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1.0 Factorial Experiment Design by Block ...... 3 1.1 Factorial Experiment in Incomplete Block ...... 3 1.2 Factorial Experiment with Two Blocks ...... 3 1.3 Factorial Experiment with Four Blocks ...... 5 Example 1 ...... 6 2.0 Fractional Factorial Experiment ...... 8 2.1 Half Duplicate Type of One Half Fractional Factorial Design ...... 9 2.2 Quarter Duplicate Type of One Half Fractional Factorial Design .... 11 2.3 Designing Fractional Factorial Experiment ...... 13 Example 2 ...... 14 Example 3 ...... 15
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Figure 1: Experimental design of 23 factorial design...... 4 Figure 2: Table showing “-“ and “+” of defined contrast ABC ...... 4 Figure 3: Design table of 24 factorial experiments with defining contrast ACD ... 5 Figure 4: Generalize interaction of 24 factorial experiment ...... 5 Figure 5: Group treatment of 24 factorial experiment ...... 6 Figure 6: Design table and blocks of 24 factorial experiments using AB and CD as defining contrasts ...... 6 Figure 7: Results of 23-1 two block experimental design ...... 6 Figure 8: Experimental block and confounded effect of 23-1 experiment ...... 7 Figure 9: Eight corner cube used to determine the block of 23-1 experiment ...... 7 Figure 10: Experimental results of two blocks for 23-1 experiment ...... 8 Figure 11: 24 factorial design showing ABCD interaction factor ...... 9 Figure 12: Design table of half duplicate blocks of 24 factorial experiments using ABCD as defining contrast ...... 10 Figure 13: Design table of half duplicate block 2 of 24 factorial experiments shown in Fig. 12 ...... 11 Figure 14: Design table and blocks of 25 factorial experiments using ABD and ACE as defining contrasts ...... 12 Figure 15: Contrasts for selected fractional factorial designs ...... 14 Figure 16: Basic 23 design for example 2 ...... 14 Figure 17: 27-4 fractional factorial design for example 2 ...... 15 Figure 18: Test factors and limits for hardness of a powdered metal component . 15 Figure 19: 27-4 fractional factorial experiment arrangement for example 3 ...... 15 Figure 20: Experimental results of the 27-4 factional factorial experiment for example 3 ...... 16 Figure 21: ANOVA results of the factors on hardness of a powdered metal component ...... 17
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1.0 Factorial Experiment Design by Block
Very often one needs to eliminate the influence of extraneous factors when running an experiment. One can do it by block method. What one’s concern is one factor in the presence of one or more unwanted factors? For example, one predicted a shift will occur while an experiment is being carried out. This can happen when one has to change to a new batch of raw material mid-way throughout the experiment due to insufficient material or limited blender capacity. Thus, the objective is to eliminate this factor influencing the data analysis.
1.1 Factorial Experiment in Incomplete Block
Let’s use a 23 factorial design to illustrate how blocking is being designed. In order to make all eight experiments in 23 2-level full factorial design, eight experiments are to be conducted under same conditions, which is as homogeneous as possible. It requires batches of raw material to be used are sufficient for completing all eight experiments. If it requires changing to new batch of material, all the eight experiments will not have “identical” material. In this case, the 23 design can arranged in two blocks of four experiments each to neutralize the effect of possible blend difference. One block uses the old batch of material and the other block uses new batch of material. It is equivalent to perform two 23-1 factorial experiment.
The disadvantage with such an experimental set-up is that certain effects are completely confounded or mixed with the blocks. As the result of blocking in which the number of effects confounded, it depends on the number of blocks.
1.2 Factorial Experiment with Two Blocks
One effect is confounded in an experiment with two blocks. Usually the highest order interaction is selected to be confounded. Thus, the three-factor interaction effect is confounded in a 23 factorial design with two blocks. In this scenario, only the main effects and two-factor interactions can be studied. The method of distributing the experimental combinations between the blocks for a 23 factorial design is shown as follows.
1. Define the effect to be confounded called the defining contrast. In this case, the logical defining contrast is the three-factor interaction ABC because ABC is the highest interaction.
2. Write all the 23 combinations in a table with “-” representing low level and “+” representing high level. The experimental combinations for this factorial design are given in Fig. 3.28.
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3. All the combinations that have the sign “-” in column ABC in Fig. 1grouped into one block, whereas the other combinations that have the sign “+” form the second block shown in Fig. 2.
4. Perform the experiments using blended material 1 for the experiments in block 1, while blended material 2 is to be used for experiments of block 2.
Experiment Factor i A B C ABC 1 - - - - 2 - - + + 3 - + - + 4 - + + - 5 + - - + 6 + - + - 7 + + - - 8 + + + + Figure 1: Experimental design of 23 factorial design
Block 1 for ABC equal to “+” Block 2 for ABC equal to “-” A B C A B C - - + - - - - + - - + + + - - + - + + + + + + - Figure 2: Table showing “-“ and “+” of defined contrast ABC
Let’s look at how to divide the experimental combinations in a 24 factorial experiment into two blocks using ACD as the defining contrast.
The experimental design is shown in Fig. 3. “*” indicates the selected experimental combination for “-” and “+” blocks.
Experiment Main Factor and Defining Contrast Block 1 Block 2 i A B C D ACD “-” “+” 1 - - - - - * 2 - - - + + * 3 - - + - + * 4 - - + + - * 5 - + - - - * 6 - + - + + * 7 - + + - + * 8 - + + + - * 9 + - - - + * 10 + - - + - *
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11 + - + - - * 12 + - + + + * 13 + + - - + * 14 + + - + - * 15 + + + - - * 16 + + + + + * Figure 3: Design table of 24 factorial experiments with defining contrast ACD
1.3 Factorial Experiment with Four Blocks
If the treatment combinations of a 2k factorial experiment are to be divided into four incomplete blocks then the experimenter can choose any two defining contrasts i.e. those effects that will be confounded with the blocks. A third effect, called the generalized interaction of the two defining contrasts, is automatically confounded with the blocks. Thus, a total of three effects will be confounded with blocks in an experiment with four incomplete blocks.
Let’s look at the procedure to divide a 24 factorial experiment into four incomplete blocks.
1. The experimenter needs to choose two defining contrasts and two effects that are to be confounded. Supposing the experimenter chooses AB and CD as the defining contrasts.
2. The third effect, which is the generalized interaction that will be confounded by multiplying both the defining contrasts and choosing the letters with odd exponent only. In this case, ABxCD = ABCD is the generalized interaction, because each of the letter A, B, C, and D has an exponent of one. More examples defining contrast and generalized interactions of 24 factorials are given in Fig. 4.
3. Group the treatment combinations into four blocks based on the signs in the defining contrasts selected is shown in Fig. 5. In this case, the table design and blocks are shown in Fig. 6.
4. The experimental observations corresponding to the treatment combinations in each block should be collected under identical conditions.
Defining Contrast Generalized Interaction AB ABC C ABD ABC CD BCD AB ACD Figure 4: Generalize interaction of 24 factorial experiment
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AB CD Block - - 1 - + 2 + - 3 + + 4 Figure 5: Group treatment of 24 factorial experiment
Main Factor Contrast Block A B C D AB CD 1 2 3 4 - - - - + + * - - - + + - * - - + - + - * - - + + + + * - + - - - + * - + - + - - * - + + - - - * - + + + - + * + - - - - + * + - - + - - * + - + - - - * + - + + - + * + + - - + + * + + - + + - * + + + - + - * + + + + + + * Figure 6: Design table and blocks of 24 factorial experiments using AB and CD as defining contrasts
Example 1 The results of 23-1 two blocks experimental design are shown in Fig. 7. Experiment 1, 4, 6, and 7 use old batch of material while, experiment 2, 3, 5, and 8 use new batch of material. Determine the significance of the factors using effect method.
