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Design and Analysis of Experiments Lecture 4.1

Design and Analysis of Experiments Lecture 4.1. Review of Lecture 3.1 Homework 3.1.1 Lenth's analysis Homework 3.1.2 Feedback on Laboratory 1 Part 1: Soybean seed germination rates Part 2: A three factor process development study. Minute Test: How Much. Minute Test: How Fast.

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Design and Analysis of Experiments Lecture 4.1

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  1. Design and Analysis of ExperimentsLecture 4.1 Review of Lecture 3.1 Homework 3.1.1 Lenth's analysis Homework 3.1.2 Feedback on Laboratory 1 Part 1: Soybean seed germination rates Part 2: A three factor process development study Diploma in Statistics Design and Analysis of Experiments

  2. Minute Test: How Much Diploma in Statistics Design and Analysis of Experiments

  3. Minute Test: How Fast Diploma in Statistics Design and Analysis of Experiments

  4. Homework 3.1.1 An experiment was run to assess the effects of three factors on the life of a cutting tool A: Cutting speed B: Tool geometry C: Cutting angle. The full 23 design was replicated three times. The results are shown in the next slide and are available in Excel file Tool Life.xls. Carry out a full analysis and report. Diploma in Statistics Design and Analysis of Experiments

  5. Results The main effects of Geometry and Cutting Angle and the Cutting SpeedxCutting Angle interaction are statistically significant. Diploma in Statistics Design and Analysis of Experiments

  6. Results Estimated Effects and Coefficients for Life (coded units) Term Effect SE Coef T P Constant 2.24 36.42 0.000 Cutting Speed 0.3 2.24 0.15 0.884 Geometry 11.33 2.24 5.05 0.000 Cutting Angle 6.83 2.24 3.05 0.008 Cutting Speed*Geometry -1.67 2.24 -0.74 0.468 Cutting Speed*Cutting Angle -8.83 2.24 -3.94 0.001 Geometry*Cutting Angle -2.83 2.24 -1.26 0.224 Cutting Speed*Geometry*Cutting Angle -2.17 2.24 -0.97 0.348 Geometry and Cutting Angle are highly significant, p < 0.0005 and p = 0.008, respectively. Cutting Speed is not significant, p = 0.88. However, the interaction between Cutting Speed and Cutting Angle is highly significant, p = 0.001. Diploma in Statistics Design and Analysis of Experiments

  7. Results Mean SE Mean Geometry - 35.17 1.586 + 46.50 1.586 Cutting Speed*Cutting Angle - - 32.83 2.242 + - 42.00 2.242 - + 48.50 2.242 + + 40.00 2.242 Diploma in Statistics Design and Analysis of Experiments

  8. Results Tool Life increases from 35.17 to 46.50 when Geometry is changed from Low to High. At Low Cutting Angle, the Cutting Speed effect is 42.00 – 32.83 = 9.17. At High Cutting Angle, the Cutting Speed effect is 40.0 – 48.5 = – 8.5. Note that these effects almost balance each other, consistent with a null Cutting Speed effect. Diploma in Statistics Design and Analysis of Experiments

  9. Lenth's analysis A process development study with four factors each at two levels Low (–)High (+) A: Catalyst Charge (lbs) 10 15 B: Temperature (C) 220 240 C: Concentration (%) 10 12 D: Pressure (bar) 50 80 Diploma in Statistics Design and Analysis of Experiments

  10. Pareto Chart,vital few versus trivial many (Juran) Diploma in Statistics Design and Analysis of Experiments

  11. Lenth's method Given several Normal values with mean 0 and given their absolute values (magnitudes, or values without signs), then it may be shown that SD(Normal values) ≈ 1.5 × median(Absolute values). Given a small number of effects with mean ≠ 0, then SD(Normal values) is a small bit bigger. Refinement: PSE ≈ 1.5 × median(Absolute values < 2.5s0) Diploma in Statistics Design and Analysis of Experiments

  12. Lenth's method illustrated Example Add 50 to 3 values, to represent 3 active effects; median will be 27, 29, 32 or 34; not much bigger, so s will be not much bigger, • provides a suitable basis for a "t"-test. Diploma in Statistics Design and Analysis of Experiments

