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: study A three factor process development Diploma in Statistics Design and Analysis of Experiments 1

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

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

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 4

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

Results Estimated Effects and Coefficients for Life (coded units) Term Constant Cutting Speed Geometry Cutting Angle Cutting Speed*Geometry Cutting Speed*Cutting Angle Geometry*Cutting Angle Cutting Speed*Geometry*Cutting Angle Effect 0. 3 11. 33 6. 83 -1. 67 -8. 83 -2. 17 SE Coef 2. 24 2. 24 T 36. 42 0. 15 5. 05 3. 05 -0. 74 -3. 94 -1. 26 -0. 97 P 0. 000 0. 884 0. 000 0. 008 0. 468 0. 001 0. 224 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 6

Results Geometry + Cutting Speed*Cutting Angle - + + + Diploma in Statistics Design and Analysis of Experiments Mean SE Mean 35. 17 46. 50 1. 586 32. 83 42. 00 48. 50 40. 00 2. 242 7

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 8

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 9

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

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. 5 s 0) Diploma in Statistics Design and Analysis of Experiments 11

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 12

Application, via Excel Term Effect Coef A B C D A*B A*C A*D B*C B*D C*D A*B*C A*B*D A*C*D B*C*D A*B*C*D -8. 000 24. 000 -5. 500 -0. 250 1. 000 -0. 000 0. 750 4. 500 -1. 250 -0. 250 0. 500 -0. 750 -0. 250 -4. 000 12. 000 -2. 750 -0. 125 0. 500 -0. 000 0. 375 2. 250 -0. 625 -0. 125 0. 250 -0. 375 -0. 125 Diploma in Statistics Design and Analysis of Experiments 13

Application, via Excel From Excel, find median(Absolute Values) = 0. 75, so initial SE is s 0 = 1. 5 × 0. 75 = 1. 125. 4 values exceed 2. 5 × s 0 = 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 14

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 15

Estimating s PSE = 0. 75 is the (pseudo) standard error of an estimated effect. SE(effect) = (s 2/8 + s 2/8) = s/2. s ≈ 2 × 0. 75 = 1. 5 Diploma in Statistics Design and Analysis of Experiments 16

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 17

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 Charge*Temp Charge*Con Temp*Con Charge*Temp*Con -5. 500 1. 000 -0. 000 4. 500 0. 500 -2. 750 0. 500 -0. 000 2. 250 0. 3307 -8. 32 1. 51 -0. 00 6. 80 0. 76 0. 000 0. 169 1. 000 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 18

Homework 3. 1. 2 Catalyst Charge 10 15 Temperature*Concentration 220 10 240 10 220 12 240 12 Diploma in Statistics Design and Analysis of Experiments Mean SE Mean 76. 25 68. 25 0. 4677 65. 25 84. 75 55. 25 83. 75 0. 6614 19

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 20

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 21

Best operating conditions Mean SE Mean Catalyst Charge*Temperature*Concentration 10 240 10 Confidence interval: 88. 50 0. 9354 88. 5 2. 31 × 0. 9354 Next best: 10 240 12 87. 00 0. 9354 not statistically significantly different. Confidence interval: Diploma in Statistics Design and Analysis of Experiments 87 2. 31 × 0. 9354 22

Comparison of fits All effect estimates are the same; SE's vary. 2 4: 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 23

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

Soybean seed germination rates Graphical analysis Diploma in Statistics Design and Analysis of Experiments 25

Soybean seed germination rates Graphical 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 26

Soybean seed germination rates Graphical 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 27

Soybean seed germination rates Numerical analysis Analysis of Variance for Failures, using Adjusted SS for Tests Source Treatment Block Error Total DF 4 4 16 24 Seq SS 83. 840 49. 840 86. 560 220. 240 Adj SS 83. 840 49. 840 86. 560 Adj MS 20. 960 12. 460 5. 410 F 3. 87 2. 30 P 0. 022 0. 103 Conclusions • Treatment differences are statistically significant, • Block differences are not. Diploma in Statistics Design and Analysis of Experiments 28

