KETTERING MATH 602 - APPLIED STATISTICS
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LECTURE 10 1Winter 1999MATH602: APPLIED STATISTICSDr. Srinivas R. ChakravarthyDepartment of Science and MathematicsKETTERING UNIVERSITYFlint, MI 48504-4898Lecture 10 Winter 1999LECTURE 10 2Winter 1999FRACTIONAL FACTORIAL DESIGNS• Complete factorial designs cannot always be conducted.Why?• 7 factors at 2 levels requires 128 runs, a significantlylarge number. In some cases, it impossible to run allthese 128 combinations.• In some cases, the commercial production may have tobe stopped for the duration of the study.LECTURE 10 3Winter 1999• In any case, the expenses related to such experimentsare prohibitively large.• Often• higher order interactions are insignificant.• no distinguishable effects are noticed whenmoderately large number of factors are used.• Fractional factorial designs (FFD) are significantalternatives to the above situations. The fraction is acarefully selected subset of all combinations.LECTURE 10 4Winter 1999• In many cases, the experiments that involve manyfactors are routinely conducted as fractional factorialsto identify factor-level combinations for a future fullstudy using complete factorial designs.• It is important to note that1. the analysis of a FFD is relatively simple andstraightforward; and2. the use of a FFD doesn’t stop us from completing afull factorial design.LECTURE 10 5Winter 1999CONFOUNDING IN FFDIn a complete factorial design involving k factors at twolevels, we have 2k units, which and the analysis of such adesign results in estimating k main effects and (2k-k-1)interaction effects. Suppose we look at a fractionalfactorial design, say, the fraction 1/2p. That is, there willbe 2k-p experimental units. Obviously, some effects aregoing to be confounded with one another.LECTURE 10 6Winter 1999What is confounding?Two or more effects are said to be confounded ifcalculated effects can only be attributed to their jointinfluence on the response, and not to their individualones. That is to say that the contrasts of the effects willbe the same (except possibly for the sign).LECTURE 10 7Winter 1999Construction and Analysis of FFDHere we will illustrate how to construct and analyzefractional factorial designs through a number ofexamples.Defining Relation: FFD’s are generated using one ormore generators. The relation that defines the generator iscalled the defining relation. This is the key to theconfounding pattern.LECTURE 10 8Winter 1999Analysis: The analysis of a FFD is very similar to whatwe saw earlier in the case of a factorial design. However,here one has to be even more careful in interpreting theresults as some factor effects are confounded with oneanother.LECTURE 10 9Winter 1999DESIGN RESOLUTIONIn practice, an important tool that will be used inselecting a FFD out of many possible ones, is the conceptof design resolution. This identifies the order ofconfounding of the main effects and the interactions.Design Resolution: A design is said to be a design ofresolution R if no p-factor effect is confounded with anyother effect containing less than R-p factors.LECTURE 10 10Winter 1999In general, the resolution of a 2-level FFD is the length ofthe shortest word in the defining relation.Note: The resolution of a design is denoted byappropriate Roman letter appended as a subscript.• For example, RIII refers to a design of resolution R = 3.• Also, another notation that is commonly used to denotea FFD is 2IIIk-p. This a p-fraction 2-level, k-factor FFDof resolution III.LECTURE 10 11Winter 1999• To further grasp the idea of design resolution, let uslook at the following examples:(a) A design resolution of III does not confound maineffects with one another but does confound the maineffects with two-factor interactions.(b) A design resolution of IV does not confound maineffects and two-factor interactions but does confound thetwo-factor interactions with the other two-factorinteractions.LECTURE 10 12Winter 1999(c) A design resolution of V does not confound maineffects and two-factor interactions but does confound thetwo-factor interactions with the other three-factorinteractions, and so on.HOW TO SELECT A FFD?Having seen how to construct a FFD using appropriategenerator(s), and also know how to analyze a FFD, thequestion now is how to select a generator in practice?LECTURE 10 13Winter 1999That is, how to select an appropriate FFD for a problemunder study in practice? Recall that “no matter how bestyou choose a statistical technique for analysis, it will notdo much good if the design is not properly chosen”.Hence, it is imperative that one is able to identify aproper design for the problem under study.After identifying the important factors that influence theresponse variable for the problem under study, useLECTURE 10 14Winter 1999(a) the existing tables to choose a FFD(b) computer to generate a FFD(c) trial and error method to identify a FFD.LECTURE 10 15Winter 1999EXAMPLE 5: (24-1 design - Problem 7.18):• A chemical product is produced in a pressure vessel• Four factors at two levels each• A – Temperature• B – Pressure• C – Concentration of Formaldehyde• D – Stirring Rate• Response is the filtration rate.LECTURE 10 16Winter 1999EXAMPLE 6: (26-2design - Example 7.9):• Shrinkage (Y) of parts in injection molding process.• Six factors at two levels each• A – Molding Temperature• B – Screw Speed• C – Holding Time• D – Cycle Time• E – Gate Size• F – Holding PressureLECTURE 10 17Winter 1999 Use E = ABC and F = BCD as generatorsDefining relation is: I = ABCE = BCDF = ADEFAlias Structure : I + ABCE + ADEF + BCDFA + BCE + DEF + ABCDFB + ACE + CDF + ABDEFC + ABE + BDF + ACDEFD + AEF + BCF + ABCDEE + ABC + ADF + BCDEFF + ADE + BCD + ABCEFABD + ACF + BEF + CDEABF + ACD + BDE + CEFAB + CE + ACDF + BDEFAC + BE + ABDF + CDEFAD + EF + ABCF + BCDEAE + BC + DF + ABCDEFAF + DE + ABCD + BCEFBD + CF + ABEF + ACDEBF + CD + ABDE + ACEFLECTURE 10 18Winter 1999 CONSTRUCTION OF A FFD• A 2k-1 FFD of highest resolution construction.• Write down a full FD for k-1 factors.• Add the k-th factor by identifying its high and lowlevels to those of the highest order-interaction.[Note: We can use any order interaction to assign the k-thfactor, but will not get the highest resolution].LECTURE 10 19Winter 1999• A 2k-2 FFD is constructed as follows• Write down a full FD with k-2 factors.• Choose two generators, say, P and Q.Design Generators: E = ABC and


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KETTERING MATH 602 - APPLIED STATISTICS

Course: Math 602-
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