Design and Analysis of Experiments Lecture 6.1

Transcription

1 Lecture Review of split unit experiments Why split? Why block? 2. Review of Laboratory 2 Cambridge grassland experiment Soup mix packet filling 3. Extending plot and treatment structures Wood stain experiment 4. Robust Product Design 5. An interesting interaction? Lecture 6.1 1

2 arise when Split units experiments one set of treatment factors is applied to experimental units, a second set of factors is applied to sub units of these experimental units. Originated in agriculture where they are referred to as split plot experiments. Whole units may be regarded as blocks "Most industrial experiments are... split plot in their design. C. Daniel (1976) p. 175 Lecture 6.1 2

3 Why split? Adding another factor after the experiment started Cambridge grassland Changing one factor is more difficult more expensive more time consuming Component lifetimes Water resistance than changing others Some factors require better precision than others Corrosion resistance Lecture 6.1 3

4 Why block? Blocking is useful when there are known external factors (covariates) that affect variation between plots. Blocking reduces bias arising due to block effects disproportionately affecting factor effects due to levels disproportionally allocated to blocks. Neighbouring plots are likely to be more homogeneous than separated plots, so that blocking reduces variation affecting comparisons when treatments are compared within blocks (precision is increased when results are combined across blocks). Lecture 6.1 4

5 Block or Not? Not blocking when there is a block effect implies reduced power for treatment effects test; because Error term includes block variation. Blocking when there is no block effect implies reduced power for treatment effects test; because Error degrees of freedom reduced Lecture 6.1 5

6 Lecture Review of split unit experiments Why split? Why block? 2. Review of Laboratory 2 Cambridge grassland experiment Soup mix packet filling 3. Extending plot and treatment structures Wood stain experiment 4. Robust Product Design 5. An interesting interaction? Lecture 6.1 6

7 Laboratory 2, Exercise 1 Cambridge Grassland Experiment 3 grassland treatments Rejuvenator Harrow no treatment R H C randomly allocated to 3 neighbouring plots, replicated in 6 neighbouring blocks 4 fertilisers Farmyard manure Straw Artificial fertiliser no fertiliser F S A C randomly allocated to 4 sub plots within each plot. Lecture 6.1 7

8 Cambridge Grassland Experiment Blocks Whole Plots Treatments H C R H R C C H R H R C C H R C R H Sub Plot 1 C A A C F F A A A A F F F C A F F C Sub Plot 2 A S C A S A C C F F A S S A S A S S Sub Plot 3 F C F F C C S F S C S A C S C C C F Sub Plot 4 S F S S A S F S C S C C A F F S A A Lecture 6.1 8

9 Experimental results, Yields in pound (lbs) Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 C H R C H R C H R C H R C H R C H R A C F S Lecture 6.1 9

10 Treatment yields vs Layout yields (Block 1) Block 1 C H R A C F S Block 1 Whole Plot Treatment H C R Sub Plot 1 C A A Sub Plot 2 A S C Sub Plot 3 F C F Sub Plot 4 S F S Lecture

11 3-Step Decomposition of Total Variation Step 1: Two components of total variation Step 2: Analysis of whole plot total variation Step 3: Analysis of subplot total variation Lecture

12 Plot Structure Units Blocks Whole Plots Subplots Lecture

13 Step 1: Two components of total variation Mintab model: Plot Subplot(Plot) Source DF SS MS F P Plot Subplot(Plot) ** Error 0 * * Total Note: DF for Plot Variation: 18 1 = 17 DF for Subplot Variation: (4 1) x 18 = 54 Minitab model: Plot Source DF SS MS F P Plot Error Total Lecture

14 Step 2: Analysis of whole plot total variation Treatment Factors Whole Plot and Treatment Structure Units ANOVA Blocks MS(Blocks) Treatment Whole Plots MS(Treatments) MS(Whole Plot Error) Subplots Lecture

15 Step 2: Analysis of whole plot total variation Minitab model: Block Treatment Source DF SS MS F P Block Treatment Error Total Minitab model: Plot (see Slide 13) Source DF SS MS F P Plot Error Total Plot Error DF = = 10 Plot Error SS = = 4681 Lecture

16 Classwork 1 From the values on Slide 15, construct an analysis of variance table for whole plots variation. Lecture

17 Whole Plot and Treatment Structure Treatment Factors Units Blocks ANOVA MS(Blocks) Treatment Whole Plots Subplots MS(Treatments) MS(Whole Plot Error) B x T Lecture

18 Step 2: Analysis of whole plot total variation Minitab model: Block Treatment Block*Treatment Source DF SS MS F P Block Treatment Block*Treatment Error Total Lecture

19 Step 3: Split Plot Analysis Plot and Treatment Structure Treatment Factors Units Blocks ANOVA MS(Blocks) Treatment Whole Plots MS(Treatments) MS(Whole Plot Error) B x T Fertiliser Subplots MS(Fertiliser) MS(Interactions) MS(Subplot Error) Lecture

20 Split Plots Analysis Minitab model: B + T + B*T + F + T*F + B*F Random effect(s) B Fixed effects T F Source DF SS MS F P B x T B*T F T*F B*F Error Total Lecture

21 Expected Mean Squares Source Expected Mean Square Block S Treatment Treatment effect Plot + 4 Fertiliser Fertiliser effect Treatment*Fertiliser 2 S 2 S 2 S 2 S Block*Fertiliser S B F 2 P 2 P 2 B F P 2 B ( i ) J I 1 + Treatment x Fertiliser effect 2 B F 2 Error / Subplot 2 S Lecture

22 Classwork 2 Identify the mean squares and F-ratios for testing treatment effects, fertiliser effects and treatment by fertiliser interaction effects. Confirm the values of the F-ratios Lecture

23 Split Plots Analysis Minitab model: B + T + B*T + F + T*F + B*F Random effect(s) B Fixed effects T F Source DF SS MS F P B T B*T F T*F Error Total Lecture

24 Expected Mean Squares Source Expected Mean Square 2 2 Block S S Treatment Treatment effect 2 S Plot + 4 P 2 P 2 P 2 B Fertiliser 2 S + Fertiliser effect Treatment*Fertiliser 2 S + Treatment x Fertiliser effect Error / Subplot 2 S Lecture

25 Decomposition Summary Step 1 Source DF SS Plot Total Subplot Total Total Lecture

26 Decomposition Summary Step 2 Source DF SS MS F P Block Treatment Plot Error Plot Total Subplot Total Total Lecture

27 Decomposition Summary Step 3 Source DF SS MS F P Block Treatment Plot Error Plot Total Fertiliser T*F Subplot Error Subplot Total Total Lecture

28 Split Plots Analysis Whole Plots Source DF SS MS F P Block Treatment Plot Error Plot Total Fertiliser T*F Subplot Error Subplot Total Total Lecture

29 Split Plots Analysis Sub plots Source DF SS MS F P Block Treatment Plot Error Plot Total Fertiliser T*F Subplot Error Subplot Total Total Lecture

30 Decomposition Summary Source DF SS MS F P Block Treatment Plot Error Plot Total Fertiliser T*F Subplot Error Subplot Total Total Lecture

31 Subplots Residuals vs Fitted Values Lecture

32 Same diagnostic, Different interpretation? Lecture

33 Subplots Residuals Normal Plot Lecture

34 Whole Plots Residuals vs Fitted Values Lecture

35 Whole Plots Residuals Normal Plot 2 Deleted Residual Score 1 2 Lecture

36 Check Interactions Lecture

37 Check Interactions Lecture

38 Check Interactions Lecture

39 Check Interactions Lecture

40 Interaction plots for Grassland experiment Treatments Lecture

41 Lecture

42 Lecture Review of split unit experiments Why split? Why block? 2. Review of Laboratory 2 Cambridge grassland experiment Soup mix packet filling 3. Extending plot and treatment structures Wood stain experiment 4. Robust Product Design 5. An interesting interaction? Lecture

43 Laboratory 2, Exercise 2: Soup mix packet filling machine Questions: What factors affect soup powder fill variation? How can fill variation be minimised? Potential factors A: Number of ports for adding oil, 1 or 3, B: Mixer vessel temperature, ambient or cooled, C: Mixing time, 60 or 80 seconds, D: Batch weight, 1500 or 2000 lbs, E: Delay between mixing and packaging, 1 or 7 days. Response: Spread of weights of 5 sample packets Lecture

44 Minitab analysis Lecture

45 Minitab analysis Normal plot vs Pareto Principle vs Lenth? Lecture

46 Alias analysis Estimated Effects Term Effect Alias E E + A*B*C*D B*E B*E + A*C*D D*E D*E + A*B*C E is aliased with or confounded with A*B*C*D Lecture

47 Graphical and numerical summaries E B D E Lecture

48 Best conditions Best conditions: Temp Low, Weight High, Delay High. Best conditions with Delay Low: Temp High, Weight Low. Lecture

49 Reduced model Fit model using active terms: B + D + E + BE + DE DE confirmed as active. Lecture

50 Diagnostics 2 Diagnostic Plot Deleted Residual Fitted Value Lecture

51 Diagnostics 3 Normal Probability Plot Deleted Residual Score 1 2 Lecture

52 Delete Design point 5, iterate analysis Effect estimates similar Interaction patterns similar s = 0.15, df = 9 ( = 14 5 ) Mean SE Mean B*D*E / 2 = to / 2 = 0.45 to 0.93 Lecture

53 Lecture Review of split unit experiments Why split? Why block? 2. Review of Laboratory 2 Cambridge grassland experiment Soup mix packet filling 3. Extending plot and treatment structures Wood stain experiment 4. Robust Product Design 5. An interesting interaction? Lecture

54 Water Resistance of Wood Stains Testing water resistance of four wood stains Pretreatments applied to whole boards Pretreated boards cut into 4 panels Stains applied to panels Replicated 3 times Lecture

55 Results Pretreatment 1 Pretreatment 2 Board Panels Stain Stain Stain Stain Lecture

56 Extending the unit structure Suppose the 6 boards were in 3 blocks of 2 e.g. 2 boards selected from each of 3 production runs, or 2 boards treated on each of 3 successive days Block Board Pretreatment Note: Boards nested in Blocks Lecture

57 Unit / Treatment Structure Diagram Factor Units Blocks Pretreatment Boards Stain Panels Lecture

58 Results of Water Resistance Experiment Block Board Pretreatment Stain Lecture

59 Extended Unit / Treatment Structure and Analysis of Variance Factor Units Blocks MS(Blocks) ANOVA Pretreatment Boards MS(Pretreatment) MS(Boards Residuals) Stain Panels MS(Stain) MS(P x S) MS(Panels Residuals) Lecture

60 Analysis of Variance for Water Resistance Minitab model: Block Pretreat Block * Pretreat Stain Pretreat * Stain Source DF SS MS F P Block Pretreat Block*Pretreat Stain Pretreat*Stain Error Total Lecture

61 Expected Mean Squares Lecture

62 Analysis of Variance for Water Resistance Minitab model: Block Pretreat Block * Pretreat Stain Pretreat * Stain Source DF SS MS F P Block Pretreat Boards Block*Pretreat Stain Pretreat*Stain Error Total Lecture

63 Analysis ignoring blocks Minitab model: Pretreat Board(Pretreat) Stain Pretreat * Stain Source DF SS MS F P Pretreat Board(Pretreat) Stain Pretreat*Stain Error Total Lecture

64 Block or Not? Not blocking when there is a block effect implies reduced power for treatment effects test; because Error term includes block variation. Blocking when there is no block effect implies reduced power for treatment effects test; because Error degrees of freedom reduced Lecture

65 Extending the treatment structure Suppose the four Stain levels are combinations of two 2-level factors: Stain type, 1 or 2, number of Coats applied, 1 or 2. Factor Units Blocks Pretreatment Boards Stain x Coats Panels Lecture

