EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
1, 2, 3
Department of Industrial Administration, Tokyo University of Science
2641 Yamazaki, Noda, Chiba, 278-8510, JAPAN
1
j7407608@ed.noda.tus.ac.jp
3
suzuki@ia.noda.tus.ac.jp
ABSTRACT
It is effective to use the orthogonal arrays in fractional factorial designs. The linear graph was developed by
Taguchi to allocate the factor to the orthogonal arrays easily. The linear graphs are prepared for each orthogonal
array beforehand. In three-level orthogonal arrays, the degree of freedom for a two factor interaction is four. Two
factor interactions will appear across two columns. Due to this reason, the number of factors that can be
considered is limited. Actually, there are only two prepared linear graphs for L27. The confounding happens if we
try to design the experiment other than the two prepared linear graphs. In this paper, we consider the
experimental design that cannot design in the prepared linear graphs for L27. We investigate the experimental
design to which a partial confounding is allowed. We propose the practical design of experiment to allocate more
factors, and evaluate it by simulation.
Keywords: Orthogonal array, Linear graph, Confounding
INTRODUCTION
The experimental design is widely used in various fields including industry, medicine and psychology. The fractional factorial designs are
effective when the number of factors considered is large and when it is difficult to experiment all the combinations. It is effective to use
the orthogonal arrays in fractional factorial designs. The linear graph was developed by Taguchi to allocate the factor to the orthogonal
arrays systematically and easily. The linear graph is prepared for each orthogonal array beforehand. The orthogonal arrays which will be
used changes according to the number of factors and the number of levels considered. Two-level orthogonal arrays such as L8 and L16 are
used when factors of two levels are considered. Three-level orthogonal arrays such as L9 and L27 are used when factors of three levels are
considered. Their use and subsequent analysis are almost the same, but there is great difference in degrees of freedom. In three-level
orthogonal arrays, the degree of freedom a two factor interaction is four. Two factor interactions will appear across two columns. Due to
this reason, the number of factors that can be considered is limited. Actually, there are only two prepared linear graphs for L27. The
confounding happens if we try to design the experiment other than the two prepared linear graphs. In this paper, In this paper, we
consider the experimental design that cannot design in the prepared linear graphs for L27. We investigate the experimental design to
1
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
which a partial confounding is allowed. We propose the practical design of experiment to allocate more factors, and evaluate it by
simulation.
ORTHOGONAL ARRAYS EXPERIMENT
A lot of factors are often taken up at the same time at the early stage of the problem solving. The experiment frequency increases in the
factorial experiment on which it experiments by all the level combinations when the number of factors taken up in the experiment
increases. Then, the obtaining necessary information can be done by an experiment frequency that is less than the factorial experiment by
providing experimental conditions by using the table that is called an orthogonal array.
L27 orthogonal array
In the factorial experiment, it is necessary to experiment by combining the levels of the factor taken up all. The total experiment
frequency increases rapidly when the number of factors increases, and the inconvenience in the experiment is caused. For this case, the
orthogonal array experiment that is the method of not the experiment on all the level combinations but conducting only a part of the
experiment is useful. When the levels of all the factor taken up are three, the three-level orthogonal array is used. Table 1 shows the L27
orthogonal array.
Table 1. L27 orthogonal array
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Component
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 2 2 2 2 2 2 2 2 2
1 1 1 1 3 3 3 3 3 3 3 3 3
1 2 2 2 1 1 1 2 2 2 3 3 3
1 2 2 2 2 2 2 3 3 3 1 1 1
1 2 2 2 3 3 3 1 1 1 2 2 2
1 3 3 3 1 1 1 3 3 3 2 2 2
1 3 3 3 2 2 2 1 1 1 3 3 3
1 3 3 3 3 3 3 2 2 2 1 1 1
2 1 2 3 1 2 3 1 2 3 1 2 3
2 1 2 3 2 3 1 2 3 1 2 3 1
2 1 2 3 3 1 2 3 1 2 3 1 2
2 2 3 1 1 2 3 2 3 1 3 1 2
2 2 3 1 2 3 1 3 1 2 1 2 3
2 2 3 1 3 1 2 1 2 3 2 3 1
2 3 1 2 1 2 3 3 1 2 2 3 1
2 3 1 2 2 3 1 1 2 3 3 1 2
2 3 1 2 3 1 2 2 3 1 1 2 3
3 1 3 2 1 3 2 1 3 2 1 3 2
3 1 3 2 2 1 3 2 1 3 2 1 3
3 1 3 2 3 2 1 3 2 1 3 2 1
3 2 1 3 1 3 2 2 1 3 3 2 1
3 2 1 3 2 1 3 3 2 1 1 3 2
3 2 1 3 3 2 1 1 3 2 2 1 3
3 3 2 1 1 3 2 3 2 1 2 1 3
3 3 2 1 2 1 3 1 3 2 3 2 1
3 3 2 1 3 2 1 2 1 3 1 3 2
a
a a
a a
a a
a a
b b b2
b b b2 b b2 b
c c c2 c c c2 c2 c c2
The feature of the three-level orthogonal array that can be read from Table 1 is as follows.
1.
Every column has the figure of “1”, “2” and “3” the same number of times.
2.
There is a combination of (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2) and (3,3) the same number of times when two
2
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
arbitrary columns are chosen.
The design of experiment using the three-level orthogonal array is basically the same as the case of two-level orthogonal array. The
analysis procedure like the allocation of factor, the analysis of variance, the estimation, and the forecast, etc. is almost the same. However,
we should take care that the degree of freedom of two-factor interaction A×B becomes AB = 2  2 = 4 because the degree of freedom of
three levels factors A and B are A = 3 – 1 = 2 and B = 3 – 1 = 2 respectively. In other words, the point "the interaction appears across
two columns though the main effect is allocated to one column" is a point remarkably different from the case of two-level orthogonal
arrays.

Linear graph
The relation between the factor and the interaction becomes complex when number of considered factors and interactions increase, and
allocating factors becomes difficult to orthogonal array by using component letters. For this case, the method of using a figure that is
called a linear graph is convenient. A linear graph is figure where the column of the orthogonal array is expressed by vertices and edges,
and the relation between the factor and the two-factor interaction is shown. This linear graph of each orthogonal array is prepared
beforehand. As for the linear graph of the L27 orthogonal array, two kinds of Figure 1 are prepared.
1
2
3,4
3,4
6,7
6,7
5
1
2
9
10
12
9,10
5
8,11
8
12,13
13
11
Figure 1. Prepared linear graph
The linear graph is made from the following rules.

The vertex shows one column

The edge shows the one column for two-level orthogonal array, and shows the column any two for the three-level
orthogonal array

The edge where the vertex and the vertex are connected shows the column to which the two-factor interaction between
columns that those two vertices show appears.

The figure written in the vertex and the edge shows the column index.
Moreover, the procedure of the allocation that uses the linear graph is as follows.
1.
The factor and the two-factor interaction considered by the experiment are shown in the vertex and edge
respectively, and the relation between the factor and the two-factor interaction is expressed in figure. The linear
graph at this time is called “necessary (required) linear graph”.
2.
The necessary (required) linear graph is built into “prepared linear graph”. As a result, it is decided that the
allocation of the factor is an allocation of the two-factor interaction.
3.
A factor not related to the two-factor interaction is allocated to the column of the vertex or the edge that has
become vacant.
4.
The column of a remaining vertex or edge allocates the error.
3
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
MOTIVATION OF THIS RESERCH
In the three-level orthogonal array, because the degree of freedom of one column is allocated by two, the degree of freedom of the
two-factor interaction becomes four. Therefore, the two-factor interaction will be allocated in two specific columns. Therefore, it may be
difficult to allocate so that the confounding does not happen to factorial effects. Actually, there are only two prepared linear graphs for
L27. The confounding always happens if we try to design the experiment other than the two prepared linear graphs.
Here, we think about the simple allocation shown in Figure 2. This linear graph cannot be built into the prepared linear graphs.
Therefore, in the L27 orthogonal array, if we try to allocate factors to the linear graph as shown in Figure 2, the confounding always
happens and the design becomes impossible. The coverage of the L27 orthogonal array where two-factor interactions are allocated is
narrow. Also, because the number of factors that we can consider compared with frequency of experiment is not many, this design may
be regarded as an fairly inefficient experimental design.
Figure 2. One example of linear graph that cannot be allocated
Then, when paying attention to “two-factor interaction appears across two columns”, if the confounding has not happened to the other
column though the confounding has happened to one of columns, it seems that some effect can be estimated by using the column in
which confounding has not happened.
In this paper, about the case where we cannot design by the prepared linear graphs for L27, we investigate the experimental design to
which a partial confounding is allowed. We propose the practical design of experiment to allocate more factors, and evaluate it by
simulation.
If this idea can be applied, the design that factor cannot be allocated shown in Figure 3 becomes possible. And applicable scope in the
L27 orthogonal array may be expands.
Figure 3. Example of linear graphs those become possible allocating
TWO-FACTOR INTERACTIONS IN THREE-LEVEL ORTHOGONAL ARRAYS
In three-level orthogonal arrays, it is a common knowledge that the two-factor interaction appears across two specific columns. However,
“At what rate is the sum of squares of two-factor interaction distributed to two columns?” is not clear. Therefore, there might be a
necessity for clarifying this before advancing this research.
This chapter considers it from two approaches of a theoretical viewpoint and a practical viewpoint. L9 orthogonal array is used because
4
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
it is basically the same even if becoming L27 orthogonal array.
Theoretical approach
Factor A and B of three levels are allocated to the 1st column and the 2nd column of the L9 orthogonal array respectively as shown in
Table 2. From the linear graph of Figure 4, Two-factor interaction A×B appears to the 3rd column and the 4th column. It is assumed that
the data of a, b, c, d, e, f, g, h and i was obtained as a result of experimenting nine times in total according to the orthogonal array.
Table 2. L9 orthogonal array that allocated two factors
factor
No.
1
2
3
4
5
6
7
8
9
A
[1]
1
1
1
2
2
2
3
3
3
B
[2]
1
2
3
1
2
3
1
2
3
A×B
[3]
1
2
3
2
3
1
3
1
2
A×B
[4]
1
2
3
3
1
2
2
3
1
data
a
b
c
d
e
f
g
h
i
1
2
3,4
Figure 4. Linear graph of L9 orthogonal array
From this data, the sum of squares of the 3rd column and the 4th column is calculated to the following.
1
S[3]  (2a 2  2ab  2b 2  2ac  2bc  2c 2  2ad  4bd  2cd  2d 2  2ae  2be 
9
4ce  2de  2e 2  4af  2bf  2cf  2df  2ef  2 f 2  2ag  2bg 
(1)
4cg  2dg  4eg  2 fg  2 g 2  4ah  2bh  2ch  2dh  2eh  4 fh 
2 gh  2h 2 2ai  4bi  2ci  4di  2ei  2 fi  2 gi  2hi  2i 2 )
S[ 4] 
1
(2a 2  2ab  2b 2  2ac  2bc  2c 2  2ad  2bd  4cd  2d 2  4ae  2be 
9
2ce  2de  2e 2  2af  4bf  2cf  2df  2ef  2 f 2  2ag  4bg 
(2)
2cg  2dg  2eg  4 fg  2 g 2  2ah  2bh  4ch  4dh  2eh  2 fh 
2 gh  2h 2 4ai  2bi  2ci  2di  4ei  2 fi  2 gi  2hi  2i 2 )
From eq. (1) and eq. (2), the following two equations can be derived.
S [ 3]  S [ 4 ] 
S [ 3]
S [ 3]  S [ 4 ]
2
(bd  cd  ae  ce  af  bf  bg  cg  eg
3
 fg  ah  ch  dh  fh  ai  bi  di  ei)
(3)

{3(c  e  g ) 2  3(a  f  h) 2  3(b  d  i) 2  (a  b  c  d  e  f  g  h  i) 2 }
{3(c  e  g )  3(b  f  g ) 2  3(c  d  h) 2  3(a  f  h) 2  3(b  d  i) 2  3(a  e  i) 2  2(a  b  c  d  e  f  g  h  i ) 2 }
2
(4)
The following relation is approved.
