Chapter 6 The 2 k Factorial Design

1. Chapter 6 The 2k Factorial Design

2. 6.1 Introduction • The special cases of the general factorial design (Chapter 5) • 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. 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. 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 208)

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. 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:

7. The interaction effect: • In that example, A = 8.33, B = -5.00 and AB = 1.67 • Analysis of Variance • The total effects:

8. Sum of squares:

9. Response:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares] Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 291.67 3 97.22 24.82 0.0002A208.331208.3353.19< 0.0001B75.00175.0019.150.0024AB8.3318.332.130.1828 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. Table of plus and minus signs:

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.

12. Use least square method to get the estimations of the coefficients • For that example, • Model adequacy: residuals (Pages 213~214)

13. Response surface plot: • Figure 6.3

14. 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

16. Estimate main effect: • Estimate two-factor interaction: the difference between the average A effects at the two levels of B

17. 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

19. 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. 218)

20. Example 6.1 A = gap, B = Flow, C = Power, y = Etch Rate

22. The regression model and response surface: • The regression model: • Response surface and contour plot (Figure 6.7)

25. 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

26. 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

28. 6.5 A Single Replicate of the 2k Design • These are 2k factorial designs with oneobservation at each corner of the “cube” • An unreplicated 2k factorial design is also sometimes called a “singlereplicate” of the 2k • If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data

29. 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)

30. 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

32. 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 • Estimates of the effects

33. The normal probability plot of the effects

36. 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)

38. Response:Filtration Rate ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares] Sum ofMeanFSourceSquaresDFSquareValueProb >F Model 5535.81 5 1107.16 56.74 < 0.0001A1870.5611870.5695.86< 0.0001C390.061390.0619.990.0012D855.561855.5643.85< 0.0001AC1314.0611314.0667.34< 0.0001AD1105.5611105.5656.66< 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

39. 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 • The regression model: • Residual Analysis (P. 235) • Response surface (P. 236)

42. Half-normal plot: the absolute value of the effect estimates against the cumulative normal probabilities.

43. 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

44. The normal probability plot of the effect estimates

45. Residual analysis

46. 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,

47. Three main effects are large No indication of large interaction effects What happened to the interactions?

48. Response:adv._rateTransform:Natural logConstant: 0.000 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares]Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 7.11 3 2.37 164.82 < 0.0001B5.3515.35371.49< 0.0001C1.3411.3493.05< 0.0001D0.4310.4329.920.0001 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

49. Final Equation in Terms of Coded Factors: Ln(adv._rate) = +1.60 +0.58 * B +0.29 * C +0.16 * D • Following Log transformation