  13. Application, via Excel Term Effect Coef A -8.000 -4.000 B 24.000 12.000 C -5.500 -2.750 D -0.250 -0.125 A*B 1.000 0.500 A*C -0.000 -0.000 A*D 0.750 0.375 B*C 4.500 2.250 B*D -1.250 -0.625 C*D -0.250 -0.125 A*B*C 0.500 0.250 A*B*D -0.750 -0.375 A*C*D -0.250 -0.125 B*C*D -0.750 -0.375 A*B*C*D -0.250 -0.125 Diploma in Statistics Design and Analysis of Experiments

  14. Application, via Excel From Excel, find median(Absolute Values) = 0.75, so initial SE is s0 = 1.5 × 0.75 = 1.125. 4 values exceed 2.5× s0 = 2.8125. The median of the remaining 11 is 0.5. Hence, PSE = 1.5 × 0.5 = 0.75. Check Slide 10 Diploma in Statistics Design and Analysis of Experiments

  15. Assessing statistical significance Critical value for effect is t.05,df× PSE df ≈ (number of effects)/3 t.05,5 = 2.57 PSE = 0.75 Critical value = 1.93 Check Slide 10 Diploma in Statistics Design and Analysis of Experiments

  16. Estimating s PSE = 0.75 is the (pseudo) standard error of an estimated effect. SE(effect) = (s2/8 + s2/8) = s/2. s ≈ 2 × 0.75 = 1.5 Diploma in Statistics Design and Analysis of Experiments

  17. Homework 3.1.2 Design Projection Since Pressure is not statistically significant, it may be treated as an "inert" factor and the design may be treated as a 23 with duplicate observations. Analyze these data accordingly. Compare results with the Lenth method and the Reduced Model method. Diploma in Statistics Design and Analysis of Experiments

  18. Homework 3.1.2 Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 72.250 0.3307 218.46 0.000 Charge -8.000 -4.000 0.3307 -12.09 0.000 Temp 24.000 12.000 0.3307 36.28 0.000 Con -5.500 -2.750 0.3307 -8.32 0.000 Charge*Temp 1.000 0.500 0.3307 1.51 0.169 Charge*Con -0.000 -0.000 0.3307 -0.00 1.000 Temp*Con 4.500 2.250 0.3307 6.80 0.000 Charge*Temp*Con 0.500 0.250 0.3307 0.76 0.471 S = 1.32288 Catalyst Charge, Temperature and Concentration main effects and the Temperature by Concentration interaction are all highly statistically significant. Diploma in Statistics Design and Analysis of Experiments

  19. Homework 3.1.2 Mean SE Mean Catalyst Charge 10 76.25 0.4677 15 68.25 0.4677 Temperature*Concentration 220 10 65.25 0.6614 240 10 84.75 0.6614 220 12 55.25 0.6614 240 12 83.75 0.6614 Diploma in Statistics Design and Analysis of Experiments

  20. Homework 3.1.2 The effect of changing Catalyst Charge from 10 to 15 lbs is to change Yield from 76.75 to 68.75, a decrease of 8, with standard error 0.66, 95% confidence interval: 8  1.5 = 6.5 to 9.5. The effect of changing Concentration from 10% to 12% at high Temperature is to change Yield from 84.75 to 83.75, a decrease of 1, with standard error 0.935, not statistically significant. At low Temperature, the change is from 65.25 to 55.25, a change of 10, with standard error 0.935, 95% confidence interval 10  2.2 = 7.8 to 12.2. Diploma in Statistics Design and Analysis of Experiments

  21. Best operating conditions Mean SE Mean Catalyst_Charge*Temperature*Concentration 10 220 10 69.50 0.9354 15 220 10 61.00 0.9354 10 240 10 88.50 0.9354 15 240 10 81.00 0.9354 10 220 12 60.00 0.9354 15 220 12 50.50 0.9354 10 240 12 87.00 0.9354 15 240 12 80.50 0.9354 Diploma in Statistics Design and Analysis of Experiments