S [ 3 ]  S[ 4 ]
and
S[ 3 ]
S[ 3 ]  S [ 4 ]
5
 constant
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
From this relation, it is not necessarily the case that two-factor interaction is distributed in two columns at an equal rate.
Practical approach
From the result of foregoing section, the rate of the distribution of two-factor interaction effect to two columns fluctuates with changes
on the experimental data. Then, we think about the typical situation that can be assumed for two-factor interaction. The factor of three
levels that is continuous and is equal intervals is considered. And we are assumed that situation in which two-factor interaction exists or
not between those levels. Under such a situation, we investigate between the change in those conditions and the relationship that
distribution of two-factor interaction effect to two columns.
Vision
Before the survey, we should think about the relation between three level values of the considered factor. Because, from the difference of
level value considered factor, there is a possibility that the difference is in the distribution of two-factor interaction on the 3rd column and
the 4th column. Then, we think separately by the difference of the level considered. Generally, main effects can approximate by linear
expression or quadratic expression. Factors A and B are assumed three-level factor consecutive and equal distance. For this case, four
patterns are thought as for how to take three level values (See to Figure 5). We think about the combination by 16 kinds of because factor
A and B are four patterns.
1
2
3
1
2
3
1
2
3
1
2
3
Figure 5. Relations between levels
The flow of the investigation is shown as follows.
1.
In each situation from Figure 5, data where two-factor interaction does not exist is made.
2.
To think about the situation where two-factor interaction exists between each level, the data of 1 is doubled.
Here, there is possibility that the difference comes out in the distribution of two-factor interaction effect to two columns
depending on the number of doubling data. Consequently, if the situation division is done to the number of doubling
data (changed data), it becomes combinations of 511.
6
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
Therefore, we investigate combinations of 8176 in total ( 16 511).
Result of the survey
From 8176 investigations, the following three patterns were able to be found.
1.
The sum of squares is distributed equally to two columns.
2.
All the sum of squares is distributed in a column of two columns.
3.
The sum of squares in not distributed equally to two columns.
“1” and “2” with an interesting feature are considered.
Pattern 1
There were 880 cases where the sum of squares is distributed equally to two columns. It was able to divide into four groups according to
the feature in graphs.
B1
B2
A1
A1
A2
A2
A3
A3
B3
B1
Graphs where two lines or more are parallel among three lines
B1
B2
B2
B3
Graphs that are ass straight lines though three lines are not parallel
A1
A1
A2
A2
A3
A3
B3
B1
Graphs where the right half or the left half of three lines is parallel
B2
B3
Other graphs
Figure 6. Graphs where the sum of squares is distributed equally to two columns
Pattern 2
There were 112 cases where all the sum of squares is distributed to the 3rd column or the 4th column. A part of seen feature is considered
to those graphs.
7
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
B3
B1
A3
A3
B2
A2
B2
A2
B1
B3
A1
A1
Graph of shape like sofa
Graph of shape like hat
Figure 7. Graphs where all the sum of squares is distributed to the 3rd column
B1
B3
B3
A3
A3
B2
B2
A2
A1
A2
B1
Graph of shape like chair
B2
A1
B3
A2
B1
A1
A3
Graph of shape like “Z”
Graph of shape like paper crane
Figure 8. Graphs where all the sum of squares is distributed to the 4th column
PROPOSED METHOD
Figure 9 shows the partial confounding linear graph. In Figure 9, the figure to which star is attached shows the confounding column. In a
word, the confounding has happened to two factor alternate action AB and CD by the third column. Both the fourth column and the
13th column to remain clear, which means each column is not confounded.
C5
A×B
3*
・
13
3*
・
4
C ×D
1A
D9
2B
Figure 9. Partial confounding linear graph
We propose 3 ways of calculating sum of squares in ANOVA.
[a] The column in which the confounding has happened is used as-is. In a word, the 3rd column and the 4th column for A×B
and the 3rd column and the 13th column for CD are used respectively as sum of squares. By the way, degree of freedom
is assumed to be 4.
8
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
[b] The column in which the confounding has happened is excluded. In a word, the 4th column for A×B and the 13th column
for C×D are used respectively as sum of squares. By the way, degree of freedom is assumed to be 2.
[c] The square harmony of the row in which the confounding has happened is adjusted to 1/2, and distributes in two rows. In a
word, the 4th column and half of the 3rd column for A×B and the 13th column and half of the 3rd column for C×D are
used respectively as sum of squares. By the way, degree of freedom is assumed to be 3.
Hereafter, three action methods to the sum of squares will be written by proposed method [a] or [b] or [c] respectively.
Procedure of simulation
1.
To correspond to the level combination of nine kinds, the population means are configured (See to Table 3).
2.
Each factorial effect is calculated from two way table. And, these are assumed to be a theoretical value.
3.
27 data is derived based on the statistical model shown in eq. (5) and design matrix (See to Table 4).
4.
Based on the derived data, the simulation is carried out 10000 times by the method of analyzing three patterns, i.e. [a],
[b] and [c].
Table 3. Two way table
B1
a
d
g
A1
A2
A3
B2
b
e
h
B3
c
f
i
y ijklm     i   j   k   l    ij    kl   ijklm
(5)
 ijklm ~ N 0,  2 
Table 4. Example of design
factor
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
A
[1]
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
B
[2]
1
1
1
2
2
2
3
3
3
1
1
1
2
2
2
3
3
3
1
1
1
2
2
2
3
3
3
C
[5]
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
9
D
[9]
1
2
3
2
3
1
3
1
2
2
3
1
3
1
2
1
2
3
3
1
2
1
2
3
2
3
1
data
171
98
201
51
77
104
108
239
151
129
155
181
85
215
128
125
51
154
29
160
72
79
5
108
103
129
156
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
In the simulation, the following assumptions are set.
[i]
Expected value in one specific combination of levels is greatly different from the linear relation (See Figure 10).
[ii]
Location of the peak for one factor with the quadratic relation is affected by the other factor (See Figure 11).
[iii]
Location height and width of the peak for one factor with the quadratic is affected by the other factor (see Figure 12).
y
B1
B2
B1
B3
B2
B3
A2
A1
A3
Figure 10. Assumption [i]
Figure 11. Assumption [ii]
B1
B2
B3
Figure 12. Assumption [iii]
We evaluate results of simulation to think about distribution of expectation variance. Furthermore, we compare proposed methods and
general “method”. Here, “general” method means the cases when no confounding occur.
SIMULATION RESULTS AND DISCUSSION
The simulation result of each assumption is shown respectively. Subsequently, we discuss those results.
Assumption [i]
In the beginning, we consider the result of the simulation on assumption [i]. Table 5 is one example of the simulation result. The average
and standard deviation of expectation of variance are shown each two-factor interaction and proposed method. The theoretical value is
written in the last row on the table.
Table 5. One of simulation results when assumption[i]
A×B
C×D
interaction
proposed methods
[a]
[b]
[c]
[a]
[b]
[c]
average
59.86432 14.87831 44.86898 69.54371 34.23711 57.77484
standard deviation 8.936742
6.5026 7.014479 9.709138 9.631566 8.557056
theoretical value 14.41906 14.41906 14.41906 38.37565 38.37565 38.37565
10
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
The accuracy of proposed method [b] can be perceived to be good from Table 5. The distribution of the expectation of variance is
compared in the difference between proposed method and general method.
500
450
two-factor interaction: A×B
proposed method: [b]
n = 10000
average = 14.88
standard deviation = 6.50
theoretical value = 14.42
400
350
frequency
300
250
200
150
100
50
9.
4
11
.8
14
.3
16
.7
19
.2
21
.6
24
.1
26
.5
29
.0
31
.4
33
.9
36
.3
38
.8
41
.2
43
.7
46
.1
48
.6
6.
9
4.
5
2.
0
-0
.4
0
expectation of variance
500
450
two-factor interaction: A×B
general method
n = 10000
average = 14.42
standard deviation = 4.48
theoretical value = 14.42
400
frequency
350
300
250
200
150
100
50
9.
