1. Chapter 13 Design of Experiments

2. Introduction “Listening” or passive statistical tools: control charts. “Conversational” or active tools: Experimental design. –Planning of experiments –A sequence of experiments

3. 13.1 A Simple Example of Experimental Design Principles The objective is to compare 4 different brands of tires for tread wear using 16 tires (4 of each brand) and 4 cars in an experiment. Illogical Design: –Randomly assign the 16 tires to the four cars –Assign each car will have all 4 tires of a given brand (confounded with differences between cars, drivers, and driving conditions) –Assign each car will have one tire of each brand Wheel Position Car 1234 LFABAB RFBABA LRDCDC RRCDCD (poor design because brands A and B would be used only on the front of each car, and brands C and D would be used only on the rear positions. Brand effect would be confounded with the position effect.

4. 13.1 A Simple Example of Experimental Design Principles Logical Design: –Each brand is used once at each position, as well as once with each car. Wheel Position Car 1234 LFABCD RFBADC LRCDAB RRDCBA

5. 13.2 Principles of Experimental Design The need to have processes in a state of statistical control when designed experiments are carried out. It is desirable to use experimental design and statistical process control methods together. General guidelines on the design of experiments: 1.Recognition of and statement of the problem 2.Choice of factors and levels 3.Selection of the response variable(s) 4.Choice of experimental design 5.Conduction of the experiment 6.Data analysis 7.Conclusions and recommendations The levels of each factor used in an experimental run should be reset before the next experimental run.

6. 13.3 Statistical Concepts in Experimental Design: Example Assume that the objective is to determine the effect of two different levels of temperature on process yield, where the current temperature is 250  F and the experimental setting is 300  F. Assume that temperature is the only factor that is to be varied.

7. 13.3 Statistical Concepts in Experimental Design: Example Day250F300F M2.42.6 Tu2.72.4 W2.22.8 Th2.5 F22.2 M2.52.7 Tu2.82.3 W2.93.1 Th2.42.9 F2.12.2

8. 13.3 Statistical Concepts in Experimental Design: Example Observations: Neither temperature setting is uniformly superior to the other over the entire test period. The fact that the lines are fairly close together would suggest that increasing temperature may not have a perceptible effect on the process yield. The yield at each temperature setting is the lowest on Friday of each week. There is considerable variability within each temperature setting.

9. 13.4 t-Tests (13.1)

10. 13.4.1 Exact t-Test (13.2)

11. 13.4.1 Exact t-Test Example 250F300F Mean2.452.57 Variance0.08720.0934 H 0 :  1 =  2 H 1 :  1 <  2

12. 13.4.1 Assumptions for Exact t-Test

13. 13.4.2 Approximate t-Test (13.3)

14. 13.4.3 Confidence Intervals for Differences

15. 13.5 Analysis of Variance (ANOVA) for One Factor Experimental Variable: Factor (e.g. Temperature) Values of Experimental Variable: Levels (250, 300) Output Variable: Effect (yield) Distinguish “between” variation from “within” variation

16. 13.5 Analysis of Variance (ANOVA) for One Factor: Example Day250F300FSS(Within) M2.42.60.00250.0009 Tu2.72.40.06250.0289 W2.22.80.06250.0529 Th2.5 0.00250.0049 F2.02.20.20250.1369 M2.52.70.00250.0169 Tu2.82.30.12250.0729 W2.93.10.20250.2809 Th2.42.90.00250.1089 F2.12.20.12250.1369 0.7850.8411.626 Avg.2.452.570.0036 0.0072

17. 13.5 Analysis of Variance (ANOVA) for One Factor: Example Anova: Single Factor SUMMARY GroupsCountSumAverageVariance 250F1024.52.450.087222222 300F1025.72.570.093444444 ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups0.07210.07270.7970.38384.4139 Within Groups1.626180.0903 Total1.69819 Output from Excel

18. 13.5 Analysis of Variance (ANOVA) for One Factor: Example Output from Minitab One-way ANOVA: Yield versus Temp Source DF SS MS F P Temp 1 0.0720 0.0720 0.80 0.384 Error 18 1.6260 0.0903 Total 19 1.6980 S = 0.3006 R-Sq = 4.24% R-Sq(adj) = 0.00% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +---------+---------+---------+--------- 250 10 2.4500 0.2953 (------------*-------------) 300 10 2.5700 0.3057 (------------*-------------) +---------+---------+---------+--------- 2.25 2.40 2.55 2.70 Pooled StDev = 0.3006

