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1 STA 536 – Fractional Factorial Experiments at Two Levels Fractional Factorial Experiments at Two Levels Source : sections , part of 5.6 Effect aliasing, resolution, minimum aberration criteria. Analysis. Techniques for resolving ambiguities in aliased effects. Choice of designs, use of design tables. Blocking in 2 kp designs.

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2 STA 536 – Fractional Factorial Experiments at Two Levels Fractional Factorial Experiments at Two Levels A 2 k full factorial design requires 2 k runs to be performed. Full factorial designs are rarely used in practice for large k (say, k > 7). A fractional factorial design is a subset or fraction of a full factorial design. For economic reasons, fractional factorial designs are commonly used

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3 STA 536 – Fractional Factorial Experiments at Two Levels 5.1 A Leaf Spring Experiment An experiment to improve a heat treatment process on truck leaf springs. The heat treatment which forms the camber (or curvature) in leaf springs consists of heating in a high temperature furnace, processing by a forming machine, and quenching in an oil bath. Response: the height of an unloaded spring, known as free height. Goal is to make the variation about the target 8.0 as small as possible. A half fraction of a 2 5 full factorial design is used to study 5 factors. A 2 51 design The quench oil temperature (Q) could not be controlled precisely.

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4 STA 536 – Fractional Factorial Experiments at Two Levels Factors and Levels, Leaf Spring Experiment Factor Level + - B. high heat temperature (F) C. heating time (seconds)2325 D. transfer time (seconds)1012 E. hold down time (seconds)23 Q. quench oil temperature (F)

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5 STA 536 – Fractional Factorial Experiments at Two Levels Leaf Spring Experiment: Design Matrix and Data

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6 STA 536 – Fractional Factorial Experiments at Two Levels 5.2 Fractional Factorial Designs: Effect Aliasing and the Criteria of Resolution and Minimum Aberration If a 2 5 design is used for the experiment, its 31 degrees of freedom would be allocated as follows: Using effect hierarchy principle, one would argue that 4fis, 5fis and even 3fis are not likely to be important. There are = 16 such effects, half of the total runs! Using a 2 5 design can be wasteful (unless 32 runs cost about the same as 16 runs.) A half fraction of a 2 5 full factorial design has 16 runs and can estimate 15 factorial effects. Ideally, estimate 5 MEs and 10 2fis. Why Using Fractional Factorial Designs?

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7 STA 536 – Fractional Factorial Experiments at Two Levels Leaf Spring Experiment: Use of a FF design instead of full factorial design is usually done for economic reasons. Since there is no free lunch, what price to pay? What factorial effects can be estimated in the leaf spring experiment? Notice that column E equals the product of the columns B, C and D. E=BCD Consequence: the column for E is used for estimating the main effect of E and also for the interaction effect between B, C and D, i.e.,

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8 STA 536 – Fractional Factorial Experiments at Two Levels Leaf Spring Experiment: The data from such a design is not capable of distinguishing the estimate of E from the estimate of BCD. The main effect E is aliased with the B × C × D interaction. Assuming BCD interaction is negligible, we can estimate E. A 2 51 has 16 runs and can estimate 15 pairs of aliased factorial effects. The replicates can be used for estimating error or dispersion effects.

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9 STA 536 – Fractional Factorial Experiments at Two Levels Leaf Spring Experiment: E = BCD or I = BCDE I = column of +s is the identity element in the group of multiplications. I = BCDE is the defining relation for the 2 51 design. It implies all the 15 effect aliasing relations : B =CDE, C = BDE, D = BCE, E = BCD, BC = DE, BD =CE, BE =CD, Q = BCDEQ, BQ =CDEQ, CQ = BDEQ, DQ = BCEQ, EQ = BCDQ, BCQ = DEQ, BDQ =CEQ, BEQ =CDQ.