66 Extending the Minitab model Block Pretreatment Block*Pretreatment Stain Coat Stain*Coat Pretreatment*Stain Pretreatment*Coat Pretreatment*Stain*Coat Lecture

67 Analysis of Variance Source DF SS MS F P Block Pretreatment Block*Pretreatment Stain Coat Stain*Coat Pretreatment*Stain Pretreatment*Coat Pretreatment*Stain*Coat Error Total Lecture

68 Lecture Review of split unit experiments Why split? Why block? 2. Review of Laboratory 2 Cambridge grassland experiment Soup mix packet filling 3. Extending plot and treatment structures Wood stain experiment 4. Robust Product Design 5. An interesting interaction? Lecture

69 Robustness Studies Seek optimal settings of experimental factors that remain optimal, irrespective of uncontrolled environmental factors. Run the experimental design, the inner array, at fixed settings of the environmental variables, the outer array. Popularised by Taguchi. Improved by Box et al Lecture

70 Study of Detergent Robustness Detergent performance affected by Temperature of wash water, T ( + or ) Hardness of wash water, H ( + or ) concentration of detergent in water, R ( + or ) Key product design factors: amount of Ingredient 1 A ( + or ) amount of Ingredient 2 B ( + or ) process version 1 C ( + or ) process version 2 D ( + or ) Response: Whiteness, measured by reflectometer Lecture

71 Study of Detergent Robustness Design points in a fractional factorial plan used to produce batches of 8 variants of the detergent; Design points in a fractional factorial plan used to set up 4 wash conditions; Samples of each detergent assessed under each of the 4 wash conditions Lecture

72 Results Product Version Environmental factors T + + H + + Design factors R + + A B C D i ii iii iv Mean Range Lecture

73 Unit / Treatment Structure Diagram Treatment Factors Design factors Experimental Units Detergent Types Environmental factors Detergent Samples Lecture

74 2 7 2 Estimated Effects Term Effect Term Effect T -2.5 R*A 0 H R*B R 0.25 R*C A R*D 0 B A*B C A*C D A*D 0 T*A -0.5 T*A*B T*B T*A*C T*C T*A*D 0.75 T*D H*A*B H*A H*A*C H*B H*A*D 0 H*C R*A*B H*D 0.5 R*A*C R*A*D Lecture

75 Split plots model analysis B significant, positive, set at high (+) level T and TC interaction significant Lecture

76 Split plots model analysis At low C, whiteness is highly sensitive to T. At high C, whiteness is relatively insensitive to T. Lecture

77 Conclusion Set B and C to high levels, A and D as convenient Environmental factors T + + H + + Design factors R + + Product A B C D i ii iii iv Mean Range Lecture

78 Lecture Review of split unit experiments Why split? Why block? 2. Review of Laboratory 2 Cambridge grassland experiment Soup mix packet filling 3. Extending plot and treatment structures Wood stain experiment 4. Robust Product Design 5. An interesting interaction? Lecture

79 Interaction between Factors Case study: Emotional Arousal Male and female subjects presented with four different visual stimuli, pictures of an infant a landscape a male nude a female nude Levels of subjects' emotional arousal were measured Arousal.xls Lecture

80 Infant Landsdcape Nude Female Nude Male Infant Landsdcape Nude Female Nude Male Interaction between Factors Case study: Emotional Arousal Levels of Arousal of Males and Females to Different Visual Stimuli Male Female Pictures Pictures Lecture

81 Interaction between Factors Case study: Emotional Arousal Picture Main Effects Plot Interaction Plot Gender F M Mean Arousal Level Mean Arousal Level I L Picture NF NM I L Picture NF NM Lecture

82 Minute test How much did you get out of today's class? How did you find the pace of today's class? What single point caused you the most difficulty? What single change by the lecturer would have most improved this class? Lecture

83 Reading Lecture Notes: Split Units Design and Analysis Lab 2 Feedback (BHH 13.1 to p. 544) Lecture