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Design & Analysis of Experiments: 2k Factorial Design

Explore the 2k factorial design in the Design & Analysis of Experiments book, understand factor effects, and analyze the chemical process example.

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Design & Analysis of Experiments: 2k Factorial Design

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  1. Design & Analysis of Experiments 8E 2012 Montgomery STT 511-STT411:DESIGN OF EXPERIMENTS AND ANALYSIS OF VARIANCEDr. Cuixian Chen Chapter 6: The 2k Factorial Design

  2. Design & Analysis of Experiments 8E 2012 Montgomery

  3. Review: The Battery Design Experiment: ANOVA for two-factor CRD What is the name of this design? This design is sometimes called a 3^2 factorial design. Power=2 represents two factors; Base=3 represents three levels. Design & Analysis of Experiments 8E 2012 Montgomery

  4. Review: Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 189 Then basically 2^2 factorial design is a special case of Ch5 with a=b=2. What is the deal? Design & Analysis of Experiments 8E 2012 Montgomery

  5. Review: Some Basic Definitions Parallel Definition of a factor effect: The change in the mean response when the factor is changed from low to high The main effect of a factor is the difference b/w average at low level and the average at high level. Lack of Interaction effect b/w Factor A & Factor B Main effect of Factor A Main effect of Factor B Design & Analysis of Experiments 8E 2012 Montgomery

  6. The 22 Factorial Design

  7. Design of Engineering Experiments– The 2k Factorial Design • Text reference, Chapter 6 • Special case of general factorial design; k factors, all at two levels • The two levels are usually called low and high (they could be either quantitative or qualitative) • Very widely used in industrial experimentation • Form a basic “building block” for other very useful experimental designs • Special (short-cut) methods for analysis • We will make use of Design-Expert Design & Analysis of Experiments 8E 2012 Montgomery

  8. Analysis Procedure for a Factorial Design • These designs are widely used in screening experiments. • Estimate factor effects • Formulate model • With replication, use full model • Statistical testing (ANOVA) • Refine the model • Analyze residuals (graphical) • Interpret results Design & Analysis of Experiments 8E 2012 Montgomery

  9. Chemical Process Example: Simplest Case: The 22 A=Reactant Concentration: Low=15%; High=25%; B=Amount of catalyst. Low=1 pound; High=2 pounds. • Consider an investigation into the effect of the concentration of reactant and amount of catalyst on conversion (yield) in a chemical process. The objective of the experiment was to determine if adjustments to either of these two factors would increase yield. Let reactant concentration be factor A and let the two levels of interest be 15 and 25 percent. The catalyst is factor B, with the high level denoting the use of 2 pounds of the catalyst and the low level denoting the use of only 1 pound. The experiment is replicated three times, so there are 12 runs. The order in which the runs are made is random, so this is a completely randomized experiment. Design & Analysis of Experiments 8E 2012 Montgomery

  10. Chemical Process Example: Simplest Case: The 22 A=Reactant Concentration: Low=15%; High=25%; B=Amount of catalyst. Low=1 pound; High=2 pounds. A = reactant concentration, B = catalyst amount, y = recovery The appearance of the data? Design & Analysis of Experiments 8E 2012 Montgomery

  11. The Simplest Case: The 22 Note: a represents treat combination of A at high level and B at low level, b represents A at low level and B at high level, and ab represents both factors at high level. By convention, (1) is used to denote both factors at low level. Also (1), a, b, and ab represent total of response observation at all n replicates taken at treatment combination Design & Analysis of Experiments 8E 2012 Montgomery “-” and “+” denote the low and high levels of a factor, respectively • Low and high are arbitrary terms • Geometrically, the four runs form the corners of a square • Factors can be quantitative or qualitative, although their treatment in the final model will be different

  12. Estimation of Factor Effects See textbook, pg. 235-236 for manual calculations The effect estimates are: A = 8.33, B = -5.00, AB = 1.67 Practical interpretation? Design-Expertanalysis Definition of a factor effect: Change in mean response when factor is changed from low to high. The main effect of a factor is difference b/w average at low level and average at high level. Design & Analysis of Experiments 8E 2012 Montgomery

  13. The effect estimates are: A = 8.33, B = -5.00, AB = 1.67 Practical interpretation? Why do we consider Factor Effects? • The effect of A (reactant concentration) is positive; this suggests that increasing A from the low level (15%) to the high level (25%) will increase the yield. • The effect of B (catalyst) is negative; this suggests that increasing the amount of catalyst added to the process will decrease the yield. • The interaction effect appears to be small relative to the two main effects. • In experiments involving 2k designs, it is always important to examine the magnitude and direction of the factor effects to determine which variables are likely to be important. The analysis of variance can generally be used to confirm this interpretation (t-tests could be used too). • Effect magnitude and direction should always be considered along with the ANOVA, because the ANOVA alone does not convey this information. Design & Analysis of Experiments 8E 2012 Montgomery