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10 STA 536 – Fractional Factorial Experiments at Two Levels B =CDE, C = BDE, D = BCE, E = BCD, BC = DE, BD =CE, BE =CD, Q = BCDEQ, BQ =CDEQ, CQ = BDEQ, DQ = BCEQ, EQ = BCDQ, BCQ = DEQ, BDQ =CEQ, BEQ =CDQ. Each of the four main effects, B, C, D and E, is estimable if the respective three-factor interaction alias is negligible, which is usually a reasonable assumption. The B×C interaction is estimable only if prior knowledge would suggest that the D × E interaction is negligible. The Q main effect is estimable under the weak assumption that the B × C × D × E × Q interaction is negligible. The B ×Q interaction is estimable under the weak assumption that the C × D × E × Q interaction is negligible.

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11 STA 536 – Fractional Factorial Experiments at Two Levels Definition: Clear Effects A main effect or two-factor interaction is clear if none of its aliases are main effects or two-factor interactions and strongly clear if none of its aliases are main effects, two-factor or three-factor interactions. Therefore a clear effect is estimable under the assumption of negligible 3-factor and higher interactions and a strongly clear effect is estimable under the weaker assumption of negligible 4-factor and higher interactions. For the leaf spring experiment, Clear effects: B, C, D, E, Q, B × Q, C × Q, D × Q and E × Q. Strongly clear effects: Q, B × Q, C × Q, D × Q and E × Q.

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12 STA 536 – Fractional Factorial Experiments at Two Levels An Alternate Design with defining relation: I = BCDEQ. The 15 aliasing relations are: B=CDEQ, C=BDEQ, D=BCEQ, E=BCDQ, Q=BCDE BC=DEQ, BD=CEQ, BE=CDQ, BQ=CDE, CD=BEQ, CE=BDQ, CQ=BDE, DE=BCQ, DQ=BCE, EQ=BCD Clear effects: all 5 MEs and 10 2fis. Strongly clear effects: all 5 MEs.

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13 STA 536 – Fractional Factorial Experiments at Two Levels Which of the two fractional plans is better? I=BCDE vs I=BCDEQ The 2nd plan is generally preferred. (Why? Because each of the 15 degrees of freedom is either clear or strongly clear) The 1st plan is preferred if there is special interest on factor Q and 2fis involving Q. The reason for adopting the 1st plan will be given in Chapter 11 in the context of robust parameter design.

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14 STA 536 – Fractional Factorial Experiments at Two Levels A 2 kp fractional factorial design has k factors, each at 2 levels, consisting of 2 kp runs. is a 2p -th fraction of the 2 k full factorial design is determined by p defining words Letters are the names of the factors: 1, 2,..., k or A,B,.... The number of letters in a word is its wordlength. The group formed by the p defining words is called the defining contrast subgroup. The group consists of 2 p -1 words plus the identity element I.

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15 STA 536 – Fractional Factorial Experiments at Two Levels Example: A 2 62 design d defining relations: 5 = 12 and 6 = 134. The two independent defining words: I = 125 and I = Then I = 125×1346 = defining contrast subgroup {I,125,1346,23456} or I = 125 = 1346 =

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16 STA 536 – Fractional Factorial Experiments at Two Levels Example: A 2 62 design d 5 = 12 and 6 = 134 I = 125 = 1346 = 23456, 1 = 25 = 346 = , 2 = 15 = = 3456, 3 = 1235 = 146 = 2456, 4 = 1245 = 136 = 2356, 5 = 12 = = 2346, 6 = 1256 = 134 = 2345, 13 = 235 = 46 = 12456, 14 = 245 = 36 = 12356, 16 = 256 = 34 = 12345, 23 = 135 = 1246 = 456, 24 = 145 = 1236 = 356, 26 = 156 = 1234 = 345, 35 = 123 = 1456 = 246, 45 = 124 = 1356 = 236, 56 = 126 = 1345 = 234. clear effects: 3, 4, 6, and 23, 24, 26, 35, 45, 56. wordlength pattern: W = (1, 1, 1, 0). resolution: III

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17 STA 536 – Fractional Factorial Experiments at Two Levels WordLength Pattern and Resolution A i = number of words of length i in its defining contrast subgroup. wordlength pattern W = (A 3,...,A k ) resolution=the length of the shortest word in the defining contrast subgroup, i.e., smallest r such that A r 1. A design of resolution I or II is useless. (Why? No main effect is allowed to be aliased with another main effect.)

