Design and Analysis of Experiments Lecture 4 · Design and Analysis of Experiments Lecture 4.1 1. ... • General pattern of increasing rates from Block 1 to ... in a 24 design run

Lecture 4.1 1

© 2016 Michael Stuart

Design and Analysis of Experiments

Lecture 4.1

1. Review of Lecture 3.1 / Laboratory 1

2. Introduction to

– Fractional Factorial Designs

– Blocking factorial designs

3. Introduction to Split Plot designs

– Fisher on Potatoes

– Water resistance of wood stains

– Cambridge Grassland Experiment

Postgraduate Certificate in Statistics


Lecture 4.1 2


Review of Laboratory 1:

Soybean seed germination rates

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

.



Lecture 4.1 3


Block

Failu

res

54321

14

12

10

8

6

4

2

Treatment

Fermate

Semesan

Spergon

Arasan

Check

Failure Profiles for Five Treatments


Graphical analysis



Lecture 4.1 4



Graphical analysis



Lecture 4.1 5


Block

Failu

res

54321

14

12

10

8

6

4

2

Treatment

Fermate

Semesan

Spergon

Arasan

Check

Failure Profiles for Five Treatments


Graphical analysis



Lecture 4.1 6


• Treatments appear almost universally better than

no treatment (Check)

• General pattern of increasing rates from Block 1 to

Block 4, reducing for Block 5

– broadly consistent with

homogeneity within blocks

and

differences between blocks,

as desired

Summary



Lecture 4.1 7


• Important exceptions, including

– high rates for Fermate in Blocks 1 and 2,

otherwise Fermate is best

– low rates for Spergon in Blocks 3 and 4

Best variety?

• Fermate best in Blocks 3, 4, 5

Arasan and Semesan best in Blocks 1, 2

Summary



Lecture 4.1 8


• Further investigation of Fermate in Blocks 1 and 2

indicated

– potential for gain in understanding

• Possibly investigate Spergon in Blocks 3 and 4

Next steps?



Lecture 4.1 9


Analysis of Variance for Failures

Source DF Adj SS Adj MS F P

Treatment 4 83.840 20.960 3.87 0.022

Block 4 49.840 12.460 2.30 0.103

Error 16 86.560 5.410

Total 24 220.240

Analysis of Variance for Failures


Treatment 4 83.840 20.960 3.07 0.040

Error 20 136.400 6.820

Total 24 220.240

Was blocking effective?



Lecture 4.1 10



Treatment 4 113.400 28.350 10.92 0.000

Block 4 84.650 21.162 8.15 0.001

Error 15 38.950 2.597

Total 23 217.958

S = 1.61142


Treatment 4 94.358 23.590 3.63 0.023

Error 19 123.600 6.505

Total 23 217.958

S = 2.55054


Exceptional case deleted:



Lecture 4.1 11





Lecture 4.1 12


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 4.1 13


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,

same as unreplicated 24 design (Lecture 3.1, Part 3)

UNLESS no interaction effect.



Lecture 4.1 14


Review of Laboratory 1, Part 2

An unreplicated 24 experiment:

A process improvement study to reduce impurity

• Lenth's method

• Reduced model

• Design projection

– which model?

• Optimum conditions



Lecture 4.1 15


2-Level Factorial Experiments

are important because they

• are relatively simple to set up

• are relatively simple to analyse

• permit several factors to be investigated in relatively

few experimental runs,

• permit even more factors to be investigated by using

carefully chosen subsets of a full experiment,

• provide clues to seeking better operating conditions.



Lecture 4.1 16


Process improvement study to

reduce impurity

Chemical manufacturing:

impurity levels 55 - 65 gms per Kg

target ≤ 35 gms per Kg

Key input factors:

catalyst concentration (%), 5 and 7,

concentration of NaOH (%), 40 and 45,

agitation speed (rpm), 10 and 20,

temperature (°F), 150 and 180.



Lecture 4.1 17


Design Point

Run Order

Catalyst Concentration

Sodium Hydroxide

Concentration

Agitation Speed

Temperature Impurity

1 2 5 40 10 150 38 2 6 7 40 10 150 40 3 12 5 45 10 150 27 4 4 7 45 10 150 30 5 1 5 40 20 150 58 6 7 7 40 20 150 56 7 14 5 45 20 150 30 8 3 7 45 20 150 32 9 8 5 40 10 180 59

10 10 7 40 10 180 62 11 15 5 45 10 180 53 12 11 7 45 10 180 50 13 16 5 40 20 180 79 14 9 7 40 20 180 75 15 5 5 45 20 180 53 16 13 7 45 20 180 54

Impurity levels in gm. per Kg. resulting from

varying levels of four two level factors

in a 24 design run in completely random order

.



Lecture 4.1 18



reduce impurity

20

10

0

-10

-20

210-1-2

Eff

ect

Score

A CatCon

B NaOHCon

C Speed

D Temp

Factor Name

BC

D

C

B

Normal Effects Plot

Lenth's PSE = 1.125



Lecture 4.1 19



reduce impurity



Lecture 4.1 20



reduce impurity



Lecture 4.1 21


Apply Lenth's analysis to

soybean seed treatments?

• Effects of 2-level factors, including interactions,

summarized in a set of independent contrasts

• Main effects of 5-level factors summarised as

5 correlated deviations from mean, with 4 df,

• Interaction effects summarised as

25 correlated deviations from mean, with 16 df.



Lecture 4.1 22


Process improvement study

Visualising the results



Mean

Temperature Impurity

150 38.88

180 60.63

Speed 10 20

40 49.75 67.00 NaOH

45 40.00 42.25

Lecture 4.1 23




Speed 10 20

40 49.75 67.00 NaOH

45 40.00 42.25



Lecture 4.1 24




2010

70

65

60

55

50

45

40

Speed

Im

pu

rity

40

45

NaOHCon

Interaction Plot

Speed 10 20

40 49.75 67.00 NaOH

45 40.00 42.25



Lecture 4.1 25


Process improvement study:

Summarising results

• Reducing temperature will reduce impurities

• Increasing concentration of NaOH will reduce

impurities

• Under those conditions, changing either catalyst

concentration or agitation speed will have little

effect

– use cheapest or most convenient levels



Lecture 4.1 26


Full model Effect Estimate

CatCon 0.25

NaOHCon -17.25

Speed 9.75

Temp 21.75

CatCon*NaOHCon 0.50

CatCon*Speed -1.00

CatCon*Temp -1.00

NaOHCon*Speed -7.50

NaOHCon*Temp 1.00

Speed*Temp -0.50

CatCon*NaOHCon*Speed 1.75

CatCon*NaOHCon*Temp -0.75

CatCon*Speed*Temp 0.25

NaOHCon*Speed*Temp 0.25

CatCon*NaOHCon*Speed*Temp 1.00



Lecture 4.1 27


Reduced model

NaOHCon -17.25

Speed 9.75

Temp 21.75

NaOHCon*Speed -7.50

s = 1.74, df = 11 = 15 − 4



Lecture 4.1 28


Projected model

NaOHCon -17.25

Speed 9.75

Temp 21.75

NaOHCon*Speed -7.50

NaOHCon*Temp 1.00

Speed*Temp -0.50

NaOHCon*Speed*Temp 0.25

s = 1.87, df = 8 = 15 − 7



Lecture 4.1 29


Comparison of fits of

Full, Reduced, Projected models All effect estimates are the same; SE's vary.

Lenth: s = 2.25, df = 3, PSE(effect) = 1.125

Reduced: s = 1.74, df = 11, SE(effect) = 0.870

Projected: s = 1.87, df = 8, SE(effect) = 0.940

"Projected" model includes 3 interactions not

included in the "Reduced" model.

Adding null effects (chance variation) to a model

may increase or decrease s, depending on chance.



Ref: Models for Experiments (Extra Notes)

Lab 1 Feedback

Lecture 4.1 30


Degrees of freedom

"Error" degrees of freedom relevant for t

– Lenth's formula

– check ANOVA table

– count estimated effects (Slides 27, 28)

– use replications (Lecture 3.1,

Slides 9, 34)

t3, .05 = 2.57

t8, .05 = 2.31

t11,.05 = 2.20

s = 2.25

s = 1.87

s = 1.74 Ref: Models for Experiments (Extra Notes)

Lab 1 Feedback



Lecture 4.1 31



Lecture 4.1


2. Introduction to









Lecture 4.1 32


Several 2-level factorsfactors: how many design points?

Factors Design points

2 22 = 4

3 23 = 8

4 24 = 16

5 25 = 32

6 26 = 64

7 27 = 128

Part 2: Introduction to

Fractional Factorial Designs



Lecture 4.1 33


Problems with big experiments

Many experimental units (plots, runs)

– large area (long time)

• inhomogeneous conditions?

