Many-Factor Studies With 2-Level FactorsDefining EffectsMain Effects2-Factor Interactions3-Way InteractionsThe Yates Algorithm for Computing Fitted EffectsFitted/Predicted Mean ResponsesJudging the "Statistical Significance" of Fitted Factorial EffectsConfidence Limits for EffectsEffect Sparsity and Normal Plotting of Fitted effectsIE 361 Module 22Design and Analysis of Experiments: Part 3(p-Way Studies and Analyses With 2-Level Factors)Reading: Section 6.3, Statistical Quality Assurance Methods forEngineersProf. Steve Vardeman and Prof. Max MorrisIow a State UniversityVardeman and Morris (I owa State University) IE 3 61 Module 22 1 / 28Complete p-Factor Studies (2-Level Factors)We wish now to think about experimentation and subsequent analysis forsystems that have many (p) factors potentially a¤ecting a response, y.We begin with full/complete p-way factorial studies (where allcombinations of some levels of these factors are represented in the dataset) and concentrate on 2  2      2 or 2pstudies (ones where each ofthe p factors has only 2 levels). There are two reasons for mostlyspecializing to 2-level factors. The …rst is that there is some specialnotation and structure that make their analysis most transparent, and thesecond is that as a practical matter, one can rarely a¤ord p-factor factorialexperimentation with many (more than 2) levels of the factors.Vardeman and Morri s (Iowa State University) IE 361 Modu le 22 2 / 28p-Factor NotationExample 22-1As our motivating example, we will use data from a 23chemical processpilot plant study taken from Statistics for Experimenters by Box, Hunter,and Hunter. The response of interest was a yield variable (y in units ofg). 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 onelevel of each factor a …rst or "low" level and the other level the second or"high" level.Vardeman and Morri s (Iowa State University) IE 361 Modu le 22 3 / 28p-Factor NotationExample 22-1 continuedIn the pilot plant study, there were m = 2 runs of the pilot plant made ateach 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 tableon panel 5 along with some additional notation for this 23factorialcontext. (The "ijk" notation above is very helpful when one needs toindicate various averages of the sample means. For example, ¯yi..is theaverage of all sample means for level i of Factor A, ¯y.jkis the average ofall sample means for level j of Factor B and level k of Factor C, etc.)Vardeman and Morri s (Iowa State University) IE 361 Modu le 22 4 / 28p-Factor Notation (2-Level Factors)Example 22-1 continuedA B C 2pname i j k y’s ¯yijks2ijk   (1) 1 1 1 59,61 60 2+   a 2 1 1 74,70 72 8 +  b 1 2 1 50,58 54 32+ +  ab 2 2 1 69,67 68 2  + c 1 1 2 50,54 52 8+  + 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 factorsby those factors appearing at their second or high level) are special to thecase where every factor has only 2 levels.Vardeman and Morri s (Iowa State University) IE 361 Modu le 22 5 / 28A Useful 3-Factor GraphicExample 22-1 continuedIt is helpful for picturing the results of a 23factorial study to plot thesample means obtained on the corners of a cube as shown in the …gurebelow.Figure: The 23Sample Mean Yields ( g) in the Pilot Plant StudyVardeman and Morri s (Iowa State University) IE 361 Modu le 22 6 / 28De…ning 3-Way (and Higher-Way) Fitted Factorial E¤ectsMain e¤ectsWe proceed to de…ne …tted e¤ects for 3- (and by analogy, higher-) waystudies, beginning with …tted main e¤ects. These areai= ¯yi.. ¯y...= ( the Factor A level i average ¯y)  (the grand average ¯y )= the (…tted) main e¤ect of the ith level of Factor Abj= ¯y.j. ¯y...= ( the Factor B level j average ¯y )  (the grand average ¯y )= the (…tted) main e¤ect of the jth level of Factor Bandck= ¯y..k ¯y...= ( the Factor C level k average ¯y)  ( the grand average ¯y )= the (…tted) main e¤ect of the kth level of Factor CVardeman and Morri s (Iowa State University) IE 361 Modu le 22 7 / 28De…ning 3-Way (and Higher-Way) Fitted Factorial E¤ectsMain e¤ectsMain e¤ects in a 23study are di¤erences between "face" average meansand the grand average mean on a plot like that on panel 6. As portrayedon that panela2= (the right face average ¯y )  (the grand average ¯y)andb2= (the top face average ¯y )  (the grand average ¯y)andc2= (the back face average ¯y )  (the grand average ¯y)Just as in 2-way studies, the …tted main e¤ects for any factor add to 0over levels of that factor (so that a …tted main e¤ect for one level in a 2pstudy is just minus one times that for the other level).Vardeman and Morri s (Iowa State University) IE 361 Modu le 22 8 / 28Fitted Main E¤ectsExample 22-1 continuedIn the pilot plant study, it is straightforward to see thata1= 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 = 2.5andc1= 63.5  64.25 = .75 and c2= 65.0  64.25 = .75The relative sizes of the A,B, and C …tted main e¤ects quantify what is tosome extent already obvious on panel 6: The left-to-right di¤erences inmeans on the corners of the cube are bigger than the top-to-bottom orfront-to-back di¤erences.Vardeman and Morri s (Iowa State University) IE 361 Modu le 22 9 / 28De…ning 3-Way (and Higher-Way) Fitted Factorial E¤ects2-Way Interactions (in a 3-Way Study)Fitted 2 factor interactions in a 3-way factorial can be thought of in atleast two ways. First, they are measures of lack of parallelism that wouldbe appropriate after averaging out over levels of the 3rd factor. Second,they represent what can be explained about a response mean if one thinksof factors acting jointly in pairs beyond what is explainable in terms ofthem acting separately. The de…nitions of these