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18 STA 536 – Fractional Factorial Experiments at Two Levels Resolution criterion Maximum Resolution criterion, proposed by Box and Hunter (1961), chooses the 2 kp design with maximum resolution. justified by the hierarchical ordering principle (Section 3.5). Rules for Resolution: Resolution R implies that no effect involving i factors is aliased with effects involving less than R i factors. In a resolution III design, some MEs are aliased with 2fis, but not with other MEs. In a resolution IV design, some MEs are aliased with 3fis and some 2fis are aliased with other 2fis. (all MEs are clear.) In a resolution V design, some MEs are aliased with 4fis and some 2fis are aliased with 3fis. (all MEs are strongly clear and all 2fis are clear.)

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19 STA 536 – Fractional Factorial Experiments at Two Levels Rules for Resolution IV and V Designs: In any resolution IV design, the main effects are clear. In any resolution V design, the main effects are strongly clear and the two-factor interactions are clear. Among the resolution IV designs with given k and p, those with the largest number of clear two-factor interactions are the best.

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20 STA 536 – Fractional Factorial Experiments at Two Levels A Projective Rationale for Resolution A projection on to any n ( R 1) factors is a full 2 n factorial design (replicated if n k p). Fig. 4.1: projections of 2 31 designs. In analysis, if there are at most R1 important factors, then all factorial effects among the important factors can be estimated. Projective property of a design

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21 STA 536 – Fractional Factorial Experiments at Two Levels Minimum Aberration Criterion Example: two 2 72 designs: d1 : I = 4567 = = 12357, d2 : I = 1236 = 1457 = Both designs have resolution IV. wordlength patterns: W(d1) = (0, 1, 2, 0, 0) and W(d2) = (0, 2, 0, 1, 0).

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22 STA 536 – Fractional Factorial Experiments at Two Levels d1 : I = 4567 = = 12357, d2 : I = 1236 = 1457 = d1 has less aberration than d2. d1 has minimum aberration (by complete search.) aliased 2fis: d1 : 45 = 67, 46 = 57, 47 = 56, d2 : 12 = 36, 13 = 26, 16 = 23, 14 = 57, 15 = 47, 17 = 45 d1 is preferred to d2.

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23 STA 536 – Fractional Factorial Experiments at Two Levels Create FFD in SAS /*FFD 2^(8-3) design using 6=135 7=235 8=1234*/ proc factex; factors x1-x5; output out = SavedDesign; data SavedDesign1; set SavedDesign; x6=x1*x3*x5; x7=x2*x3*x5; x8=x1*x2*x3*x4; run; proc print; run; /*FFD 2^(8-3) design using 6=12 7=13 8=34*/ data SavedDesign2; set SavedDesign; x6=x1*x2; x7=x1*x3; x8=x3*x4; run; proc print; run;

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24 STA 536 – Fractional Factorial Experiments at Two Levels FFD by SAS Macro %macro createFFD; proc factex; factors x1-x&k /NLEV=&s; output out = SavedDesign; proc IML; use SavedDesign; read all into D; def=&def; if &s=2 then D=(D+1)/2; else if &s=3 then D=D+1; DD=D || mod(D*def`, &s); create ffddsn from DD; append from DD; quit; %mend; /*FFD 2^(8-3) with 6=135, 7=235, 8=1234*/ %let k=5; %let q=3; %let s=2; %let def={ , , }; %createFFD; proc print; run; /*FFD 2^(8-3) with 6=12, 7=13, 8=45*/ %let k=5; %let q=3; %let s=2; %let def={ , , }; %createFFD; proc print; run;

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25 STA 536 – Fractional Factorial Experiments at Two Levels d1 : I = 4567 = = 12357, d2 : I = 1236 = 1457 = /*d1 : I = 4567 = = 12357*/ %let k=5; %let s=2; %let def={ , }; %createFFD; /*d2 : I = 1236 = 1457 = */ %let k=5; %let s=2; %let def={ , }; %createFFD;

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26 STA 536 – Fractional Factorial Experiments at Two Levels 5.3 Analysis - the leaf spring experiment defining relation: I = BCDE aliasing relations: B = CDE, C = BDE, D = BCE, E = BCD, BC = DE, BD = CE, BE = CD, Q = BCDEQ, BQ = CDEQ, CQ = BDEQ, DQ = BCEQ, EQ = BCDQ, BCQ = DEQ, BDQ = CEQ, BEQ = CDQ. 9 clear effects: B, C, D, E, Q, BQ, CQ, DQ and EQ 6 pairs of aliased effects: BC = DE, BD = CE, BE = CD, BCQ =DEQ, BDQ = CEQ, BEQ = CDQ an effect is estimable if its alias is negligible. Same strategy as in full factorial experiments except for the interpretation and handling of aliased effects.

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27 STA 536 – Fractional Factorial Experiments at Two Levels data leaf; input B C D E Q y1-y3 m s2 lns; BQ=B*Q; CQ=C*Q; DQ=D*Q; EQ=E*Q; BC=B*C; BD=B*D; CD=C*D; BCQ=B*C*Q; BDQ=B*D*Q; BEQ=B*E*Q; cards; ;

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28 STA 536 – Fractional Factorial Experiments at Two Levels SAS code PROC REG DATA=LEAF outest=effects; MODEL M=B C D E Q BQ CQ DQ EQ BC BD CD BCQ BDQ BEQ; RUN; %halfnormalplot(effects, m); PROC REG DATA=LEAF outest=effects; MODEL LNS=B C D E Q BQ CQ DQ EQ BC BD CD BCQ BDQ BEQ; RUN; %halfnormalplot(effects, LNS);

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29 STA 536 – Fractional Factorial Experiments at Two Levels SAS Macro halfnormalplot %macro halfnormalplot(effectdata, response); /*effectdata is from the output of regression*/ data effect2; set &effectdata; drop &response intercept _RMSE_; proc transpose data=effect2 out=effect3; data effect4; set effect3; effect=ABS(col1*2); proc sort data=effect4; by effect; proc print data=effect4; proc rank data=effect4 out=effect5; var effect; ranks nefff; run; proc sql; select * from effect5; quit; data effect5; set effect5; neff=quantile('NORMAL', *(nefff-0.5)/&sqlobs); proc gplot data=effect5; plot effect*neff=_NAME_; run; quit; %mend;

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30 STA 536 – Fractional Factorial Experiments at Two Levels Factorial Effects, Leaf Spring Experiment

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31 STA 536 – Fractional Factorial Experiments at Two Levels For the location effects, visually one may judge that Q,B,C,CQ and possibly E,BQ are significant. One can apply the studentized maximum modulus test (see chapter 4, not covered in class) to confirm that Q,B,C are significant at 0.05 level. The B×Q and C×Q plots show that they are synergystic.

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32 STA 536 – Fractional Factorial Experiments at Two Levels For illustration, we use the model For the dispersion effects (based on z i = lns 2 i values), the half-normal plot is given in next slide. Visually only effect B stands out. This is confirmed by applying the studentized maximum modulus test. For illustration, we will include B,DQ,BCQ in the following model, (1)

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33 STA 536 – Fractional Factorial Experiments at Two Levels For the dispersion effects

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34 STA 536 – Fractional Factorial Experiments at Two Levels

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35 STA 536 – Fractional Factorial Experiments at Two Levels Two-Step Procedure for Optimization Minimize dispersion. Choose x B = 1, x D x Q = 1 and x B x C x Q = +1. Use interaction plots to determine level settings of C, D, Q. Should choose (B,C,D,Q) = (,+,+,). Bring location to target (t = 8.0). E is an adjustment factor. After chosen (B,C,D,Q) = (,+,+,). ˆy = (1) x E (+1) (1) (1)(1) (+1)(1) = x E. By solving ˆy = 8.0, x E should be set at ( )/ = Note: The uncoded value for factor E is (0.5) = 3.77 seconds. Caution: there are no experimental data around 3.77 seconds.

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36 STA 536 – Fractional Factorial Experiments at Two Levels 5.4 Techniques for Resolving the Ambiguities in Aliased Effects Among the three factorial effects that feature in model (1), B is clear and DQ is strongly clear. However, the term x B x C x Q is aliased with x D x E x Q. The following three techniques can be used to resolve the ambiguities. Subject matter knowledge may suggest some effects in the alias set are not likely to be significant (or does not have a good physical interpretation). Or use effect hierarchy principle to assume away some higher order effects. Or use a follow-up experiment to de-alias these effects. Three methods are given in section 5.4 of WH. Two are considered here.

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37 STA 536 – Fractional Factorial Experiments at Two Levels Fold-over Technique Suppose the original experiment is based on a design with generators d1 : 4 = 12, 5 = 13, 6 = 23, 7 = 123. None of its main effects are clear. To de-alias them, we can choose another 8 runs (no in Table 4.7) with reversed signs for each of the 7 factors. This follow-up design d2 has the generators d2 : 4 = 12,5 = 13,6 = 23,7 = 123 With the extra degrees of freedom, we can introduce a new factor 8 for run number 1-8, and -8 for run number See Table 4.7. The combined design d1+d2 is a design and thus all main effects are clear. (Its defining contrast subgroup is on p.227 of WH).

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38 STA 536 – Fractional Factorial Experiments at Two Levels Table 5.6: Augmented Design Matrix Using Fold-Over Technique 7=123 8=124 8=135 8=236 => 5=138=13*124=234 6=238=23*124=134 7=123 8=124 Without 8, I=2345=1346=1237 =1256=1457=3567 =3567

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39 STA 536 – Fractional Factorial Experiments at Two Levels Suppose one factor, say 5, is very important. We want to de-alias 5 and all 2fis involving 5. Choose, instead, the following design Then the combined design d1+d3 is a design with the generators Since 5 does not appear in (5), 5 is strongly clear and all 2fis involving 5 are clear. However, other main effects are not clear (see Table 5.7 of WH for d1+d3). Choice between d2 and d3 depends on the priority given to the effects {I=124=236=1237=1346=347=167=2467} Block Factor 8=135. Thus {1358=I=124=236=1237= 1346= 347=167=2467 =23458=12568=2578=4568= 14578=35678= } Fold-over Technique: Version Two

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40 STA 536 – Fractional Factorial Experiments at Two Levels Critique of Fold-over Technique Fold-over technique is not an efficient technique. It requires doubling of the run size and can only de-alias a specific set of effects. In practice, after analyzing the first experiment, a set of effects will emerge and need to be de-aliased. It will usually require much fewer runs to de-alias a few effects. A more efficient technique that does not have these deficiencies is the optimum design approach given in Section

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41 STA 536 – Fractional Factorial Experiments at Two Levels Optimal Design Approach for Follow- Up Experiments This approach add runs according to a particular optimal design criterion. The D and D s criteria shall be discussed. Optimal design criteria depend on the assumed model. In general, the model should contain: All effects and their aliases (except those judged unimportant a priori or by the effect hierarchy principle) identified as significant in the initial experiment. A block variable that accounts for differences in the average value of the response over different time periods. An intercept. In the leaf spring experiment, we specify the model:

