Design and Analysis of Experiments Lecture PDF Free Download

Lecture 6.1 1. 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

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

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

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

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

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

Cambridge Grassland Experiment Blocks 1 2 3 4 5 6 Whole Plots 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 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

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 266 213 208 210 222 266 220 184 184 216 178 207 202 175 184 169 142 151 C 165 127 155 150 167 163 155 118 153 159 125 135 147 118 98 132 104 69 F 198 180 200 247 203 228 190 168 174 225 149 162 184 175 144 164 145 116 S 184 127 150 188 167 157 140 128 141 174 107 113 154 112 113 116 89 101 Lecture 6.1 9

Treatment yields vs Layout yields (Block 1) Block 1 C H R A 266 213 208 C 165 127 155 F 198 180 200 S 184 127 150 Block 1 Whole Plot 1 2 3 Treatment H C R Sub Plot 1 C A A 127 266 208 Sub Plot 2 A S C 213 184 155 Sub Plot 3 F C F 180 165 200 Sub Plot 4 S F S 127 198 150 Lecture 6.1 10

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 6.1 11

Plot Structure Units Blocks Whole Plots Subplots Lecture 6.1 12

Step 1: Two components of total variation Mintab model: Plot Subplot(Plot) Source DF SS MS F P Plot 17 54577 3210.4 2.63 0.004 Subplot(Plot) 54 65896 1220.3 ** Error 0 * * Total 71 120473 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 17 54577 3210 2.63 0.004 Error 54 65896 1220 Total 71 120473 Lecture 6.1 13

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 6.1 14

Step 2: Analysis of whole plot total variation Minitab model: Block Treatment Source DF SS MS F P Block 5 37425 7485 6.79 0.000 Treatment 2 12471 6236 5.65 0.005 Error 64 70577 1103 Total 71 120473 Minitab model: Plot (see Slide 13) Source DF SS MS F P Plot 17 54577 3210 2.63 0.004 Error 54 65896 1220 Total 71 120473 Plot Error DF = 17 5 2 = 10 Plot Error SS = 54577 37425 12471 = 4681 Lecture 6.1 15

Classwork 1 From the values on Slide 15, construct an analysis of variance table for whole plots variation. Lecture 6.1 16

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 6.1 17

Step 2: Analysis of whole plot total variation Minitab model: Block Treatment Block*Treatment Source DF SS MS F P Block 5 37425 7485 6.13 0.000 Treatment 2 12471 6236 5.11 0.009 Block*Treatment 10 4681 468 0.38 0.949 Error 54 65896 1220 Total 71 120473 Lecture 6.1 18

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 6.1 19

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 5 37425 7485 21.37 0.002 x T 2 12471 6236 13.32 0.002 B*T 10 4681 468 1.94 0.079 F 3 56023 18674 151.24 0.000 T*F 6 782 130 0.54 0.774 B*F 15 1852 123 0.51 0.914 Error 30 7240 241 Total 71 120473 Lecture 6.1 20

Expected Mean Squares Source Expected Mean Square Block S + 3 + 4 + 12 Treatment + 4 + Treatment effect Plot + 4 Fertiliser + 3 + Fertiliser effect Treatment*Fertiliser 2 S 2 S 2 S 2 S Block*Fertiliser + 3 2 2 2 2 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 6.1 21

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 6.1 22

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 5 37425 7485 15.99 0.000 T 2 12471 6236 13.32 0.002 B*T 10 4681 468 2.32 0.027 F 3 56023 18674 92.43 0.000 T*F 6 782 130 0.64 0.594 Error 45 7240 202 Total 71 120473 Lecture 6.1 23

Expected Mean Squares Source Expected Mean Square 2 2 Block S + 4 + 12 2 S Treatment + 4 + 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 6.1 24

Decomposition Summary Step 1 Source DF SS Plot Total 17 54577 Subplot Total 54 65896 Total 71 120473 Lecture 6.1 25

Decomposition Summary Step 2 Source DF SS MS F P Block 5 37425 7485 15.99 0.000 Treatment 2 12471 6236 13.32 0.002 Plot Error 10 4681 468 2.32 0.027 Plot Total 17 54577 Subplot Total 54 65896 Total 71 120473 Lecture 6.1 26

Decomposition Summary Step 3 Source DF SS MS F P Block 5 37425 7485 15.99 0.000 Treatment 2 12471 6236 13.32 0.002 Plot Error 10 4681 468 2.32 0.027 Plot Total 17 54577 3210 2.63 0.004 Fertiliser 3 56023 18674 92.43 0.000 T*F 6 782 130 0.64 0.694 Subplot Error 45 9092 202 Subplot Total 54 65896 Total 71 120473 Lecture 6.1 27

Split Plots Analysis Whole Plots Source DF SS MS F P Block 5 37425 7485 15.99 0.000 Treatment 2 12471 6236 13.32 0.002 Plot Error 10 4681 468 2.32 0.027 Plot Total 17 54577 3210 2.63 0.004 Fertiliser 3 56023 18674 92.43 0.000 T*F 6 782 130 0.64 0.694 Subplot Error 45 9092 202 Subplot Total 54 65896 Total 71 120473 Lecture 6.1 28

Split Plots Analysis Sub plots Source DF SS MS F P Block 5 37425 7485 15.99 0.000 Treatment 2 12471 6236 13.32 0.002 Plot Error 10 4681 468 2.32 0.027 Plot Total 17 54577 3210 2.63 0.004 Fertiliser 3 56023 18674 92.43 0.000 T*F 6 782 130 0.64 0.694 Subplot Error 45 9092 202 Subplot Total 54 65896 Total 71 120473 Lecture 6.1 29

Decomposition Summary Source DF SS MS F P Block 5 37425 7485 15.99 0.000 Treatment 2 12471 6236 13.32 0.002 Plot Error 10 4681 468 2.32 0.027 Plot Total 17 54577 3210 2.63 0.004 Fertiliser 3 56023 18674 92.43 0.000 T*F 6 782 130 0.64 0.694 Subplot Error 45 9092 202 Subplot Total 54 65896 Total 71 120473 Lecture 6.1 30

Subplots Residuals vs Fitted Values Lecture 6.1 31

Same diagnostic, Different interpretation? Lecture 6.1 32

Subplots Residuals Normal Plot Lecture 6.1 33

Whole Plots Residuals vs Fitted Values Lecture 6.1 34

Whole Plots Residuals Normal Plot 2 Deleted Residual 1 0-1 -2-3 -2-1 0 Score 1 2 Lecture 6.1 35

Check Interactions Lecture 6.1 36

Check Interactions Lecture 6.1 37

Check Interactions Lecture 6.1 38

Check Interactions Lecture 6.1 39

Interaction plots for Grassland experiment Treatments Lecture 6.1 40

Lecture 6.1 41

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 6.1 43

Minitab analysis Lecture 6.1 44

Minitab analysis Normal plot vs Pareto Principle vs Lenth? Lecture 6.1 45

Alias analysis Estimated Effects Term Effect Alias E -0.470 E + A*B*C*D B*E 0.405 B*E + A*C*D D*E -0.315 D*E + A*B*C E is aliased with or confounded with A*B*C*D Lecture 6.1 46

Graphical and numerical summaries E B D + + 1.71 1.22 1.31 1.60 E + 0.83 1.15 + 1.17 0.82 Lecture 6.1 47

Best conditions Best conditions: Temp Low, Weight High, Delay High. Best conditions with Delay Low: Temp High, Weight Low. Lecture 6.1 48

Reduced model Fit model using active terms: B + D + E + BE + DE DE confirmed as active. Lecture 6.1 49

Diagnostics 2 Diagnostic Plot Deleted Residual 1 0-1 -2-3 0.8 1.0 1.2 1.4 Fitted Value 1.6 1.8 Lecture 6.1 50

Diagnostics 3 Normal Probability Plot Deleted Residual 2 1 0-1 -2-3 -2-1 0 Score 1 2 Lecture 6.1 51

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 - - - 1.700 0.153 + - - 1.205 0.108 - + - 1.975 0.108 + + - 1.225 0.108 - - + 0.975 0.108 + - + 1.360 0.108 - + + 0.690 0.108 + + + 0.940 0.108 1.205 2.26 0.15/ 2 = 0.965 to 1.445 0.69 2.26 0.15/ 2 = 0.45 to 0.93 Lecture 6.1 52

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 6.1 54

Results Pretreatment 1 Pretreatment 2 Board 1 2 3 4 5 6 Panels 1-4 5-8 9-12 13-16 17-20 21-24 Stain 1 43.0 57.4 52.8 46.6 52.2 32.1 Stain 2 51.8 60.9 59.2 53.5 48.3 34.4 Stain 3 40.8 51.1 51.7 35.4 45.9 32.2 Stain 4 45.5 55.3 55.3 32.5 44.6 30.1 Lecture 6.1 55

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 1 2 3 1 1 4 2 2 1 5 2 3 1 6 2 Note: Boards nested in Blocks Lecture 6.1 56

Unit / Treatment Structure Diagram Factor Units Blocks Pretreatment Boards Stain Panels Lecture 6.1 57

Results of Water Resistance Experiment Block Board Pretreatment Stain 1 2 3 4 1 1 1 43.0 51.8 40.8 45.5 4 2 46.6 53.5 35.4 32.5 2 2 1 57.4 60.9 51.1 55.3 5 2 52.2 48.3 45.9 44.6 3 3 1 52.8 59.2 51.7 55.3 6 2 32.1 34.4 32.2 30.1 Lecture 6.1 58

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 6.1 59

Analysis of Variance for Water Resistance Minitab model: Block Pretreat Block * Pretreat Stain Pretreat * Stain Source DF SS MS F P Block 2 376.99 188.49 0.95 0.514 Pretreat 1 782.04 782.04 3.93 0.186 Block*Pretreat 2 398.38 199.19 15.67 0.000 Stain 3 266.01 88.67 6.98 0.006 Pretreat*Stain 3 62.79 20.93 1.65 0.231 Error 12 152.52 12.71 Total 23 2038.72 Lecture 6.1 60