4
11
.8
14
.3
16
.7
19
.2
21
.6
24
.1
26
.5
29
.0
31
.4
33
.9
36
.3
38
.8
41
.2
43
.7
46
.1
48
.6
6.
9
4.
5
2.
0
-0
.4
0
expectation of variance
Figure 13. Comparison between proposed method and general method about AB in assumption [i]
11
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
450
400
two-factor interaction: C×D
proposed method: [b]
n = 10000
average = 34.24
standard deviation = 9.63
theoretical value = 38.38
350
frequency
300
250
200
150
100
50
7.
3
10
.8
14
.2
17
.7
21
.2
24
.7
28
.2
31
.7
35
.2
38
.7
42
.2
45
.7
49
.2
52
.7
56
.1
59
.6
63
.1
66
.6
70
.1
73
.6
77
.1
0
expectation of variance
450
400
two-factor interaction: C×D
general method
n = 10000
average = 38.30
standard deviation = 7.24
theoretical value = 38.38
350
frequency
300
250
200
150
100
50
7.
3
10
.8
14
.2
17
.7
21
.2
24
.7
28
.2
31
.7
35
.2
38
.7
42
.2
45
.7
49
.2
52
.7
56
.1
59
.6
63
.1
66
.6
70
.1
73
.6
77
.1
0
expectation of variance
Figure 14. Comparison between proposed method and general method about CD in assumption [i]
Assumption [ii]
Next, we consider the result of the simulation on assumption [ii]. Table 6 is one example of the simulation results. The view in the table
is the same as Table 5.
Table 6. One of simulation results when assumption [ii]
A×B
C×D
interaction
proposed methods
[a]
[b]
[c]
[a]
[b]
[c]
average
57.4127 100.8054 71.87694 44.23939 74.4588 54.31253
standard deviation 8.897399 16.61214 11.28259 7.723863 14.13491 9.645327
theoretical value 116.8491 116.8491 116.8491 70.96354 70.96354 70.96354
The accuracy of proposed method [b] can be perceived to be good from Table 6. The distribution of the expectation of variance is
compared in the difference between proposed method and general method.
12
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
500
450
400
two-factor interaction: A×B
proposed method: [b]
n = 10000
average = 100.81
standard deviation = 16.61
theoretical value = 116.85
frequency
350
300
250
200
150
100
50
49
.2
55
.6
62
.0
68
.4
74
.9
81
.3
87
.7
94
.1
10
0.
6
10
7.
0
11
3.
4
11
9.
8
12
6.
3
13
2.
7
13
9.
1
14
5.
5
15
2.
0
15
8.
4
16
4.
8
17
1.
2
17
7.
7
0
expectation of variance
500
450
two-factor interaction: A×B
general method
n = 10000
average = 116.74
standard deviation = 12.58
theoretical value = 116.85
400
frequency
350
300
250
200
150
100
50
49
.2
55
.6
62
.0
68
.4
74
.9
81
.3
87
.7
94
.1
10
0.
6
10
7.
0
11
3.
4
11
9.
8
12
6.
3
13
2.
7
13
9.
1
14
5.
5
15
2.
0
15
8.
4
16
4.
8
17
1.
2
17
7.
7
0
expectation of variance
Figure 15. Comparison between proposed method and general method about AB in assumption [ii]
450
400
two-factor interaction: C×D
proposed method: [b]
n = 10000
average = 74.46
standard deviation = 14.14
theoretical value = 70.96
frequency
350
300
250
200
150
100
50
30
.8
36
.0
41
.1
46
.2
51
.3
56
.5
61
.6
66
.7
71
.8
77
.0
82
.1
87
.2
92
.3
97
.5
10
2.
6
10
7.
7
11
2.
8
11
8.
0
12
3.
1
12
8.
2
13
3.
3
0
expectation of variance
450
400
two-factor interaction: C×D
general method
n = 10000
average = 70.92
standard deviation = 9.84
theoretical value = 70.96
350
frequency
300
250
200
150
100
50
30
.8
36
.0
41
.1
46
.2
51
.3
56
.5
61
.6
66
.7
71
.8
77
.0
82
.1
87
.2
92
.3
97
.5
10
2.
6
10
7.
7
11
2.
8
11
8.
0
12
3.
1
12
8.
2
13
3.
3
0
expectation of variance
Figure 16. Comparison between proposed method and general method about CD in assumption [ii]
13
Syuhei Okada1, Yasuhiro Itoh2, Tomomichi Suzuki3
Assumption [iii]
Finally, we consider the result of the simulation on assumption [iii]. It differed from assumption [i] and [ii], the result came out various
patterns. The following four tables show the part.
Table 7. Simulation result 1 when assumption [iii]
A×B
C×D
interaction
proposed methods
[a]
[b]
[c]
[a]
[b]
[c]
average
81.43182 8.807125 57.22359 101.0725 48.08855 83.4112
standard deviation 10.53986 5.024544 7.614161 11.70653 11.45278 10.22662
theoretical value 5.932884 5.932884 5.932884 92.89509 92.89509 92.89509
Table 8. Simulation result 2 when assumption [iii]
A×B
C×D
interaction
proposed methods
[a]
[b]
[c]
[a]
[b]
[c]
average
43.5979 30.43377 39.20986 36.96178 17.16153 30.3617
standard deviation 7.697881 9.136014 7.345312 7.222255 7.03476 6.332047
theoretical value 45.76141 45.76141 45.76141 47.71001 47.71001 47.71001
Table 9. Simulation result 3 when assumption [iii]
A×B
C×D
interaction
proposed methods
[a]
[b]
[c]
[a]
[b]
[c]
average
71.04239 129.6875 90.59077 8.010358 3.623449 6.548055
standard deviation 9.774784 18.6415 12.57809 3.457904 3.451917 3.054701
theoretical value 84.88985 84.88985 84.88985 5.830469 5.830469 5.830469
Table 10. Simulation result 4 when assumption [iii]
A×B
C×D
interaction
proposed methods
[a]
[b]
[c]
[a]
[b]
[c]
average
83.95075 0.397627 56.09971 89.50047 11.49707 63.49933
standard deviation 10.6663 1.771825 7.196521 11.04718 5.752556 8.111556
theoretical value 40.79969 40.79969 40.79969 58.08946 58.08946 58.08946
When these tables are seen, it is clear that the presumption of the effect doesn't go well if we always use the same proposal technique.
Discussion
We consider about assumption [i] and [ii]. From Figures 13, 14, 15, and 16, the proposed method may be possible for estimation of
two-factor interaction at some level though dispersion is broad measurably.
Subsequently, we consider about assumption [iii]. A big difference is caused for the simulation results by situations and proposed
methods unlike assumption [i] or [ii] from Table 7, 8, 9 and 10. Consequently, it might be difficult to use the proposed method like
assumption [iii].
In fact, the sum of squares is distributed almost equally to two columns at assumption [i] and [ii]. Due to this reason, the estimation
went well by using proposed method [b]. On the other hand, because the distribution of the sum of squares to two columns was not
constant, it became a result that the estimation did not go well for assumption [iii].
CONCLUSION
We investigate the experimental design to which a partial confounding is allowed when it is not possible to design the experiment using
the prepared linear graph for L27. We set three typical patterns of two-factor interaction as assumption, and evaluated each of three
14
EXPERIMENTAL DESIGN TO ALLOCATE MORE FACTORS TO L27
proposed methods by the simulation.
Under the assumption [i] and [ii], there is a possibility that the estimation of two-factor interaction goes well. But, under the
assumption [iii], the difference is caused in the result of each proposed methods, and the estimation of two-factor interaction may fail. In
the future, we should consider other patterns.
REFERENCES
Y. Ojima, T. Suzuki and S. Yasui (2001), An Alternative Expression of the Fractional Factorial Designs for Two-level and Three-level Factors, Frontiers in
Statistical Quality Control, 309-316, 2004
C. F. Jeff Wu and Michael Hamada (2000), Experiments – Planning, Analysis, and Parameter Design Optimization, John Wiley & Sons, New York
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