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Cornelius Gordon Banks
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1 1.0 ial Experiment Design by Block ial Experiment in Incomplete Block ial Experiment with Two Blocks ial Experiment with Four Blocks... 5 Example Fractional ial Experiment....1 Half Duplicate Type of One Half Fractional ial Design Quarter Duplicate Type of One Half Fractional ial Design Designing Fractional ial Experiment Example Example

2 Figure 1: Experimental design of 3 factorial design... 4 Figure : Table showing - and + of defined contrast ABC... 4 Figure 3: Design table of 4 factorial experiments with defining contrast ACD... 5 Figure 4: Generalize interaction of 4 factorial experiment... 5 Figure 5: Group treatment of 4 factorial experiment... 6 Figure 6: Design table and blocks of 4 factorial experiments using AB and CD as defining contrasts... 6 Figure 7: Results of 3-1 two block experimental design... 6 Figure : Experimental block and confounded effect of 3-1 experiment... 7 Figure 9: Eight corner cube used to determine the block of 3-1 experiment... 7 Figure 10: Experimental results of two blocks for 3-1 experiment... Figure 11: 4 factorial design showing ABCD interaction factor... 9 Figure 1: Design table of half duplicate blocks of 4 factorial experiments using ABCD as defining contrast Figure 13: Design table of half duplicate block of 4 factorial experiments shown in Fig Figure 14: Design table and blocks of 5 factorial experiments using ABD and ACE as defining contrasts... 1 Figure 15: Contrasts for selected fractional factorial designs Figure 16: Basic 3 design for example Figure 17: 7-4 fractional factorial design for example Figure 1: Test factors and limits for hardness of a powdered metal component. 15 Figure 19: 7-4 fractional factorial experiment arrangement for example Figure 0: Experimental results of the 7-4 factional factorial experiment for example Figure 1: ANOVA results of the factors on hardness of a powdered metal component

3 1.0 ial 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 ial Experiment in Incomplete Block Let s use a 3 factorial design to illustrate how blocking is being designed. In order to make all eight experiments in 3 -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 3 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 3-1 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. ial 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 3 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 3 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.. Write all the 3 combinations in a table with - representing low level and + representing high level. The experimental combinations for this factorial design are given in Fig

4 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.. 4. Perform the experiments using blended material 1 for the experiments in block 1, while blended material is to be used for experiments of block. Experiment i A B C ABC Figure 1: Experimental design of 3 factorial design Block 1 for ABC equal to + Block for ABC equal to - A B C A B C Figure : Table showing - and + of defined contrast ABC Let s look at how to divide the experimental combinations in a 4 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. Main and Defining Contrast Block 1 A B C D ACD * * * * * * * * * * Experiment i Block

5 * * * * * * Figure 3: Design table of 4 factorial experiments with defining contrast ACD 1.3 ial Experiment with Four Blocks If the treatment combinations of a k 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 4 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.. 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 4 factorials are given in Fig 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 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 4 factorial experiment - 5 -

6 AB CD Block Figure 5: Group treatment of 4 factorial experiment Main Contrast Block A B C D AB CD * * * * * * * * * * * * * * * * Figure 6: Design table and blocks of 4 factorial experiments using AB and CD as defining contrasts Example 1 The results of 3-1 two blocks experimental design are shown in Fig. 7. Experiment 1, 4, 6, and 7 use old batch of material while, experiment, 3, 5, and use new batch of material. Determine the significance of the factors using effect method. # A B C Results Figure 7: Results of 3-1 two block experimental design - 6 -

7 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.. # Confounded Effect A B C ABC Block Results Figure : Experimental block and confounded effect of 3-1 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, 3, 5, and, which are the black corner are designated for block. Figure 9: Eight corner cube used to determine the block of 3-1 experiment The response of two blocks are shown in Fig

