Diploma in Statistics Design and Analysis of Experiments Lecture 5.11 © 2010 Michael Stuart Lecture 5.1 Part 1 "Split Plot" experiments 1.Review of –randomised

Diploma in StatisticsDesign and Analysis of Experiments

Lecture 5.1 1© 2010 Michael Stuart

Lecture 5.1 Part 1"Split Plot" experiments

1. Review of– randomised block designs– hierarchical / nested designs

2. Examples

3. Analysis of Whole Units

4. Analysis of Sub Units

5. Split plot analysis

6. Expected Mean Squares– Error terms for tests

7. Interactions



Minute Test: How Much

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Minute Test: How Fast

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Randomised Blocks, Again

An experiment was conducted to assess the effects of applying four chemicals to soybean seeds with a view to improving germination rates.

Each treatment was applied to 100 seeds planted in adjacent plots. As a check, another plot was planted with 100 seeds which received no treatment.

The experiment was replicated in five blocks of five plots each, with each treatment being assigned to plots at random within each block.

The number of failures in each plot was recorded.



Testing for Interaction with Blocks

Analysis of Variance for Failures

Source DF Seq SS Adj SS Adj MS F PBlock 4 49.8400 49.8400 12.4600 2.30 0.103Treatment 4 83.8400 83.8400 20.9600 3.87 0.022Block*Treatment 16 86.5600 86.5600 5.4100 **Error 0 * * *Total 24 220.2400

** Denominator of F-test is zero.

Without a valid reference term, it is not possible to have an F test for interaction.

To check for interaction,

– replicate each design point within each block,

– replicates provide estimate of pure error



Assumption of No Interaction

• With no replication, use block by treatment interaction mean square as error mean square.

• With interaction present, this means

– estimate of is inflated,

– power of the F test for treatment effects is reduced.



Blocking as Random Effect

Source Expected Mean Square

1 Block (3) + 5.0000 (1)

2 Treatment (3) + Q[2]

3 Error (3)



Values

Values

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S

T

eB

eS

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eT

e = eB + eS + eT

Hierarchy of components of variation

Batchvariation

Samplingvariation

Testingvariation

y

•

•



Hierarchical Design forEstimating Components of Variation

Batch 1 2 3 4 5 Sample 1 2 3 4 5 6 7 8 9 10 Test 40 39 30 30 26 28 25 26 29 28 14 15 30 31 24 24 19 20 17 17 Batch 6 7 8 9 10 Sample 11 12 13 14 15 16 17 18 19 20 Test 33 32 26 24 23 24 32 33 34 34 29 29 27 27 31 31 13 16 27 24 Batch 11 12 13 14 15 Sample 21 22 23 24 25 26 27 28 29 30 Test 25 23 25 27 29 29 31 32 19 20 29 30 23 23 25 25 39 37 26 28

60 measurementsnested in 30 samplesnested in 15 batches



Part 2 ExamplesExample 1

3 varieties of wheat are planted in a homogeneous block of 3 plots, with varieties randomly assigned to plots;

the experiment is replicated 4 times, with separate randomisations in each block, as follows:

Block 1 Block 2

V2 V1 V3 V3 V1 V2

Block 3 Block 4

V1 V3 V2 V2 V3 V1



ALBH AHBL

ALBL AHBH

Example 1

Block 1 Block 2

V2 V1 V3 V3 V1 V2

Block 3 Block 4

V1 V3 V2 V2 V3 V1

Following planting, it was decided to try two new fertilisers. Each plot was divided in four subplots and a 22 was implemented in each, with the possible combinations being assigned at random to subplots within each plot, as shown for one plot below.



Plot structure

48 subplots

nested in 12 whole plots

nested in 4 blocks

Treatment structure

3 varieties randomly allocated to whole plots within blocks

22 = 4 fertiliser combinations randomly allocated to subplots within whole plots

least variation

in between variation

most variation



Example 2

Electronic components are baked in an oven at a set temperature for a set time. Two factors thought to influence the life times of the components were the oven temperature and the bake time. Trial settings for these factors were chosen as follows:

Oven Temperature (T), °F, 580, 600, 620, 640,

Baking time (B), min, 5, 10, 15.