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42 STA 536 – Fractional Factorial Experiments at Two Levels D-Criterion In Table 5.8, the columns B, C, D, E, and Q comprise the design matrix while the columns B, block, BCQ, DEQ, DQ comprise the model matrix. Two runs are to be added to the original 16-run experiment. There are 4 5 = 1024 possible choices of factor settings for the follow up runs (runs 17 and 18) since each factor can take on either the + or - level in each run. For each of the 1024 choices of settings for B, C, D, E, Q for runs 17 and 18, denote the corresponding model matrix by X d, d = 1,...,1024. We may choose the factor settings d that maximizes the D-criterion, i.e. Maximizing the D criterion minimizes the volume of the confidence ellipsoid for all model parameters b. 64 choices of d attain the maximum D value of 4,194,304. Two are: d1 : (B,C,D,E,Q) = (++++) and (+++) d2 : (B,C,D,E,Q) = (++++) and (++++)

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43 STA 536 – Fractional Factorial Experiments at Two Levels Table 5.8: Augmented Design Matrix and Model Matrix, Leaf Spring Experiment

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44 STA 536 – Fractional Factorial Experiments at Two Levels /*Optimal Design Approach for Follow-Up Experiments*/ proc IML; use leaf; read all; x={ , }; res={ }; kk=0; do i1=1 to 4; B1=B // x[,i1]; do i2=1 to 4; C1=C // x[,i2]; do i3=1 to 4; D1=D // x[,i3]; do i4=1 to 4; E1=E // x[,i4]; do i5=1 to 4; Q1=Q // x[,i5]; kk=kk+1; DD=j(18,1,1)|| (j(16,1,-1)//{1,1}) || B1 || D1#Q1 || B1#C1#Q1 || D1#E1#Q1; tmp=det(DD`*DD); res=res // (kk || i1 || i2 || i3 || i4 || i5 || tmp); end; call sort(res, { }, {7}); print x, res; quit;

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45 STA 536 – Fractional Factorial Experiments at Two Levels d1 : (B,C,D,E,Q) = (++++) and (+++) d2 : (B,C,D,E,Q) = (++++) and (++++)

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46 STA 536 – Fractional Factorial Experiments at Two Levels

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47 STA 536 – Fractional Factorial Experiments at Two Levels 5.5 Selection of 2 kp Designs Using Minimum Aberration and Related Criteria Questions: How to choose a good 2 kp design? What criteria should be used to judge goodness? Example: Two 2 94 IV designs from Table 5A.3 (P254) Design d1 with generators 6=123, 7=124, 8=125, 9=1345 Defining contrast subgroup: I = 1236 = 1247 = 1258 = = 3467 = 3568 = = 4578 = = = = = = = Resolution: IV Wordlength pattern: W = (0, 6, 8, 0, 0, 1, 0) Clear effects: all 9 MEs, any 2fis involving 9 Strongly clear effects: 9

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48 STA 536 – Fractional Factorial Experiments at Two Levels Comparing two design Design d2 with generators 6=123, 7=124, 8=134, 9=2345 Defining contrast subgroup: I = 1236 = 1247 = 1348 = = 3467 = 2468 = = 2378 = = = 1678 = = = = Resolution: IV Wordlength pattern: W=(0,7,7,0,0,0,1) Clear effects: all 9 MEs, any 2fis involving 5 or 9 Strongly clear effects: 5 and 9 Which design is better? Answer depends.

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49 STA 536 – Fractional Factorial Experiments at Two Levels Comparing two design d1 has less aberration than d2 (A 4 (d1)=6

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50 STA 536 – Fractional Factorial Experiments at Two Levels Tables 5A.1-5A.7 2 kp fractional factorial designs.

