ST 524
NCSU - Fall 2008
Factorial Experiments
Example: Study of the effects of five cowpea varieties and method of cultivation on yield. Yield, in lb. per plot
of 1 100 morgen1, Cowpea varieties: A, B, C, D; Method of cultivation: 1, 2, 3.
Methods of Cultivation correspond to the factor “spacing in the rows” (S) with three levels: 4” (Method 1), 8”
(Method 3), and 12” (Method 2).
1. Analysis of Variance: Factorial Experiment RCBD, 5 Variety × 3 Row Spacing × 4 Blk


Fixed effects model,
Model, RCBD: yijk     k   i   j   ij   ijk

k
 0,
k

i

0,
j
 0,
j
i
  
ij
 0 ,  ijk ~ N  0,  2 
i, j
------------------------------------------------------------------------------------------------original scale yield in lb/plot
1
The GLM Procedure
Class Level Information
Class
Levels
Values
blk
4
1 2 3 4
Variety
5
A B C D E
spacing
3
4 8 12
Number of Observations Read
Number of Observations Used
60
60
------------------------------------------------------------------------------------------------original scale yield in lb/plot
2
The GLM Procedure
Dependent Variable: yield
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
17
2711.900000
159.523529
12.59
<.0001
Error
42
532.100000
12.669048
Corrected Total
59
3244.000000
Source
blk
Variety
spacing
Variety*spacing
1
R-Square
Coeff Var
Root MSE
yield Mean
0.835974
6.244492
3.559361
57.00000
DF
Type I SS
Mean Square
F Value
Pr > F
3
4
2
8
638.400000
1089.166667
109.200000
875.133333
212.800000
272.291667
54.600000
109.391667
16.80
21.49
4.31
8.63
<.0001
<.0001
0.0198
<.0001
South African unit of measure equal to about 2 acres.
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
1
ST 524
NCSU - Fall 2008
Factorial Experiments
Source
blk
Variety
spacing
Variety*spacing
DF
Type III SS
Mean Square
F Value
Pr > F
3
4
2
8
638.400000
1089.166667
109.200000
875.133333
212.800000
272.291667
54.600000
109.391667
16.80
21.49
4.31
8.63
<.0001
<.0001
0.0198
<.0001
2. Hypothesis for main effect of Variety
H o :  A   B  C   D   E  0
H1 :  i  0, for some i =A, B,
H o : 4"  8"  12"  0
3. Hypothesis for main effect of Row Spacing
,E
H1 :  j  0, for some j=1, 2, 3
4. Hypothesis for interaction effects Variety* Row Spacing
H o :   A1    A2      E 2    E 3  0
H1 :  ij  0, for some i =A, ,E; j =1,2,3
5. Since all P values (Pr > F) are lower than the significance level α = 0.05, we can reject each null
hypothesis and conclude that the effects of variety on yield is dependent on the row spacing selected
(Variety*Spacing is significant, P < 0.0001).
6. Main effect of Variety is significant, (P < 0.0001 ). There are significant differences on the response
(yield) to the distinct Varieties on average of Row Spacing.
7. Main effect of Row Spacing is significant, (P = 0.0198 < 0.05 =  There are significant differences
on the response (yield) to the distinct Row Spacing on average of Varieties.
8. Main effects of Variety and Row Spacing are less important since their interaction is significant.
Means
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

original scale yield in lb/plot
var_
Variety
spacing
mn_yield
yield
A
B
C
D
E
A
A
A
B
B
B
C
C
C
D
D
D
E
E
E
.
4
8
12
.
.
.
.
.
4
8
12
4
8
12
4
8
12
4
8
12
4
8
12
57.0000
55.3000
57.1000
58.6000
51.3333
56.1667
55.4167
57.6667
64.4167
47.5000
50.7500
55.7500
57.5000
56.7500
54.2500
53.2500
55.2500
57.7500
62.2500
58.5000
52.2500
56.0000
64.2500
73.0000
54.9831
44.1158
39.5684
81.3053
48.9697
10.3333
40.0833
23.3333
73.1742
33.6667
42.2500
57.5833
7.0000
10.2500
12.9167
46.9167
50.2500
36.2500
4.9167
3.6667
8.9167
28.6667
14.9167
32.0000
4
stderr_
yield
0.95728
1.48519
1.40656
2.01625
2.02010
0.92796
1.82764
1.39443
2.46938
2.90115
3.25000
3.79418
1.32288
1.60078
1.79699
3.42479
3.54436
3.01040
1.10868
0.95743
1.49304
2.67706
1.93111
2.82843
Analysis of Variety Main Effect
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
2
ST 524
NCSU - Fall 2008
Factorial Experiments

Analysis of Row Spacing Main Effect and Row Spacing*Variety Interaction Effect
Graphical representation of the mean response (yield)
Row Spacing
y
ij .
Variety
4”
8”
12”
A
B
C
D
E
y. j.
47.5000
50.7500
55.7500
51.333
57.5000
56.7500
54.2500
56.167
53.2500
55.2500
57.7500
55.417
62.2500
58.5000
52.2500
57.667
56.0000
64.2500
73.0000
64.417
55.300
57.100
58.600
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
yi..
y...
= 57.000
3
ST 524
NCSU - Fall 2008
Factorial Experiments
Orthogonal Contrasts – Orthogonal Polynomial Coefficients


Equally spaced quantitative treatments or levels of a quantitative factor.
Their use allows for the analysis of the independent computation of the contribution of a given power
of the independent variable (factor), X, X2, X3, . . .