  22. Best operating conditions Mean SE Mean Catalyst Charge*Temperature*Concentration 10 240 10 88.50 0.9354 Confidence interval: 88.5  2.31 × 0.9354 Next best: 10 240 12 87.00 0.9354 not statistically significantly different. Confidence interval: 87  2.31 × 0.9354 Diploma in Statistics Design and Analysis of Experiments

  23. Comparison of fits All effect estimates are the same; SE's vary. 24: s = 1.5, PSE = 0.75 Reduced: s = 1.314, SE(effect) = 0.6572 Projected: s = 1.323, SE(effect) = 0.6614 Diploma in Statistics Design and Analysis of Experiments

  24. Lab Part 1: Soybean seed germination rates Diploma in Statistics Design and Analysis of Experiments

  25. Soybean seed germination ratesGraphical analysis Diploma in Statistics Design and Analysis of Experiments

  26. Soybean seed germination ratesGraphical analysis: Summary • Treatments appear almost universally better than no treatment • General pattern of increasing rates from Block 1 to Block 4, reducing for Block 5 • consistent with homogeneity within blocks and differences between blocks, as desired • Important exceptions, including • high rates for Fermate in Blocks 1 and 2, otherwise Fermate is best • low rates for Spergon in Blocks 3 and 4 Diploma in Statistics Design and Analysis of Experiments

  27. Soybean seed germination ratesGraphical analysis: Indications • Arasan and Semesan uniformly better than no treatment • Spergon better apart from Block 2, Fermate better apart from Block 1 • Fermate best in Blocks 3, 4, 5 Arasan and Semesan best in Blocks 1, 2 • Further investigation of Fermate in Blocks 1 and 2 indicated • potential for gain in understanding • Possibly investigate Spergon in Blocks 3 and 4 Diploma in Statistics Design and Analysis of Experiments

  28. Soybean seed germination ratesNumerical analysis Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment 4 83.840 83.840 20.960 3.87 0.022 Block 4 49.840 49.840 12.460 2.30 0.103 Error 16 86.560 86.560 5.410 Total 24 220.240 Conclusions • Treatment differences are statistically significant, • Block differences are not. Diploma in Statistics Design and Analysis of Experiments

  29. Soybean seed germination ratesWas blocking effective? Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment 4 83.840 83.840 20.960 3.87 0.022 Block 4 49.840 49.840 12.460 2.30 0.103 Error 16 86.560 86.560 5.410 Total 24 220.240 Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment 4 83.840 83.840 20.960 3.07 0.040 Error 20 136.400 136.400 6.820 Total 24 220.240 Diploma in Statistics Design and Analysis of Experiments

  30. Soybean seed germination ratesEffects plots Diploma in Statistics Design and Analysis of Experiments

  31. Soybean seed germination ratesFactor Means Least Squares Means for Failures Treatment Mean SE Mean Arasan 6.2 1.04 Check 10.8 1.04 Fermate 5.8 1.04 Semesan 6.6 1.04 Spergon 8.2 1.04 Block 1 5.2 1.04 2 7.6 1.04 3 8.4 1.04 4 9.4 1.04 5 7.0 1.04 Diploma in Statistics Design and Analysis of Experiments

  32. Soybean seed germination ratesFactor Means, sorted Least Squares Means for Failures Treatment Mean SE Mean Fermate 5.8 1.04 Arasan 6.2 1.04 Semesan 6.6 1.04 Spergon 8.2 1.04 Check 10.8 1.04 Block 1 5.2 1.04 5 7.0 1.04 2 7.6 1.04 3 8.4 1.04 4 9.4 1.04 Diploma in Statistics Design and Analysis of Experiments

  33. Soybean seed germination ratesDiagnostics Diploma in Statistics Design and Analysis of Experiments

  34. Soybean seed germination ratesNumerical analysis: first iteration Exceptional case deleted: Analysis of Variance for Failures, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Treatment 4 94.358 113.400 28.350 10.92 0.000 Block 4 84.650 84.650 21.162 8.15 0.001 Error 15 38.950 38.950 2.597 Total 23 217.958 • Treatment differences and Block differences statistically significant Diploma in Statistics Design and Analysis of Experiments

  35. Soybean seed germination ratesNumerical analysis: first iteration Diagnostics satisfactory Diploma in Statistics Design and Analysis of Experiments