  14. Chemical Process Example: Simplest Case: The 22 A=Reactant Concentration: Low=15%; High=25%; B=Amount of catalyst. Low=1 pound; High=2 pounds. ## 2^2 factorial design ## A=c(rep(15, 6), rep(25, 6)); B=c(rep(1, 3), rep(2, 3), rep(1, 3), rep(2, 3)); y=c(28, 25, 27, 18, 19, 23, 36, 32, 32, 31, 30, 29); anova(lm(y~factor(A)+factor(B)+factor(A)*factor(B))) ##as.factor works the same as factor ## anova(lm(y~factor(A)*factor(B))) ## “*” sign here means all combination of Factor A, B and interaction AB ## ## Or use ## tab6.1<-read.table("http://people.uncw.edu/chenc/STT411/dataset%20backup/Chemical-Process-Yield.txt",header = TRUE); A<- tab6.1$Conc.; B<- tab6.1$Catalyst; y<- tab6.1$Yield anova(lm(y~factor(A)*factor(B))); ## But it is NOT the same as ## anova(lm(y~factor(A*B))); ## Design & Analysis of Experiments 8E 2012 Montgomery

  15. Statistical Testing - ANOVA The F-test for the “model” source is testing the significance of the overall model; That is, is either A, B, or AB or some combination of these effects important? Design & Analysis of Experiments 8E 2012 Montgomery

  16. Estimation of Factor Effects form ANOVA table Term EffectSumSqr % Contribution Model Intercept Model A 8.33333 208.333 64.4995 Model B -5 75 23.2198 Model AB 1.66667 8.33333 2.57998 Model Error 31.3333 9.70072 Model Total 323 n=3, SSA= 208.333, abs(effect A) =(SSA/n)^0.5 =(208.333/3)^0.5 = 8.3333 abs(effect B) =(SSB/n)^0.5 =(75 /3)^0.5 = 5 abs(effect AB) =(SSAB/n)^0.5 =(8.33333 /3)^0.5 =1.666666 Design & Analysis of Experiments 8E 2012 Montgomery

  17. Statistical Testing - ANOVA Reviews: Q1: what is the absolute value of the effect of factor A? Q2: what is the absolute value of the effect of factor B? Q3: what is the absolute value of the interaction effect of A and B? Q4: what is R^2, the goodness of fit? Q5: What are the THREE hypotheses? Shall we reject all the null hypotheses? Q6: If we consider model selection, what would be more appropriate model to consider? Q7: How many observations in total and how many replicates for each combination of the factors? R^2 =SSModel/SST =1- SSE/SST Design & Analysis of Experiments 8E 2012 Montgomery

  18. Residuals and Diagnostic Checking Figure 6.2 presents a normal probability plot of these residuals and a plot of the residuals versus the predicted yield. These plots appear satisfactory, so we have no reason to suspect that there are any problems with the validity of our conclusions. Design & Analysis of Experiments 8E 2012 Montgomery

  19. Example: Another 23 factorial design Design & Analysis of Experiments 8E 2012 Montgomery

  20. Example: Another 23 factorial design dat5.3<-read.table("http://people.uncw.edu/chenc/STT411/dataset%20backup/BHH2-Data/tab0503.dat",header = TRUE); anova(lm(y~factor(K)+factor(T)+factor(K)*factor(T), data=dat5.3)); anova(lm(y~factor(K)+factor(T)+factor(K*T), data=dat5.3)); anova(lm(y~factor(K)*factor(T), data=dat5.3)); 4*(mean(dat5.3$y[dat5.3$T==1])-mean(dat5.3$y[dat5.3$T==-1]) )^2 ## to find SST=2116 ## 4*(mean(dat5.3$y[dat5.3$K==1])-mean(dat5.3$y[dat5.3$K==-1]) )^2 ## to find SSK=9 ## 4*(mean(dat5.3$y[dat5.3$T*dat5.3$K==1])-mean(dat5.3$y[dat5.3$T*dat5.3$K==-1]) )^2 ## to find SSTK=400 ## Design & Analysis of Experiments 8E 2012 Montgomery

  21. Example: Another 23 factorial design Q1: what is the absolute value of effect of K ? Q2: what is the absolute value of effect of factor T? Q3: what is the absolute value of the interaction effect of T and K? Q4: what is R^2, the goodness of fit? Q5: Shall we reject all the null hypotheses? Q6: If we consider model selection, what would be the more appropriate model to consider? Q7: How many observations in total and how many replicates for each combination of the factors? Design & Analysis of Experiments 8E 2012 Montgomery

  22. The 23 Factorial Design

  23. The 23 Factorial Design Definition of a factor effect: Change in mean response when factor is changed from low to high. The main effect of a factor is difference b/w average at low level and average at high level. Design & Analysis of Experiments 8E 2012 Montgomery

  24. Effects in The 23 Factorial Design Analysis done via computer Design & Analysis of Experiments 8E 2012 Montgomery

  25. An Example of a 23 Factorial Design • A 2^3 factorial design was used to develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between electrodes, the gas flow (C2F6 is used as the reactant gas), and the RF power applied to the cathode (see Figure 3.1 for a schematic of the plasma etch tool). Each factor is run at two levels, and design is replicated twice. The response variable is the etch rate for silicon nitride (Å/m). The etch rate data are shown in Table 6.4, and the design is shown geometrically in Figure 6.6. Design & Analysis of Experiments 8E 2012 Montgomery

  26. The 23 Factorial Design: finding factor effect Design & Analysis of Experiments 8E 2012 Montgomery

  27. The 23 Factorial Design: finding factor effect Design & Analysis of Experiments 8E 2012 Montgomery

  28. The 23 Factorial Design: finding factor effect Design & Analysis of Experiments 8E 2012 Montgomery

  29. Table of – and + Signs for the 23 Factorial Design (pg. 218) The appearance of the data? Design & Analysis of Experiments 8E 2012 Montgomery

  30. Properties of the Table • Except for column I, every column has an equal number of + and – signs • The sum of the product of signs in any two columns is zero. • Multiplying any column by I leaves that column unchanged (identity element) • The product of any two columns yields a column in the table: • Orthogonal design • Orthogonality is an important property shared by all factorial designs Design & Analysis of Experiments 8E 2012 Montgomery

  31. An Example of a 23 Factorial Design Details of Low and High levels. A = gap, B = Flow, C = Power, y = Etch Rate tab6.4<-read.table("http://people.uncw.edu/chenc/STT411/dataset%20backup/nitride-etch.txt",header = TRUE); anova(lm(tab6.4$y~factor(tab6.4$A)*factor(tab6.4$B)*factor(tab6.4$C))); anova(lm(tab6.4$y~factor(tab6.4$A)+factor(tab6.4$C)+factor(tab6.4$A*tab6.4$C))); 4*(mean(tab6.4$y[factor(tab6.4$A*tab6.4$B*tab6.4$C)==1])-mean(tab6.4$y[factor(tab6.4$A*tab6.4$B*tab6.4$C)==-1]) )^2 ## SSABC = 126.5625 ## Design & Analysis of Experiments 8E 2012 Montgomery

  32. ANOVA Summary – Full Model Design & Analysis of Experiments 8E 2012 Montgomery

  33. Factorial design in SAS /*############## Factorial design in SAS ######*/ data tab5p3; infile '\\bearsrv\classrooms\Math\wangy\stt4511\BHH2-Data\tab0503.dat' firstobs=2; input run T C K y; run; procprint data=tab5p3; run; procglm data=tab5p3; /*classes K T C; this is only used if you do not need the estimate!*/ model y = K T C K*T K*C T*C K*T*C ; lsmeans K T C K*T; run; Design & Analysis of Experiments 8E 2012 Montgomery

  34. Design & Analysis of Experiments 8E 2012 Montgomery

  35. Estimation of Factor Effects Verify the relationship between the effect and the sum of square. n=2, SSA= 41310.5625, abs(effect A) = (SSA/(2*n))^0.5 =(41310.5625/(2*2))^0.5 = 101.625 abs(effect B) = (SSB/(2*n))^0.5 =(217.5625/(2*2))^0.5 = 7.375 abs(effect C) = (SSC/(2*n))^0.5 =(374850.0625/(2*2))^0.5 =306.125 abs(effect AB) = (SSAB/(2*n))^0.5 =(2475.0625/(2*2))^0.5 =24.875 abs(effect AC) = (SSAC/(2*n))^0.5 =(94402.5625/(2*2))^0.5 =153.625 abs(effect BC) = (SSBC/(2*n))^0.5 =(18.0625/(2*2))^0.5 =2.125 abs(effect ABC) = (SSABC/(2*n))^0.5 =(126.5625/(2*2))^0.5 =5.625 Design & Analysis of Experiments 8E 2012 Montgomery

  36. ANOVA Summary – Full Model & refined model Design & Analysis of Experiments 8E 2012 Montgomery

  37. ANOVA Summary – Full Model Review: Q1: what is R^2, the goodness of fit? Q2: State clearly all SEVEN hypotheses. Shall we reject all the null hypotheses? Q3: If we consider model selection, what would be the more appropriate model to consider? Based on the previous PPT slide, may we build the refined model by hand? Why? Q4: How many observations in total and how many replicates for each combination of the factors? Design & Analysis of Experiments 8E 2012 Montgomery

  38. Model Summary Statistics for Reduced Model • R2 for FULL model to measure the proportion of total variability explained by the model. A potential problem with this statistic is that it always increases as factors are added to the model, even if these factors are not significant. • The adjusted R2 is a statistic that is adjusted for the “size” of the model, that is, the number of factors. The adjusted R2 can actually decrease if non-significant terms are added to a model. Design & Analysis of Experiments 8E 2012 Montgomery

  39. Model Summary Statistics for Reduced Model • R2for prediction (based on PRESS). • The PRESS statistic is a measure of how well the model will predict new data. (PRESS is actually an acronym for prediction error sum of squares, and it is computed as the sum of the squared prediction errors obtained by predicting the i-th data point with a model that includes all observations except the i-th one.) • A model with a small valueof PRESS indicates that the model is likely to be a good predictor. Design & Analysis of Experiments 8E 2012 Montgomery

  40. Design & Analysis of Experiments 8E 2012 Montgomery

  41. A revisit of the Pilot Plant investigation by considering all factors. dat5.3<-read.table("http://people.uncw.edu/chenc/STT411/dataset%20backup/BHH2-Data/tab0503.dat",header = TRUE); #anova(lm(y~factor(K)+factor(T)+factor(C)+factor(K*T)+factor(T*C)+factor(K*C)+factor(K*T*C),data=dat5.3)); anova(lm(y~factor(K)+factor(T)+factor(C)+factor(K*T)+factor(T*C)+factor(K*C)+factor(K*T*C),data=dat5.3)); ## Simplified model ## anova(lm(y~factor(K)*factor(T)*factor(C),data=dat5.3)); Design & Analysis of Experiments 8E 2012 Montgomery

  42. Q1: what is the absolute value of the effect of factor T ? • Q2: what is the absolute value of the effect of factor C? • Q3: What is the absolute value of the interaction effect of KTC? • Q4: what is R^2, the goodness of fit? • Q5: Shall we reject all the null hypotheses? • Q6: If we consider model selection, what would be the more appropriate model to consider? • Q7: How many observations in total and how many replicates for each combination of the factors? Design & Analysis of Experiments 8E 2012 Montgomery

  43. How do we analysis exe5.3 in SAS data exe5p3; infile '\\bearsrv\classrooms\Math\wangy\stt4511\BHH2-Data\exe0503.dat' firstobs=2; input test depth depthLevel watering wateringLevel$ type typeLevel$ rep1 rep2 rep3; run; procprint data=exe5p3; run; data rep1; set exe5p3; keep depth watering type rep1; run; procprint data=rep1; run; data rep2; set exe5p3; keep depth watering type rep1 rep2; rep1=rep2; drop rep2; run; procprint data=rep2; run; data rep3; set exe5p3; keep depth watering type rep1 rep3; rep1=rep3; drop rep3; run; procprint data=rep3; run; data new; set rep1 rep2 rep3; run; procprint data=new; run; procglm data=new; model rep1 = depth watering type depth*watering depth*type watering*type depth*watering*type ; run; Design & Analysis of Experiments 8E 2012 Montgomery

  44. The general 2k Factorial Design (skipped)

  45. The General 2kFactorial Design • Section 6-4, pg. 253, Table 6-9, pg. 25 • There will be k main effects, and Design & Analysis of Experiments 8E 2012 Montgomery

  46. The General 2kFactorial Design: example 6.2 • A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product. The four factors are temperature (A), pressure (B), concentration of formaldehyde (C), and stirring rate (D). Each factor is present at two levels. The design matrix and the response data obtained from a single replicate of the 24 experiment are shown in Table 6.10 and Figure 6.10. The 16 runs are made in random order. The process engineer is interested in maximizing the filtration rate. Current process conditions give filtration rates of around 75 gal/h. The process also currently uses the concentration of formaldehyde, factor C, at the high level. The engineer would like to reduce the formaldehyde concentration as much as possible but has been unable to do so because it always results in lower filtration rates. Design & Analysis of Experiments 8E 2012 Montgomery

  47. The General 2kFactorial Design: example 6.2 Design & Analysis of Experiments 8E 2012 Montgomery

  48. The General 2kFactorial Design: example 6.2 Design & Analysis of Experiments 8E 2012 Montgomery

  49. Estimates of the Effects The normal probability plot of factor effects is shown in Figure 6.11. All of effects that lie along the line are negligible, whereas the large effects are far from the line. The important effects that emerge from this analysis are the main effects of A, C, and D and the AC and AD interactions. Design & Analysis of Experiments 8E 2012 Montgomery

  50. The Half-Normal Probability Plot of Effects Design & Analysis of Experiments 8E 2012 Montgomery

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