1. 1 The 2 3 Factorial Design Standard order: (1), a, b, ab, c, ac, bc, abc

2. 2 Effects in The 2 3 Factorial Design Analysis done via computer

3. 3 An Example of a 2 3 Factorial Design A = carbonation (%), B = pressure (psi), C = speed (bpm), y = fill deviation (mm) A (%)

4. 4 Factorial Effect Treatment Combination I ABABCACBCABC (1) = -4+--+-++- a = 1 ++----++ b = -1 +-+--+-+ ab = 5++++---- c = -1+--++--+ ac = 3++--++-- bc = 2+-+-+-+- abc = 11++++++++ Contrast 2418614244 Effect 3.002.250.751.750.250.50 Table of – and + Signs for the 2 3 Factorial Design

5. 5 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: A x B = AB A x A = B x B = C x C = A 2 = B 2 = C 2 = I AB x BC = AB 2 C = AC Orthogonal design Orthogonality is an important property shared by all factorial designs

6. 6 Estimation of Factor Effects TermEffectSumSqr% Contribution A33646.1538 B2.2520.2525.9615 C1.7512.2515.7051 AB0.752.252.88462 AC0.250.250.320513 BC0.511.28205 ABC0.511.28205 LOF0 P Error 56.41026

7. 7 ANOVA Summary – Full Model Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model73.00710.4316.690.0003 A36.00136.0057.60< 0.0001 B20.25120.2532.400.0005 C12.25112.2519.600.0022 AB2.2512.253.600.0943 AC0.2510.250.400.5447 BC1.0011.001.600.2415 ABC1.0011.001.600.2415 Pure Error5.0080.63 Cor Total78.0015 Std. Dev.0.79R-Squared0.9359 Mean1.00Adj R-Squared0.8798 C.V.79.06Pred R-Squared0.7436 PRESS20.00Adeq Precision13.416

8. 8 Model Coefficients – Full Model Coefficient Standard95% CI 95% CI Factor EstimateDFErrorLowHighVIF Intercept 1.0010.200.541.46 A-Carbonation 1.5010.201.041.96 1.00 B-Pressure 1.1310.200.671.58 1.00 C-Speed 0.8810.200.421.33 1.00 AB 0.3810.20 -0.0810.83 1.00 AC 0.1310.20 -0.330.58 1.00 BC 0.2510.20-0.210.71 1.00 ABC 0.2510.20-0.210.71 1.00

9. 9 Refine Model – Remove Nonsignificant Factors Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model70.75417.6926.84< 0.0001 A36.00136.0054.62< 0.0001 B20.25120.2530.720.0002 C12.25112.2518.590.0012 AB2.2512.253.410.0917 Residual7.25110.66 LOF2.2530.751.200.3700 Pure E5.0080.63 C Total78.0015 Std. Dev.0.81R-Squared0.9071 Mean1.00Adj R-Squared0.8733 C.V.81.18Pred R-Squared0.8033 PRESS15.34Adeq Precision15.424

10. 10 Model Coefficients – Reduced Model Coefficient Standard 95% CI 95% CI Factor EstimateDFErrorLowHigh Intercept1.0010.200.551.45 A-Carbonation1.5010.201.051.95 B-Pressure1.1310.200.681.57 C-Speed0.8810.200.431.32 AB 0.3810.20-0.0720.82

11. 11 Model Summary Statistics R 2 and adjusted R 2 R 2 for prediction (based on PRESS)

12. 12 Model Summary Statistics Standard error of model coefficients Confidence interval on model coefficients

13. 13 The Regression Model Final Equation in Terms of Coded Factors: Fill-deviation = +1.00 +1.50 * A +1.13 * B +0.88 * C +0.38 * A * B Final Equation in Terms of Actual Factors: Fill-deviation = +9.62500 -2.62500 * Carbonation -1.20000 * Pressure +0.035000 * Speed +0.15000 * Carbonation * Pressure

14. 14 Residual Plots are Satisfactory

15. 15 Model Interpretation Moderate interaction between carbonation level and pressure

16. 16 Model Interpretation Cube plots are often useful visual displays of experimental results

17. 17 Contour & Response Surface Plots – Speed at the High Level

18. 18 Example 6-1: Development of a Nitride Etching Process (page 215)

19. 19 Example 6-1: Development of a Nitride Etching Process