1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay High (2 pounds ) Low (1pound) B=60 ( ) Ab=90 ( ) (1)=80 ( ) A=100 ( ) Amount of catalyst, B Reactant concentration A “-” 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

2 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery Factor Treatment combinationReplicate AB123Total --Alow,Blow Ahigh Blow Alow Bhigh Ahigh,Bhigh

3 Prof. Indrajit Mukherjee, School of Management, IIT Bombay 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? Analysis of variance for the experiment Source of variation Sum of squares Degrees of freedom Mean squareF0F0 P-value A B AB Error Total

4 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Effects in The 2 3 Factorial Design (a)Geometric View High Low - + Factor A Factor B Factor C (1)a b c ab ac bc abc - + High Low + High Low - The 2 3 Factorial Design Run Factor ABC (b) Design Matrix

5 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Analysis done via computer Geometric presentation of contrasts corresponding To the main effects And Interactions in the 2 3 design (a)Main Effects (b)Two-Factor interaction (c)Three-Factor interaction Effects in The 2 3 Factorial Design

6 Prof. Indrajit Mukherjee, School of Management, IIT Bombay An Example of a 2 3 Factorial Design A = gap, B = Flow, C = Power, y = Etch Rate The Plasma Etch Experiment Coded FactorsEtch RateFactor Levels RunABCReplicate 1Replicate 2TotalLow(-1)High(+!) (1)=1154A(Gap, cm) a=1319B (C2F6 flow, SCCM) b=1234C(Power, W) ab= c= ac= bc= abc=1589

7 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Estimation of Factor Effects Effect Estimate Summary FactorEffect estimateSum of squares Percent Contribution A , B C , AB AC , BC ABC

8 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ANOVA Summary – Full Model Analysis of variance for the Plasma Etching Experiment Source of variation Sum of squaresDegrees of freedom Mean squareF0F0 P-value Gap (A) 41, Gas Flow (B) Power (C) 374, AB AC 94, BC ABC Error 18, Total 531,

9 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Coefficients – Full Model Factor Coefficient EstimatedDF Standard Error 95% CI Low 95% CI HighVIF Inter A B C AB AC BC ABC

10 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Coefficients – Reduced Model Factor Coefficient EstimatedDF Standard Error 95% CI Low 95% CI HighVIF Inter A Gap C Power AC

11 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Interpretation Cube plots are often useful visual displays of experimental results 200 sccm 275w Gap A C 2 F 6 Flow Power C C=2089 ab ac=1617 bc=2138abc= w 125 sccm 1.20cm 0.80cm a=1319 b=1234ab=1277 (1)= The 2 3 Design for the Plasma etch experiment

12 Prof. Indrajit Mukherjee, School of Management, IIT Bombay 200 sccm 225w Gap A C 2 F 6 Flow Power C R=15 ab R=119 R=12R= w 125 sccm 1.20cm 0.80cm R=19 R=32 R=7 R= Ranges of etch rates What do the large ranges when gap and power are at the high level tell you? Cube Plot of Ranges

13 Prof. Indrajit Mukherjee, School of Management, IIT Bombay 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 best

14 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Resin Plant Experiment Pilot Plant Filtration Value Experiment Run Number FactorsRun label Filtration Rate (gal/h) ABCD 1----(1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd96

15 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Resin Plant Experiment B C D Data from the Pilot Plant Filtration rate Experiment

16 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ABABCACBCABCDADBDABDCDACDBCDABCD a b ab c ac bc abc d ad bd abd cd acd bcd abcd++++++=+-++++r++ Contrast constant for the 2 4 design

17 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Factor Effect Estimates and sums of Squares for the 2 4 Factorial Model Term Effect Estimate Sum of Squares Percent Contribution A B C D AB AC AD BC BD CD ABC ABD ACD BCD (ABCD)

18 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Design Projection: ANOVA Summary for the Model as a 2 3 in Factors A, C, and D Analysis of variance for the Pilot Plant filtration Rate Experiment in A, C, and D Source of variation Sum of squares Degrees of freedom Mean squareF0F0 P-value A < C < D < AC < AD < CD <1 ACD <1

19 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Interpretation – Main Effects and Interactions Response Response Response Response Response 40 --

20 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Drilling Experiment Example 6.3 A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill B C D Data from the Drilling Experiment

21 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Box-Cox Method A log transformation is recommended The procedure provides a confidence interval on the transformation parameter lambda If unity is included in the confidence interval, no transformation would be needed

22 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Effect Estimates Following the Log Transformation Three main effects are large No indication of large interaction effects What happened to the interactions?

23 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ANOVA Following the Log Transformation Analysis of variance for the Log Transformation Source of variation Sum of squares Degrees of freedom Mean squareF0F0 P-value B < C < D < Error Total

24 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Adjusted multipliers for Lenth’s method Suggested because the original method makes too many type I errors, especially for small designs (few contrasts) Simulation was used to find these adjusted multipliers Lenth’s method is a nice supplement to the normal probability plot of effects JMP has an excellent implementation of Lenth’s method in the screening platform Number of Contrasts71531 Original ME Adjusted ME Original SME Adjusted SME

25 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The least squares estimate of β is The matrix is diagonal – consequences of an orthogonal design The regression coefficient estimates are exactly half of the ‘usual” effect estimates The “usual” contrasts

26 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Fraction of Design Space Std Err Meal Min StdErr Mean Max StdErr Mean Cubical Radius = 1 Points = 10000

27 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The hypotheses are: This sum of squares has a single degree of freedom

28 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ANOVA for Example 6.6 (A Portion of Table 6.22) Analysis for the Reduce Model Source of variation Sum of squares Degrees of freedom Mean squareF0F0 P-value Model <0.000 A < C D <0.000 AC <0.000 AD <0.000

29 Prof. Indrajit Mukherjee, School of Management, IIT Bombay 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 x2x2 (a)Two Factors(b)Three Factors x1x1 x1x1 Central Composite Designs x3x3 x2x2

30 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Catalyst Type Time Temperature A 2 3 factorial design with one qualitative factor and center points Center Points and Qualitative Factors