Factor # Results A B C 1 - 1 - 1 - 1 34 2 - 1 - 1 + 1 62 3 - 1 + 1 - 1 43 4 - 1 + 1 + 1 58 5 + 1 - 1 - 1 56 6 + 1 - 1 + 1 51 7 + 1 + 1 - 1 58 8 + 1 + 1 + 1 54 Figure 7: Results of 23-1 two block experimental design
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Solution Using the block method mentioned the text which is using effect to be confounded and naturally it is the highest interaction ABC. Thus, the experimental block are shown in Fig. 8.
Factor Confounded Effect # Block Results A B C ABC 1 - 1 - 1 - 1 - 1 1 34 2 - 1 - 1 + 1 + 1 2 62 3 - 1 + 1 - 1 + 1 2 43 4 - 1 + 1 + 1 - 1 1 58 5 + 1 - 1 - 1 + 1 2 56 6 + 1 - 1 + 1 - 1 1 51 7 + 1 + 1 - 1 - 1 1 58 8 + 1 + 1 + 1 + 1 2 54 Figure 8: Experimental block and confounded effect of 23-1 experiment
Alternatively, the blocks can be determined from the eight corner cube shown in Fig. 9. Experiment 1, 4, 6, and 7, which are the white corner are designated for block 1, while experiment 2, 3, 5, and 8, which are the black corner are designated for block 2.
Figure 9: Eight corner cube used to determine the block of 23-1 experiment
The response of two blocks are shown in Fig. 10.
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Factor # Results A B C 1 - 1 - 1 - 1 34 4 - 1 + 1 + 1 58 6 + 1 - 1 + 1 51 7 + 1 + 1 - 1 58 (a) Experimental results of block 1
Factor # Results A B C 2 - 1 - 1 + 1 62 3 - 1 + 1 - 1 43 5 + 1 - 1 - 1 56 8 + 1 + 1 + 1 54 (b) Experimental results of block 2 Figure 10: Experimental results of two blocks for 23-1 experiment
Using block 1 results, the effect due to factor A is (51 + 58)/2 - (34 + 58)/2 = 8.5 The effect due factor B is (58 +58)/2 - (34 + 51)/2 = 15.5. The effect due to factor C is (58 + 51)/2 - (58 + 34)/2 = 8.5.
Using block 2 results, the effect due to factor A is (56 + 54)/2 - (62 + 43)/2 = 2.5 The effect due factor B is (43 +54)/2 - (62 + 56)/2 = - 10.6. The effect due to factor C is (62 + 54)/2 - (53 + 54)/2 = 4.5.
Both blocks show that factor B has significant effective, while the significant effect of factor A and C are about the same.
2.0 Fractional Factorial Experiment
According to Glossary & Tables for Statistical Quality Control published by The American Society of Quality Control ASQC 1983, it defines fractional factorial design as “A factorial experiment in which only an adequately chosen fraction of the treatment combinations required for the complete factorial experiment is selected to be run”.
The 2k factorial experiment can become quite large and involve large resource if k value is large. In many experimental situations, certain higher order interactions are assumed to negligible or even though they are not negligible. It would be a waste of experimental effort to use the complete factorial experiment. Thus, when k is large, the experimenter can make use of a fractional factorial experiment whereby only one half, one fourth, or even one eighth of the total factorial experimental design is actually carried out. It is desired that the chosen fractional factorial designs experiments have the desirable properties of being both balanced and orthogonal.
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Alternative for k value larger than five, Plackett-Burman design is also a better choice. In 1946, R. L. Plackett and J. P. Burman published their famous paper entitled “The Design of Optimal Multifactorial Experiments” which described the construction of very economical design with the run number a multiple of four instead of a power of two. Plackett-Burman design is a very efficient screening design when only main effects are the interested factors.
In fractional factorial experiment, there is confounded effect where the main factors are used to estimate another main factor from the estimate of the interaction effect between two or more main factors. This would mean confounding lead to the loss of ability to estimate some effects and/or interactions.
2.1 Half Duplicate Type of One Half Fractional Factorial Design
The construction of a half duplicate design is same as the allocation of a 2k 2-level factorial experiment into two blocks. Firstly, a defining contrast is selected to be confounded then the two blocks are constructed with either one of them can be selected as the design to be carried out the experiment.
Let’s consider a 24 factorial experimental design as shown in Fig. 11 showing four factor ABCD interaction.
Experiment Factor i A B C D ABCD 1 - - - - + 2 - - - + - 3 - - + - - 4 - - + + + 5 - + - - - 6 - + - + + 7 - + + - + 8 - + + + - 9 + - - - - 10 + - - + + 11 + - + - + 12 + - + + - 13 + + - - + 14 + + - + - 15 + + + - - 16 + + + + + Note that column ABCD is obtaining by multiplying the sign of column A, B, C, and D Figure 11: 24 factorial design showing ABCD interaction factor
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If one wishes to use a half duplicate design with the chosen defining contrast ABCD then based on the 24 factorial design showing ABCD interaction factor shown in Fig. 11, the two block experimental designs can be formulated and is shown in Fig. 12.
Block 1 for ABCD equal to “-” Block 2 for ABCD equal to “+” A B C D A B C D - - - + ------+ - - - + + - + - - - + - + - + + + - + + - + - - - + - - + + - + + + - + - + + - + + + - - + + + - + + + + Figure 12: Design table of half duplicate blocks of 24 factorial experiments using ABCD as defining contrast
From Fig. 12, either block can be selected for experiment. If one selects block 2 then experimental data needs to be collected following experimental combinations shown Fig. 13 that contains eight combinations, with all possible main factors and interactions in a 24 full factorial experimental design. Even though there are two or more duplications like AD and BC, it allows us to calculate an explicit sum of square for error with no increase in the number of sum of square due to main factors or interactions. The number of sum of square with the above data is 8 - 1 = 7. The total number of possible effects i.e. main factors and their interactions in a 24 experiment is 15, out of which interaction ABCD is not present in block 2, because all the combinations in this block have the same sign “+”. This leaves out 14 effects that are present in the experiment, which means that each of the seven sum of squares is shared by two effects. It can be seen in Fig. 3.40 that there are seven pairs of effects i.e. main factors and interactions such that the effects in each pair have the same “−” and “+” signs and the same sum of squares. All the pairs are A&BCD, B&ACD, C&ABD, D&ABC, AB&CD AC&BD, and AD&BC. The effects in a pair are called aliases. The aliases in each group can be obtained by deleting the letters with an even exponent from the product of the effects i.e. main factor or interaction and the defining contrast. For example, the alias of A is AxABCD = A2BCD = BCD. The aliases in this one half fractional factorial design are (A + BCD), (B + ACD), (C + ABD), (D + ABC), (AB + CD), (AC + BD), and (AD + BC).
In summary, in a one half fractional factorial design, the sum of squares of the defining contrast cannot be calculated. In addition, there are exactly two effects, which are the main factors and/or interactions are in each alias group. If the test statistic obtained from the sum of squares of an alias group is significant, one cannot determine which one of the members of that group is the significant factor without supplementary statistical evidence. However, fractional factorial designs have their
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Main Factor Interaction Factor A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD - - - - + + + + + + - - - - + - - + + + - - - - + + + - - + - + - + - + - - + - + - + - + - + + - - - + + - - - + + - + + - - + - - + + - - + - - + + + - + - - + - - + - - + - + + + + - - + - - - - + - - + + + + + + + + + + + + + + + + + + Figure 13: Design table of half duplicate block 2 of 24 factorial experiments shown in Fig. 12
2.2 Quarter Duplicate Type of One Half Fractional Factorial Design
The construction of a quarter duplicate design is identical to the allocation of a 2k factorial experiment into four blocks. Two defining contrasts are specified to partition the 2k combinations into four blocks. Any one of the four blocks can be selected to perform the experiment and analysis. In this design, the defining contrasts and the generalized interaction are not present because each of these will have the same sign “−” or “+” in any block selected.
Let’s consider a one quarter fractional design of a 25 factorial design, constructed using ABD and ACE as the defining contrasts. The generalized interaction is BCDE. The experimental design combinations and assignment of blocks are shown in Fig. 14.
Defining Experiment Main Factor Block Assignment Contrasts i A B C D E ABD ACE 1 2 3 4 1 ------* 2 - - - - + - + * 3 - - - + - + - * 4 - - - + + + + * 5 - - + - - - + * 6 - - + - + - - * 7 - - + + - + + * 8 - - + + + + - * 9 - + - - - + - * 10 - + - - + + + *
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11 - + - + - - - * 12 - + - + + - + * 13 - + + - - + + * 14 - + + - + + - * 15 - + + + - - + * 16 - + + + + - - * 17 + - - - - + + * 28 + - - - + + - * 19 + - - + - - + * 20 + - - + + - - * 21 + - + - - + - * 22 + - + - + + + * 23 + - + + - - - * 24 + - + + + - + * 25 + + - - - - + * 26 + + - - + - - * 27 + + - + - + + * 28 + + - + + + - * 29 + + + - - - - * 30 + + + - + - + * 31 + + + + - + - * 32 + + + + + + + * Figure 14: Design table and blocks of 25 factorial experiments using ABD and ACE as defining contrasts
In this design, ABD, ACE, and BCDE are not present because each of these factors will have the same “−” or “+” sign in any of the four blocks. This leaves out 25 - 1 - 3 = 28 effects, which consist of five main factors and 23 interactions factors in this design.
1 5 Since the total number of experimental combinations of the design is /4 (2 ), which is 8, only seven (8 - 1) sums of square can be calculated. This means that each sum of squares is shared by 28/7 = 4 effects, which are main factors and interaction factors. Thus, there are four aliases in each group. The aliases in each group can be obtained by deleting the letters with even exponents from the products of any one effect i.e. main factor or interaction with each defining contrast and the generalized interaction. For example, the aliases of factor A are AxABD = A2BD = BD, AxACE = A2CE = CE, and AxBCDE = ABCDE.
This means that factor A, and interaction BD, CE and ABCDE share the same sum of square, mean square, and test statistics.
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2.3 Designing Fractional Factorial Experiment
The type of alias relationship presents in a fractional factorial design of experiment is defined by its resolution, which are resolution III, IV, and V.
Resolution III design: In this type of design, no main factor is aliased with any other main factor. The main factors are aliased with two-factor interactions and the two-factor interactions are aliased with other two-factor interactions. Examples are 23-1 and 25-2 designs.
Resolution IV design: It is a design where no main factor is aliased with either another main factor or a two-factor interaction. Two-factor interactions are aliased with other two-factor interactions. Examples are 24-1 and 26-2 designs.
Resolution V design: In this design, no main factor is aliased with either another main factor or a two-factor interaction. No two-factor is aliased with other two-factor interactions and two-factor interactions are aliased with three-factor interactions. Examples are 25-1 and 26-1 designs.
Figure 15 contains recommended defining contrasts for selected fractional factorial designs and their resolutions. A basic design is a 2a full factorial design where a = k - q. For example, the basic design of a 27-3 fractional factorial design is a 24 full factorial design. The number of rows, which are treatment combinations in a 2k-q fractional factorial design is equal to the number of rows which are treatment combinations in the associated basic design.
Number Fractional Experiment/Treatment Defining of Factor Resolution Design 2k-q Combination Contract k 3 23-1 (1/2) III 4 ABC 4 24-1 (1/2) IV 8 ABCD 25-2 (1/4) III 8 ABD, ACE 5 25-1 (1/2) V 16 ABCDE 26-3 (1/8) III 8 ABD, ACE, BCF 6 26-2 (1/4) IV 16 ABCE, BCDF ABD, ACE, BCF, 27-4 (1/16) III 8 ABCG 7 ABCE, BCDF, 27-3 (1/8) IV 16 ACDG 27-2 (1/4) IV 32 ABCDF, ABDEG BCDE, ACDF, 28-4 (1/16) IV 16 ABCG, ABDH 8 ABCF, ABDG, 28-3 (1/8) IV 32 BCDEH
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ABCE, BCDF, 29-5 (1/32) III 16 ACDG, ABDH, ABCDJ 9 BCDEF, ACDEG, 29-4 (1/16) IV 32 ABDEH, ABCEJ ABCDG, ACEFH, 29-3 (1/8) IV 64 CDEFJ Figure 15: Contrasts for selected fractional factorial designs
Example 2 An experiment is to be conducted to test the effect of seven factors on some response variables. The experimenter is satisfied with Resolution III. It is a 27-4 fractional design.
Solution From Fig. 15, the recommended defining contrasts for this example are ABD, ACE, BCF, and ABCG.
Start with the basic design, which is a 2a full factorial design where a = k - q. Since k = 7 and q = 4, thus, the basic design is 23 full factorial design, which contains factor A, B, and C and its orthogonal array is shown in Fig. 16.
Experiment Main Factor Interaction Factor i A B C AB AC BC ABC 1 - - - + + + - 2 - - + + - - + 3 - + - - + - + 4 - + + - - + - 5 + - - - - + + 6 + - + - + - - 7 + + - + - - - 8 + + + + + + + Figure 16: Basic 23 design for example 2
Using the alias relationship, identify the columns for the remaining q factors, which are D, E, F, and G. One sees the following aliases.
One alias of D is DxABD = AB, which means D and AB share the same column. One alias of E is ExACE = AC, which means E and ACE share the same column. One alias of F is FxBCF = BC, which means F and BCF share the same column. One alias of G is GxABCG = ABC, which means G and ABC share the same column.
Since they are aliases, they can be replaced. Thus, after replacing interaction factor AB, AC, BC, and ABC of Fig. 16 with main factor D, E, F, and G respectively, Fig. 17 shown the modified experimental combinations for the final 27-4 fractional
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Experiment Factor i A B C D E F G 1 - - - + + + - 2 - - + + - - 3 - + - - + - + 4 - + + - - + - 5 + - - - - + + 6 + - + - + - - 7 + + - + - - - 8 + + + + + + + Figure 17: 27-4 fractional factorial design for example 2
Example 3 Figure 18 contains test factor and limit for conducting experiments to test the effects of seven factors on the hardness of a powdered metal component by analysis of variance of all the seven factors.
Factor Test Limits Designator Components Lower Limit Upper Limit A Material composition 5% (-) 10% (+) B Binder type 1 (-) 2 (+) C Position in the basket Bottom (-) Top (+) D Temperature of heat treatment 800oF (-) 900oF (+) E Quenching bath medium Water (-) Oil (+) F Annealing temperature 300oF (-) 400oF (+) Speed of conveyor belt in G 2ft/min (-) 4ft/min (+) annealing oven Figure 18: Test factors and limits for hardness of a powdered metal component
27-4 fractional factorial experiment arrangement for example 3 is shown in Fig. 19.
G C D E A B F Speed of Position Temperature Quenching Material Binder Annealing Conveyor Belt in the of Heat Bath Composition Type Temperature in Annealing Basket Treatment Medium Oven 5 1 Bottom 900 Oil 400 2 5 1 Top 900 Water 300 4 5 2 Bottom 800 Oil 300 4 5 2 Top 800 Water 400 2 10 1 Bottom 800 Water 400 4 10 1 Top 800 Oil 300 2 10 2 Bottom 900 Water 300 2 10 2 Top 900 Oil 400 4 Figure 19: 27-4 fractional factorial experiment arrangement for example 3
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The experimental results – the hardness test including the mean of duplicate and mean of the average of the duplicate for each experiment are shown in Fig. 20.
Duplicate 2 1 2 Experiment Factor y yik yik ik i 2 i1 A B C D E 1 2 i1 1 - - - + + 71 72 143 71.5 2 - - + + - 106 100 206 103.0 3 - + - - + 59 62 121 60.5 4 - + + - - 91 94 185 92.5 5 + - - - - 122 119 241 120.5 6 + - + - + 91 94 185 92.5 7 + + - + - 131 119 250 125.0 8 + + + + + 85 69 154 77.0 Grand Total 1,485.0 Figure 20: Experimental results of the 27-4 factional factorial experiment for example 3
Solution The analysis of variance ANOVA begins with calculating the sum of square for all the factor A, B, C, D, and E. 2 23 2 y 23 2 ik 2 j1 k1 Total sum of square is SS T yik 3 i1 k1 2 x2
2 2 2 2 2 2 2 2 2 2 2 2 SS T 71 72 106 100 59 62 91 94 122 119 91 94 14852 1312 1192 852 692 145,373137,826.6 7,546.4. 16
The sum of square due to factor A 143 206 1211852 241185 250 1542 14852 is SSA = = 139,740.6 - 137,826.6 8 8 16 = 1,914.0.
The sum of square due to factor B 143 206 2411852 121185 250 1542 14852 is SSB = 8 8 16 = 138,090.6 - 137,826.6 = 264.1.
The sum of square due to interaction of factor C 143121 241 2502 206 1851851542 14852 is SSC = 8 8 16 = 137,865.6 - 137,826.6 = 39.1.
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The sum of square due to factor D 121185 2411852 143 206 250 1542 14852 is SSD = 8 8 16 = 137,854.1 - 137,826.6 = 27.6.
The sum of square due to interaction of factor E 206 185 241 2502 1431211851542 14852 is SSE = 8 8 16 = 142,691.6 - 137,8276.6 = 4,865.1.
The analysis of variance ANOVA table for the hardness test is shown in Fig. 21.
F-value Sum of Degree of Mean Calculated from F- Factor Square Freedom of Square of F-value for p-value Table for of Factor Factor Factor Factor = 0.05 F (1, 10) A 1,914.0 1 1,914.0 43.8 0.05 < 0.001 = 4.96 F (1, 10) B 264.1 1 264.1 6.0 0.05 < 0.050 = 4.96 F (1, 10) C 39.1 1 39.1 0.9 0.05 > 0.100 = 4.96 F (1, 10) D 27.6 1 27.6 0.6 0.05 > 0.100 = 4.96 F (1, 10) E 4,865.1 1 4,865.1 111.3 0.05 < 0.001 = 4.96 Error 436.5 10 43.7 - - - Total 7,546.4 15 - - - - Figure 21: ANOVA results of the factors on hardness of a powdered metal component
Results show that factor A, B, and E are significant at = 0.05. They have effects on the hardness of a powdered metal component.
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A P Analysis of variance ...... 16, 17 Plackett, R. L...... 9 Plackett-Burman design ...... 9 B p-value ...... 17 Burman, J. P...... 9 R C Resolution III design ...... 13 Confounding ...... 9 Resolution IV design...... 13 Resolution V design D ...... 13 Defining contrast ...... 3 S Sum of square of error F ...... 10 Factorial Experiment design by blocking ...... 3 T Fractional factorial experiment ...... 8 The American Society of Quality Control ...... 8 G Generalized interaction ...... 5
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