4. 7-1 The Strategy of Experimentation Every experiment involves a sequence of activities: 1.Conjecture – the original hypothesis that motivates the experiment. 2.Experiment – the test performed to investigate the conjecture. 3.Analysis – the statistical analysis of the data from the experiment. 4.Conclusion – what has been learned about the original conjecture from the experiment. Often the experiment will lead to a revised conjecture, and a new experiment, and so forth.

5. 7-2 Factorial Experiments

9. 7-3 2 k Factorial Design

10. 7-3.1 2 2 Example

11. 7-3 2 k Factorial Design 7-3.1 2 2 Example

12. 7-3 2 k Factorial Design 7-3.1 2 2 Example

13. 7-7 Factorial Experiments with More than Two Levels

14. 7-3 2 k Factorial Design 7-3.1 2 2 Example

15. 7-3 2 k Factorial Design

17. 7-3.4 2 k Design for k  3 Factors Consider a 2 3 Design

18. 7-3.4 2 k Design for k  3 Factors

19. Main Effects

20. 7-3.4 2 k Design for k  3 Factors AB Interaction Effect

21. 7-3.4 2 k Design for k  3 Factors

24. 7-7Factorial Experiments with More than Two Levels

25. 7-7 Factorial Experiments with More than Two Levels

26. 7-7 Factorial Experiments with More than Two Levels

27. 7-7 Factorial Experiments with More than Two Levels

28. 7-7 Factorial Experiments with More than Two Levels Sum of Squares partition: Degrees of freedom partition:

29. 7-7 Factorial Experiments with More than Two Levels Mean Squares:

30. 7-7 Factorial Experiments with More than Two Levels

31. 7-7 Factorial Experiments with More than Two Levels

32. 7-7 Factorial Experiments with More than Two Levels

33. 7-7 Factorial Experiments with More than Two Levels

34. 7-7 Factorial Experiments with More than Two Levels

35. 7-7 Factorial Experiments with More than Two Levels

36. Model Adequacy 7-7 Factorial Experiments with More than Two Levels

37. Model Adequacy 7-7 Factorial Experiments with More than Two Levels

38. Model Adequacy 7-7 Factorial Experiments with More than Two Levels

39. Computer Output 7-7 Factorial Experiments with More than Two Levels

40. Example 7-11 OPTIONS NOOVP NODATE NONUMBER LS=80; DATA ex711; DO obs= 1 to 3; DO type=1 to 3; DO method='Dipping', 'Spraying'; INPUT force @@; OUTPUT; END; END;END; CARDS; 4 5.4 5.6 5.8 3.8 5.5 4.5 4.9 4.9 6.1 3.7 5 4.3 5.6 5.4 6.3 4 5 PROC GLM DATA=ex711; CLASS type method; MODEL force= type method type*method; MEANS type method type*method/snk; OUTPUT out=new p=phat r=resid; TITLE 'Two-way ANOVA'; DATA MEANS; INPUT Dipping Spraying type @@; CARDS; 4.26 5.30 1 5.30 6.07 2 3.83 5.17 3 PROC PLOT data=means; PLOT Dipping*type='0' Spraying*type='1'/overlay vaxis=3.5 to 6.5 by 0.5; TITLE 'Interaction Plot'; PROC PLOT data=new; PLOT resid*phat/vaxis=-0.4 to 0.35 by 0.05; /* Residual Plot */ PLOT resid*type/vaxis=-0.4 to 0.35 by 0.05; /* Residual by Primer Type */ PLOT resid*method/vaxis=-0.4 to 0.35 by 0.05; /* Residuals by Application Method */ TITLE ‘Residual Plot'; RUN; QUIT; 7-7 Factorial Experiments with More than Two Levels

41. Two-way ANOVA The GLM Procedure Dependent Variable: force Sum of Source DF Squares Mean Square F Value Pr > F Model 5 9.73111111 1.94622222 23.67 <.0001 Error 12 0.98666667 0.08222222 Corrected Total 17 10.71777778 R-Square Coeff Var Root MSE force Mean 0.907941 5.747656 0.286744 4.988889 Source DF Type I SS Mean Square F Value Pr > F type 2 4.58111111 2.29055556 27.86 <.0001 method 1 4.90888889 4.90888889 59.70 <.0001 type*method 2 0.24111111 0.12055556 1.47 0.2693 Source DF Type III SS Mean Square F Value Pr > F type 2 4.58111111 2.29055556 27.86 <.0001 method 1 4.90888889 4.90888889 59.70 <.0001 type*method 2 0.24111111 0.12055556 1.47 0.2693 7-7 Factorial Experiments with More than Two Levels

