Main effects are determined from the marginal means
Simple effects refer to differences among treatment means
at a single level of another factor
Factorial Experiments
If there are interactions, we should be able to measure and
test them.
– We cannot do this if we vary only one factor at a time
We can combine two or more factors at two or more levels
of each factor
– Each level of every factor occurs together with each level of every
other factor
– Total number of treatments = the product of the levels of each factor
This has to do with the selection of treatments
– Can be used in any design - CRD, RBD, Latin Square - etc.
– “Designs” generally refer to the layout of replications or blocks in an
experiment
– A “factorial” refers to the treatment combinations
Advantages and Disadvantages
Advantages - IF the factors are independent
– Results can be described in terms of the main effects
– Hidden replication - the other factors become
replications of the main effects
Disadvantages
– As the number of factors increase, the experiment
becomes very large
– Can be difficult to interpret when there are
interactions
Uses for Factorial Experiments
When you are charting new ground and you
want to discover which factors are important and
which are not
When you want to study the relationship among
a number of factors
When you want to be able to make
recommendations over a wide range of
conditions
How to set up a Factorial Experiment
The Field Plan
– Choose an appropriate experimental design
– Make sure treatments include combinations of all factors at
all levels
– Set up randomization appropriate to the chosen design
Data Analysis
–
–
–
–
–
Construct tables of means and deviations
Complete an ANOVA table
Perform significance tests
Compute appropriate means and standard errors
Interpret the analysis and report the results
Two-Factor Experiments
Four spacings at two nitrogen levels (2x4=8
treatments) in three blocks
Block
I
II
III
Tables of Means
Spacing
Nitrogen
Mean
Block
Mean
T11
T12
T13
T14
A1.
T21
T22
T23
T24
A2.
B.1
B.2
B.3
B.4
Y..
I
R1
II
R2
III
R3
Mean
Y..
ANOVA for a Two-Factor Experiment
(fixed model)
Source
df
Total
rab-1
Block
r-1
SSR
A
a-1
SSA
B
b-1
SSB
AB
SS
MS
F
MSR=
SSR/(r-1)
MSA=
SSA/(a-1)
MSB=
SSB/(b-1)
MSAB=
SSAB/(a-1)(b-1)
MSE=
SSE/(r-1)(ab-1)
FR=
MSR/MSE
FA=
MSA/MSE
FB=
MSB/MSE
FAB=
MSAB/MSE
SSTot
(a-1)(b-1) SSAB
Error (r-1)(ab-1) SSE=
SSTot-SSR-SSA
-SSB-SSAB
Note: F tests may be different if any of the factors are random effects
Definition formulae
SSTot i j k Yijk Y
SSR abk Y..k Y
SSA rbi Yi.. Y
2
SStreatment = SSA + SSB + SSAB
2
SSB ra j Y.j. Y
2
2
SSAB r i j Yij. Yi.. Y.j. Y
2
Means and Standard Errors
Factor
A
Factor
B
Treatment
AB
Standard Error
MSE/rb
MSE/ra
MSE/r
Std Err Diff
2MSE/rb
2MSE/ra
2MSE/r
t statistic
(Y1 Y2 ) / StdErrDiff
Interpretation
If the AB interaction is significant:
– the main effects may have no meaning whether or not
they test significant
– summarize in a two-way table of means for the
various AB combinations
If the AB interaction is not significant:
– test the independent factors for significance
– summarize in a one-way table of means for the
significant main effects
Interactions
V1
No interaction
•
Main effects
for varieties
Tests for main
effects are meaningful
because differences are
constant across all levels
of factor B
Avg for V1
V2
Avg for V2
20 40 60 80
Interaction
•
•
Tests for main effects may
be misleading
In this case the test would
show no differences between
varieties, when in fact their
response to factor B is very
different
V1
V2
Avg for V1
Avg for V2
20 40 60 80
Factor B
Factorial Example
To study the effect of row spacing and phosphate on
the yield of bush beans
– 3 spacings: 45 cm, 90 cm, 135 cm
– 2 phosphate levels: 0 and 25 kg/ha
S2P1
60
S1P1
65
S3P2
66
S3P1
59
S1P2
56
S2P2
62
S1P2
45
S3P1
55
S3P2
57
S1P1
58
S2P2
50
S2P1
59
S1P1
55
S3P1
51
S1P2
43
S2P1
54
S2P2
45
S3P2
50
Tables of Means
Treatment Means
Spacing
Phosphate
S1
S2
S3
Mean
P1
59.3 57.7 55.0
57.3
P2
48.0 52.3 57.7
52.7
Mean
53.7 55.0 56.3
55.0
Block Means
Block
Mean
I
II
III
61.3 54.0 49.7
Mean
55.0
ANOVA
Source
df
SS
Total
17
752.00
Block
2
417.33
208.67
31.00**
Spacing
2
21.33
10.67
1.58
Phosphate
1
98.00
98.00
14.56**
SXP
2
148.00
74.00
11.00**
Error
10
67.33
6.73
** Significant at the 1% level.
CV = 4.7%
StdErr Spacing Mean = 1.059
StdErr Phosphate Mean = 0.865
StdErr Treatment (SxP) Mean = 1.498
MS
F
Report of Statistical Analysis
Spacing
Phosphate
45 cm
90 cm
135 cm
None
59.33
57.67
55.00
25 kg/ha
48.00
52.33
57.67
Yield response depends on whether or not phosphate was
supplied
If no phosphate - yield decreases as spacing increases
If phosphate is added - yield increases as spacing increases
Blocking was effective
Presentation of Results