Design and Analysis of Experiments Lecture 5 · Lecture 5.1 3 © 2016 Michael Stuart How Fast...

Lecture 5.1 1

© 2016 Michael Stuart

Design and Analysis of Experiments

Lecture 5.1

1. Review of Lecture 4.2

2. Fisher's Potato Study

− Decomposing the split plot analysis

− Multi level diagnostics

3. Split Plots in Blocks

− Cambridge Grassland Study; Review

− Baking Electronic Components; Analysis

− Block*Treatment Interaction as Error

− Split plot analysis

− An unexpected interaction

Postgraduate Certificate in Statistics


Lecture 5.1 2


How Much



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Lecture 5.1 3


How Fast



5432

Lecture 5.1 4


Review of Lecture 4.2

• Unit structure with no treatments

– Hierarchical Nesting vs Crossing

– Fixed and Random Factor Effects

– Implications for design and analysis

• Split Units Analysis

– right and wrong models



Lecture 5.1 5


Values

Values

m

•

sS

sT

eP

eS

sP

eT

e = eP + eS + eT

Model for variation in moisture content

Process

variation

Sampling

variation

Testing

variation

y

•

•



Lecture 5.1 6


Model for variation in moisture content

Basic model:

Y = m + eP + eS + eT

Components of variance:

2

T

2

S

2

P

2

Y ssss

2

T

2

S

2

PY ssss



Lecture 5.1 7


Aside on Nesting and Crossing

• Two factors are crossed when

each level of one factor occurs with

every level of the other factor.

– usually concerns treatment factors

• A factor is nested in another factor (the nesting

factor) if

each level of the nested factor occurs with

exactly one level of the nesting factor.

– usually concerns unit / plot factors.



Lecture 5.1 8


Recall Iron Content of Food study

• 4 samples of each of 3 food types were cooked in

each of 3 pot types,

• iron content in each sample measured in

milligrams of iron per 100 grams of cooked food.

Pot Type Food Type

Meat Legumes Vegetables

Aluminium 1.77 2.36 1.96 2.14 2.40 2.17 2.41 2.34 1.03 1.53 1.07 1.30

Clay 2.27 1.28 2.48 2.68 2.41 2.43 2.57 2.48 1.55 0.79 1.68 1.82

Iron 5.27 5.17 4.06 4.22 3.69 3.43 3.84 3.72 2.45 2.99 2.80 2.92



Lecture 5.1 9


Hierachical / Nested Structure

Diploma in Statistics


Batches

Samples

Subsamples

(Tests)

1

1

1 2

2 3 4

2

5

3 4

6 7 8

Lecture 5.1 10


Aside on Fixed and Random Factor Effects

Experimental factor variation usually has fixed effect,

e.g., Temperature, Pressure, Pot Type, Food Type,

Experimental unit variation usually has random effect,

e.g., Batch, Sample, Board, Panel,

(with exceptions)

Implications for Analysis of Variance



Lecture 5.1 12


Units

Batches

Samples

Tests

Unit / Treatment Structure Diagram

Factor

Process

Sampling

method

Test method

ANOVA

MS(Process)

MS(Batch variation)

MS(Sampling method)

MS(P x S)

MS(Sampling variation)

MS(Test method)

MS(Interactions)

MS(Testing error) Postgraduate Certificate in Statistics


Lecture 5.1 13


Split units design:

Right and Wrong Analysis

• Testing water resistance of four wood stains

• Stains applied to four panels cut from a board

• Boards are pretreated with one of two treatments.



Lecture 5.1 14


Assessing variation

• Variation between boards due to

– chance

– Pretreatments?

• Variation between panels due to

– chance

– stains?

– pretreatment by stain interaction?