42. Two-way ANOVA The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 0.082222 Number of Means 2 3 Critical Range 0.3607064 0.4416697 Means with the same letter are not significantly different. SNK Grouping Mean N type A 5.6833 6 2 B 4.7833 6 1 B 4.5000 6 3 Two-way ANOVA The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 0.082222 Number of Means 2 Critical Range 0.2945156 Means with the same letter are not significantly different. SNK Grouping Mean N method A 5.5111 9 Sprayin B 4.4667 9 Dipping 7-7 Factorial Experiments with More than Two Levels

43. Two-way ANOVA The GLM Procedure Level of Level of ------------force------------ type method N Mean Std Dev 1 Dipping 3 4.26666667 0.25166115 1 Sprayin 3 5.30000000 0.36055513 2 Dipping 3 5.30000000 0.36055513 2 Sprayin 3 6.06666667 0.25166115 3 Dipping 3 3.83333333 0.15275252 3 Sprayin 3 5.16666667 0.28867513 7-7 Factorial Experiments with More than Two Levels

44. 7-7 Factorial Experiments with More than Two Levels ods graphics off; symbol v=plus; title 'Normal Probability-Probability Plot'; proc capability data=new noprint; /* normaltest option 을 사용하면 normality check 가능 (noprint 제거 ) */ ppplot resid / normal square; run;

45. Interaction Plot Dipping*type 도표. 사용된 기호 : '0'. Spraying*type 도표. 사용된 기호 : '1'. Dipping | 6.5 + | | 1 6.0 + | 5.5 + | |1 0 | | 1 5.0 + | 4.5 + | |0 | 4.0 + | 0 | 3.5 + | -+----------------------------------+----------------------------------+ 1 2 3 type Residual Plot resid*phat 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 0.35 + | A 0.30 + B | 0.25 + | A A 0.20 + | A 0.15 + | 0.10 + B | 0.05 + | A A 0.00 + | A -0.05 + | -0.10 + | A -0.15 + | B -0.20 + | -0.25 + | A A -0.30 + | -0.35 + | -0.40 + B | -+-----------+-----------+-----------+-----------+-----------+-----------+ 3.5 4.0 4.5 5.0 5.5 6.0 6.5 phat 7-7 Factorial Experiments with More than Two Levels

46. Residual Plot resid*type 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 0.35 + | A 0.30 +A A | 0.25 + | A A 0.20 + | A 0.15 + | 0.10 +A A | 0.05 + | A A 0.00 + | A -0.05 + | -0.10 + | A -0.15 + | B -0.20 + | -0.25 + | A A -0.30 + | -0.35 + | -0.40 +A A | -+-----------------------------------+-----------------------------------+ 1 2 3 type Residual Plot resid*method 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 0.35 + | A 0.30 + A A | 0.25 + | A A 0.20 + | A 0.15 + | 0.10 + A A | 0.05 + | A A 0.00 + | A -0.05 + | -0.10 + | A -0.15 + | B -0.20 + | -0.25 + | A A -0.30 + | -0.35 + | -0.40 + A A | ---+------------------------------------------------+-- Dipping Sprayin method 7-7 Factorial Experiments with More than Two Levels

47. Speed MaterialFeed100220475715870 B100.004122 110 108 85 108 60 66 50 80 60 0.008332 330 276 310 248 295 248 275 276 310 0.014640 500 612 500 543 450 612 610 696 610 V100.004192 170 136 130 122 85 108 75 136 75 0.008386 365 333 330 318 330 473 350 499 390 0.014810 725 779 670 810 750 893 890 1820 890 Thrust Forces in Drilling (3-Way Factorial) 7-7 Factorial Experiments with More than Two Levels

