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Chapter 8Design & Analysis of Experiments1Chapter 8: Two-Level fractional factorial designThe one-half fraction of the 2k designThe general 2k-p fractional factorial design

In the two sample test, by the CLT you can use the normal

You use the t because you cannot use the normal (small sample size, unknown variance)

At the end, they have to perform an experiment (with randomization). Choose something within subjects. The card training game. The one you use to teach in work analysis for performance rating1Chapter 8Design & Analysis of Experiments2The one half fraction of the 2k designMotivation for fractional factorials is obvious; as the number of factors becomes large enough to be interesting, the size of the designs grows very quickly.

Emphasis is on factor screening; efficiently identify the factors with large effects

There may be many variables (often because we dont know much about the system)

Almost always run as unreplicated factorials, but often with center points

2Chapter 8Design & Analysis of Experiments3The one half fraction of the 2k designThe sparsity of effects principleThere may be lots of factors, but few are importantSystem is dominated by main effects, low-order interactions

The projection propertyEvery fractional factorial contains full factorials in fewer factors

Sequential experimentationCan add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation

3Chapter 8Design & Analysis of Experiments4The one half fraction of the 2k designNotation: because the design has 2k/2 runs, its referred to as a 2k-1 Consider a really simple case, the 23-1. The idea is to run 23-1 combinations instead of the 24 necessary for the full factorial.

ABC is called the generator or wordI is the defining relation = ABC is the defining relation. The defining relation of a fractional factorial is the set of columns that are equal to the identity column I. For a 2^(3-1), I=ABC

4Chapter 8Design & Analysis of Experiments5The one half fraction of the 2k designConsider the signs in ABC. If we select the 4 treatment combinations with the plus sign in ABCThe A and the BC contrasts are the same : A and BC are aliasedB is aliased with AC, and C is aliased with AB

ABC is called the generator or wordI is the defining relation = ABC is the defining relation. The defining relation of a fractional factorial is the set of columns that are equal to the identity column I. For a 2^(3-1), I=ABC

5Chapter 8Design & Analysis of Experiments6The one half fraction of the 2k designIn this case I=ABC is called the defining relation of the design. ABC can be also called a word or generatorA = BC, B = AC, C = AB (or me = 2 fi)Aliases can be found from the defining relation I = ABC by multiplication:AI = A(ABC) = A2BC = BCBI =B(ABC) = ACCI = C(ABC) = AB

me=main effectsfi: Factor interactions6Chapter 8Design & Analysis of Experiments7The one half fraction of the 2k designSuppose that I = -ABC is the defining relation (i.e., selecting the minus signs in the ABC factor)Implies slightly different aliases: A = -BC, B= -AC, and C = -ABBoth designs belong to the same family, defined by

Suppose that after running the principal fraction, the alternate fraction was also runThe two groups of runs can be combined to form a full factorial in two blocks of four runs each an example of sequential experimentation

7Chapter 88The one half fraction of the 2k designA design is of resolution R if no p-factor effect is aliased with another effect containing less than R-p factors

Resolution III Designs:me = 2fi example Resolution IV Designs:2fi = 2fiexampleResolution V Designs:2fi = 3fiexample

8Chapter 8Design & Analysis of Experiments9The one half fraction of the 2k designDESIGN GENERATION

Organize the columns of the first k-1 factors in Yates orderSelect the generator that provides the maximum resolutionMultiply the k factor by the generator used in the defining relationUse the resulting factors of the multiplication to create the column k

EXERCISE: Generate the design for the half fraction of the 24 factorial design9Chapter 8Design & Analysis of Experiments10The one half fraction of the 2k designEXAMPLE: PLASMA ETCH

Factorial designs can be used in developing a nitride etch process on a single wafer plasma etcher. The process uses C2F6 as the reactant gas. It is possible to vary the gas flow, the power applied to the cathode, the pressure in the reactor chamber, and the spacing between the anode and the cathode (gap).

LevelA (Gap, cm)B (Pressure, mTorr)C (C2F6 Flow, SCCM)D (Power, w)Low (-)0.8450125275High (+)1.255020032510Chapter 6Design & Analysis of Experiments11The one half fraction of the 2k designEXAMPLE: PLASMA ETCH DATA

The data was obtained using a design with I=ABCDHow do we generate the design in MINITAB?

ABCDEtch rate-1-1-1-15501-1-11749-11-11105211-1-1650-1-11110751-11-1642-111-16011111729Explain the way you would generate the table11Chapter 8Design & Analysis of Experiments12The one half fraction of the 2k designEXAMPLE: MINITAB DESIGN STRUCTURE

12Chapter 8Design & Analysis of Experiments13The one half fraction of the 2k designMINITAB OUTPUT

13Chapter 8Design & Analysis of Experiments14The one half fraction of the 2k designNORMAL AND HALF NORMAL PLOTS OF EFFECTS

14Chapter 8Design & Analysis of Experiments15The one half fraction of the 2k designMINITAB OUTPUT (REDUCED)

15Chapter 8Design & Analysis of Experiments16The one half fraction of the 2k designMINITAB OUTPUT (RESIDUALS)

16Chapter 8Design & Analysis of Experiments17The one half fraction of the 2k designMINITAB OUTPUT (RESIDUALS)

This residuals look weird. However, note that the residuals from the original experiment were not weird. This is a makeup example

If it were real, the interpretation is the one that really creates the significant differences17Chapter 8Design & Analysis of Experiments18The one half fraction of the 2k designMINITAB OUTPUT (FACTOR PLOTS)

18Chapter 6Design & Analysis of Experiments19The general 2k-p fractional factorial design2k-1 = one-half fraction, 2k-2 = one-quarter fraction, 2k-3 = one-eighth fraction, , 2k-p = 1/ 2p fraction

Add p columns to the basic design; select p independent generators

The defining relation for the design consists of the p generators initially chosen and the 2p p -1 generalized interactions

The alias structure may be found by multiplying each effect column by the defining relation

19Chapter 6Design & Analysis of Experiments20The general 2k-p fractional factorial designImportant to select generators so as to maximize resolution (e.g., it is usually better to choose a design where main effects are confounded with 3-way interactions (Resolution IV) over a design where main effects are confounded with 2-way interactions (Resolution III).

Projection a design of resolution R contains full factorials in any R 1 of the factors

Effects calculation:

20Chapter 6Design & Analysis of Experiments21EXAMPLE: INJECTION MOLDING DATA

Parts manufactured in an injection molding process are showing excessive shrinkage, which is causing problems in assembly operations upstream from the injection molding area. In an effort to reduce the shrinkage, a quality improvement team has decided to use a designed experiment to study the injection molding process. The team investigates six factors mold temperature (A), screw speed (B), holding time (C), cycle time (D), gate size (E), and holding pressure (F) each at two levels, with the objective of learning how each factor affects shrinkage and obtaining preliminary information about how the factors interact. Management will only approve at most 20 runs as it is necessary to stop the production for running the experiment. Suggest an experimental plan

The general 2k-p fractional factorial design21Chapter 6Design & Analysis of Experiments22EXAMPLE: INJECTION MOLDING DATA

The data was obtained using a designHow do we generate the design in MINITAB?

ABCDEFShrinkage-1-1-1-1-1-161-1-1-11-110-11-1-1113211-1-1-1160-1-11-11141-11-1-1115-111-1-1-126111-11-160-1-1-11-1181-1-111112-11-111-13411-11-1-160-1-1111-1161-111-1-15-1111-113711111152

The general 2k-p fractional factorial design22Chapter 623EXAMPLE: MINITAB DESIGN STRUCTURE

The general 2k-p fractional factorial design

23Chapter 824NORMAL AND HALF NORMAL PLOTS OF EFFECTS

The general 2k-p fractional factorial design

2425MINITAB OUTPUT

The general 2k-p fractional factorial design

25Chapter 626MINITAB OUTPUT (RESIDUALS)

The general 2k-p fractional factorial design

26Chapter 627MINITAB OUTPUT (RESIDUALS)

The general 2k-p fractional factorial design

27Chapter 628The general 2k-p fractional factorial design

28Chapter 629MINITAB OUTPUT (INTERACTION PLOTS)

The general 2k-p fractional factorial design

Interpret the significant ones29Chapter 630MINITAB OUTPUT (ABC FULL FACTORIAL DESIGN PROJECTION)

The general 2k-p fractional factorial design

Interpret the significant ones30Chapter 631MINITAB OUTPUT (FULL FACTORIAL ANALYSIS FOR ABC)

The general 2k-p fractional factorial design

Interpret the significant ones31Chapter 8Design & Analysis of Experiments32Appendix: Yates orderIt is also called the standard order

It standardizes the treatment combination arrangements for the 2k factorial designs

For full factorial designs you need to arrange the factors into k columns. The kth column consists of 2k-1 minus signs followed by 2k-1 plus signs

EXERCISE: For the 22 and 23 designs, arrange the treatment combinations using the standard or Yates order.

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