Two level factorial designs (chap6)

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  • 1. 1 Chapter 6 The 2k Factorial Design
  • 2. 2 6.1 Introduction • The special cases of the general factorial design • k factors and each factor has only two levels • Levels: – quantitative (temperature, pressure,…), or qualitative (machine, operator,…) – High and low – Each replicate has 2 × … × 2 = 2k observations
  • 3. 3 • Assumptions: (1) the factor is fixed, (2) the design is completely randomized and (3) the usual normality assumptions are satisfied • Wildly used in factor screening experiments
  • 4. 4 6.2 The 22 Factorial Design • Two factors, A and B, and each factor has two levels, low and high. • Example: the concentration of reactant v.s. the amount of the catalyst (Page 219)
  • 5. 5 • “-” 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
  • 6. 6 • Average effect of a factor = the change in response produced by a change in the level of that factor averaged over the levels if the other factors. • (1), a, b and ab: the total of n replicates taken at the treatment combination. • The main effects: −+ −= + − + = −−+=−+−= AA yy n b n aab baab n abab n A 2 )1( 2 )]1([ 2 1 )]}1([]{[ 2 1 −+ −= + − + = −−+=−+−= BB yy n a n bab abab n baab n B 2 )1( 2 )]1([ 2 1 )]}1([]{[ 2 1
  • 7. 7 • The interaction effect: • In that example, A = 8.33, B = -5.00 and AB = 1.67 • Analysis of Variance • The total effects: n ab n ab baab n abab n AB 22 )1( ])1([ 2 1 )]}1([]{[ 2 1 + − + = −−+=−−−= baabContrast ababContrast baabContrast AB B A −−+= −−+= −−+= )1( )1( )1(
  • 8. 8 • Sum of squares: ABBATE i j n k ijkT AB B A SSSSSSSSSS n y ySS n abab SS n abab SS n baab SS −−−= −= −−+ = −−+ = −−+ = ⋅⋅⋅ = = = ∑∑∑ 4 4 ])1([ 4 )]1([ 4 )]1([ 22 1 2 1 1 2 2 2 2
  • 9. 9 Response:Conversion ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 291.67 3 97.22 24.82 0.0002 A 208.33 1 208.33 53.19 < 0.0001 B 75.00 1 75.00 19.15 0.0024 AB 8.33 1 8.33 2.13 0.1828 Pure Error 31.33 8 3.92 Cor Total 323.00 11 Std. Dev. 1.98 R-Squared 0.9030 Mean 27.50 Adj R-Squared 0.8666 C.V. 7.20 Pred R-Squared 0.7817 PRESS 70.50 Adeq Precision 11.669 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?
  • 10. 10 • Table of plus and minus signs: I A B AB (1) + – – + a + + – – b + – + – ab + + + +
  • 11. 11 • The regression model: – x1 and x2 are coded variables that represent the two factors, i.e. x1 (or x2 ) only take values on –1 and 1. – Use least square method to get the estimations of the coefficients – For that example, – Model adequacy: residuals (Pages 224~225) and normal probability plot (Figure 6.2) εβββ +++= 22110 xxy 21 2 00.5 2 33.8 5.27ˆ xxy       − +      +=
  • 12. 12 6.3 The 23 Design • Three factors, A, B and C, and each factor has two levels. (Figure 6.4 (a)) • Design matrix (Figure 6.4 (b)) • (1), a, b, ab, c, ac, bc, abc • 7 degree of freedom: main effect = 1, and interaction = 1
  • 13. 13
  • 14. 14 • Estimate main effect: • Estimate two-factor interaction: the difference between the average A effects at the two levels of B ])1([ 4n 1 4 )1( 4 abcacaba ])1([ 4 1 bccbabcacaba n bccb n yy bcabccacbaba n A AA −−−−+++= +++ − +++ = −= −+−+−+−= −+ n aacbbc n cababc acacbabbcabc n AB 44 )1( )]1([ 4 1 +++ − +++ = +−+−−+−=