48. OPTIONS NOOVP NODATE NONUMBER LS=80; DATA threeway; INFILE 'C:\users\myung\Documents\Teaching\threeway.dat'; INPUT material$ feed speed thrust @@; PROC ANOVA data=threeway; CLASS material feed speed; MODEL thrust = material | feed | speed; MEANS material | feed | speed/snk; TITLE 'Three-way ANOVA'; DATA means1; INPUT b10 v10 feed @@; CARDS; 84.9 122.9 0.004 290 377.3 0.008 577.3 903.7 0.014 PROC PLOT data=means1; PLOT b10*feed='B' v10*feed='V'/overlay; TITLE 'Interaction Plot for Material*Feed'; DATA means2; INPUT b10 v10 speed @@; CARDS; 339 441.3 100 315.2 396.3 220 284 402.5 475 310.2 464.7 715 338.7 635 870 PROC PLOT data=means2; PLOT b10*speed='B' v10*speed='V'/overlay; TITLE 'Interaction Plot for Speed*Material'; DATA means3; INPUT fd1 fd2 fd3 speed @@; CARDS; 148.5 353.3 668.3 100 114.8 312.3 640.3 220 93.8 297.8 638.3 475 74.8 336.3 751.3 715 87.8 368.8 1004 870 PROC PLOT data=means3; PLOT fd1*speed='1' fd2*speed='2' fd3*speed='3'/overlay; TITLE 'Interaction Plot for Feed*Speed'; RUN; QUIT; 7-7 Factorial Experiments with More than Two Levels B10.004 100 122 B10.004 100 110 B10.004 220 108 B10.004 220 85 B10.004 475 108 B10.004 475 60 B10.004 715 66 B10.004 715 50 B10.004 870 80 B10.004 870 60 B10.008 100 332 B10.008 100 330

49. Three-way ANOVA The ANOVA Procedure Dependent Variable: thrust Sum of Source DF Squares Mean Square F Value Pr > F Model 29 5343804.483 184269.120 11.30 <.0001 Error 30 489376.500 16312.550 Corrected Total 59 5833180.983 R-Square Coeff Var Root MSE thrust Mean 0.916105 32.52509 127.7206 392.6833 Source DF Anova SS Mean Square F Value Pr > F material 1 340054.817 340054.817 20.85 <.0001 feed 2 4157143.633 2078571.817 127.42 <.0001 material*feed 2 237956.433 118978.217 7.29 0.0026 speed 4 152472.233 38118.058 2.34 0.0781 material*speed 4 88303.433 22075.858 1.35 0.2735 feed*speed 8 255092.367 31886.546 1.95 0.0880 material*feed*speed 8 112781.567 14097.696 0.86 0.5565 Three-way ANOVA The ANOVA Procedure Class Level Information Class Levels Values material 2 B10 V10 feed 3 0.004 0.008 0.014 speed 5 100 220 475 715 870 Number of Observations Read 60 Number of Observations Used 60 ------------------------------------------------------------------------------- 7-7 Factorial Experiments with More than Two Levels

50. Three-way ANOVA The ANOVA Procedure Student-Newman-Keuls Test for thrust NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 30 Error Mean Square 16312.55 Number of Means 2 Critical Range 67.348663 Means with the same letter are not significantly different. SNK Grouping Mean N material A 467.97 30 V10 B 317.40 30 B10 Three-way ANOVA The ANOVA Procedure Student-Newman-Keuls Test for thrust NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 30 Error Mean Square 16312.55 Number of Means 2 3 Critical Range 82.48493 99.569343 Means with the same letter are not significantly different. SNK Grouping Mean N feed A 740.50 20 0.014 B 333.65 20 0.008 C 103.90 20 0.004 7-7 Factorial Experiments with More than Two Levels

51. Three-way ANOVA The ANOVA Procedure Level of Level of ------------thrust----------- material feed N Mean Std Dev B10 0.004 10 84.900000 25.631794 B10 0.008 10 290.000000 30.430248 B10 0.014 10 577.300000 75.938645 V10 0.004 10 122.900000 38.922858 V10 0.008 10 377.300000 62.241555 V10 0.014 10 903.700000 330.551408 ----------------------------------------------------------------------------- Three-way ANOVA The ANOVA Procedure Student-Newman-Keuls Test for thrust NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 30 Error Mean Square 16312.55 Number of Means 2 3 4 5 Critical Range 106.48759 128.54347 141.77902 151.24267 Means with the same letter are not significantly different. SNK Grouping Mean N speed A 486.83 12 870 A 390.17 12 100 A 387.42 12 715 A 355.75 12 220 A 343.25 12 475 7-7 Factorial Experiments with More than Two Levels

52. Three-way ANOVA The ANOVA Procedure Level of Level of ------------thrust----------- material speed N Mean Std Dev B10 100 6 339.000000 207.933643 B10 220 6 315.166667 209.632456 B10 475 6 284.000000 188.258333 B10 715 6 310.166667 250.363269 B10 870 6 338.666667 264.809869 V10 100 6 441.333333 268.722658 V10 220 6 396.333333 271.457302 V10 475 6 402.500000 309.417356 V10 715 6 464.666667 362.389661 V10 870 6 635.000000 649.672533 Level of Level of ------------thrust----------- feed speed N Mean Std Dev 0.004 100 4 148.50000 38.897301 0.004 220 4 114.75000 23.200216 0.004 475 4 93.75000 27.183022 0.004 715 4 74.75000 24.459150 0.004 870 4 87.75000 33.270357 0.008 100 4 353.25000 27.097048 0.008 220 4 312.25000 26.234519 0.008 475 4 297.75000 36.206583 0.008 715 4 336.25000 100.260910 0.008 870 4 368.75000 99.113992 0.014 100 4 668.75000 132.185161 0.014 220 4 640.25000 116.339661 0.014 475 4 638.25000 169.800618 0.014 715 4 751.25000 161.953440 0.014 870 4 1004.00000 556.462637 Level of Level of Level of ------------thrust----------- material feed speed N Mean Std Dev B10 0.004 100 2 116.00000 8.485281 B10 0.004 220 2 96.50000 16.263456 B10 0.004 475 2 84.00000 33.941125 B10 0.004 715 2 58.00000 11.313708 B10 0.004 870 2 70.00000 14.142136 B10 0.008 100 2 331.00000 1.414214 B10 0.008 220 2 293.00000 24.041631 B10 0.008 475 2 271.50000 33.234019 B10 0.008 715 2 261.50000 19.091883 B10 0.008 870 2 293.00000 24.041631 B10 0.014 100 2 570.00000 98.994949 B10 0.014 220 2 556.00000 79.195959 B10 0.014 475 2 496.50000 65.760931 B10 0.014 715 2 611.00000 1.414214 B10 0.014 870 2 653.00000 60.811183 V10 0.004 100 2 181.00000 15.556349 V10 0.004 220 2 133.00000 4.242641 V10 0.004 475 2 103.50000 26.162951 V10 0.004 715 2 91.50000 23.334524 V10 0.004 870 2 105.50000 43.133514 V10 0.008 100 2 375.50000 14.849242 V10 0.008 220 2 331.50000 2.121320 V10 0.008 475 2 324.00000 8.485281 V10 0.008 715 2 411.00000 86.267027 V10 0.008 870 2 444.50000 77.074639 V10 0.014 100 2 767.50000 60.104076 V10 0.014 220 2 724.50000 77.074639 V10 0.014 475 2 780.00000 42.426407 V10 0.014 715 2 891.50000 2.121320 V10 0.014 870 2 1355.00000 657.609307 7-7 Factorial Experiments with More than Two Levels

53. Interaction Plot for Material*Feed b10*feed 도표. 사용된 기호 : 'B'. v10*feed 도표. 사용된 기호 : 'V'. b10 | 900 + V | 800 + | 700 + | 600 + | B | 500 + | 400 + | V | 300 + | B | 200 + | | V 100 + | B | 0 + | ---+-------------+-------------+-------------+-------------+-------------+-- 0.004 0.006 0.008 0.010 0.012 0.014 feed Interaction Plot for Speed*Material b10*speed 도표. 사용된 기호 : 'B'. v10*speed 도표. 사용된 기호 : 'V'. b10 | 650 + | V | 600 + | 550 + | 500 + | | V | 450 + | V | 400 + V | V | 350 + | B B | | B 300 + | | B | 250 + ---+-------------+-------------+-------------+-------------+-------------+-- 0 200 400 600 800 1000 speed Interaction Plot for Feed*Speed fd1*speed 도표. 사용된 기호 : '1'. fd2*speed 도표. 사용된 기호 : '2'. fd3*speed 도표. 사용된 기호 : '3'. fd1 | | 1000 + 3 | 900 + | 800 + | | 3 700 + | 3 | 3 3 600 + | 500 + | 400 + | 2 2 | 2 300 + 2 | 200 + | 1 100 + 1 | 1 1 | 0 + | --+-------------+-------------+-------------+-------------+-------------+-- 0 200 400 600 800 1000 speed 7-7 Factorial Experiments with More than Two Levels

54. The Latin square design is used to eliminate two nuisance sources of variability; that is, it systemically allows blocking two dimensions. Thus, the rows and columns actually represent two restrictions on randomization. In a Latin square design, there are p treatments and p levels of each of the two blocking variables. Each treatment level appears in each row and column once. The arrangement should be randomly selected from all possible arrangement. For instance, there is only 1 3x3, 4 4x4, 56 5x5, and 9408 6x6 Latin squares. The Latin Square Design

