1. 1 The General 2 k Factorial Design Section 6-4, pg. 224, Table 6-9, pg. 225 There will be k main effects, and

2. 2 General Procedure for a 2 k Factorial Design Estimate factor effects (sign & magnitude) From initial model (usually the full model) Perform statistical testing (ANOVA) Refine model (removing nonsignificant variables) Analyze residuals (model adequacy/assumptions checking) Refine model if it’s necessary Interpret results (main/interaction effects plots, contour plots, response surfaces)

3. 3 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner of the “cube” An unreplicated 2 k factorial design is also sometimes called a “single replicate” of the 2 k These designs are very widely used Risks…if there is only one observation at each corner, is there a chance of unusual response observations spoiling the results? Modeling “noise”?

4. 4 Spacing of Factor Levels in the Unreplicated 2 k Factorial Designs If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data More aggressive spacing is usually better

5. 5 Unreplicated 2 k Factorial Designs 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 –Normal probability plotting of effects (Daniels, 1959) –Other methods…see text, pp. 234

6. 6 Example of an Unreplicated 2 k Design A 2 4 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 = mole ratio/concentration, D= stirring rate Experiment was performed in a pilot plant

7. 7 The Resin Plant Experiment

8. 8

9. 9 Estimates of the Effects TermEffectSumSqr% Contribution Model Intercept Error A21.6251870.5632.6397 Error B3.12539.06250.681608 Error C9.875390.0626.80626 Error D14.625855.56314.9288 Error AB0.1250.06250.00109057 Error AC-18.1251314.0622.9293 Error AD16.6251105.5619.2911 Error BC2.37522.56250.393696 Error BD-0.3750.56250.00981515 Error CD-1.1255.06250.0883363 Error ABC1.87514.06250.245379 Error ABD4.12568.06251.18763 Error ACD-1.62510.56250.184307 Error BCD-2.62527.56250.480942 Error ABCD1.3757.56250.131959

10. 10 The Normal Probability Plot of Effects

11. 11 The Half-Normal Probability Plot

12. 12 ANOVA Summary for the Model Response:Filtration Rate ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb >F Model5535.8151107.1656.74< 0.0001 A1870.5611870.5695.86< 0.0001 C390.061390.0619.990.0012 D855.561855.5643.85< 0.0001 AC1314.0611314.0667.34< 0.0001 AD1105.5611105.5656.66< 0.0001 Residual195.121019.51 Cor Total5730.9415 Std. Dev.4.42R-Squared0.9660 Mean70.06Adj R-Squared0.9489 C.V.6.30Pred R-Squared0.9128 PRESS499.52Adeq Precision20.841

13. 13 The Regression Model 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

14. 14 Model Residuals are Satisfactory

15. 15 Model Interpretation – Interactions

16. 16 Model Interpretation – Cube Plot If one factor is dropped, the unreplicated 2 4 design will project into two replicates of a 2 3 A unreplicated 2 k design, if h (h a full two-level factorial 2 k-h with 2 h replicates. Design projection is an extremely useful property, carrying over into fractional factorials

17. 17 The Resin Plant Experiment

18. 18 The Resin Plant Experiment – Projected Design Analysis

19. 19 Model Interpretation – Response Surface Plots With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates