n
2
Factorial Experiment
4 Factor used to Remove Chemical
Oxygen demand from Distillery Spent
Wash
R.K. Prasad and S.N. Srivastava (2009). “Electrochemical degradation of distillery
spent wash using catalytic anode: Factorial design of experiments,” Chemical
Engineering Journal, Vol. 146, pp. 22-29.
Data Description
• Response: Y = % Chemical Oxygen Demand Removed
from Distillery Spent wash
• Factors and Levels:




A: Current Density (mA/cm2) – 14.285, 42.857
B: Dilution (%) – 10, 30
C: Time (hrs) – 2, 5
D: pH – 4, 9
• Experimental Runs: 16 – All 24 Combinations of levels
of A,B,C,D
Data – Normal Order
Run
10
9
4
15
3
5
8
6
12
1
7
14
2
11
16
13
CurrDens Dilution
14.285
10
42.857
10
14.285
30
42.857
30
14.285
10
42.857
10
14.285
30
42.857
30
14.285
10
42.857
10
14.285
30
42.857
30
14.285
10
42.857
10
14.285
30
42.857
30
Time
2
2
2
2
5
5
5
5
2
2
2
2
5
5
5
5
pH
4
4
4
4
4
4
4
4
9
9
9
9
9
9
9
9
Label
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
y
28.265
32.520
32.230
36.210
28.265
38.560
34.562
40.230
65.230
56.680
62.135
64.430
71.210
68.720
72.270
68.260
For the Label, any factor at its high level appears in lower case form.
(1) Corresponds to the case when all factors are at their low levels.
Table of Contrasts - I
• Create a Column for the intercept (I), one for each
Main Effect and each Interaction (A,…,D, AB,…,CD,
ABC,…,BCD, ABCD), and one for the response (y). If
there were multiple replicates per treatment, replace
y with the mean of those r replicates
• Create a row for each experimental run (treatment),
using the Labels from the previous slide.
• For the Intercept Column, put +1 in each row
• For all Main Effects, Put +1 if that factor was at its
high level, -1 if at its low level (Note: Books use +/-)
Table of Contrasts - II
Trt
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
y
28.265
32.520
32.230
36.210
28.265
38.560
34.562
40.230
65.230
56.680
62.135
64.430
71.210
68.720
72.270
68.260
For Interactions, multiply the coefficients in each row for the Main Effects that make
up that Interaction.
For Row 1 and Column AB: A has coefficient -1, B has -1, so AB has (-1)(-1) = +1
For Row 1 and Column ABC: (-1)(-1)(-1) = -1
For Row 1 and Column ABCD: (-1)(-1)(-1)(-1) = +1
An Interaction will have a coefficient of +1 if it has an even number of its Main Effects
at their low levels, -1 if an odd number.
Table of Contrasts - III
Trt
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
AB
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
AC
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
AD
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
BD
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
CD
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
ABC
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
ABD
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
ACD
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
BCD
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
ABCD
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
y
28.265
32.52
32.23
36.21
28.265
38.56
34.562
40.23
65.23
56.68
62.135
64.43
71.21
68.72
72.27
68.26
Create 4 Rows below this “matrix”: Contrast, Divisor, Effect, Sum of Squares
2n
Contrast   ki y i
i 1
ki  1 In EXCEL, you can take the SUMPRODUCT of each Column with y
 2n for the Intercept (I) Column
Divisor   n 1
2 for all other columns
Contrast
Effect 
Divisor
r
2
Sum of Squares  SS  n  Contrast 
Sum of Squares is not typically computed for Intercept
2
Table of Contrasts - IV
Trt
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
Divisor
Contrast
Effect
SumSq
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
16
799.777
49.986
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
8
11.443
1.430
8.184
8
20.877
2.610
27.241
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
8
8
44.377 258.093
5.547
32.262
123.082 4163.250
AB
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
8
4.423
0.553
1.223
AC
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
8
7.483
0.935
3.500
AD
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
8
-36.953
-4.619
85.345
BC
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
8
-3.743
-0.468
0.876
BD
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
8
-10.367
-1.296
6.717
CD
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
ABC
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
ABD
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
8
19.593
2.449
23.993
8
-16.717
-2.090
17.466
8
14.227
1.778
12.650
ACD
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
8
-7.973
-0.997
3.973
BCD
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
8
-4.367
-0.546
1.192
ABCD
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
y
28.265
32.52
32.23
36.21
28.265
38.56
34.562
40.23
65.23
56.68
62.135
64.43
71.21
68.72
72.27
68.26
8
-8.013
-1.002
4.013
Effect A: y A,High  y A,Low 
32.52  36.21  38.56  40.23  56.68  64.43  68.72  68.26 28.265  32.23  28.265  34.562  65.23  62.135  71.21  72.27 405.61  394.167


