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1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay + - + - High (2 pounds ) Low (1pound) B=60 (18+19+23) Ab=90 (31+30+29) (1)=80 (28+25+27) A=100.

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Presentation on theme: "1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay + - + - High (2 pounds ) Low (1pound) B=60 (18+19+23) Ab=90 (31+30+29) (1)=80 (28+25+27) A=100."— Presentation transcript:

1 1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay + - + - High (2 pounds ) Low (1pound) B=60 (18+19+23) Ab=90 (31+30+29) (1)=80 (28+25+27) A=100 (36+32+32) 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 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,Blow28252780 +-Ahigh Blow3632 100 -+Alow Bhigh18192360 ++Ahigh,Bhigh31302990

3 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 208.33 1 53.150.0001 B 75.00 1 19.130.0024 AB 8.33 1 2.130.1826 Error 24.84 83.92 Total 323.00 11

4 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 1--- 2+-- 3-+- 4++- 5--+ 6+-+ 7-++ 8+++ (b) Design Matrix

5 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 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 550604(1)=1154A(Gap, cm) 0.801.2 21 669650a=1319B (C2F6 flow, SCCM) 125200 31 633601b=1234C(Power, W) 275325 411642635ab=1277 5 110371052c=2089 611749868ac=1617 71110751063bc=2138 8111729860abc=1589

7 7 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Estimation of Factor Effects Effect Estimate Summary FactorEffect estimateSum of squares Percent Contribution A-101.62541,310.56257.7736 B7.375217.56250.0406 C306.125374,850.062570.5373 AB-24.8752475.06250.4657 AC-153.62594,402.562517.7642 BC-2.12518.06250.0034 ABC5.625126.56250.0238

8 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) 41,310.5625 1 18.340.0027 Gas Flow (B) 217.5625 1 0.100.7639 Power (C) 374,850.0625 1 166.410.0001 AB 2475.0625 1 1.100.3252 AC 94,402.5625 1 41.910.0002 BC 18.0625 1 0.010.9308 ABC 126.5625 1 0.060.8186 Error 18,020.5000 8 2252.5625 Total 531,420.9375 15

9 9 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Coefficients – Full Model Factor Coefficient EstimatedDF Standard Error 95% CI Low 95% CI HighVIF Inter776.06111.87748.7803.42 A-50.81111.87-78.17-23.451 B3.69111.87-23.6731.051 C153.06111.87125.7180.421 AB-12.44111.87-39.814.921 AC-76.81111.87-104.17-49.451 BC-1.06111.87-28.4226.31 ABC2.81111.87-24.5530.171

10 10 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Coefficients – Reduced Model Factor Coefficient EstimatedDF Standard Error 95% CI Low 95% CI HighVIF Inter776.06110.42753.35798.77 A Gap-50.81110.42-73.5228.101 C Power153.06110.42130.35175.771 AC-76.81110.42-99.52-54.101

11 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=1589 325w 125 sccm 1.20cm 0.80cm a=1319 b=1234ab=1277 (1)=1154 + - + - + - The 2 3 Design for the Plasma etch experiment

12 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=131 325w 125 sccm 1.20cm 0.80cm R=19 R=32 R=7 R=34 + - + - + - 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 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 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)45 2+---a71 3-+--b48 4++--ab65 5--+-c68 6+-+-ac60 7-++-bc80 8+++-abc65 9---+d43 10+--+ad100 11-+-+bd45 12++-+abd104 13--++cd75 14+-++acd86 15-+++bcd70 16++++abcd96

15 15 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Resin Plant Experiment B C D - + 6548 68 60 65 71 45 80 10445 75 86 96 100 43 70 Data from the Pilot Plant Filtration rate Experiment

16 16 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ABABCACBCABCDADBDABDCDACDBCDABCD 1--+-++--++-+--+ a+----++--++++-- b-+--+-+-+-++-+- ab+++--------++++ c--++--+-++--++- ac+--++----++--++ bc-+-+-+--+-+-+-+ abc+++++++-------- d--+-++-++-+-++- ad+----+++-----++ bd-+--+-++++--+-+ abd+++----+-++---- cd--++--+++-++--+ acd+--++--+---++-- bcd-+-+-+-+++-+-+- abcd++++++=+-++++r++ Contrast constant for the 2 4 design

17 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 A21.6251870.56332.6397 B3.12539.06250.681608 C9.875390.06256.80626 D14.625855.562514.9288 AB0.1250.06250.00109057 AC-18.1251314.06322.9293 AD16.6251105.56319.2911 BC2.37522.56250.393696 BD-0.3750.56250.00981515 CD-1.1255.06250.0883363 ABC1.87514.06250.45379 ABD4.12568.06251.18763 ACD-1.62510.56250.184307 BCD-2.62527.56250.480942 (ABCD)1.3757.56250.131959

18 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 1870.5625 1 83.36<0.0001 C 390.0625 1 17.38<0.0001 D 855.5625 1 38.13<0.0001 AC 1314.0625 1 58.56<0.0001 AD 1105.5625 1 49.27<0.0001 CD 5.06 1 <1 ACD 10.56 1 <1

19 19 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Model Interpretation – Main Effects and Interactions 90 80 70 60 + Response 50 - 90 80 70 60 + Response 50 - 90 80 70 60 + Response 50 - 100 90 80 70 60 50 + Response 40 100 90 80 70 60 50 + Response 40 --

20 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 - + 5.704.98 3.24 3.44 9.07 1.98 1.68 9.97 9.437.77 4.09 4.53 16.03 2.44 2.07 11.75 Data from the Drilling Experiment

21 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 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 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 5.345 1 381.79<0.0001 C 1.339 1 95.64<0.0001 D 0.431 1 30.79<0.0001 Error 0.173 120.014 Total 7.288 15

24 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 ME3.7642.5712.218 Adjusted ME2.2952.142.082 Original SME9.0085.2194.218 Adjusted SME4.8914.1634.03

25 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 26 Prof. Indrajit Mukherjee, School of Management, IIT Bombay 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Fraction of Design Space Std Err Meal Min StdErr Mean 0.500 Max StdErr Mean 1.000 Cubical Radius = 1 Points = 10000

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

28 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 5535.81 51107.1659.02<0.000 A 1870.5625 1 99.71<0.0001 C 390.0625 1 20.790.0005 D 855.5625 1 45.61<0.000 AC 1314.0625 1 70.05<0.000 AD 1105.5625 1 58.93<0.000

29 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 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

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