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19741975.
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2 :
SOME
FEATURES OF THE DESIGN OF EXPERIMENTS
Dong To, D. Sc.
ABSTRACT:
The School of Design of
Experiments (DOE) results from theoretical works in the
early thirties of the twentieth century by a british
statistician, Ronald A. Fischer. It can be defined as the
science of acquisition and evaluation of information by
experiments. Among significant contributors to the area are
Frank Yates, George E. P. Box, William G. Cochran (1, 2, 3).
A practical approach based on cost/benefit analysis and
signal/noise ratio has been developed after the second world
war by a japanese engineer, Genechi Taguchi (4). Hundred
thousand of engineers receive training every year in Taguchi
method, giving rapid and dominant success achievements to
Japan. In USA, applications of experimental design in
various industries to improve product performance save many
millions of dollars per year.
In DOE, the details of an
experiment must be planned in advance to gain as much useful
information as possible. Randomization is a critical
process to eliminate various sources of error. Latin
squares, graecolatin squares are used to control one, two
or three sources of variability in assays. Complete
randomization of variables in experiments could be achieved
using factorial design, but randomization using magic
squares is very simple due to the high symmetry of the
"orthogonal" magic square. The later eliminates the
possibility of errors due to the complexities of the
classical "blocking" technique or the "linear graph" method
of Taguchi. In fact, the random integers of a magic square
correlate strongly to the Yates' standard order.
STEPS OF AN EXPERIMENT:
Planning:
The
DOE plays an enormous role in encouraging teamwork for
process/product development and requires many steps. Members
of various areas of Sales/Marketing/QAQC/R&D of a company
should participate in the planning of experiment. The list
of "effects" and "responses" as measurable quality
characteristics of the process/product should be made. Then,
the key quality characteristics/specifications must be
identified with a margin of acceptable variability. To keep
the design as simple as possible, the number of causes,
"factors" or "variables" influencing the responses should be
limited to the most important ones. In many events, only a
small percentage (20%) of the factors could contribute to
the majority (80%) of the effects, according to the Pareto
distribution. Therefore, the choice of key factors and their
"levels" to be tested depends on theoretical knowledge and
practical experience.
Selection of Experimental Design:
This
involves sample size, number of replicates, definition of
factors or system inputs, definition of the responses or
system outputs, etc....
Full
factorial designs require many runs to estimate the main
effects and all interactions between factors. Fractional
factorial designs require far fewer runs than full factorial
designs, but can only estimate the main effects with none or
all interactions. Several software programs using various
design types such as PlackettBurman, BoxBehnken, Taguchi,
Response Surface Modeling are available for this purpose. In
experiments with qualitive factors or with constraints on
the factor settings, Doptimal designs are very useful (1).
Running the experiment:
It
is necessary to avoid errors on the procedures, for example
perform the experiment carefully according to an inner array
to evaluate the "control factors". Issues to be addressed
are how many times (replications), what test order
(randomization) should each experimental condition be
performed. Replication can be arranged in an outer array to
estimate the "noise factors" due to technician competence,
to material/equipment variation and to operation difference.
A data collection should be developed and used. The cost of
running the experiment should be calculated.
Organizing/analyzing the data:
In
some screening experiments, Yates' algorithm or simple
graphical method can be useful to obtain reliable
conclusions. Otherwise, statistical methods given as
software packages should be used to organize and analyze the
data. Hypothesis testing, analysis of variance and signal to
noise ratios can be readily calculated by these computer
programs for an objective evaluation.
Evaluation of results:
Good
technical knowledge and common sense are required in this
step. According to Taguchi, the cost to society is least
when a product is closest to its desired mean value, and
that as it deviates from its specifications, the cost to
society increases in a quadratic function. Therefore, the
cost/benefit analysis is necessary to develop a quality
product in a costeffective way.
CONDITIONS OF ORTHOGONALITY:
Orthogonality provides conditions to minimize the variances
of the regression coefficients of the fitted models and to
obtain reliable factorial effects. Experiments arranged in
orthogonal equations, matrices, arrays lead to a balance of
experimental factors, an accurate identification and an
efficient evaluation of factorial effects.