For a quadratic curve, orthogonal polynomial curve, Y  Y  b11  b22 , can be analyzed using the
coefficients in table, where 1 ,  2 are the orthogonal transformation of X, X2.

Main effect is partitioned in a set of mutually orthogonal effects, each with one degree of freedom,
and associated test for the null hypothesis that the polynomial term equal 0,


H o : b1  0
H o : b2  0
Sequentially, each sum of squares is the additional contribution due to fitting a curve one degree
higher.
Similar decomposition may be used to study significant interactions between factors.
Table of coefficients – orthogonal polynomial contrasts – 1 degree of freedom
Nº of levels
Factor
Divisor
Order
1
2
3
4
5
c
2
i
i
-1
+1
1
-1
0
+1
2
2
+1
-2
+1
6
1
-3
-1
+1
+3
20
2
+1
-1
-1
+1
4
3
-1
+3
-3
+1
20
1
-2
-1
0
+1
+2
10
2
+2
-1
-2
-1
+2
14
3
-1
+2
0
-2
+1
10
4
+1
-4
+6
-4
+1
70
2
3
4
5
2
Analysis of Spacing Main Effects - Orthogonal polynomial
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
4
ST 524
NCSU - Fall 2008
Factorial Experiments
In example, factor “Spacing in the rows” (S) is a quantitative factor with three levels: 4”, 8” and 12”.

Slinear is used to analyzed whether response (yield) to increasing levels of spacing presents a linear
trend.

Sdev. linear allows us to test whether response (yield) to increasing levels of spacing is not simply

linear, but may require a higher degree polynomial.
Table of Means and Totals for each Method (Spacing) with associated coefficients for orthogonal
polynomial contrasts
c1
c2
r  cij2
c3
div
i. j
yc
i i
div
i
Means
55.3000 57.10
58.60
Totals
1106
1142
1172
Slinear
-1
0
1
40
-2
+1
120
Sdev. linear +1
2
1.650
4
-0.075
Concave curve
Number of repetitions for each mean is 4*5 = 20
Sum of squared coefficients = 2 (linear)
= 6 (dev.from linear)

Estimated value of the linear combination of means related to Slinear  C linear  and Sdev. linear  C dev.linear 



C linear 




 -1  55.30   0  57.10  1  58.60   3.30  1.65
2

C dev.linear 
2
1  55.30   2  57.10  1  58.60   0.3  0.075
4
4
Decomposition of the Sum of Squares for Spacing in SS(Slinear) and SS(Sdev. linear)
SS(Spacing) = SS(due to linear effect of Spacing) + SS(deviation from linear effect of Spacing)
SS(deviation from linear effect of Spacing) is what is remaining after fitting the linear effect of Spacing if
this term is significant, then the effect of Spacing on the response (yield) is not just linear, it may be
necessary to run a new experiment with more levels of spacing to get a better idea of the trend of the
response to increasing levels of spacing.
Working with Totals for each spacing level:
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
5
ST 524
NCSU - Fall 2008
Factorial Experiments
Cˆ

   -1 1106   0 1142  1 1172

2
SS  Slinear
2
linear
r  cij2
 4  5   -1   0   1
2
i, j

SS Sdev linear

Cˆ


dev linear
r  cij2

2
2
662
 108.9
40



108.9 + 0.3 = 109.2 = SS (Spacing)
2
62

 0.3
120
i, j
Contrast
DF
Contrast SS
Mean Square
F Value
Pr > F
1
1
108.9000000
0.3000000
108.9000000
0.3000000
8.60
0.02
0.0054
0.8784
Row Spacing linear
Row Spacing dev linear
Estimate
Dependent Variable: yield
Estimate
Standard
Error
t Value
Pr > |t|
1.65000000
-0.07500000
0.56278432
0.48738552
2.93
-0.15
0.0054
0.8784
Parameter
Row Spacing linear
Row Spacing dev linear
(-2.93)2 = 8.60
Conclusion: Linear Main effect of Row Spacing is highly significant, while Dev. From Linear is not
significant. A linear trend is adequate to represent the trend on yield response to increasing levels of Row
Spacing.
Analysis Interaction effects of Variety*Row Spacing
Variety
Spacing
A
4”
8”
12”
B
4”
8”
12”
C
4”
8”
12”
D
4”
8”
12”
E
4”
8”
12”
c11
c12
c13
c21
c22
c23
c31
c32
c33
c41
c42
c43
c51
c52
c53
means
47.50 50.75 55.75 57.50 56.75 54.25 53.25 55.25 57.75 62.25 58.50 52.25 56.00 64.25 73.00
Totals
190
-1
Slinear
Sdev. linear -1
203
223
230
227
217
213
221
231
249
234
209
224
257
292
0
2
1
-1
-1
-1
0
2
1
-1
-1
-1
0
2
1
-1
-1
-1
0
2
1
-1
-1
-1
0
2
1
-1
Contrasts Sum of Squares within each cultivar
SS  Slinear , A 
C

