Design and Analysis of Experiments Lecture 6.1

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Design and Analysis of ExperimentsLecture 6.1

1. Review of split unit experiments

2. Review of Laboratory 2

3. Review of special topics (part)

Diploma in StatisticsDesign and Analysis of Experiments

Minute Test: How Much

Minute Test: How Fast

Split units experiments

arise when– 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 designs.

"Most industrial experiments are ... split plot in their design.“ C. Daniel (1976) p. 175

Reasons for using split units

• Adding another factor after the experiment started

• Changing one factor is– more difficult– more expensive– more time consuming

• than changing others• Some factors require better precision than others

Blocks

Whole units

Subunits

Recognising Plot and Treatment Structure

Factor

Whole unitTreatment

SubunitTreatment

MS(Blocks)

MS(WTreatments)MS(Whole units)MS(B x WT)

MS(STreatments)MS(Interactions)MS(Subunits)

Illustration: Water resistance of wood stains

Boards

Panels

Plot structure, Treatment structure

Factor

Pretreatment

24 subunits (panels) nested in 6 whole units (boards).2 pretreatments allocated to whole units.4 stains allocated to subunits within whole units.

Assessing variation

• Variation between boards due to– chance– Pretreatments?

• Variation between panels due to– chance– stains?– pretreatment by stain interaction?

Boards

Panels

Analysis of Variance

Factor

Pretreatment

MS(Pretreatment)MS(Boards)

MS(Stain)MS(Interaction)MS(Panels)

Minitab modelPretreatment Board(Pretreatment)Stain Pretreatment*Stain

Source DF SS MS F P

Pretreatment 1 782.04 782.04 4.03 0.115Board(Pretreatment) 4 775.36 193.84 15.25 0.000

Stain 3 266.00 88.67 6.98 0.006Pretreatment*Stain 3 62.79 20.93 1.65 0.231Error 12 152.52 12.71

Total 23 2038.72

Justifying the ANOVA Source Expected Mean Square1 Pretreat (5) + 4.0000 (2) + Q[1]2 Board(Pretreat) (5) + 4.0000 (2)3 Stain (5) + Q[3]4 Pretreat*Stain (5) + Q[4]5 Error (5)

Alternative notation:

Source Expected Mean Square

1 Pretreat + 4 + Pretreatment effect

2 Board(Pretreat) + 4

3 Stain + Stain effect

4 Pretreat*Stain + interaction effect

5 ErrorDiploma in StatisticsDesign and Analysis of Experiments

Extending the unit structureSuppose the 6 boards were in 3 blocks of 2

e.g. 2 boards selected from 3 production runs,

e.g. 2 boards treated on 3 successive days

Block 1 Block 2 Block 3B P S R B P S R B P S R1 1 1 43.0 2 1 1 57.4 3 1 1 52.81 1 2 51.8 2 1 2 60.9 3 1 2 59.21 1 3 40.8 2 1 3 51.1 3 1 3 51.71 1 4 45.5 2 1 4 55.3 3 1 4 55.3

4 2 1 46.6 5 2 1 52.2 6 2 1 32.14 2 2 53.5 5 2 2 48.3 6 2 2 34.44 2 3 35.4 5 2 3 45.9 6 2 3 32.24 2 4 32.5 5 2 4 44.6 6 2 4 30.1

Blocks

Boards

Panels

Expanded Unit and Treatment Structure

Factor

Pretreatment

MS(Blocks)

MS(Pretreatment)MS(Boards)MS(B x PT)

MS(Stain)MS(Interactions)MS(Panels)

Analysis of VarianceMinitab model

BlockPretreatment Block*PretreatmentStain Block*Stain Pretreatment*StainSource DF SS MS F P

Block 2 376.99 188.49 0.95 0.514

Pretreatment 1 782.04 782.04 3.93 0.186Block*Pretreatment 2 398.38 199.19 15.67 0.000

Stain 3 266.01 88.67 6.98 0.006Pretreatment*Stain 3 62.79 20.93 1.65 0.231Error 12 152.52 12.71

Total 23 2038.72

Extending the treatment structure

4 Stain levels ↔ two 2-level factors:

Stain type 1 or 2

number of Coats applied 1 or 2

Blocks

Boards

Panels

Expanded Unit and Treatment Structure

Factor

Pretreatment

Stain x Coat

MS(Blocks)

MS(Pretreatment)MS(B x PT)

MS(Stain)MS(Coat)MS(Interactions)MS(Panels)

Minitab model

Pretreatment Block*Pretreatment

Stain Coat Stain*Coat Pretreatment*Stain Pretreatment*CoatPretreatment*Stain*Coat

Source DF SS MS F P

Block 2 376.99 188.49 0.95 0.514Pretreatment 1 782.04 782.04 3.93 0.186Block*Pretreatment 2 398.38 199.19 15.67 0.000

Stain 1 38.00 38.00 2.99 0.109Coat 1 214.80 214.80 16.90 0.001Stain*Coat 1 13.20 13.20 1.04 0.328

Pretreatment*Stain 1 43.20 43.20 3.40 0.090Pretreatment*Coat 1 18.38 18.38 1.45 0.252Pretreatment*Stain*Coat 1 1.21 1.21 0.10 0.762Error 12 152.52 12.71

Total 23 2038.72

Laboratory 2, Exercise 1: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

Minitab analysis

Normal plot vs Pareto Principle vs Lenth?

