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trường Đại Học Dược Khoa Sài Gòn
1974-1975.    Bài 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, graeco-latin 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/QA-QC/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 Plackett-Burman, Box-Behnken, Taguchi, Response Surface Modeling are available for this purpose. In experiments with qualitive factors or with constraints on the factor settings, D-optimal 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 cost-effective 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:

L1 = C1 Y1+ C2 Y2+ ..............+ Cn Ln

L2 = 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 (C1 C'1+ C2 C'2+ ..............+ Cn 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=Lf.

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 = 23 = 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 2f, 2 levels, f factors or variables, we can calculate:

* f main effects * 1  f-factor interaction effect

Fractional Factorial Design:

To test 5 factors at 2 levels, the number of runs in a full factorial design would be only Lf = 25 = 32 and it is achievable. If the number of factors increases, for example to 8, we would have to perform 28 = 128 runs, or if the number of levels increases, for example to 5, we would have to perform 35 = 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 2f-pof 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,                     one-eight fraction           8 blocks

The following table compares the number of runs between full and fractional-factorial design:

Full                               Fractional

Lf                 Runs           Model                   Blocks        Runs:

24                16               L8                2                 24/2   =       24-1    = 23

215               32768                   L16               2048           215/211=       215-11  = 24

34                81               L9                9                 34/3   =       34-2    = 32

313               1594323     L27               59049                   313/310=       313-10  = 33

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 many-variables-at-a-time 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, graeco-latin squares, hyper graeco-latin 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

 A B C B C A C A B

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.

GRAECO-LATIN 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 GRAECO-LATIN 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 odd-magic square, make supplemental grids to the n-by-n 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 3-by-3 magic square:

 3 2 6 2 7 6 1 5 9 = 9 5 1 4 8 4 3 8 7

3-by-3

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).

Even-magic squares require more trial and error than odd ones. In the 4-by-4, 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

4-by-4

MAGIC SQUARE  II

The sum of integers in any row, column, diagonal of an n-by-n magic square equals to n(1+n2)/2.  The sum in any four diametrically opposite cells amounts to 2(1+n2). The latter condition defines orthogonality of a magic square. Using a computer program, the number of distinct 4-by-4 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: Design-Ease, Echip, RS/Discover and RS/Explore, Statistical Analysis System SAS etc....

REFERENCES:

1) Box, G.E.P., Draper, N.R. (1987): Empirical Model-Building 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 McGraw-Hill, Inc, NY

5)   Kurosaka, R.T. (1985): Magic Squares - Byte, 10:383-388

6)  Reiner, B.S. (1981): Magic Squares and Matrices, The Mathematical Gazette, 81: 250-252

7)   Sonneborn III, H. (1988):  Magic Squares and Textile Designs,  Access, 7: 10-16

8)   Ross, P. (1988): Taguchi Techniques for Quality Engineering, McGraw-Hill, Inc.  NY

9)  Taguchi, G. (1987): System of Experimental Design, Vol 1, American Supplier Institute, Inc,   Dearborn, MI  Giáo Sư Tô Đồng
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1974-1975.

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