Soybean seed germination rates Was blocking effective? Analysis of Variance for Failures, using Adjusted SS for Tests Source Treatment Block Error Total DF 4 4 16 24 Seq SS 83. 840 49. 840 86. 560 220. 240 Adj SS 83. 840 49. 840 86. 560 Adj MS 20. 960 12. 460 5. 410 F 3. 87 2. 30 P 0. 022 0. 103 Analysis of Variance for Failures, using Adjusted SS for Tests Source Treatment Error Total DF 4 20 24 Seq SS 83. 840 136. 400 220. 240 Diploma in Statistics Design and Analysis of Experiments Adj SS 83. 840 136. 400 Adj MS 20. 960 6. 820 F 3. 07 29 P 0. 040

Soybean seed germination rates Effects plots Diploma in Statistics Design and Analysis of Experiments 30

Soybean seed germination rates Factor Means Least Squares Means for Failures Treatment Arasan Check Fermate Semesan Spergon Block 1 2 3 4 5 Diploma in Statistics Design and Analysis of Experiments Mean 6. 2 10. 8 5. 8 6. 6 8. 2 SE Mean 1. 04 5. 2 7. 6 8. 4 9. 4 7. 0 1. 04 31

Soybean seed germination rates Factor Means, sorted Least Squares Means for Failures Treatment Fermate Arasan Semesan Spergon Check Block 1 5 2 3 4 Diploma in Statistics Design and Analysis of Experiments Mean 5. 8 6. 2 6. 6 8. 2 10. 8 SE Mean 1. 04 5. 2 7. 0 7. 6 8. 4 9. 4 1. 04 32

Soybean seed germination rates Diagnostics Diploma in Statistics Design and Analysis of Experiments 33

Soybean seed germination rates Numerical analysis: first iteration Exceptional case deleted: Analysis of Variance for Failures, using Adjusted SS for Tests Source Treatment Block Error Total DF 4 4 15 23 Seq SS 94. 358 84. 650 38. 950 217. 958 Adj SS 113. 400 84. 650 38. 950 Adj MS 28. 350 21. 162 2. 597 F 10. 92 8. 15 P 0. 000 0. 001 • Treatment differences and Block differences statistically significant Diploma in Statistics Design and Analysis of Experiments 34

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

Soybean seed germination rates Comparisons 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 --+---------+-----+---(-----*----) (-----*-----) (--------*---------) --+---------+-----+---- Diploma in Statistics Design and Analysis of Experiments -9. 0 -6. 0 -3. 0 36 0. 0

Soybean seed germination rates Multiple comparisons Tukey 95. 0% Simultaneous Confidence Intervals Response Variable Failures All Pairwise Comparisons among Levels of Treatment = Arasan Treatment subtracted from: 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 Diploma in Statistics Design and Analysis of Experiments 0. 0 4. 0 37 8. 0

Soybean seed germination rates Multiple comparisons Treatment = Fermate Treatment subtracted from: Lower Center Upper Semesan -1. 112 2. 600 6. 312 Spergon 0. 488 4. 200 7. 912 -----+---------+-----+(--------*---------) -----+---------+-----+-4. 0 Treatment = Semesan Treatment Spergon 0. 0 4. 0 8. 0 subtracted from: Lower Center Upper -1. 837 1. 600 5. 037 -----+---------+-----+(----*----) -----+---------+-----+- Diploma in Statistics Design and Analysis of Experiments -4. 0 0. 0 38 4. 0 8. 0

Soybean seed germination rates Further exploratory analysis Diploma in Statistics Design and Analysis of Experiments 39

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

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

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 42

Include interaction in model? Analysis of Variance for Rate, using Adjusted SS for Tests Source Block Treatment Block*Treatment Error Total DF 4 4 16 0 24 Seq SS 49. 8400 83. 8400 86. 5600 * 220. 2400 Adj SS 49. 8400 83. 8400 86. 5600 * Adj MS 12. 4600 20. 9600 5. 4100 * ** Denominator of F-test is zero. S = * Check Slide 27 Diploma in Statistics Design and Analysis of Experiments 43 F ** ** ** P

Include interaction in model? Recall F-test logic: MS(Error) ≈ s 2 MS(Effect) ≈ s 2 + 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 44

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

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