Lecture 5.1 15


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 5.1 16


Minitab analysis

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 5.1 17


Results


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 5.1 18


Results


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 5.1 19


The wrong analysis

Assuming each Pretreatment / Stain combination run

4 times in random order

Analysis of Variance for Resistance

Source DF SS MS F P

Pretreat 1 782.04 782.04 13.49 0.002

Stain 3 266.01 88.67 1.53 0.245

Pretreat*Stain 3 62.79 20.93 0.36 0.782

Error 16 927.88 57.99

Total 23 2038.72



Lecture 5.1 20


Right and Wrong

Right: 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

Wrong: Source DF SS MS F P

Pretreat 1 782.04 782.04 13.49 0.002

Stain 3 266.01 88.67 1.53 0.245

Pretreat*Stain 3 62.79 20.93 0.36 0.782

Error 16 927.88 57.99

Total 23 2038.72



Lecture 5.1 21


Right and Wrong

Right: 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

Wrong: Source DF SS MS F P

Pretreat 1 782.04 782.04 13.49 0.002

Stain 3 266.01 88.67 1.53 0.245

Pretreat*Stain 3 62.79 20.93 0.36 0.782

Error 16 927.88 57.99

Total 23 2038.72



Lecture 5.1 22


The wrong analysis

F(Pretreat) has smaller denominator, MS(Error);

F(Pretreat) is now highly significant.

MS(Error) is increased, F(Stain) is reduced;

F(Stain) is now not statistically significant



Lecture 5.1 23



Lecture 5.1













Lecture 5.1 24


Fisher's Potato Study

• Twelve varieties of potatoes planted in 36 plots

– each variety planted in three plots "scattered

over the area"

• Each plot divided into three subplots,

– each subplot fertilised with one of three

fertilisers.



Lecture 5.1 25


Fisher's Potato Study

Varieties:

Ajax

Arran Comrade

British Queen

Duke of York

Epicure

Great Scott

Iron Duke

King of Kings

Kerr's Pink

Nithsdale

Tinwald Perfection

Up-to-Date

Fertilisers:

Basal manure dressing

Manure with added

Potassium Sulphate

Manure with added

Potassium Chloride.



Lecture 5.1 26


Results: Yield (lbs per plant)

Variety Sulphate Chloride Basal

Ajax 3.20 4.00 3.86 2.55 3.04 4.13 2.82 1.75 4.71

Arran Comrade 2.25 2.56 2.58 1.96 2.15 2.10 2.42 2.17 2.17

British Queen 3.21 2.82 3.82 2.71 2.68 4.17 2.75 2.75 3.32

Duke of York 1.11 1.25 2.25 1.57 2.00 1.75 1.61 2.00 2.46

Epicure 2.36 1.64 2.29 2.11 1.93 2.64 1.43 2.25 2.79

Great Scot 3.38 3.07 3.89 2.79 3.54 4.14 3.07 3.25 3.50

Iron Duke 3.43 3.00 3.96 3.33 3.08 3.32 3.50 2.32 3.29

K. of K. 3.71 4.07 4.21 3.39 4.63 4.21 2.89 4.20 4.32

Kerr's Pink 3.04 3.57 3.82 2.96 3.18 4.32 2.00 3.00 3.88

Nithsdale 2.57 2.21 3.58 2.04 2.93 3.71 1.96 2.86 3.56

Tinwald Perfection 3.46 3.11 2.50 2.83 2.96 3.21 2.55 3.39 3.36

Up-to-Date 4.29 2.93 4.25 3.39 3.68 4.07 4.21 3.64 4.11



Lecture 5.1 27


Plot structure

108 subplots

grouped in 36 plots

Two layers of plot structure:

subplots

in

whole plots

Treatment Structure 12 varieties assigned haphazardly to whole plots

3 fertilisers assigned systematically to subplots

within whole plots

less variation

more variation



Lecture 5.1 28



Factors Units

Variety Whole Plots

↓

Fertiliser Subplots



Lecture 5.1 29


Analysis of Split Plots design

• Varieties vary between whole plots,

– variety effects evaluated with reference to

chance variation between whole plots

• Treatments vary between subplots

– treatment effects evaluated with reference to

chance variation between subplots

• Two sources of chance variation

• Implications for Analysis of Variance



Lecture 5.1 30


Minitab analysis

Model: Variety WholePlot(Variety)