 1.430
8
8
8
Effect AB: y AB,High  y AB,Low 
28.265  36.21  28.265  40.23  65.23  64.43  71.21  68.26 32.52  32.23  38.56  34.562  56.68  62.135  68.72  72.27 402.1  397.677


 0.553
8
8
8
Alternative Approach:
36.21  40.23  64.43  68.26 32.23  34.562  62.135  72.27 209.13  201.197
Effect A @ B  1:


 1.98325
4
4
4
32.52  38.56  56.68  68.72 28.265  28.265  65.23  71.21 196.48  192.97
Effect A @ B  1:


 0.8775
4
4
4
1
Effect AB:
1.98325  0.8775  0.553
2
Analysis of Variance
SSA  r  2
 r 2
n 1
r  2n 
2 2 n 1
22
n 1

  y A,Low  y
 y
y
  A,Low 2 A,High

 Contrast A
•
•
Source
A
B
C
D
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
df
SS
1
8.184
1 27.241
1 123.082
1 4163.250
1
1.223
1
3.500
1 85.345
1
0.876
1
6.717
1 23.993
1 17.466
1 12.650
1
3.973
1
1.192
1
4.013
•
2


y
 y A,High
n 1 

 y A,High  y
 r  2   y A,Low  A,Low


2

 
2
2

2
  y A,High  y A,Low
 
 
2
 




2
2
 
y A,Low  y A,High
   y A,High 
 
2
 




2




2
 r  2n 1  (2)
r  2n   1
2


Contrast A  
 Effect A 


4
4  2n 1


r
1
2
2
Contrast A   11.43  8.184
n 
2
16
SSA 
r
2
Contrast A 
n 
2
Notes: Factor D (pH) has by far the largest effect on the outcome.
With all mean effects and interactions, there are no error degrees of
freedom, and no tests can be conducted
Consider dropping interactions with small sums of squares to obtain an
error term (Authors dropped: AB, AC, BC, and BCD)
Source
A
B
C
D
AD
BD
CD
ABC
ABD
ACD
ABCD
Error
df
SS
1
8.184
1
27.241
1 123.082
1 4163.250
1
85.345
1
6.717
1
23.993
1
17.466
1
12.650
1
3.973
1
4.013
4
6.790
MS
F_obs
F(.05)
P-value
8.184
4.821
7.709
0.093
27.241
16.048
7.709
0.016
123.082
72.509
7.709
0.001
4163.250 2452.601
7.709
0.000
85.345
50.278
7.709
0.002
6.717
3.957
7.709
0.118
23.993
14.134
7.709
0.020
17.466
10.289
7.709
0.033
12.650
7.452
7.709
0.052
3.973
2.341
7.709
0.201
4.013
2.364
7.709
0.199
1.697
Regression Approach
 1 if A is at High Level
Let: X 1  
Similarly defined X 2 ( B), X 3 (C ), X 4 ( D)

1
if
A
is
at
Low
Level

E Y    0  1 X 1   2 X 2  3 X 3   4 X 4  12 X 1 X 2  13 X 1 X 3  14 X 1 X 4   23 X 2 X 3   24 X 2 X 4  34 X 3 X 4 
 123 X 1 X 2 X 3  124 X 1 X 2 X 4  134 X 1 X 3 X 4   234 X 2 X 3 X 4  1234 X 1 X 2 X 3 X 4
^
Note:  0  Effect I
^
1
 1   Effect A  since Effect A  y A,High  y A,Low  2  1
2
^
Full Model:
^
Y  49.986  0.715 X 1  1.305 X 2  2.774 X 3  16.131X 4  0.276 X 1 X 2  0.468 X 1 X 3
2.310 X 1 X 4  0.234 X 2 X 3  0.648 X 2 X 4  1.225 X 3 X 4 
1.045 X 1 X 2 X 3  0.889 X 1 X 2 X 4  0.498 X 1 X 3 X 4  0.273 X 2 X 3 X 4  0.501X 1 X 2 X 3 X 4
Reduced Model (Coefficients do not change due to orthogonal design):
^
Y  49.986  0.715 X 1  1.305 X 2  2.774 X 3  16.131X 4
2.310 X 1 X 4  0.648 X 2 X 4  1.225 X 3 X 4 
1.045 X 1 X 2 X 3  0.889 X 1 X 2 X 4  0.498 X 1 X 3 X 4  0.501X 1 X 2 X 3 X 4
and so on...
Further Model Reduction (Simplification)
• When testing the effects after removing the
Interactions with the smallest effects, we find BD,
ACD, and ABCD all have P-values that are > 0.10. Now
we remove them for a simpler model.
Source
A
B
C
D
AD
CD
ABC
ABD
Error
^
df
SS
1
8.184
1
27.241
1 123.082
1 4163.250
1
85.345
1
23.993
1
17.466
1
12.650
7
21.493
MS
F_obs
F(.05)
P-value
8.184
2.665
5.591
0.147
27.241
8.872
5.591
0.021
123.082
40.086
5.591
0.000
4163.250 1355.908
5.591
0.000
85.345
27.796
5.591
0.001
23.993
7.814
5.591
0.027
17.466
5.688
5.591
0.049
12.650
4.120
5.591
0.082
3.070
This model has:
8.184   12.650
R2 
 0.9952
8.184   12.650  21.493
Y  49.986  0.715 X 1  1.305 X 2  2.774 X 3  16.131X 4  2.310 X 1 X 4  1.225 X 3 X 4  1.045 X 1 X 2 X 3  0.889 X 1 X 2 X 4
Normal Probability Plot of Factor & Interaction Effects
Under hypothesis of no main effects or interactions, estimated effects should be
approximately normally distributed with mean 0. Construct a normal probability plot of
estimated effects
Effect
std.norm
-4.619
-1.739
-2.090
-1.245
-1.296
-0.946
-1.002
-0.714
-0.997
-0.515
-0.546
-0.335
-0.468
-0.165
0.553
0.000
0.935
0.165
1.430
0.335
1.778
0.515
2.449
0.714
2.610
0.946
5.547
1.245
32.262
1.739
Normal Probability Plot of Estimated Factor
Effects and Interactions
2.000
1.500
1.000
Std. Normal Quantiles
Trt
AD
ABC
BD
ABCD
ACD
BCD
BC
AB
AC
A
ABD
CD
B
C
D
0.500
0.000
-0.500
-1.000
-1.500
Clearly, several effects
fall well away from
central line
-2.000
-10.000
-5.000
0.000
5.000
10.000
15.000
Effect
20.000
25.000
30.000
35.000
A Simple Test for Effects & Interactions
• Method described by Lenth (1989):
 Obtain s0 = 1.5*median(|Effects|)
 Compute: pseudo standard error:
PSE = median(|Effects|*Indicator(|Effect| < 2.5*s0))
 Compute Simultaneous Margin of Error:
SME = t(.05/(2*Cm),d)*PSE
where m = # of Effects, Cm=m(m-1)/2, d=m/3
s0
 Consider effect significant if |Effect| > SME
Based on this criteria, only pH main effect is
significant. When not making adjustment for
multiple tests (ME), 3 effects are significant or
very close
2.5xs0
PSE
d
t(.975;d)
ME
gamma
SME
2.145563
5.363906
1.943813
5
2.570582
4.996729
0.998293
10.14408
Estimates of Contrasts Simultaneous Margin of Error = 10.14
35
30
25
20
15
10
5
0
A
-5
-10
B
C
D
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD