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ISU IE 361 - Module 22

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IE 361 Module 22Design and Analysis of Experiments: Part 3Prof.s Stephen B. Vardeman and Max D. MorrisReading: Section 6.3, Statistical Quality Assurance Methods for Engineers1In this module, we consid er the analysis of p-way factorial studies, and inparticular ones where each of the p factors has only 2 levels. These are the2 × 2 ×···×2 or 2pstudies.Basic 2pNotationWe wish now to think about experimentation and subsequent analysis for sys-tems that have many (p) factors potentially affecting a response, y. We beginwith full factorial studies (where all combinations of some levels of these factorsare represented in the data set) and concentrate on 2pstudies for two reasons.The first is that there is some special notation and structure that make their2analysis most transparent, and the second is that as a practical matter, one canrarely afford p-factor factorial experimentation with many levels of each factor.Example 22-1 As our motivating example, we will use data from a 23chem-ical process pilot plant study taken from Statistics for Experimenters by Box,Hunter, and Hunter. The response of interest was a yield variable (y in unitsof g). Factors and levels were as in the following table.Factor "Low" (−) and "High" LevelsA-Temperature 160◦C vs 180◦CB-Concentration 20% vs 40%C-Catalyst #1 vs #2Note that it is typical in 2pstudies to make an arbitrary designation of one levelof each factor a first or "low" level and the other level the second or "high"3level. In the pilot plant study, there were m =2runs of the pilot plant madeat each combination of levels of these 3 factors. We’ll let¯yijk= the sample mean yield at level i of A, level j of B, and level k of Candsijk= the sample standard deviation of yield atlevel i of A, level j of B, and level k of CThe catalyst data and some summary statistics are then given in the table belowalong with some additional notation for this 23factorial context.4ABC2pname ijk y’s ¯yijks2ijk−−− (1) 1 1 1 59,61 60 2+ −− a21174,70728− + − b12150,585432++− ab 2 2 1 69,67 68 2−−+ c11250,54528+ − + ac 2 1 2 81,85 83 8− ++ bc 1 2 2 46,44 45 2+++ abc 2 2 2 79,81 80 2While the "ijk" notation is perfectly general and could be applied to anyI × J × K factorial, the +/− notation used in the table and the special "2pname" convention (that names a combination of levels of the 3 factors by thosefactors appearing at their second or high level) are special to the case whereevery factor has only 2 levels. The "ijk" notation is helpful when one needsto indicate various averages of the sample means. For example, ¯yi..is the5average of all sample means for level i of Factor A, ¯y.jkistheaverageofallsample means for level j of Factor B and level k of Factor C, etc.It is helpful for picturing the results of a 23factorial study to plot the samplemeans obtained on the corners of a cube as shown in Figure 1.6Figure 1: The 23Sample Mean Yields ( g) in the Pilot Plant Study7Defining and Computing 3- (and by Analogy Higher-)WayFittedEffectsJust as it was helpful to define fitted effects in 2-way factorials, it is useful tomake similar definitions for 3- and higher-way studies. we begin (as we did for2-way studies) with fitted main effects.Theseareai=¯yi..− ¯y...=(theFactorAleveli average ¯y) − (the grand average ¯y)= the (fitted) main effect of the ith level of Factor Aandbj=¯y.j.− ¯y...=(the Factor B level j average ¯y) − (the grand average ¯y)= the (fitted) main effect of the jth level of Factor B8andck=¯y..k− ¯y...=(theFactorClevelk average ¯y) − (the grand average ¯y)= the (fitted) main effect of the kth level of F actor CThese are differences between "face" average means and the grand averagemean on a plot like Figure 1. For example, as portrayed on Figure 1,a2=(the right face average ¯y) − (the grand average ¯y)Example 22-1 continued In the pilot plant study, it is straightforward to seethata1=52.75 − 64.25 = −11.5 and a2=75.75 − 64.25 = 11.5andb1=66.75 − 64.25 = 2.5 and b2=61.75 − 64.25 = −2.59andc1=63.5 − 64.25 = −.75 and c2=65.0 − 64.25 = .75The relative sizes of the A,B, and C fitted main effects quantify what is tosome extent already obvious in Figure 1: The left-to-right differences in meanson the corners of the cube are bigger than the top-to-bottom or front-to-backdifferences.Just as in 2-way studies, the fitted main effectsforanyfactoraddto0overlevels of that factor (so that the fitted main effect for one level is just minusonetimesthatfortheotherlevel).Fitted 2 factor interactions in a 3-way factorial can be thought of in at leasttwo different ways. In the first place, they are measures of lack of parallelismthat would be appropriate after averaging out over levels of the 3rd factor.10Another is that they represent the what can be explained about a responsemean if one thinks of factors acting jointly in pairs beyond what is explainablein terms of them acting separately. The definitions of these look exactly likethe d efinitions from two-factor studies, except that an extra dot appears oneach ¯y. That isabij=¯yij.−³¯y...+ ai+ bj´andacik=¯yi.k− (¯y..+ ai+ ck)andbcjk=¯y.jk−³¯y...+ bj+ ck´It is a consequence of these definitions that fitted 2-factor interactions add to0 over levels of any one of the factors involved. In a 2pstudy, this allows one11to compute a single one of these fitted interactions of a given type and obtainthe other three by simple sign changes.Example 22-1 continued In the pilot plant study,ab11=¯y11.− (¯y...+ a1+ b1)=56− (64.25 + (−11.5) + 2.5) = .75so thatab12= −.75 and ab21= −.75 and ab22= .75Similarly,ac11=¯y1.1− (¯y...+ a1+ c1)=57− (64.25 + (−11.5) + (−.75)) = 5.0so thatac12= −5.0 and ac21= −5.0 and ac22=5.012And finally,bc11=¯y.11− (¯y...+ b1+ c1)=66− (64.25 + (2.5) + (−.75)) = 0so thatbc12=0 and bc21=0 and bc22=0The last of these says, for example, that after averaging over levels of Factor A,there would be perfect parallelism on an interaction plot of means ¯y.jk.Ontheother hand, the fairly large size of the AC two-factor interactions is consistentwith the clear lack of parallelism on Figure 2 , that is an interaction plot ofaverages of ¯y’s top to bottom (over levels of Factor B) from Figure 1.13Figure 2: Interaction Plot for Pilot Plant Study After Averaging over Levels ofB (Showing Strong AC Two-Factor Interaction/Lack of Parallelism)14Main effects and two-wa y interactions


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