  36. Soybean seed germination ratesComparisons with Control Dunnett 95.0% Simultaneous Confidence Intervals Response Variable Failures Comparisons with Control Level Treatment = Check subtracted from: Treatment Lower Center Upper --+---------+---------+---------+---- Arasan -7.385 -4.600 -1.815 (---------*--------) Fermate -9.720 -6.725 -3.730 (---------*---------) Semesan -6.985 -4.200 -1.415 (--------*--------) Spergon -5.385 -2.600 0.185 (--------*---------) --+---------+---------+---------+---- -9.0 -6.0 -3.0 0.0 Diploma in Statistics Design and Analysis of Experiments

  37. Soybean seed germination ratesMultiple comparisons Tukey 95.0% Simultaneous Confidence Intervals Response Variable Failures All Pairwise Comparisons among Levels of Treatment Treatment = Arasan subtracted from: Treatment Lower Center Upper -----+---------+---------+---------+- Fermate -5.912 -2.200 1.512 (---------*--------) Semesan -3.037 0.400 3.837 (--------*--------) Spergon -1.437 2.000 5.437 (--------*--------) -----+---------+---------+---------+- -4.0 0.0 4.0 8.0 Diploma in Statistics Design and Analysis of Experiments

  38. Soybean seed germination ratesMultiple comparisons Treatment = Fermate subtracted from: Treatment Lower Center Upper -----+---------+---------+---------+- Semesan -1.112 2.600 6.312 (--------*---------) Spergon 0.488 4.200 7.912 (--------*---------) -----+---------+---------+---------+- -4.0 0.0 4.0 8.0 Treatment = Semesan subtracted from: Treatment Lower Center Upper -----+---------+---------+---------+- Spergon -1.837 1.600 5.037 (--------*--------) -----+---------+---------+---------+- -4.0 0.0 4.0 8.0 Diploma in Statistics Design and Analysis of Experiments

  39. Soybean seed germination ratesFurther exploratory analysis Diploma in Statistics Design and Analysis of Experiments

  40. Soybean seed germination ratesFurther exploratory analysis Sorted by seed Diploma in Statistics Design and Analysis of Experiments

  41. Soybean seed germination ratesFurther exploratory analysis Subset and repeat analysis, to anticipate improved results • Next: investigate block inhmogeneity Diploma in Statistics Design and Analysis of Experiments

  42. Homework 4.1.1 Inspection of the original profile plot suggests that four treatments, Check, Arasan, Semesan and Fermate, show a consistent pattern in three blocks, Blocks 3, 4 and 5. Use the Subset Worksheet command of the Data menu to create a subset of the corresponding data; select "Specify which rows to exclude", select "Rows that match", click "condition", use the dialog box tools to enter " 'Block' <= 2 Or 'Treatment'="Spergon" " as the condition, click Ok, Ok. Repeat the full analysis as above. Report in detail. Diploma in Statistics Design and Analysis of Experiments

  43. Include interaction in model? Analysis of Variance for Rate, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Block 4 49.8400 49.8400 12.4600 ** Treatment 4 83.8400 83.8400 20.9600 ** Block*Treatment 16 86.5600 86.5600 5.4100 ** Error 0 * * * Total 24 220.2400 ** Denominator of F-test is zero. S = * Check Slide 27 Diploma in Statistics Design and Analysis of Experiments

  44. Include interaction in model? Recall F-test logic: MS(Error) ≈ s2 MS(Effect)≈ s2 + effect contribution F = MS(Effect) / MS(Error) ≈ 1 if effect absent, >>1 if effect present If Block by Treatment interaction is absent, use MS(Interaction) as MS(Error) Diploma in Statistics Design and Analysis of Experiments

  45. Part 2 a four factor process improvement study Low (–)High (+) A: catalyst concentration (%), 5 7, B: concentration of NaOH (%), 40 45, C: agitation speed (rpm), 10 20, D: temperature (°F), 150 180. The current levels are 5%, 40%, 10rpm and 180°F, respectively. Diploma in Statistics Design and Analysis of Experiments

  46. Design and Results Diploma in Statistics Design and Analysis of Experiments

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