Fertiliser Variety* Fertiliser

Fixed: Variety, Fertiliser

Random: WholePlot

Source DF SS MS F P

Variety 11 43.64 3.97 5.46 0.000

WholePlot(Variety) 24 17.44 0.73 4.32 0.000

Fertiliser 2 0.35 0.17 1.04 0.362

Variety*Fertiliser 22 2.19 0.10 0.59 0.909

Error 48 8.10 0.17

Total 107 71.70



Lecture 5.1 31


Minitab analysis

Model: Variety WholePlot(Variety)

Fertiliser Variety* Fertiliser

Fixed: Variety, Fertiliser

Random: WholePlot

Source DF SS MS F P

Variety 11 43.64 3.97 5.46 0.000


Fertiliser 2 0.35 0.17 1.04 0.362


Error 48 8.10 0.17

Total 107 71.70



Lecture 5.1 32



Lecture 5.1













Lecture 5.1 33


Decomposing the Split Plot Analysis

• Analysis of Varieties in Whole Plots

– Fertilisers irrelevant,

– analyse averages in whole plots

• Analysis of Fertilisers in subplots

– allowing for differences between whole plots

– randomised blocks structure



Lecture 5.1 34


Analysis of Varieties in Whole Plots



Variety Whole Plot Yields

Ajax 2.86 2.93 4.23

Arran Comrade 2.21 2.29 2.28

British Queen 2.89 2.75 3.77

Duke of York 1.43 1.75 2.15

Epicure 1.97 1.94 2.57

Great Scot 3.08 3.29 3.84

Iron Duke 3.42 2.80 3.52

King of King 3.33 4.30 4.25

Kerr's Pink 2.67 3.25 4.01

Nithsdale 2.19 2.67 3.62

Tinwald Perfection 2.95 3.15 3.02

Up-to-Date 3.96 3.42 4.14

Lecture 5.1 35


"One Way"Analysis of

Varieties in Whole Plots

Minitab model: Variety

Source DF SS MS F P

Variety 11 14.55 1.32 5.46 0.000

Error 24 5.81 0.24

Total 35 20.36

Split Plot Analysis

Source DF SS MS F P

Variety 11 43.64 3.97 5.46 0.000


Fertiliser 2 0.35 0.17 1.04 0.362


Error 48 8.10 0.17

Total 107 71.70



Lecture 5.1 36


"One Way"Analysis of

Varieties in Whole Plots

Minitab model: Variety

Source DF SS MS F P

Variety 11 14.55 1.32 5.46 0.000

Error 24 5.81 0.24

Total 35 20.36

Split Plot Analysis

Source DF SS MS F P

Variety 11 43.64 3.97 5.46 0.000


Fertiliser 2 0.35 0.17 1.04 0.362


Error 48 8.10 0.17

Total 107 71.70



Lecture 5.1 37


"Randomised Blocks" analysis

of Fertilisers in Subplots

Minitab model: WholePlot Fertiliser

Source DF SS MS F P

WholePlot 35 61.08 1.75 11.89 0.000

Fertiliser 2 0.35 0.17 1.19 0.310

Error 70 10.27 0.15

Total 107 71.70

Split Plot Analysis

Source DF SS MS F P

Variety 11 43.64 3.97 5.46 0.000


Fertiliser 2 0.35 0.17 1.04 0.362


Error 48 8.10 0.17

Total 107 71.70



Lecture 5.1 38


"Randomised Blocks" analysis

of Fertilisers in Subplots

Minitab model: WholePlot Fertiliser

Source DF SS MS F P

WholePlot 35 61.08 1.75 11.89 0.000

Fertiliser 2 0.35 0.17 1.19 0.310

Error 70 10.27 0.15

Total 107 71.70

Split Plot Analysis

Source DF SS MS F P

Variety 11 43.64 3.97 5.46 0.000


Fertiliser 2 0.35 0.17 1.04 0.362


Error 48 8.10 0.17

Total 107 71.70



Lecture 5.1 39



Lecture 5.1













Lecture 5.1 40


Diagnostic Plots



Lecture 5.1 41


Diagnostics for Whole Plots Analysis

Unremarkable



3

2

1

0

-1

-2

-3210-1-2

Dele

ted

Resi

du

al

Normal Score

Normal Probability Plot

Lecture 5.1 42


Refit with Ajax deleted

Source DF SS MS F P

