1. Diploma in Statistics Design and Analysis of Experiments Lecture 4.11 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

2. Diploma in Statistics Design and Analysis of Experiments Lecture 4.12 Minute Test: How Much

3. Diploma in Statistics Design and Analysis of Experiments Lecture 4.13 Minute Test: How Fast

4. Diploma in Statistics Design and Analysis of Experiments Lecture 4.14 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 2 3 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.

5. Diploma in Statistics Design and Analysis of Experiments Lecture 4.15 Results The main effects of Geometry and Cutting Angle and the Cutting SpeedxCutting Angle interaction are statistically significant.

6. Diploma in Statistics Design and Analysis of Experiments Lecture 4.16 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.

7. Diploma in Statistics Design and Analysis of Experiments Lecture 4.17 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

8. Diploma in Statistics Design and Analysis of Experiments Lecture 4.18 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.

9. Diploma in Statistics Design and Analysis of Experiments Lecture 4.19 Lenth's analysis A process development study with four factors each at two levels Low (–)High (+) A: Catalyst Charge (lbs)1015 B: Temperature (  C)220240 C: Concentration (%)1012 D: Pressure (bar)5080

10. Diploma in Statistics Design and Analysis of Experiments Lecture 4.110 Pareto Chart, vital few versus trivial many (Juran)

11. Diploma in Statistics Design and Analysis of Experiments Lecture 4.111 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 < 2.5s 0 )

12. Diploma in Statistics Design and Analysis of Experiments Lecture 4.112 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.

13. Diploma in Statistics Design and Analysis of Experiments Lecture 4.113 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 Application, via Excel

14. Diploma in Statistics Design and Analysis of Experiments Lecture 4.114 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

15. Diploma in Statistics Design and Analysis of Experiments Lecture 4.115 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

16. Diploma in Statistics Design and Analysis of Experiments Lecture 4.116 Estimating  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

17. Diploma in Statistics Design and Analysis of Experiments Lecture 4.117 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 2 3 with duplicate observations. Analyze these data accordingly. Compare results with the Lenth method and the Reduced Model method.

18. Diploma in Statistics Design and Analysis of Experiments Lecture 4.118 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.

19. Diploma in Statistics Design and Analysis of Experiments Lecture 4.119 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

20. Diploma in Statistics Design and Analysis of Experiments Lecture 4.120 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.

21. Diploma in Statistics Design and Analysis of Experiments Lecture 4.121 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

22. Diploma in Statistics Design and Analysis of Experiments Lecture 4.122 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

23. Diploma in Statistics Design and Analysis of Experiments Lecture 4.123 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

24. Diploma in Statistics Design and Analysis of Experiments Lecture 4.124 Lab Part 1:Soybean seed germination rates

25. Diploma in Statistics Design and Analysis of Experiments Lecture 4.125 Soybean seed germination rates Graphical analysis

26. Diploma in Statistics Design and Analysis of Experiments Lecture 4.126 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 Soybean seed germination rates Graphical analysis: Summary

27. Diploma in Statistics Design and Analysis of Experiments Lecture 4.127 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 Soybean seed germination rates Graphical analysis: Indications

28. Diploma in Statistics Design and Analysis of Experiments Lecture 4.128 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. Soybean seed germination rates Numerical analysis

29. Diploma in Statistics Design and Analysis of Experiments Lecture 4.129 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 Soybean seed germination rates Was blocking effective?

30. Diploma in Statistics Design and Analysis of Experiments Lecture 4.130 Soybean seed germination rates Effects plots

31. Diploma in Statistics Design and Analysis of Experiments Lecture 4.131 Soybean seed germination rates Factor 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

32. Diploma in Statistics Design and Analysis of Experiments Lecture 4.132 Soybean seed germination rates Factor 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

33. Diploma in Statistics Design and Analysis of Experiments Lecture 4.133 Soybean seed germination rates Diagnostics

34. Diploma in Statistics Design and Analysis of Experiments Lecture 4.134 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 Soybean seed germination rates Numerical analysis: first iteration

35. Diploma in Statistics Design and Analysis of Experiments Lecture 4.135 Diagnostics satisfactory Soybean seed germination rates Numerical analysis: first iteration

36. Diploma in Statistics Design and Analysis of Experiments Lecture 4.136 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 Soybean seed germination rates Comparisons with Control

37. Diploma in Statistics Design and Analysis of Experiments Lecture 4.137 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 Soybean seed germination rates Multiple comparisons

38. Diploma in Statistics Design and Analysis of Experiments Lecture 4.138 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 Soybean seed germination rates Multiple comparisons

39. Diploma in Statistics Design and Analysis of Experiments Lecture 4.139 Soybean seed germination rates Further exploratory analysis

40. Diploma in Statistics Design and Analysis of Experiments Lecture 4.140 Soybean seed germination rates Further exploratory analysis Sorted by seed

41. Diploma in Statistics Design and Analysis of Experiments Lecture 4.141 Subset and repeat analysis, to anticipate improved results Next:investigate block inhmogeneity Soybean seed germination rates Further exploratory analysis

42. Diploma in Statistics Design and Analysis of Experiments Lecture 4.142 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.

43. Diploma in Statistics Design and Analysis of Experiments Lecture 4.143 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

44. Diploma in Statistics Design and Analysis of Experiments Lecture 4.144 Include interaction in model? Recall F-test logic: MS(Error) ≈  2 MS(Effect) ≈  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)

45. Diploma in Statistics Design and Analysis of Experiments Lecture 4.145 Part 2 a four factor process improvement study Low (–)High (+) A: catalyst concentration (%),57, B: concentration of NaOH (%),4045, C: agitation speed (rpm), 1020, D: temperature (°F), 150180. The current levels are 5%, 40%, 10rpm and 180°F, respectively.

46. Diploma in Statistics Design and Analysis of Experiments Lecture 4.146 Design and Results