linear , A
r  cij2

2
  -1 190   0   203  1  223 

 4    -1   0   1
2
i, j
Contrast
Row Spacing
Row Spacing
Row Spacing
Row Spacing
Row Spacing
Row Spacing
Row Spacing
Row Spacing
Row Spacing
Row Spacing
linear in A
linear in B
linear in C
linear in D
linear in E
dev linear in
dev linear in
dev linear in
dev linear in
dev linear in
A
B
C
D
E
DF
1
1
1
1
1
1
1
1
1
1
2
2
2


Contrast SS
136.1250000
21.1250000
40.5000000
200.0000000
578.0000000
2.0416667
2.0416667
0.1666667
4.1666667
0.1666667
Tuesday October 21, 2008 Orthogonal Polynomial contrasts

332
 136.125
8
Mean Square
136.1250000
21.1250000
40.5000000
200.0000000
578.0000000
2.0416667
2.0416667
0.1666667
4.1666667
0.1666667
F Value
10.74
1.67
3.20
15.79
45.62
0.16
0.16
0.01
0.33
0.01
Pr > F
0.0021
0.2037
0.0810
0.0003
<.0001
0.6901
0.6901
0.9092
0.5694
0.9092
*
ns
ns
*
*
ns
ns
ns
ns
ns
6
ST 524
NCSU - Fall 2008
Factorial Experiments
Sum =
984.33
= (109.2+875.13) = [SS (S) + SS (V*S)]
Estimated coefficients – orthogonal polynomial contrasts
Dependent Variable: yield
Parameter
Row
Row
Row
Row
Row
Spacing
Spacing
Spacing
Spacing
Spacing
linear
linear
linear
linear
linear
in
in
in
in
in
A
B
C
D
E
Estimate
Standard
Error
t Value
Pr > |t|
4.12500000
-1.62500000
2.25000000
-5.00000000
8.50000000
1.25842400
1.25842400
1.25842400
1.25842400
1.25842400
3.28
-1.29
1.79
-3.97
6.75
0.0021
0.2037
0.0810
0.0003
<.0001
Row Spacing linear in A =

C linear , A 
 -1  47.50   0  50.75  1  55.75  8.25  4.125
2
2
Row Spacing linear in B

C linear , B 
 -1  57.50   0  56.75  1  54.25  3.25  1.625
2
The GLM Procedure
2
Dependent Variable: yield
Parameter
Row
Row
Row
Row
Row
Spacing
Spacing
Spacing
Spacing
Spacing
linear
linear
linear
linear
linear
dev
dev
dev
dev
dev
in
in
in
in
in
A
B
C
D
E
Estimate
Standard
Error
t Value
Pr > |t|
0.43750000
-0.43750000
0.12500000
-0.62500000
0.12500000
1.08982715
1.08982715
1.08982715
1.08982715
1.08982715
0.40
-0.40
0.11
-0.57
0.11
0.6901
0.6901
0.9092
0.5694
0.9092
Row Spacing Dev. from linear in A =

C dev.linear , A 
1  47.50   2  50.75  1  55.75  1.75  0.4375
4
4
Row Spacing Dev. from linear in B =

1  57.50   2  56.75  1  54.25  1.75  0.4375
C dev.linear , B 
4
4
Analysis of Simple effects

Analyze Variety effect within eah spacing level
The GLM Procedure
Least Squares Means
Variety*spacing Effect Sliced by spacing for yield
Sum of
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
7
ST 524
NCSU - Fall 2008
Factorial Experiments
spacing
4
8
12

DF
Squares
Mean Square
F Value
Pr > F
4
4
4
474.700000
387.800000
1101.800000
118.675000
96.950000
275.450000
9.37
7.65
21.74
<.0001
0.0001
<.0001
Analyze Row Spacing Effect within each Variety
The GLM Procedure
Least Squares Means
Variety*spacing Effect Sliced by Variety for yield
Variety
A
B
C
D
E
DF
Sum of
Squares
Mean Square
F Value
Pr > F
2
2
2
2
2
138.166667
23.166667
40.666667
204.166667
578.166667
69.083333
11.583333
20.333333
102.083333
289.083333
5.45
0.91
1.60
8.06
22.82
0.0078
0.4086
0.2130
0.0011
<.0001
Tuesday October 21, 2008 Orthogonal Polynomial contrasts
8