Equations/Matrices:
The
two following linear equations:
L_{1
}= C_{1} Y_{1}+ C_{2} Y_{2}+
..............+ C_{n}
L_{n}
L_{2
}= C'_{1} Y'_{1}+ C'_{2} Y'_{2}+
..............+ C'_{n}
Y'_{n}
are
said orthogonal when the sum of the product of the
corresponding coefficients is zero, e.g.:
3
(C_{1} C'_{1}+ C_{2} C'_{2}+
..............+ C_{n}
C'_{n })= 0
Arrays:
When
running a full factorial design, the number of experiments n
representing every possible combination of factors is the
number of level L of each factor raised to the power of the
total number of factors f, n=L^{f}.
For
example, in a design testing three factors (f=3) of an
enzymatic reaction: pH, concentration of the substrate,
temperature and at two levels (L=2): low,  or 1 and high, +
or 2, the number of runs = 2^{3 }= 8 can be arranged
according to the following orthogonal array:
Run #
pH Conc Temp
Standard order
1
1 1 1
2
2 1 1
3
1 2 1
4
2 2 1
5
1 1 2
6
2 1 2
7
1 2 2
8
2 2 2
Note
that the frequency of appearance of levels 1, 2 and of their
combinations 11, 12, 21, 22 is the same in all columns.
The
data results of replicates of this array can be analyzed to
calculate the effects of pH, concentration and temperature
as well as interactions of those factors on the performance
of the enzymatic reaction. Therefore, the optimum conditions
can be selected.
Full Factorial Design:
In
the preceding example of full factorial design 2^{f},
2 levels, f factors or variables, we can calculate:
* f
main effects
* 1 ffactor interaction
effect
Fractional Factorial Design:
To
test 5 factors at 2 levels, the number of runs in a full
factorial design would be only L^{f} = 2^{5 }
= 32 and it is achievable. If the number of factors
increases, for example to 8, we would have to perform 2^{8
}= 128 runs, or if the number of levels increases, for
example to 5, we would have to perform 3^{5 }= 243
runs. Therefore, it is almost impossible to perform
experiments covering all possible combinations. In the real
world, although the number of factors that must be tested is
large, some of them do not interact or are negligible, and
it is possible to use fractional factorial design to reduce
the number of experiments to an acceptable range. Many
techniques are used in the blocking of full factorial design
to create a fractional factorial design. For example, one
can use a full matrix to run a fractional 2^{fp}_{R
}of equal or greater number of factors: p is the number
of blocks, R is the resolution defining the blocking:
p=1,
half
fraction 2 blocks
p=2,
quarter fraction 4 blocks
p=3,
oneeight fraction 8 blocks
The
following table compares the number of runs between full and
fractionalfactorial design:
Full
Fractional
L^{f }
Runs Model
Blocks Runs:
2^{4 }
16 L_{8}
2 24/2 = 2^{41} = 2^{3}
2^{15 }
32768 L_{16}
2048 2^{15}/2^{11}= 2^{1511}
= 2^{4}
3^{4 }
81 L_{9}
9 3^{4}/3 = 3^{42}
= 3^{2}
3^{13 }
1594323 L_{27}
59049 3^{13}/3^{10}=
3^{1310} = 3^{3}
A
schematic example:
Randomization of experimental conditions:
To
randomize the order of the individual runs to be performed
is relatively simple and will not be described here. On the
contrary, the equal assignment of experimental conditions to
balance the effects of uncontrolled variables, the
performing of manyvariablesatatime strategy and the
pairing of combinatory analyses and statistical mathematics
are fundamental tasks of the design scientist. In the
following, we will assess only the role of squares in the
DOE.
The
use of latin squares, graecolatin squares, hyper
graecolatin squares in the design in order to control one,
two or three sources of variables is classical. The
condition of orthogonality can be defined as the appearance
only once in a row/column of any factors or treatments A, B,
C...
",$,(...
I, II, III... etc.
LATIN SQUARES
The
number of simple orthogonal latin squares is limited to two
squares 3x3, three squares 4x4, four squares 5x5, one square
6x6, two squares 7x7 and two squares 8x8.
GRAECOLATIN SQUARES
Aα 
Bß 
Cτ 
Bτ 
Cα 
Aß 
Cß 
Aτ 
Bα 
They
come from superimposition of two 3x3 designs. The
orthogonality is provided when there is appearance only once
of any combination of the two factors.
HYPER GRAECOLATIN
SQUARES
IAα 
IIBß 
IIICτ 
IVDδ 
IIIBδ 
IVAτ 
IDß 
IICα 
IVCß 
IIIDα 
IIAδ 
IBτ 
IIDτ 
ICδ 
IVBα 
IIIAß 
They
result from superimposition of three 4x4 designs. The
orthogonality is observed when there is appearance once of
any combination of three factors: latin character: A, B, C,
D, greek alphabet:
α,
ß,
τ,
δ
and
roman numeral: I, II, III, IV
MAGIC SQUARES IN THE DOE:
A
magic square is an array of consecutive numbers arranged in
a square grid so that the sum of every row, column, diagonal
add up to the same amount. Its construction can be found in
literature (5, 6, 7).
Squares with an odd number
of elements are not hard to make, and random integers are
found paired in diametrically opposite cells. To construct
an oddmagic square, make supplemental grids to the nbyn
square, arrange integers in sequence, and then move the
integers from supplemental grids to the symmetrical grids
inside the square area. See, for example, the 3by3 magic
square:




3 

















2 

6 








2 
7 
6 




1 

5 

9 



= 



9 
5 
1 





4 

8 








4 
3 
8 






7 














3by3
MAGIC SQUARE I
Two other general
construction techniques were found in literature. Reiner's
method consists of two cyclic permutations perpendicular to
each other followed by a final diagonalization process (6).
Kurosaka's program uses an elegant algorithm move path (5).
Evenmagic squares require
more trial and error than odd ones. In the 4by4, place the
numbers from 1 through 16 from left to right and top to
bottom only in the cells lying on the two main diagonals.
Then start backward from 1 through 16 from right to left and
bottom to top in the remaining cells (5):
1 


4 
% 

15 
14 

' 
1 
15 
14 
4 

6 
7 

12 


9 
12 
6 
7 
9 

10 
11 

8 


5 
8 
10 
11 
5 
13 


16 

3 
2 

13 
3 
2 
16 
4by4
MAGIC SQUARE II
The sum of integers in any
row, column, diagonal of an nbyn magic square equals to
n(1+n^{2})/2. The sum in any four diametrically
opposite cells amounts to 2(1+n^{2}). The latter
condition defines orthogonality of a magic square. Using a
computer program, the number of distinct 4by4 magic
squares generated is 880 (7). Most of them, however, are
probably not orthogonal.
Randomization using magic
squares in design of experiments can be very simple due to
the high symmetry of "orthogonal" magic squares. We found
that the random integers from a magic square row or column
correlate strongly to the Yates' standard order and provides
a more powerful randomization of control factors than other
blocking techniques. This can be demonstrated in the
following three examples using full and fractional factorial
designs.
Software:
DesignEase, Echip,
RS/Discover and RS/Explore, Statistical Analysis System SAS
etc....
REFERENCES:
1) Box, G.E.P., Draper, N.R.
(1987): Empirical ModelBuilding and Response Surfaces,
John Wiley & Sons, NY.
2) Box, G.E.P., Hunter, W.G.,
Hunter, J.S. (1978): Statistics for Experimenters,
Introduction to Design, Data Analysis and Model Building,
John Wiley & Sons, NY.
3) Montgomery, D.C. (1997):
Design and Analysis of Experiments Fourth Edition, John Wiley
& Sons, NY.
4) Ross, P. (1988):
Taguchi Techniques for Quality Engineering McGrawHill,
Inc, NY
5) Kurosaka, R.T. (1985):
Magic Squares  Byte, 10:383388
6) Reiner, B.S. (1981):
Magic Squares and Matrices, The Mathematical Gazette,
81: 250252
7) Sonneborn III, H.
(1988): Magic Squares and Textile Designs, Access,
7: 1016
8) Ross, P. (1988):
Taguchi Techniques for Quality Engineering, McGrawHill,
Inc. NY
9) Taguchi, G. (1987):
System of Experimental Design, Vol 1, American Supplier
Institute, Inc, Dearborn, MI
Giáo Sư
Tô
Đồng
Khoa trưởng
trường Đại Học Dược Khoa Sài
Gòn
19741975.
www.ninhhoa.com 