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51 STA 536 – Fractional Factorial Experiments at Two Levels Table 5A.2: 16-Run 2 kp FFD (k p = 4)

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52 STA 536 – Fractional Factorial Experiments at Two Levels Table 5A.3: 32 Run 2 kp FFD (kp=5, 6k11)

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53 STA 536 – Fractional Factorial Experiments at Two Levels Research problems: How to construct minimum aberration designs with large run sizes (>128)? What is the relationship between minimum aberration and maximum #of clear effects?

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54 STA 536 – Fractional Factorial Experiments at Two Levels Example: Two 2 62 designs in Table 5A.2 design d1 with generators 5 = 123, 6 = 124 I = 1235 = 1246 = 3456 resolution IV 6 clear effects: 6 MEs. design d2 with generators 5 = 12, 6 = 134 I = 125 = 1346 = resolution III 9 clear effects: 3, 4, 6, 23, 24, 26, 35, 45, 56. d1 is generally preferred. d2 is useful when some 2fis are important.

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55 STA 536 – Fractional Factorial Experiments at Two Levels Choice of Fractions and Avoidance of Specific Combinations A 2 74 III design with 4=12, 5=13, 6=23 and 7=123. There are 16 fractions (2 74 ) by using + or signs in the 4 generators. 4 = ±12, 5 = ±13, 6 = ±23, 7 = ±123. Most tables present the standard fraction with all + signs. All 16 fractions are equivalent by changing signs of some columns. Other fractions are useful to avoid some treatment combinations (If specific combinations (e.g., (+++) for high pressure, high temperature, high concentration) are deemed undesirable or even disastrous, they can be avoided by choosing a fraction that does not contain them. Example on p.237 of WH)

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56 STA 536 – Fractional Factorial Experiments at Two Levels Factor assignment: 2 41 : 4= : 4=12, 5= : 4=12, 5=13, 6= : 4=12, 5=13, 6=23, 7=123 Table 5A.1 8-run designs

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57 STA 536 – Fractional Factorial Experiments at Two Levels 5.6 Blocking in Fractional Factorial Designs Example: To arrange a 2 62 design in 4 blocks in Table 4B.2. design generators 5 = 123, 6 = 124 treatment defining contrast subgroup: I = 1235 = 1246 = 3456 block generators B 1 = 134,B 2 = 234 block defining contrast subgroup: B 1 = 134,B 2 = 234,B 1 B 2 = 12

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58 STA 536 – Fractional Factorial Experiments at Two Levels Aliasing relations: I = 1235 = 1246 = 3456 B 1 = 134 = 245 = 236 = 156 B 2 = 234 = 145 = 136 = 256 B 1 B 2 = 12 = 35 = 46 = = 235 = 246 = = 135 = 146 = = 125 = = = = 126 = = 123 = = = = 124 = = 25 = 2346 = = 26 = 2345 = = 23 = 2456 = = 24 = 2356 = = 56 = 1245 = = 45 = 1256 = 1234 Clear effects: all 6 MEs Can estimate 3 block effects, 6 clear MEs and 6 pairs of aliased 2fis.

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59 STA 536 – Fractional Factorial Experiments at Two Levels Definition: A main effect or two-factor interaction is said to be clear if it is not aliased with any other main effects or two- factor interactions as well as not confounded with any block effects. The definition of strongly clear effect has the additional requirement that it is not aliased with any three-factor interactions. Minimum aberration Several different definitions by combining treatment and block defining contrast subgroups.

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60 STA 536 – Fractional Factorial Experiments at Two Levels Table 5B.1-5B.5 (p ): 2 kp designs in 2 q blocks. Run sizes 8, 16, 32, 64, 128 with k p = 3, 4, 5, 6, 7 k9 and 1qkp1. designs are chosen by # of clear effects. Two 2 62 designs with k = 6, p = 2, q = 1, 2. Research problem: How to construct optimal blocked 2 kp designs with large run sizes (128)?

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