To save on costly runs, three components were baked together at each temperature, with one withdrawn at each of the set times. This plan was replicated 3 times. The results follow.



Example 2

Results of accelerated life time testsfor electronic components

Baking Time (min.)

Replicate Temperature of Oven (°F)

5 10 15

1 580 217 233 175 600 158 138 152 620 229 186 155 640 223 227 156

2 580 188 201 195 600 126 130 147 620 160 170 161 640 201 181 172

3 580 162 170 213 600 122 185 180 620 167 181 182 640 182 201 199



Example 2

What are the whole units?

What are the whole unit treatments?

What are the sub units?

What are the sub unit treatments?

What is the plot structure?

What is the treatment structure?



Example 2

What are the whole units?

What are the whole unit treatments?

What are the sub units?

What are the sub unit treatments?

oven load of 3 components

oven temperatures

single components

baking times



Unit and Treatment Structures

Baking Time (min.)

Replicate Temperature of Oven (°F)

5 10 15

1 580 217 233 175 600 158 138 152 620 229 186 155 640 223 227 156

2 580 188 201 195 600 126 130 147 620 160 170 161 640 201 181 172

3 580 162 170 213 600 122 185 180 620 167 181 182 640 182 201 199



Unit structure

36 sub units

nested in 12 whole units

nested in 3 blocks

Treatment structure

4 temperatures randomly allocated to whole units within blocks

3 baking times randomly allocated to sub units within whole units

least variation

in between variation

most variation



Implications of unit and treatment structures

Treatment effects assessed relative to variation between units to which they are applied.



Case study

Paper manufactured in two stages:

pulp prepared in large batches, long process,

batches divided into small parts, each of which is put through a short cooking process.

Experiment to investigate effects of

three pulp preparation methods,

four cooking temperature settings

on tensile strength of the paper.



Case study

Protocol:

batch made using one method, randomly selected,

each of four samples "cooked" at one of the four different temperatures, random order

repeated for the other two methods,

replicated on successive days, new random orderings

Results



Randomised Blocks analysis for Methods

Day Method 1 2 3

1 34.50 35.25 37.25

2 38.75 37.25 39.50

3 31.00 33.25 37.50



Randomised Blocks analysis for Methods

Analysis of Variance for Y, no interaction term

Source DF Seq SS Adj SS Adj MS F PB 2 19.389 19.389 9.694 4.28 0.102M 2 32.097 32.097 16.049 7.08 0.049Error 4 9.069 9.069 2.267Total 8 60.556

S = 1.50578

Analysis of Variance for Y, with interaction term

Source DF Seq SS Adj SS Adj MS F PB 2 19.3889 19.3889 9.6944 4.28 0.102M 2 32.0972 32.0972 16.0486 7.08 0.049B*M 4 9.0694 9.0694 2.2674 **Error 0 * * *Total 8 60.5556



Diagnostics

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Diagnostics

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Normal Score

Normal Plot



Randomised Blocks analysis for Temperature

Day 1 Day 2 Day 3 200 31.00 30.00 32.67 225 34.00 32.67 37.00 250 36.00 38.00 39.67

Temperature

275 38.00 40.33 43.00



Randomised Blocks analysis for Temperature

Analysis of Variance for Y, no interaction term

Source DF Seq SS Adj SS Adj MS F PB 2 25.879 25.879 12.940 11.30 0.009T 3 144.636 144.636 48.212 42.10 0.000Error 6 6.871 6.871 1.145Total 11 177.386

S = 1.07013

Analysis of Variance for Y, with interaction term

Source DF Seq SS Adj SS Adj MS F PB 2 25.879 25.879 12.940 11.30 0.009T 3 144.636 144.636 48.212 42.10 0.000B*T 6 6.871 6.871 1.145 **Error 0 * * *Total 11 177.386



Diagnostic analysis

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Diagnostic analysis

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Normal Plot



Response: Strength (Y)

Factors: Day (Block), BMethod, MTemperature, T

Effects to include in model:

BMB*M

TB*TM*T

Split plot analysis

assessed at whole unit level

assessed at sub unit level



Split plot analysis

Analysis of Variance for Y

Source DF Seq SS Adj SS Adj MS F PB 2 77.556 77.556 38.778 4.68 0.126 xM 2 128.389 128.389 64.194 7.08 0.049B*M 4 36.278 36.278 9.069 2.14 0.138T 3 434.083 434.083 144.694 42.01 0.000B*T 6 20.667 20.667 3.444 0.81 0.580M*T 6 75.167 75.167 12.528 2.96 0.052Error 12 50.833 50.833 4.236Total 35 822.972

x Not an exact F-test.