19. 13.5 Analysis of Variance (ANOVA) for One Factor The degrees of freedom for “Total” will always be the total number of data values minus one. The degrees of freedom for “Factor” will always be equal to the number of levels of the factor minus one. The degrees of freedom for “Within” will always be equal to (one less than the number of observations per level) multiplied by (the number of levels). The ratio of these mean squares is a random variable of an F distribution with numerator and denominator d.f. Assumptions of normality of the population and equality of the variances

20. 13.5.1 ANOVA for a Single Factor with More than Two Levels Assume the process has three temperature settings, and data were collected over 6 weeks, with 2 weeks at each temperature setting.

21. Day250F300F350F M2.42.63.2 Tu2.72.43.0 W2.22.83.1 Th2.5 2.8 F22.22.5 M 2.72.9 Tu2.82.33.1 W2.93.13.4 Th2.42.93.2 F2.12.22.6 13.5.1 ANOVA for a Single Factor with More than Two Levels: Example

23. 13.5.1 ANOVA for a Single Factor with More than Two Levels (13.4)

24. 13.5.1 ANOVA for a Single Factor with More than Two Levels

25. Output from Excel 13.5.1 ANOVA for a Single Factor with More than Two Levels: Example Anova: Single Factor SUMMARY GroupsCountSumAverageVariance 250F1024.52.450.087222 300F1025.72.570.093444 350F1029.82.980.079556 ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups1.54466720.7723338.9039280.0010723.354131 Within Groups2.342270.086741 Total3.88666729

26. Output from Minitab One-way ANOVA: Yield versus Temp Source DF SS MS F P Temp 2 1.5447 0.7723 8.90 0.001 Error 27 2.3420 0.0867 Total 29 3.8867 S = 0.2945 R-Sq = 39.74% R-Sq(adj) = 35.28% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +---------+---------+---------+--------- 250 10 2.4500 0.2953 (-------*-------) 300 10 2.5700 0.3057 (-------*------) 350 10 2.9800 0.2821 (------*-------) +---------+---------+---------+--------- 2.25 2.50 2.75 3.00 Pooled StDev = 0.2945 13.5.1 ANOVA for a Single Factor with More than Two Levels: Example

27. 13.5.2 Multiple Comparison Procedures 13.5.3 Sample Size Determination (13.5)

28. 13.5.4 Additional Terms and Concepts in One-Factor ANOVA An experimental unit is the unit to which a treatment is applied (the days). If the temperature settings had been randomly assigned to the days, it would be a “completely randomized design.” Blocks: Extraneous factors that vary and have an effect on the response, but not interested. One should “block” on factors that could be expected to influence the response variable and randomize over factors that might be influential, but that could not be “blocked”.

29. The cars were the blocks and the variation due to cars would be isolated. have one tire of each brand Wheel Position Car 1234 LFABAB RFBABA LRDCDC RRCDCD 13.5.4 Additional Terms and Concepts in One-Factor ANOVA Randomized block design Wheel Position Car 1234 LFABCD RFBADC LRCDAB RRDCBA The cars and wheel position were the blocks. Each brand is used once at each position, as well as once with each car. Latin square design

30. 13.5.4 Additional Terms and Concepts in One-Factor ANOVA (13.6) (13.7)

31. 13.5.4 Additional Terms and Concepts in One-Factor ANOVA

32. 13.5.4 Additional Terms and Concepts in One-Factor ANOVA The data in the temperature example were “balanced” in that there was the same number of obs for each level of the factor.

33. 13.6 Regression Analysis of Data from Designed Experiments Regression and ANOVA both could be used as methods of analysis. Regression provides the tools for residual analysis, and the estimation of parameters. For fixed factors, ANOVA should be supplemented or supplanted.

34. 13.6 Regression Analysis of Data from Designed Experiments (13.8)

35. 13.6 Regression Analysis of Data from Designed Experiments

36. 13.6 Regression Analysis of Data from Designed Experiments

37. 13.6 Regression Analysis of Data from Designed Experiments: Example Day250FRes.Res^2300FRes.Res^2350FRes.Res^2 M2.4-0.050.00252.60.030.00093.20.220.0484 Tu2.70.250.06252.4-0.170.02893.00.020.0004 W2.2-0.250.06252.80.230.05293.10.120.0144 Th2.50.050.00252.5-0.070.00492.8-0.180.0324 F2.0-0.450.20252.2-0.370.13692.5-0.480.2304 Sum-0.45-0.35-0.30 M2.50.050.00252.70.130.01692.9-0.080.0064 Tu2.80.350.12252.3-0.270.07293.10.120.0144 W2.90.450.20253.10.530.28093.40.420.1764 Th2.4-0.050.00252.90.330.10893.20.220.0484 F2.1-0.350.12252.2-0.370.13692.6-0.380.1444 Sum24.50.450.78525.70.350.84129.80.300.7162.342 Avg2.452.572.98

38. 13.6 Regression Analysis of Data from Designed Experiments The production is higher for the 2 nd week at each temperature setting. The production is especially high during Wednesday of the week. The more ways we look at data, the more we are apt to discover.

39. 13.6 Regression Analysis of Data from Designed Experiments

43. 13.7 ANOVA for Two Factors Example now includes two factors: “weeks” and “temperature”. In a factorial design (or cross-classified design), each level of every factor is “crossed” with each level of every other factor. (If there are a levels of one factor and b levels of a second factor, there are ab combinations of factor levels.) In a nested factor design, one factor is “nested” within another factor.

44. 13.7 ANOVA for Two Factors

45. 13.7.1 ANOVA with Two Factors: Factorial Designs Why not study each factor separately rather than simultaneously? –Interaction among factors

46. 13.7.1.1 Conditional Effects Factor effects are generally called main effects. Conditional effects (simple effects): the effects of one factor at each level of another factor.

47. 13.7.2 Effect Estimates

48. 13.7.2 Effect Estimates

49. 13.7.2 Effect Estimates

50. 13.7.3 ANOVA Table for Unreplicated Two-Factor Design When both factors are fixed, the main effects and the interaction are tested against the residual. When both factors are random, the main effects are tested against the interaction effect, and the interaction effect is tested against the residual. When one factor is fixed and the other random, the fixed factor is tested against the interaction, the random factor is tested against the residual, and the interaction is tested against the residual. ANOVA Source of VariationSSdfMSF T010 P010 TP (residual)1001 Total1003

51. 13.7.4 Yates’s Algorithm A LowHigh BLow10, 12, 168, 10, 13 High14, 12, 1512, 15, 16

52. 13.7.4 Yates’s Algorithm

53. 13.7.4 Yates’s Algorithm Treatment Combination Total(1)(2)SS 38 31 41 43 A LowHigh BLow10, 12, 168, 10, 13 High14, 12, 1512, 15, 16

54. 13.7.4 Yates’s Algorithm

55. 13.7.4 Yates’s Algorithm Treatment Combination Total(1)(2)SS 3869=38+31 3184=41+43 41-7=31-38 432=43-41 Treatment Combination Total(1)(2)SS 3869153=69+84 3184-5=-7+2 41-715=84-69 4329=2-(-7)

56. 13.7.4 Yates’s Algorithm

57. 13.7.4 Yates’s Algorithm Treatment Combination Total(1)(2)SS 3869153 3184-5(-5) 2 /(3*2 2 )=2.08 (A) 41-715(15) 2 /(3*2 2 )=18.75 (B) 4329(9) 2 /(3*2 2 )=6.75 (AB)

58. 13.7.4 Yates’s Algorithm ANOVA Source of VariationSSdfMSF A2.081 <1 B18.751 3.36 AB6.751 1.21 Residual44.6785.58 Total72.2511

59. 13.7.4 Yates’s Algorithm Two-way ANOVA: Yield versus B, A Source DF SS MS F P B 1 18.7500 18.7500 3.36 0.104 A 1 2.0833 2.0833 0.37 0.558 Interaction 1 6.7500 6.7500 1.21 0.304 Error 8 44.6667 5.5833 Total 11 72.2500 S = 2.363 R-Sq = 38.18% R-Sq(adj) = 14.99%