Minitab analysis

Estimated Effects for Y

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

Graphical and numerical summaries

1.71.61.51.41.31.21.11.00.90.8

Interaction Plot for Y Interaction Plot for Y

E E – + – +

B – 1.71 0.83

D – 1.31 1.17

+ 1.22 1.15 + 1.60 0.82

Best conditions

Best conditions:

B Low, D High, E High.

Best conditions with E Low:

B High, D Low.

1.71.61.51.41.31.21.11.00.90.8

Interaction Plot for Y Interaction Plot for Y

Reduced modelFit model using active terms:

B + D + E + BE + DE

DE confirmed as active.

76543210

Standardized Effect

Pareto Chart of the Standardized Effects

Diagnostics

1.81.61.41.21.00.8

Fitted Value

Diagnostic Plot

Diagnostics

-3210-1-2

Normal Probability Plot

Delete Design point 5, iterate analysis

• Effect estimates similar• Interaction patterns similar• s = 0.15, df = 9 ( = 14 – 5 )Least Squares Means for Y Mean SE MeanB*D*E - - - 1.7000 0.1532 + - - 1.2050 0.1083 - + - 1.9750 0.1083 + + - 1.2250 0.1083 - - + 0.9750 0.1083 + - + 1.3600 0.1083 - + + 0.6900 0.1083 + + + 0.9400 0.1083

0.69 2.26×0.15/√2 = 0.37 to 1.01

1.205 2.26×0.15/√2 = 0.965 to 1.445

Laboratory 2, Exercise 2Cambridge Grassland Experiment

3 grassland treatmentsRejuvenator RHarrow Hno treatment C

randomly allocated to 3 neighbouring plots,replicated in 6 neighbouring blocks

4 fertilisersFarmyard manure FStraw SArtificial fertiliser Ano fertiliser C

randomly allocated to 4 sub plots within each plot.

Cambridge Grassland ExperimentBlocks 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 3Treatments 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

Randomised Blocks analysis forTreatments

Source DF SS MS F P

WB 5 149700 29940 15.99 0.000

WT 2 49884 24942 13.32 0.002

WB*WT 10 18725 1872 **

Error 0 * *

Total 17 218309

Model: WB + WT + WBxWT

Diagnostics

WB x WT Interaction Plot

WT x WB Interaction Plot

Split Plots AnalysisModel: B + T + B x T

+ F + B x F + T x F

Source DF SS MS F PB 5 37425.1 7485.0 21.37 0.002 xT 2 12471.0 6235.5 13.32 0.002B*T 10 4681.1 468.1 1.94 0.079

F 3 56022.7 18674.2 151.24 0.000B*F 15 1852.1 123.5 0.51 0.914T*F 6 781.5 130.3 0.54 0.774Error 30 7239.6 241.3

Total 71 120473.3

250200150100

Fitted Value

Residuals Versus Fitted Values

Diagnostics

250200150100

Fitted Value

Residuals Versus Fitted Values

Same diagnostic, Different interpretation?

Treatment comparisons

• First step: summary statistics

Variable Treatment Count MeanY C 24 181.46 H 24 150.96 R 24 157.17

s2 = MS(B*T) = 468.1; df = 10

Compare Harrow with Control

tsignificanllystatisticaisdifference

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Laboratory 2, Exercise 3Fertiliser experiment

Fertilisers potentially affecting bean yield

Low HighDung (D): none 10 tons per acreNitrochalk (N):none 0.4 cwt per acreSuperPhosphate (P):none 0.6 cwt per acreMuriate of Potash (K): none 1.0 cwt per acre

Questions:

What factors affect bean yield?

How can bean yield be maximised?

Exercise 2, Minitab analysis

N is marginally significant.

Pareto Principle suggests adding NPK and DP.

ABDABCACDBC

Effect

A DB NC PD K

Factor Name5

-10210-1-2

ctScore

Not SignificantSignificant

Effect Type

Pareto Chart of the Effects

Lenth's PSE = 3

Normal Plot of the Effects

Term Effect CoefConstant 47.250Block -0.375D -0.750 -0.375N -8.000 -4.000P 0.250 0.125K -2.250 -1.125D*N 1.250 0.625D*P 4.500 2.250D*K 0.000 0.000N*P 2.250 1.125N*K -1.750 -0.875P*K 0.500 0.250D*N*P -2.000 -1.000D*N*K 2.000 1.000D*P*K 2.250 1.125N*P*K -5.500 -2.750

Reduced modelFit model using active terms:

D, N, P, K, DP, NP, NK, PK, NPK

Active effects confirmed.Diagnostics unremarkable

543210

Standardized Effect

A DB NC PD K

Factor Name

Pareto Chart of the Standardized Effects

3-factor interaction

Adding N reduces yield overall, but has a small positive effect at high P and low K.