  • 15. 15 • Three-factor interaction: • Contrast: Table 6.3 – Equal number of plus and minus – The inner product of any two columns = 0 – I is an identity element – The product of any two columns yields another column – Orthogonal design • Sum of squares: SS = (Contrast)2 /8n )]1([ 4n 1 )]}1([][][]{[ 4 1 −++−+−−= −+−−−−−= ababcacbcabc ababcacbcabc n ABC
  • 16. 16 Factorial Effect Treatment Combination I A B AB C AC BC ABC (1) + – – + – + + – a + + – – – – + + b + – + – – + – + ab + + + + – – – – c + – – + + – – + ac + + – – + + – – bc + – + – + – + – abc + + + + + + + + Contrast 24 18 6 14 2 4 4 Effect 3.00 2.25 0.75 1.75 0.25 0.50 0.50 Table of – and + Signs for the 23 Factorial Design (pg. 231)
  • 17. 17 • Example 6.1 A = carbonation, B = pressure, C = speed, y = fill deviation
  • 18. 18 Term Effect SumSqr % Contribution Model Intercept Error A 3 36 46.1538 Error B 2.25 20.25 25.9615 Error C 1.75 12.25 15.7051 Error AB 0.75 2.25 2.88462 Error AC 0.25 0.25 0.320513 Error BC 0.5 1 1.28205 Error ABC 0.5 1 1.28205 Error LOF 0 Error P Error 5 6.41026 Lenth's ME 1.25382 Lenth's SME 1.88156 • Estimation of Factor Effects
  • 19. 19 • ANOVA Summary – Full Model Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 73.00 7 10.43 16.69 0.0003 A 36.00 1 36.00 57.60 < 0.0001 B 20.25 1 20.25 32.40 0.0005 C 12.25 1 12.25 19.60 0.0022 AB 2.25 1 2.25 3.60 0.0943 AC 0.25 1 0.25 0.40 0.5447 BC 1.00 1 1.00 1.60 0.2415 ABC 1.00 1 1.00 1.60 0.2415 Pure Error 5.00 8 0.63 Cor Total 78.00 15 Std. Dev. 0.79 R-Squared 0.9359 Mean 1.00 Adj R-Squared 0.8798 C.V. 79.06 Pred R-Squared 0.7436 PRESS 20.00 Adeq Precision 13.416
  • 20. 20 • The regression model and response surface: – The regression model: – Response surface and contour plot (Figure 6.7) 21321 2 75.0 2 75.1 2 25.2 2 00.3 00.1ˆ xxxxxy       +      +      +      += Coefficient Standard 95% CI 95% CI Factor Estimate DF Error Low High Intercept 1.00 1 0.20 0.55 1.45 A-Carbonation 1.50 1 0.20 1.05 1.95 B-Pressure 1.13 1 0.20 0.68 1.57 C-Speed 0.88 1 0.20 0.43 1.32 AB 0.38 1 0.20 -0.072 0.82
  • 21. 21 • Contour & Response Surface Plots – Speed at the High Level F ill-deviation A: C a rb o n a tio n B:Pressure 10.00 10.50 11.00 11.50 12.00 25.00 26.25 27.50 28.75 30.00 0.5 1.375 2.25 3.125 2 2 2 2 -0.375 0.9375 2.25 3.5625 4.875 Fill-deviation 10.00 10.50 11.00 11.50 12.00 25.00 26.25 27.50 28.75 30.00 A : C a rb o n a tio n B : P re ssu re
  • 22. 22 • Refine Model – Remove Nonsignificant Factors Response: Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 70.75 4 17.69 26.84 < 0.0001 A 36.00 1 36.00 54.62 < 0.0001 B 20.25 1 20.25 30.72 0.0002 C 12.25 1 12.25 18.59 0.0012 AB 2.25 1 2.25 3.41 0.0917 Residual 7.25 11 0.66 LOF 2.25 3 0.75 1.20 0.3700 Pure E 5.00 8 0.63 C Total 78.00 15 Std. Dev. 0.81 R-Squared 0.9071 Mean 1.00 Adj R-Squared 0.8733 C.V. 81.18 Pred R-Squared 0.8033 PRESS 15.34 Adeq Precision 15.424
  • 23. 23 6.4 The General 2k Design • k factors and each factor has two levels • Interactions • The standard order for a 24 design: (1), a, b, ab, c, ac, bc, abc, d, ad, bd, abd, cd, acd, bcd, abcd two-factor interactions 2 three-factor interactions 3 1 factor interaction k k k             − M
  • 24. 24 • The general approach for the statistical analysis: – Estimate factor effects – Form initial model (full model) – Perform analysis of variance (Table 6.9) – Refine the model – Analyze residual – Interpret results • 2 ... )( 2 1 2 2 )1()1)(1( KABCkKABC KABCk KABC Contrast n SS Contrast n KABC kbaContrast    = = ±±±=
  • 25. 25 6.5 A Single Replicate of the 2k Design • These are 2k factorial designs with one observation at each corner of the “cube” • An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k • If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data
  • 26. 26 • Lack of replication causes potential problems in statistical testing – Replication admits an estimate of “pure error” (a better phrase is an internal estimate of error) – With no replication, fitting the full model results in zero degrees of freedom for error • Potential solutions to this problem – Pooling high-order interactions to estimate error (sparsity of effects principle) – Normal probability plotting of effects (Daniels, 1959)
  • 27. 27 • Example 6.2 (A single replicate of the 24 design) – A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin – The factors are A = temperature, B = pressure, C = concentration of formaldehyde, D= stirring rate
  • 28. 28
  • 29. 29 • Estimates of the effects Term Effect SumSqr % Contribution Model Intercept Error A 21.625 1870.56 32.6397 Error B 3.125 39.0625 0.681608 Error C 9.875 390.062 6.80626 Error D 14.625 855.563 14.9288 Error AB 0.125 0.0625 0.00109057 Error AC -18.125 1314.06 22.9293 Error AD 16.625 1105.56 19.2911 Error BC 2.375 22.5625 0.393696 Error BD -0.375 0.5625 0.00981515 Error CD -1.125 5.0625 0.0883363 Error ABC 1.875 14.0625 0.245379 Error ABD 4.125 68.0625 1.18763 Error ACD -1.625 10.5625 0.184307 Error BCD -2.625 27.5625 0.480942 Error ABCD 1.375 7.5625 0.131959 Lenth's ME 6.74778 Lenth's SME 13.699
  • 30. 30 • The normal probability plot of the effects Norm al plot Normal%probability E ffe ct -18.12 -8.19 1.75 11.69 21.62 1 5 10 20 30 50 70 80 90 95 99 A C D AC AD
  • 31. 31 C : C o n ce n tra tio n Interaction G raph FiltrationRate A: Te m p e ra tu re -1.00 -0.50 0.00 0.50 1.00 41.7702 57.3277 72.8851 88.4426 104 D : S tirrin g R a te Interaction G raph FiltrationRate A: T e m p e ra tu re -1.00 -0.50 0.00 0.50 1.00 43 58.25 73.5 88.75 104
  • 32. 32 • B is not significant and all interactions involving B are negligible • Design projection: 24 design => 23 design in A,C and D • ANOVA table (Table 6.13)
  • 33. 33 Response:Filtration Rate ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob >F Model 5535.81 5 1107.16 56.74 < 0.0001 A 1870.56 1 1870.56 95.86 < 0.0001 C 390.06 1 390.06 19.99 0.0012 D 855.56 1 855.56 43.85 < 0.0001 AC 1314.06 1 1314.06 67.34 < 0.0001 AD 1105.56 1 1105.56 56.66 < 0.0001 Residual 195.12 10 19.51 Cor Total 5730.94 15 Std. Dev. 4.42 R-Squared 0.9660 Mean 70.06 Adj R-Squared 0.9489 C.V. 6.30 Pred R-Squared 0.9128 PRESS 499.52 Adeq Precision 20.841
  • 34. 34 • The regression model: • Residual Analysis (P. 251) • Response surface (P. 252) Final Equation in Terms of Coded Factors: Filtration Rate = +70.06250 +10.81250 * Temperature +4.93750 * Concentration +7.31250 * Stirring Rate -9.06250 * Temperature * Concentration +8.31250 * Temperature * Stirring Rate
  • 35. 35 S tu d e n tize d R e s id u a ls Normal%probability Norm al plot of residuals -1.83 -0.96 -0.09 0.78 1.65 1 5 10 20 30 50 70 80 90 95 99
  • 36. 36 • Half-normal plot: the absolute value of the effect estimates against the cumulative normal probabilities. H a lf N o rm a l p lo t HalfNormal%probability |E ffe c t| 0.00 5.41 10.81 16.22 21.63 0 20 40 60 70 80 85 90 95 97 99 A C D AC AD
  • 37. 37 • Example 6.3 (Data transformation in a Factorial Design) A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
  • 38. 38 • The normal probability plot of the effect estimates Half Norm al plot HalfNormal%probability |E ffe ct| 0.00 1.61 3.22 4.83 6.44 0 20 40 60 70 80 85 90 95 97 99 B C D B C B D
  • 39. 39 • Residual analysis R e s id u a l Normal%probability Norm al plot of residuals -1.96375 -0.82625 0.31125 1.44875 2.58625 1 5 10 20 30 50 70 80 90 95 99 P re d icte d Residuals Residuals vs. P redicted -1.96375 -0.82625 0.31125 1.44875 2.58625 1.69 4.70 7.70 10.71 13.71
  • 40. 40 • The residual plots indicate that there are problems with the equality of variance assumption • The usual approach to this problem is to employ a transformation on the response • In this example, yy ln* =
  • 41. 41 Half Norm al plot HalfNormal%probability |E ffe ct| 0.00 0.29 0.58 0.87 1.16 0 20 40 60 70 80 85 90 95 97 99 B C D Three main effects are large No indication of large interaction effects What happened to the interactions?
  • 42. 42 Response: adv._rate Transform: Natural log Constant: 0.000 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 7.11 3 2.37 164.82 < 0.0001 B 5.35 1 5.35 371.49 < 0.0001 C 1.34 1 1.34 93.05 < 0.0001 D 0.43 1 0.43 29.92 0.0001 Residual 0.17 12 0.014 Cor Total 7.29 15 Std. Dev. 0.12 R-Squared 0.9763 Mean 1.60 Adj R-Squared 0.9704 C.V. 7.51 Pred R-Squared 0.9579 PRESS 0.31 Adeq Precision 34.391
  • 43. 43 • Following Log transformation Final Equation in Terms of Coded Factors: Ln(adv._rate) = +1.60 +0.58 * B +0.29 * C +0.16 * D
  • 44. 44 R e s id u a l Normal%probability Norm al plot of residuals -0.166184 -0.0760939 0.0139965 0.104087 0.194177 1 5 10 20 30 50 70 80 90 95 99 P re d icte d Residuals Residuals vs. P redicted -0.166184 -0.0760939 0.0139965 0.104087 0.194177 0.57 1.08 1.60 2.11 2.63
  • 45. 45 • Example 6.4: – Two factors (A and D) affect the mean number of defects – A third factor (B) affects variability – Residual plots were useful in identifying the dispersion effect – The magnitude of the dispersion effects: – When variance of positive and negative are equal, this statistic has an approximate normal distribution )( )( ln 2 2 * − + = iS iS Fi
  • 46. 46 6.6 The Addition of Center Points to the 2k Design • Based on the idea of replicating some of the runs in a factorial design • Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: 0 1 1 2 0 1 1 1 First-order model (interaction) Second-order model k k k i i ij i j i i j i k k k k i i ij i j ii i i i j i i y x x x y x x x x β β β ε β β β β ε = = > = = > = = + + + = + + + + ∑ ∑∑ ∑ ∑∑ ∑
  • 47. 47 no "curvature"F Cy y= ⇒ The hypotheses are: 0 1 1 1 : 0 : 0 k ii i k ii i H H β β = = = ≠ ∑ ∑ 2 Pure Quad ( )F C F C F C n n y y SS n n − = + This sum of squares has a single degree of freedom
  • 48. 48 • Example 6.6 5Cn = Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature
  • 49. 49 Response: yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 2.83 3 0.94 21.92 0.0060 A 2.40 1 2.40 55.87 0.0017 B 0.42 1 0.42 9.83 0.0350 AB 2.500E-003 1 2.500E-003 0.058 0.8213 Curvature 2.722E-003 1 2.722E-003 0.063 0.8137 Pure Error 0.17 4 0.043 Cor Total 3.00 8 Std. Dev. 0.21 R-Squared 0.9427 Mean 40.44 Adj R-Squared 0.8996 C.V. 0.51 Pred R-Squared N/A PRESS N/A Adeq Precision 14.234
  • 50. 50 • If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model