Expected Mean Squares Lecture 6.1 61

Analysis of Variance for Water Resistance Minitab model: Block Pretreat Block * Pretreat Stain Pretreat * Stain Source DF SS MS F P Block 2 376.99 188.49 0.95 0.514 Pretreat 1 782.04 782.04 3.93 0.186 Boards Block*Pretreat 2 398.38 199.19 15.67 0.000 Stain 3 266.01 88.67 6.98 0.006 Pretreat*Stain 3 62.79 20.93 1.65 0.231 Error 12 152.52 12.71 Total 23 2038.72 Lecture 6.1 62

Analysis ignoring blocks Minitab model: Pretreat Board(Pretreat) Stain Pretreat * Stain Source DF SS MS F P Pretreat 1 782.04 782.04 4.03 0.115 Board(Pretreat) 4 775.36 193.84 15.25 0.000 Stain 3 266.00 88.67 6.98 0.006 Pretreat*Stain 3 62.79 20.93 1.65 0.231 Error 12 152.52 12.71 Total 23 2038.72 Lecture 6.1 63

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 6.1 65

Extending the Minitab model Block Pretreatment Block*Pretreatment Stain Coat Stain*Coat Pretreatment*Stain Pretreatment*Coat Pretreatment*Stain*Coat Lecture 6.1 66

Analysis of Variance Source DF SS MS F P Block 2 376.99 188.49 0.95 0.514 Pretreatment 1 782.04 782.04 3.93 0.186 Block*Pretreatment 2 398.38 199.19 15.67 0.000 Stain 1 38.00 38.00 2.99 0.109 Coat 1 214.80 214.80 16.90 0.001 Stain*Coat 1 13.20 13.20 1.04 0.328 Pretreatment*Stain 1 43.20 43.20 3.40 0.090 Pretreatment*Coat 1 18.38 18.38 1.45 0.252 Pretreatment*Stain*Coat 1 1.21 1.21 0.10 0.762 Error 12 152.52 12.71 Total 23 2038.72 Lecture 6.1 67

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 6.1 69

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 6.1 70

Study of Detergent Robustness Design points in a 2 4 1 fractional factorial plan used to produce batches of 8 variants of the detergent; Design points in a 2 3 1 fractional factorial plan used to set up 4 wash conditions; Samples of each detergent assessed under each of the 4 wash conditions Lecture 6.1 71

Results Product Version Environmental factors T + + H + + Design factors R + + A B C D i ii iii iv Mean Range 1 88 85 88 85 86.50 3 2 + + 80 77 80 76 78.25 4 3 + + 90 84 91 86 87.75 7 4 + + 95 87 93 88 90.75 8 5 + + 84 82 83 84 83.25 2 6 + + 85 84 82 82 83.25 3 7 + + 91 93 92 92 92.00 2 8 + + + + 89 88 89 87 88.25 2 Lecture 6.1 72

Unit / Treatment Structure Diagram Treatment Factors Design factors Experimental Units Detergent Types Environmental factors Detergent Samples Lecture 6.1 73

2 7 2 Estimated Effects Term Effect Term Effect T -2.5 R*A 0 H -0.25 R*B -0.125 R 0.25 R*C -0.125 A -2.25 R*D 0 B 6.875 A*B 1.875 C 0.875 A*C 0.375 D -3.75 A*D 0 T*A -0.5 T*A*B -0.375 T*B -0.625 T*A*C -0.125 T*C 2.125 T*A*D 0.75 T*D -0.25 H*A*B 0.125 H*A -0.75 H*A*C -0.125 H*B 0.375 H*A*D 0 H*C -0.375 R*A*B 0.375 H*D 0.5 R*A*C -0.125 R*A*D -0.75 Lecture 6.1 74

Split plots model analysis B significant, positive, set at high (+) level T and TC interaction significant Lecture 6.1 75

Split plots model analysis At low C, whiteness is highly sensitive to T. At high C, whiteness is relatively insensitive to T. Lecture 6.1 76

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 1 88 85 88 85 86.50 3 2 + + 80 77 80 76 78.25 4 3 + + 90 84 91 86 87.75 7 4 + + 95 87 93 88 90.75 8 5 + + 84 82 83 84 83.25 2 6 + + 85 84 82 82 83.25 3 7 + + 91 93 92 92 92.00 2 8 + + + + 89 88 89 87 88.25 2 Lecture 6.1 77

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 6.1 79

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 25 20 15 10 25 20 15 10 Pictures Pictures Lecture 6.1 80

Interaction between Factors Case study: Emotional Arousal Picture Main Effects Plot Interaction Plot 25 25 Gender F M Mean Arousal Level 20 15 Mean Arousal Level 20 15 10 10 I L Picture NF NM I L Picture NF NM Lecture 6.1 81

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 6.1 82

Reading Lecture Notes: Split Units Design and Analysis Lab 2 Feedback (BHH 13.1 to p. 544) Lecture 6.1 83

Design and Analysis of Experiments Lecture 6.1