At low N, the P effect is negative at low K and positive at high K.

At high N, this interaction is reversed.

The best combination is no fertiliser.

Interaction Plot for YN Low

Interaction Plot for YN High

Least Squares Means for Y

Mean SE Mean N*P*K - - - 55.5 2.419 + - - 41.5 2.419 - + - 47.5 2.419 + + - 49.0 2.419 - - + 49.0 2.419 + - + 42.5 2.419 - + + 53.0 2.419 + + + 40.0 2.419

Best combination

N*P*K Mean SEBest: - - - 55.5 2.419Next best: - + + 53.0 2.419

Difference: 2.5 3.42

t: 2.5/3.42 = 0.73,

not statistically significantly different.

Review topics (part)

• Time related issues– repeated measures– cross-over designs

• Complex block structures• Analysis of Covariance• Robustness studies• Response surface designs

Time related issues1. Repeated measures

Example:

Calves fed diet supplements to improve growth

Initial weight recorded, Y0

blocks formed based on initial weight,

weights recorded at

4 weeks, Y1

8 weeks Y2

12 weeks Y3

16 weeks Y4

Split plots analysis?

• Calves are whole units• Time periods are sub units

Problems:

correlation structure

varying standard deviation

Solution:

Multivariate analysis

Time related issues2. Crossover designs

• Repeated measures designs compares diets on different calves,

• reduce variation by comparing diets on same calves,

• e.g. diet A for weeks 1 to 4

diet B for weeks 5 to 8

diet C for weeks 9 to 12

diet D for weeks 13 to 16• requires attention to order of dietsDiploma in StatisticsDesign and Analysis of Experiments

Crossover design

• Every diet occurs– once for each calf,– once in each time

period– Latin square

? correlation structure

? carry over

? experimental set up versus actual use

Calf Time Period 1 - 4 5 – 8 9 – 12 13 - 16

1 A B C D 2 B D A C 3 C A D B 4 D C B A

Complex blocking

• 2 blocking factors– calves and time periods– Latin square– Latin rectangle

• Incomplete blocks– more treatments than plots in a block– balanced incomplete blocks

Analysis of Covariance

Objective: take account of variation in uncontrolled environmental variables.

Solution: measure the environmental variables at each design point and incorporate in the analysis through regression methods (Analysis of Covariance)

Effects: reduces "error" variation, makes factor effects more significant

adjusts factor effect estimates to take account of extra variation source.

Analysis of Covariance; Illustration

Breaking strength of monofilament fibre (Y)produced by three different machines (1, 2, 3)

allowing for variation in fibre thickness (X)

Machine 1 Machine 2 Machine 3

Y X Y X Y X 36 20 40 22 35 21 41 25 48 28 37 23 39 24 39 22 42 26 42 25 45 30 34 21 49 32 44 28 32 15

Analysis of Covariance; Minitab

Analysis of Covariance; MinitabGeneral Linear Model: Y versus Machine

Source DF Seq SS Adj SS Adj MS F PX 1 305.13 178.01 178.01 69.97 0.000Machine 2 13.28 13.28 6.64 2.61 0.118Error 11 27.99 27.99 2.54Total 14 346.40

S = 1.59505

One-way ANOVA: Y versus Machine

Source DF SS MS F PMachine 2 140.4 70.2 4.09 0.044Error 12 206.0 17.2Total 14 346.4

S = 4.143

Analysis of Covariance; Minitab

32.530.027.525.022.520.017.515.0

Machine

Scatterplot of Y vs X

Covariance vs BlockingChance causes and assignable causes of variation

(W. Shewhart, 1931)

Chance causes of variation are themany individually negligible and unpredictable

butcollectively influential

factors that affect a process or system.

Assignable causes of variation are thefew individually influential and predictable effect

factors that affect a process or system.

Diploma i StatisticsDesign and Analysis of Experiments

Covariance vs Blocking

Blocking Chance causes

Covariance Assignable causes

Diploma i StatisticsDesign and Analysis of Experiments

Robustness StudiesSeek optimal settings of experimental factorsthat 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

Study of Detergent Robustness

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

Split plots model analysis

B significant, positive,

set at high (+) level

T and TC interaction

significant

Split plots model analysis

At low C, whiteness is highly sensitive to T.

At high C, whiteness is relatively insensitive to T.

Conclusion

• Set B and C to high levels, A and D as convenient

Design and Analysis of Experiments Lecture 6.1

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