Variety 10 42.66 4.27 6.78 0.000


Fertiliser 2 0.17 0.09 0.69 0.508


Error 44 5.51 0.13

Total 98 63.99

No substantial change



Lecture 5.1 43


Refit Diagnostics



4

3

2

1

0

-1

-2

-3

-43210-1-2-3

Dele

ted

Resi

du

al

Score

Normal Probability Plot

Unremarkable

Lecture 5.1 44



Lecture 5.1













Lecture 5.1 45


Split Plots in Blocks 1

Cambridge Grassland Experiment

Original plan:

Investigate two new grassland cultivation treatments:

grassland “Rejuvenator” R

conventional Harrow H

by comparison with

no treatment (Control) C

in 6 independently randomised blocks of 3 adjacent

plots each.



Lecture 5.1 46



Blocks 1 2 3 4 5 6

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



Lecture 5.1 47



Subsequent addition:

investigate 3 fertilisers

Farmyard manure F

Straw S

Artificial fertiliser A

by comparison with

no fertiliser (Control) C

allocated at random to 4 sub plots within each plot.



Lecture 5.1 48



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 5.1 49


Results

Yield was recorded in pounds (lbs) of green

produce from a single cut of each subplot made on

June 31, 1931 and are shown in the table below.

Block 1 Block 2 Block 3

C H R C H R C H R

A 266 213 208 210 222 266 220 184 184 C 165 127 155 150 167 163 155 118 153 F 198 180 200 247 203 228 190 168 174 S 184 127 150 188 167 157 140 128 141

Block 4 Block 5 Block 6

C H R C H R C H R

A 216 178 207 202 175 184 169 142 151 C 159 125 135 147 118 98 132 104 69 F 225 149 162 184 175 144 164 145 116 S 174 107 113 154 112 113 116 89 101



Lecture 5.1 50


Plot structure

72 subplots

grouped in 18 whole plots

grouped in 6 blocks

3 grassland treatments randomly assigned

to whole plots within blocks

4 fertilisers randomly assigned

to subplots within whole plots

least variation

in between variation

most variation



Lecture 5.1 51



Factors

Grassland

Treatment

Fertiliser

Units

Blocks

↓

Whole Plots

↓

Subplots



Lecture 5.1 52


Split Plots Analysis

• Treatments vary between whole plots,

– Treatment effects evaluated with reference to

chance variation between whole plots

• Fertilisers vary between subplots

– fertiliser effects evaluated with reference to

chance variation between subplots

• Whole plots nested in Blocks

• Three sources of chance variation

• Implications for Analysis of Variance



Lecture 5.1 53



Lecture 5.1













Lecture 5.1 54


Split Plots in Blocks 2

Electronic components baked in an oven.

Two factors thought to influence life times:

oven temperature

bake time.

Trial settings:

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

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

To save on costly runs,

three components baked together at each temperature,

one withdrawn at each of the set times.

This plan was replicated 3 times (in 3 blocks).



Lecture 5.1 55


Results of accelerated life time tests

for 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



Lecture 5.1 56


Classwork

What are the whole units?

What are the whole unit treatment factors?

