dB dB dB dB 1 STA 536 – Fractional Factorial Experiments at Two Levels dB dB dB dB 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 2 k−p designs.
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1 STA 536 – Fractional Factorial Experiments at Two Levels Fractional Factorial Experiments at Two Levels Source : sections 5.1-5.5, part of 5.6 Effect.
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CommandButton1CommandButton1CommandButton1CommandButton1 1STA 536 – Fractional Factorial Experiments at Two Levels
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.
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5.2 Fractional Factorial Designs: Effect Aliasing and the Criteria of Resolution and Minimum Aberration
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 10+5+1 = 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.
Why Using Fractional Factorial Designs?
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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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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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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.)
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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;
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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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. 9-16 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 9-16. 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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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
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 5.4.2.
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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 (−++++)
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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.
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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)
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