8 # A B C Results (a) Experimental results of block 1 # A B C Results (b) Experimental results of block Figure 10: Experimental results of two blocks for 3-1 experiment Using block 1 results, the effect due to factor A is (51 + 5)/ - (34 + 5)/ =.5 The effect due factor B is (5 +5)/ - ( )/ = The effect due to factor C is (5 + 51)/ - (5 + 34)/ =.5. Using block results, the effect due to factor A is ( )/ - (6 + 43)/ =.5 The effect due factor B is (43 +54)/ - (6 + 56)/ = The effect due to factor C is (6 + 54)/ - ( )/ = 4.5. Both blocks show that factor B has significant effective, while the significant effect of factor A and C are about the same..0 Fractional ial Experiment According to Glossary & Tables for Statistical Quality Control published by The American Society of Quality Control ASQC 193, 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 k 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. - -

9 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..1 Half Duplicate Type of One Half Fractional ial Design The construction of a half duplicate design is same as the allocation of a k -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 4 factorial experimental design as shown in Fig. 11 showing four factor ABCD interaction. Experiment i A B C D ABCD Note that column ABCD is obtaining by multiplying the sign of column A, B, C, and D Figure 11: 4 factorial design showing ABCD interaction factor - 9 -

10 If one wishes to use a half duplicate design with the chosen defining contrast ABCD then based on the 4 factorial design showing ABCD interaction factor shown in Fig. 11, the two block experimental designs can be formulated and is shown in Fig. 1. Block 1 for ABCD equal to - Block for ABCD equal to + A B C D A B C D Figure 1: Design table of half duplicate blocks of 4 factorial experiments using ABCD as defining contrast From Fig. 1, either block can be selected for experiment. If one selects block 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 4 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 - 1 = 7. The total number of possible effects i.e. main factors and their interactions in a 4 experiment is 15, out of which interaction ABCD is not present in block, 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 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 = A BCD = 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

11 greatest use when k is quite large and there is some previous knowledge concerning the interactions. It becomes evident that one should always be aware of what the alias structure is for a fractional experiment before finally adopting the experimental plan. Proper choice of defining contrasts and awareness of the alias structure are important considerations before an experimental design is selected. Main Interaction A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD Figure 13: Design table of half duplicate block of 4 factorial experiments shown in Fig. 1. Quarter Duplicate Type of One Half Fractional ial Design The construction of a quarter duplicate design is identical to the allocation of a k factorial experiment into four blocks. Two defining contrasts are specified to partition the k 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 5 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 Block Assignment Contrasts i A B C D E ABD ACE * * * * * * * * * *

12 * * * * * * * * * * * * * * * * * * * * * * Figure 14: Design table and blocks of 5 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 = effects, which consist of five main factors and 3 interactions factors in this design. Since the total number of experimental combinations of the design is 1 / 4 ( 5 ), which is, only seven ( - 1) sums of square can be calculated. This means that each sum of squares is shared by /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 = A BD = BD, AxACE = A CE = 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

13 .3 Designing Fractional ial 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 3-1 and 5- 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 4-1 and 6- 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 5-1 and 6-1 designs. Figure 15 contains recommended defining contrasts for selected fractional factorial designs and their resolutions. A basic design is a a full factorial design where a = k - q. For example, the basic design of a 7-3 fractional factorial design is a 4 full factorial design. The number of rows, which are treatment combinations in a k-q fractional factorial design is equal to the number of rows which are treatment combinations in the associated basic design. Number of k Fractional Design k-q Resolution Experiment/Treatment Combination Defining Contract (1/) III 4 ABC (1/) IV ABCD 5 5- (1/4) III ABD, ACE 5-1 (1/) V 16 ABCDE (1/) III ABD, ACE, BCF 6- (1/4) IV 16 ABCE, BCDF 7-4 (1/16) III ABD, ACE, BCF, ABCG ABCE, BCDF, (1/) IV 16 ACDG 7- (1/4) IV 3 ABCDF, ABDEG -4 (1/16) IV 16-3 (1/) IV 3 BCDE, ACDF, ABCG, ABDH ABCF, ABDG, BCDEH

14 9 9-5 (1/3) III 16 ABCE, BCDF, ACDG, ABDH, ABCDJ 9-4 (1/16) IV 3 BCDEF, ACDEG, ABDEH, ABCEJ 9-3 (1/) IV 64 ABCDG, ACEFH, CDEFJ Figure 15: Contrasts for selected fractional factorial designs Example 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 7-4 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 a full factorial design where a = k - q. Since k = 7 and q = 4, thus, the basic design is 3 full factorial design, which contains factor A, B, and C and its orthogonal array is shown in Fig. 16. Experiment Main Interaction i A B C AB AC BC ABC Figure 16: Basic 3 design for example 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 7-4 fractional

15 factorial design. Note that in real-life applications, signs + and in the design table are replaced by the actual levels of the factors. Experiment i A B C D E F G Figure 17: 7-4 fractional factorial design for example Example 3 Figure 1 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. Test Limits Designator Components Lower Limit Upper Limit A Material composition 5% (-) 10% (+) B Binder type 1 (-) (+) C Position in the basket Bottom (-) Top (+) D Temperature of heat treatment 00 o F (-) 900 o F (+) E Quenching bath medium Water (-) Oil (+) F Annealing temperature 300 o F (-) 400 o F (+) G Speed of conveyor belt in annealing oven ft/min (-) 4ft/min (+) Figure 1: Test factors and limits for hardness of a powdered metal component 7-4 fractional factorial experiment arrangement for example 3 is shown in Fig. 19. A Material Composition B Binder Type C Position in the Basket D Temperature of Heat Treatment E Quenching Bath Medium F Annealing Temperature G Speed of Conveyor Belt in Annealing Oven 5 1 Bottom 900 Oil Top 900 Water Bottom 00 Oil Top 00 Water Bottom 00 Water Top 00 Oil Bottom 900 Water Top 900 Oil Figure 19: 7-4 fractional factorial experiment arrangement for example 3

16 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. 0. Experiment i Duplicate yik y 1 ik y ik i 1 A B C D E 1 i Grand Total 1,45.0 Figure 0: Experimental results of the 7-4 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. Total sum of square is 3 3 yik 1 1 SS j k T y ik 3 x i 1 k 1 SS T , ,6.6 7, The sum of square due to factor A is SS A = 16 = 1, = 139, ,6.6 The sum of square due to factor B is SS B = 16 = 13, ,6.6 = The sum of square due to interaction of factor C is SS C = 16 = 137, ,6.6 =

17 The sum of square due to factor D is SS D = 16 = 137, ,6.6 = 7.6. The sum of square due to interaction of factor E is SS E = 16 = 14, ,76.6 = 4, The analysis of variance ANOVA table for the hardness test is shown in Fig. 1. Sum of Square of Degree of Freedom of Mean Square of Calculated F-value for A 1, , B C D F-value from F- Table for = 0.05 F0.05(1, 10) = 4.96 F0.05(1, 10) = 4.96 F0.05(1, 10) = 4.96 F0.05(1, 10) = 4.96 F0.05(1, 10) p-value < < > > E 4, , = 4.96 < Error Total 7, Figure 1: ANOVA results of the factors on hardness of a powdered metal component Results show that factor A, B, and E are significant at = They have effects on the hardness of a powdered metal component

18 A Analysis of variance... 16, 17 B Burman, J. P C Confounding... 9 D Defining contrast... 3 F ial Experiment design by blocking... 3 Fractional factorial experiment... G Generalized interaction... 5 P Plackett, R. L Plackett-Burman design... 9 p-value R Resolution III design Resolution IV design Resolution V design S Sum of square of error T The American Society of Quality Control

Fractional Factorials

Fractional Factorials Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 26 1 Fractional Factorials Number of runs required for full factorial grows quickly A 2 7 design requires 128