What are the whole unit treatments?

What are the sub units?

What are the sub unit treatment factors?

What are the sub unit treatments?

What is the unit structure?

What is the treatment structure?



Lecture 5.1 57


Classwork

What are the whole units?

What are the whole unit treatment

factors?

What are the whole unit treatments

What are the sub units?

What are the sub unit treatment

factors?

What are the sub unit treatments?

Bakes

Oven temperature

580, 600, 620, 640

Single components

Baking time

5, 10, 15



Lecture 5.1 58


Unit / Treatment Structure

Baking Time (min.)


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



Lecture 5.1 59


Unit structure

36 components

nested in 12 bakes

nested in 3 replicates

Treatment structure

4 temperatures randomly allocated to bakes within

replicates

3 baking times randomly allocated to components within

bakes

Classwork: Produce a Unit/Treatment Structure Diagram

least variation

in between variation

most variation



Lecture 5.1 60


Experimental

Units

Replicates

Bakes

Components


Treatment

Factors

Temperature

Baking time

ANOVA

MS(Replicates)

MS(Temperature)

MS(Bakes Error)

MS(Baking time)

MS(TxB)

MS(Components Error) Postgraduate Certificate in Statistics


↓

↓

Lecture 5.1 61



Lecture 5.1













Lecture 5.1 62


Replication (Blocking) as a Random Effects

Factor

Day to day changes in set-up conditions depend on

• environmental factors,

• staff factors,

• timing of critical tasks,

– and more,

varying more or less unpredictably



Lecture 5.1 63


Recall: Block by Treatment Interaction

as Error term

Ref: Laboratory 1, Lecture 4.1



Table 1: Numbers of failures in 25 plots of 100 soybean seeds, arranged in blocks of 5 plots, with random allocation of seed treatments to plots within blocks.

Block Treatment I II III IV V

Check 8 10 12 13 11 Arasan 2 6 7 11 5 Spergon 4 10 9 8 10 Semesan 3 5 9 10 6 Fermate 9 7 5 5 3

.

Soybean seed germination rates

Lecture 5.1 64


Block

Failu

res

54321

14

12

10

8

6

4

2

Treatment

Fermate

Semesan

Spergon

Arasan

Check

Failure Profiles for Five Treatments

Soybean seed germination rates

Graphical analysis



Lecture 5.1 65


Test for interaction?

Analysis of Variance for Rate, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

Block 4 49.8400 49.8400 12.4600 **

Treatment 4 83.8400 83.8400 20.9600 **

Block*Treatment 16 86.5600 86.5600 5.4100 **

Error 0 * * *

Total 24 220.2400

Compare with: Source DF Seq SS Adj SS Adj MS F P

Treatment 4 83.840 83.840 20.960 3.87 0.022

Block 4 49.840 49.840 12.460 2.30 0.103

Error 16 86.560 86.560 5.410

Total 24 220.240



Lecture 5.1 66


Model including interaction

Failures equals overall mean

plus

Treatment effect

plus

Block effect

plus

Treatment by Block interaction effect

plus

chance variation

No replication implies no measure of chance variation,

UNLESS no interaction effect.



Lecture 5.1 67


A Randomized Blocks Analysis for

Temperature effects


for electronic components, with oven run means Baking Time (min.)


5 10 15 Mean

1 580 217 233 175 208.3 600 158 138 152 149.3 620 229 186 155 190.0 640 223 227 156 202.0 2 580 188 201 195 194.7 600 126 130 147 134.3 620 160 170 161 163.7 640 201 181 172 184.7 3 580 162 170 213 181.7 600 122 185 180 162.3 620 167 181 182 176.7 640 182 201 199 194.0



Lecture 5.1 68



Temperature effects


for electronic components, with oven run means Baking Time (min.)