S = 2.05818



Split plot analysis

Exercise: Check calculation of F ratios for M and T and corresponding degrees of freedom; cross check with previous analyses.



4 Expected Mean Squares

Source Expected Mean Square for Each Term

1 B (7) + 3.0000 (5) + 4.0000 (3) + 12.0000 (1)

2 M (7) + 4.0000 (3) + Q[2, 6]

3 B*M (7) + 4.0000 (3)

4 T (7) + 3.0000 (5) + Q[4, 6]

5 B*T (7) + 3.0000 (5)

6 M*T (7) + Q[6]

7 Error (7)

Exercise: Translate into 2 notation.



Diagnostics

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Diagnostics

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Normal Plot



Interaction effect?

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Treatment Profile Plot



Interaction effect?

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Reasons for using split plots

• Adding another factor after the experiment started

• Some factors require better precision than others

• Changing one factor is

– more difficult– more expensive– more time consuming

than changing others



Reading

DCM §4.1, §14.4



Lecture 5.1 Part 2Further Developments

• Repeated measures

• Robust design

• Analysis of Covariance

• Non-normal error

• Strategies for experiments



Robust design

Seek optimal settings of experimental factors

that remain optimal,

irrespective of uncontrolled environmental factors.

Run the experimental design, the inner array,at a range of settings of the environmental variables, the outer array.

Popularised by Taguchi.



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 fibreproduced by three different machines,

allowing for variation in fibre thickness.

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




General 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




32.530.027.525.022.520.017.515.0

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Scatterplot of Y vs X



Further Developments

• Non-Normal errors

– transformations

– generalised linear models, including Logistic



Changing spread with log



























Why the log transform works

High spread at high X

transformed to

low spread at high Y

Low spread at low X

transformed to

high spread at low Y



Strategies for Experimenting

– Consultation

– Planning

– Resources

– Ethical issues

– Implementation of design

– Application of results





Strategy for ExperimentationShewhart's PDCA Cycle

Check

Act

Plan

Do



Strategy for ExperimentationShewhart's PDCA Cycle

• Plan: Plan a change to the process, predict its effect, plan to measure the effect

• Do: Implement the change as an experiment and measure the effect

• Check: Analyse the results to learn what effect the change had, if any

• Act: If successful, make the change permanent, proceed to plan the next improvement

or

if not, proceed to plan an alternative change



Strategy for Experimentation:new vs old manufacturing process

Plan:

• Compare defect rates for old process and new (cheaper) process

– predict reduction, or no increase, in number of defectives using new process

• Sample output over an eight week period, six days per week

– select 50 components at random per day

• Count number of defectives per sample



Do:

• Implement plan

• Record daily numbers of defectives

Assessing experimental process for manufacturing electronic components



Check:

• Analyse data

• test statistical significance of the change




Act:

• If no worse, make the change permanent,

– proceed to plan the next improvement

or

• if not, proceed to plan an alternative change




Resources

e.g. sample size

Need to know

Also, need to know €



Ethical issues

– withholding medical treatment?

– double-blind experiments,

– inadequate budget puts patients at risk for non-informative results



Strategy

When you see the credits roll at the end of a successful movie you realize there are many more things that must be attended to in addition to choosing a good script.

Similarly in running a successful experiment there are many more things that must be attended to in addition to choosing a good experimental design.

Ref: Robinson, G.K., Practical Strategies for Experimenting, Wiley, 2000.

Documents

Diploma in Statistics Design and Analysis of Experiments Lecture 5.11 © 2010 Michael Stuart Lecture 5.1 Part 1 "Split Plot" experiments 1.Review of –randomised