1. Fractional Factorial Experiments at Two Levels Source : sections 5.1-5.5, 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 2k−p designs.

2. Fractional Factorial Experiments at Two Levels
A 2k full factorial design requires 2k 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

3. 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 25 full factorial design is used to study 5 factors.
A 25−1 design
The quench oil temperature (Q) could not be controlled precisely.

4. Factors and Levels, Leaf Spring Experiment
B. high heat temperature (F)
1840
1880
C. heating time (seconds) 23 25
D. transfer time (seconds) 10 12
E. hold down time (seconds) 2 3
Q. quench oil temperature (F)

5. Leaf Spring Experiment: Design Matrix and Data

6. Why Using Fractional Factorial Designs?
5.2 Fractional Factorial Designs: Effect Aliasing and the Criteria of Resolution and Minimum Aberration
Why Using Fractional Factorial Designs?
If a 25 design is used for the experiment, its 31 degrees of freedom would be allocated as follows:
Using effect hierarchy principle, one would argue that 4fi’s, 5fi’s and even 3fi’s are not likely to be important. There are = 16 such effects, half of the total runs! Using a 25 design can be wasteful (unless 32 runs cost about the same as 16 runs.)
A half fraction of a 25 full factorial design has 16 runs and can estimate 15 factorial effects.
Ideally, estimate 5 ME’s and 10 2fi’s.

7. Leaf Spring Experiment: 25-1
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.,

8. Leaf Spring Experiment: 25-1
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 25−1 has 16 runs and can estimate 15 pairs of aliased factorial effects.
The replicates can be used for estimating error or dispersion effects.

9. Leaf Spring Experiment: 25-1
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 25−1 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.

10. 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.

11. 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.

12. 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 ME’s and 10 2fi’s.
Strongly clear effects: all 5 ME’s.

13. 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 2fi’s involving Q.
The reason for adopting the 1st plan will be given in Chapter 11 in the context of robust parameter design.

14. A 2k−p fractional factorial design
has k factors, each at 2 levels, consisting of 2k−p runs.
is a 2−p-th fraction of the 2k 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 2p-1 words plus the identity element I.

15. The two independent defining words: I = 125 and I = 1346.
Example: A 26−2 design d
defining relations: 5 = 12 and 6 = 134.
The two independent defining words: I = 125 and I = 1346.
Then I = 125×1346 =
defining contrast subgroup {I,125,1346,23456} or
I = 125 = 1346 =

16. Example: A 26−2 design d 5 = 12 and 6 = 134
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

17. WordLength Pattern and Resolution
Ai = number of words of length i in its defining contrast subgroup.
wordlength pattern W = (A3, ,Ak)
resolution=the length of the shortest word in the defining contrast subgroup, i.e., smallest r such that Ar≥1.
A design of resolution I or II is useless. (Why? No main effect is allowed to be aliased with another main effect.)

18. Resolution criterion
Maximum Resolution criterion, proposed by Box and Hunter (1961),
chooses the 2k−p 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 ME’s are aliased with 2fi’s, but not with other ME’s.
In a resolution IV design, some ME’s are aliased with 3fi’s and some 2fi’s are aliased with other 2fi’s. (all ME’s are clear.)
In a resolution V design, some ME’s are aliased with 4fi’s and some 2fi’s are aliased with 3fi’s. (all ME’s are strongly clear and all 2fi’s are clear.)

19. 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.

20. A Projective Rationale for Resolution
Projective property of a design
A projection on to any n ( ≤ R − 1) factors is a full 2n factorial design (replicated if n ≤ k − p).
Fig. 4.1: projections of 23−1 designs.
In analysis, if there are at most R−1 important factors, then all factorial effects among the important factors can be estimated.

21. Minimum Aberration Criterion
Example: two 27−2 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).

22. d1 : I = 4567 = = 12357, d2 : I = 1236 = 1457 =
d1 has less aberration than d2.
d1 has minimum aberration (by complete search.)
aliased 2fi’s:
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.

23. 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;

24. FFD by SAS Macro /*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 def={ , , };
proc print; run;
%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;

25. 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 def={ , };

26. 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.

27. 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; ;

28. 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);
MODEL LNS=B C D E Q BQ CQ DQ EQ BC BD CD BCQ BDQ BEQ;
%halfnormalplot(effects, LNS);

29. 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;

30. Factorial Effects, Leaf Spring Experiment

31. 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.

32. For illustration, we use the model
For the dispersion effects (based on zi = lns2i 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)

33. For the dispersion effects

35. Two-Step Procedure for Optimization
Minimize dispersion.
Choose xB = −1, xDxQ = −1 and xBxCxQ = +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) xE (+1) − (−1) (−1)(−1) − (+1)(−1) = xE.
By solving ˆy = 8.0, xE should be set at (8.0 − )/ = 2.54.
Note: The uncoded value for factor E is (0.5) = 3.77 seconds. Caution: there are no experimental data around 3.77 seconds.

36. 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 xBxCxQ is aliased with xDxExQ.
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.

37. 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).

38. 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
Without 8, I=2345=1346=1237 =1256=1457=3567 =3567

39. Fold-over Technique: Version Two
Suppose one factor, say 5, is very important. We want to de-alias 5 and all 2fi’s 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 2fi’s 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= }

40. 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

41. Optimal Design Approach for Follow-Up Experiments
This approach add runs according to a particular optimal design criterion. The D and Ds 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:

42. 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 45 = 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 Xd, 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 (−++++)

43. Table 5.8: Augmented Design Matrix and Model Matrix, Leaf Spring Experiment

44. /*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;

45. d1 : (B,C,D,E,Q) = (+++−+) and (++−+−)

47. 5.5 Selection of 2k−p Designs Using Minimum Aberration and Related Criteria
Questions:
How to choose a good 2k−p design?
What criteria should be used to judge “goodness”?
Example: Two 29−4IV 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 = = = = = = = 26789
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

48. Comparing two 29-4 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.

49. Comparing two 29-4 design
d1 has less aberration than d2 (A4(d1)=6<A4(d2)=7).
d1 is preferred to d2 in general.
d2 has 15 clear 2fi’s, but d1 has 8 clear 2fi’s.
d2 may be superior to d1 if prior information is available on the importance of 2fi’s.
Guideline:
Minimum aberration designs are preferred in general (by hierarchical ordering principal).
Among resolution IV designs, those with the largest number of clear 2fi’s are the best.

50. Tables 5A.1-5A.7 2k−p fractional factorial designs.

51. Table 5A.2: 16-Run 2k−p FFD (k− p = 4)

52. Table 5A.3: 32 Run 2k−p FFD (k−p=5, 6≤k≤11)

53. 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?

54. Example: Two 26−2 designs in Table 5A.2
design d1 with generators 5 = 123, 6 = 124
I = 1235 = 1246 = 3456
resolution IV
6 clear effects: 6 ME’s.
design d2 with generators 5 = 12, 6 = 134
I = 125 = 1346 = 23456
resolution III
9 clear effects: 3, 4, 6, 23, 24, 26, 35, 45, 56.
d1 is generally preferred.
d2 is useful when some 2fi’s are important.

55. Choice of Fractions and Avoidance of Specific Combinations
A 27−4III design with 4=12, 5=13, 6=23 and 7=123.
There are 16 fractions (27−4) 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)

56. Factor assignment:
Table 5A.1 8-run designs
24−1: 4=123
25−2: 4=12, 5=13
26−3: 4=12, 5=13, 6=23
27−4: 4=12, 5=13, 6=23, 7=123

57. 5.6 Blocking in Fractional Factorial Designs
Example: To arrange a 26−2 design in 4 blocks in Table 4B.2.
design generators 5 = 123, 6 = 124
treatment defining contrast subgroup:
I = 1235 = 1246 = 3456
block generators B1 = 134,B2 = 234
block defining contrast subgroup:
B1 = 134,B2 = 234,B1B2 = 12

58. Aliasing relations: Clear effects: all 6 MEs
B1 = 134 = 245 = 236 = 156
B2 = 234 = 145 = 136 = 256
B1B2 = 12 = 35 = 46 =
1 = 235 = 246 = 13456
2 = 135 = 146 = 23456
3 = 125 = = 456
4 = = 126 = 356
5 = 123 = = 346
6 = = 124 = 345
13 = 25 = 2346 = 1456
14 = 26 = 2345 = 1356
15 = 23 = 2456 = 1346
16 = 24 = 2356 = 1345
34 = 56 = 1245 = 1236
36 = 45 = 1256 = 1234
Clear effects: all 6 MEs
Can estimate 3 block effects, 6 clear ME’s and 6 pairs of aliased 2fi’s.

59. 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.

60. Table 5B.1-5B.5 (p. 260-264): 2k−p designs in 2q blocks.
Run sizes 8, 16, 32, 64, 128 with k − p = 3, 4, 5, 6, 7
k≤9 and 1≤q≤k−p−1.
designs are chosen by # of clear effects.
Two 26−2 designs with k = 6, p = 2, q = 1, 2.
Research problem:
How to construct optimal blocked 2k−p designs with large run sizes (≥128)?