5 10 15 Mean

1 580 217 233 175 208.3 600 158 138 152 149.3 620 229 186 155 190.0 640 223 227 156 202.0 2 580 188 201 195 194.7 600 126 130 147 134.3 620 160 170 161 163.7 640 201 181 172 184.7 3 580 162 170 213 181.7 600 122 185 180 162.3 620 167 181 182 176.7 640 182 201 199 194.0



Lecture 5.1 69



Temperature effects


for electronic components, with oven run means


Mean Lifetime

1 580 208.3 600 149.3 620 190.0 640 202.0 2 580 194.7 600 134.3 620 163.7 640 184.7 3 580 181.7 600 162.3 620 176.7 640 194.0



Lecture 5.1 70



Temperature effects

Mean results of accelerated life time tests

for electronic components for each oven run

Minitab model: R T

Temperature Replicate of Oven (°F) 1 2 3

580 208.3 194.7 181.7 600 149.3 134.3 162.3 620 190.0 163.7 176.7 640 202.0 184.7 194.0



Lecture 5.1 71


Minitab results

Analysis of Variance for Lifetime

Source DF SS MS F P

R 2 654.24 327.12 3.32 0.107

T 3 4164.77 1388.26 14.09 0.004

Error 6 591.31 98.55

Total 11 5410.32

Conclusions:

the effect of changing Temperature is highly statistically

significant,

blocking appears to have been effective.



Lecture 5.1 72


Minitab results

Minitab model: R T

Source DF SS MS F P

R 2 654.24 327.12 3.32 0.107

T 3 4164.77 1388.26 14.09 0.004

Error 6 591.31 98.55

Total 11 5410.32

Minitab model: R T R*T

Source DF SS MS F P

R 2 654.24 327.12 * *

T 3 4164.77 1388.26 * *

R*T 6 591.31 98.55 * *

Error * ** **

Total 11 5410.32



Lecture 5.1 73


Block by Treatment Interaction



Lecture 5.1 74


Diagnostic analysis



Lecture 5.1 75


Diagnostic analysis



Lecture 5.1 76



Lecture 5.1













Lecture 5.1 77


Split units model in Minitab

R

T R*T

B T*B R*B

• R is designated as a Random effect factor

• R*T serves as error term at Oven/Bake level

• (R*T*B could serve as error term at Component

level)



Lecture 5.1 78



Source DF SS MS F P

R 2 1962.7 981.4 0.54 0.618 x

T 3 12494.3 4164.8 14.09 0.004

R*T 6 1773.9 295.7 1.22 0.362

B 2 566.2 283.1 0.16 0.856

T*B 6 2600.4 433.4 1.79 0.185

R*B 4 7021.3 1755.3 7.23 0.003

Error 12 2912.1 242.7

Total 35 29331.0

x Not an exact F-test.



Lecture 5.1 79



Source DF SS MS F P

R 2 1962.7 981.4 0.54 0.618 x

T 3 12494.3 4164.8 14.09 0.004

R*T 6 1773.9 295.7 1.22 0.362

B 2 566.2 283.1 0.16 0.856

T*B 6 2600.4 433.4 1.79 0.185

R*B 4 7021.3 1755.3 7.23 0.003

Error 12 2912.1 242.7

Total 35 29331.0

x Not an exact F-test.



Lecture 5.1 80


Diagnostics



Lecture 5.1 81


Diagnostics



Lecture 5.1 82



Lecture 5.1













Lecture 5.1 83


Graphical and numerical summaries

Temperature 580 600 620 640

Mean Lifetime 195 149 177 194



Lecture 5.1 84


Graphical and numerical summaries

The experimenter "may want to look into his

technique and records quite carefully".

C. Daniel (1976), p. 270



Baking Time

Replicate 1 2 3

5 207 169 158 10 196 171 184 15 160 169 194

Lecture 5.1 85



Lecture 5.1













Lecture 5.1 86


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 5.1 87


Reading

Lecture Notes:

Split Units Design and Analysis

Split Plots Analysis Case Study



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