56. OPTIONS NOOVP NODATE NONUMBER LS=80; DATA latin; INPUT operator batch treat$ force @@; CARDS; 1 1 A -1 1 2 B -8 1 3 C -7 1 4 D 1 1 5 E -3 2 1 B -5 2 2 C -1 2 3 D 13 2 4 E 6 2 5 A 5 3 1 C -6 3 2 D 5 3 3 E 1 3 4 A 1 3 5 B -5 4 1 D -1 4 2 E 2 4 3 A 2 4 4 B -2 4 5 C 4 5 1 E -1 5 2 A 11 5 3 B -4 5 4 C -3 5 5 D 6 PROC GLM data=latin; CLASS operator batch treat; MODEL force = operator batch treat; MEANS operator batch treat/snk; output out=new p=phat r=resid; TITLE 'Latin Square Design'; proc plot data=new; plot resid*(phat operator batch treat); Title 'Residual plot'; RUN; QUIT; The Latin Square Design Example 5-4

57. Latin Square Design The GLM Procedure Class Level Information Class Levels Values operator 5 1 2 3 4 5 batch 5 1 2 3 4 5 treat 5 A B C D E Number of Observations Read 25 Number of Observations Used 25 Latin Square Design The GLM Procedure Dependent Variable: force Sum of Source DF Squares Mean Square F Value Pr > F Model 12 548.0000000 45.6666667 4.28 0.0089 Error 12 128.0000000 10.6666667 Corrected Total 24 676.0000000 R-Square Coeff Var Root MSE force Mean 0.810651 816.4966 3.265986 0.400000 Source DF Type I SS Mean Square F Value Pr > F operator 4 150.0000000 37.5000000 3.52 0.0404 batch 4 68.0000000 17.0000000 1.59 0.2391 treat 4 330.0000000 82.5000000 7.73 0.0025 Source DF Type III SS Mean Square F Value Pr > F operator 4 150.0000000 37.5000000 3.52 0.0404 batch 4 68.0000000 17.0000000 1.59 0.2391 treat 4 330.0000000 82.5000000 7.73 0.0025 The Latin Square Design

58. Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 10.66667 Number of Means 2 3 4 5 Critical Range 4.5005364 5.5107155 6.1325358 6.5839317 Means with the same letter are not significantly different. SNK Grouping Mean N operator A 3.600 5 2 B A 1.800 5 5 B A 1.000 5 4 B A -0.800 5 3 B -3.600 5 1 The Latin Square Design Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 10.66667 Number of Means 2 3 4 5 Critical Range 4.5005364 5.5107155 6.1325358 6.5839317 Means with the same letter are not significantly different. SNK Grouping Mean N batch A 1.800 5 2 A 1.400 5 5 A 1.000 5 3 A 0.600 5 4 A -2.800 5 1

59. Latin Square Design The GLM Procedure Student-Newman-Keuls Test for force NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 10.66667 Number of Means 2 3 4 5 Critical Range 4.5005364 5.5107155 6.1325358 6.5839317 Means with the same letter are not significantly different. SNK Grouping Mean N treat A 4.800 5 D A 3.600 5 A B A 1.000 5 E B C -2.600 5 C C -4.800 5 B The Latin Square Design

60. resid*phat 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A A A | A A -0.5 + | A -1.0 + A A A | A A -1.5 + | A -2.0 + A | -2.5 + | A A -3.0 + A | A -3.5 + | --+---------+---------+---------+---------+---------+---------+---------+- -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 phat Residual plot resid*operator 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A B | A A -0.5 + | A -1.0 + B A | B -1.5 + | A -2.0 + A | -2.5 + | A A -3.0 + A | A -3.5 + ---+--------------+--------------+--------------+--------------+-- 1 2 3 4 5 operator The Latin Square Design

61. Residual plot resid*batch 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A A A | B -0.5 + | A -1.0 + A A A | A A -1.5 + | A -2.0 + A | -2.5 + | A A -3.0 + A | A -3.5 + ---+--------------+--------------+--------------+--------------+-- 1 2 3 4 5 batch Residual plot resid*treat 도표. 범례 : A = 1 관측치, B = 2 관측치, 등. resid | 5.0 + A | A 4.5 + | A 4.0 + | 3.5 + | 3.0 + | A 2.5 + | 2.0 + A | A 1.5 + | 1.0 + A | A 0.5 + | 0.0 + A B | A A -0.5 + | A -1.0 + A B | A A -1.5 + | A -2.0 + A | -2.5 + | B -3.0 + A | A -3.5 + | ---+--------------+--------------+--------------+--------------+-- A B C D E treat The Latin Square Design