– high materials cost

– high labour costs

– difficult logistics

Solution:

choose an informative subset of design points

NB: design matrix columns key to this development



Lecture 4.1 34


A 24 with 16 design points

Design Point

A B C D

1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +

10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +



Lecture 4.1 35


The first 8 design points

Design Point

A B C D

1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +

10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +



Lecture 4.1 36


Design Point

A B C D

1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +

10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +

The middle 8 design points



Lecture 4.1 37


Another 8 design points

Design Point

A B C D

1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +

10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +



Lecture 4.1 38


Design Point

A B C D AB AC AD BC BD CD

2 + – – – – – – + + + 3 – + – – – + + – – + 5 – – + – + – + – + – 8 + + + – + + – + – – 10 + – – + – – + + – – 11 – + – + – + – – + – 13 – – + + + – – – – + 16 + + + + + + + + + +

Same 8 design points, with 2fi’s

Confounded effects:

A = BC

B = AC

C = AB



Lecture 4.1 39


Design Point

A B C D AB AC AD BC BD CD

2 + – – – – – – + + + 3 – + – – – + + – – + 5 – – + – + – + – + – 8 + + + – + + – + – – 10 + – – + – – + + – – 11 – + – + – + – – + – 13 – – + + + – – – – + 16 + + + + + + + + + +

Same 8 design points, with 2fi’s

Confounded effects:

A = BC

B = AC

C = AB



Lecture 4.1 40


Same 8 design points,

with all interactions Design Point

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

2 + – – – – – – + + + + + + – – 3 – + – – – + + – – + + + – + – 5 – – + – + – + – + – + – + + – 8 + + + – + + – + – – + – – – –

10 + – – + – – + + – – + – – + + 11 – + – + – + – – + – + – + – + 13 – – + + + – – – – + + + – – + 16 + + + + + + + + + + + + + + +

AD = BCD

BD = ACD

CD = ABD

I = ABC

Confounded effects:

A = BC

B = AC

C = AB

D = ABCD Postgraduate Certificate in Statistics


Lecture 4.1 41


Same 8 design points,

with all interactions Design Point

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

2 + – – – – – – + + + + + + – – 3 – + – – – + + – – + + + – + – 5 – – + – + – + – + – + – + + – 8 + + + – + + – + – – + – – – –

10 + – – + – – + + – – + – – + + 11 – + – + – + – – + – + – + – + 13 – – + + + – – – – + + + – – + 16 + + + + + + + + + + + + + + +

AD = BCD

BD = ACD

CD = ABD

I = ABC

Confounded effects:

A = BC

B = AC

C = AB

D = ABCD Postgraduate Certificate in Statistics


Lecture 4.1 42


Design Point

A B C Y

1 – – – Y1

2 + – – Y2 3 – + – Y3 4 + + – Y4 5 – – + Y5 6 + – + Y6 7 – + + Y7 8 + + + Y8

ABC

– + + – + – – +

Clever design

Each row gives design points for a 4-factor experiment

Fourth column estimates D main effect.

Fourth column also estimates ABC interaction effect.

In fact, fourth column estimates D + ABC in 24-1.

D=



Lecture 4.1 43


Classwork Exercise: Confirm confounding patterns

Design Point

A= BCD

B= ACD

C= ABD

D= ABC

Y

1 – – – – Y1

2 + – – + Y2

3 – + – + Y3

4 + + – – Y4

5 – – + + Y5

6 + – + – Y6

7 – + + – Y7

8 + + + + Y8

Confirm "confounding" or "aliasing" patterns shown.

Also, confirm AB = CD.

What other effects are aliased?



Lecture 4.1 44


Fractional factorial designs First half fraction

Design Point

A B C D Y

1 – – – – 70

2 + – – + 62

3 – + – + 88

4 + + – – 81

5 – – + + 60

6 + – + – 49

7 – + + – 88

8 + + + + 79

Full factorial design

Design Point

A B C D Y

1 – – – – 70 2 + – – – 60 3 – + – – 89 4 + + – – 81 5 – – + – 60 6 + – + – 49 7 – + + – 88 8 + + + – 82 9 – – – + 69

10 + – – + 62 11 – + – + 88 12 + + – + 81 13 – – + + 60 14 + – + + 52 15 – + + + 86 16 + + + + 79

Identify corresponding

design points



Lecture 4.1 45


Fractional factorial designs First half fraction

Design Point

A B C D Y

1 – – – – 70

2 + – – + 62

3 – + – + 88

4 + + – – 81

5 – – + + 60

6 + – + – 49

7 – + + – 88

8 + + + + 79

Full factorial design

Design Point

A B C D Y

1 – – – – 70 2 + – – – 60 3 – + – – 89 4 + + – – 81 5 – – + – 60 6 + – + – 49 7 – + + – 88 8 + + + – 82 9 – – – + 69

10 + – – + 62 11 – + – + 88 12 + + – + 81 13 – – + + 60 14 + – + + 52 15 – + + + 86 16 + + + + 79

Identify corresponding

design points

1

2 3

4

5

6 7

8 Postgraduate Certificate in Statistics


Lecture 4.1 46


Fractional factorial designs

Column A estimates A + BCD Column A estimates A – BCD

Full 24 design: Column A estimates ½[(A + BCD) + (A – BCD)] = A

Second half fraction

Design Point

A B C D Y

9 – – – + 69 1 + – – – 60 2 – + – – 89

12 + + – + 81 5 – – + – 60

14 + – + + 52 15 – + + + 86 8 + + + – 82

First half fraction

Design Point

A B C D Y

1 – – – – 70 10 + – – + 62 11 – + – + 88 4 + + – – 81

13 – – + + 60 6 + – + – 49 7 – + + – 88

16 + + + + 79



Lecture 4.1 47


Fractional factorial designs

With bigger designs (more factors) use smaller

fractions, e.g.

25 = 32 design points;

identify 4 ¼ fractions of 8 design points each.

Choose fractions to alias

main effects with 4-factor interactions,

2-factor interaction with 3-factor interactions.

Run one fraction.

If doubtful about a 2fi, run another appropriate

fraction to resolve the alias.



Lecture 4.1 48



Lecture 4.1


2. Introduction to









Lecture 4.1 49


In multifactor experiments

requiring

several runs in inhomogeneous conditions,

fractions may be used as blocks.

Block effects are aliased with suitable high level

interactions.

Blocking Factorials Designs



Lecture 4.1 50


Blocking a 24 experiment

Block 1: ABCD = + Block 2: ABCD = –

Second half fraction

Design Point

A B C D Y

9 – – – + 69 1 + – – – 60 2 – + – – 89

12 + + – + 81 5 – – + – 60

14 + – + + 52 15 – + + + 86 8 + + + – 82

First half fraction

Design Point

A B C D Y

1 – – – – 70 10 + – – + 62 11 – + – + 88 4 + + – – 81

13 – – + + 60 6 + – + – 49 7 – + + – 88

16 + + + + 79

ABCD effect confounded with block difference

All other effects unconfounded, estimated separately

within blocks Postgraduate Certificate in Statistics


Lecture 4.1 51



Lecture 4.1


2. Introduction to









Lecture 4.1 52


Recall Randomised Blocks Case Study

Reducing yield loss in a chemical process

• Process: chemicals blended, filtered and dried

• Problem: yield loss at filtration stage

• Proposal: adjust initial blend to reduce yield loss

• Plan:

– prepare five different blends

– use each blend in successive process runs, in

random order

– repeat at later times (blocks)



Lecture 4.1 53


Unit Structure

Block 1 Block 2 Block 3



Unit 1_1

Unit 1_2

Unit 1_3

Unit 1_4

Unit 1_5

Unit 2_1

Unit 2_2

Unit 2_3

Unit 2_4

Unit 2_5

Unit 3_1

Unit 3_2

Unit 3_3

Unit 3_4

Unit 3_5

Blocks

Units

Units nested in Blocks

15 experimental units

grouped in 3 blocks

less variation

more variation

Lecture 4.1 54


Introduction to Split Plots designs

• The first ever split plots design? (Fisher, 1925)

• Think of Broadbalk (Lecture 1.2, slide 71, Notes p.15)



© Rothamsted Research

Lecture 4.1 55


Potatoes



Lecture 4.1 56


Potatoes



Lecture 4.1 57


Experimental factors

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 4.1 58


The first ever split plots design?

• 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 4.1 59


Field layout



Lecture 4.1 60


Field layout



Lecture 4.1 61


Whole Plots Numbered



1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24

25 26 27 28

29 30 31 32

33 34 35 36

Lecture 4.1 62


Assignment of Varieties to Whole Plots

Varieties:

Ajax

Arran Comrade

British Queen

Duke of York

Epicure

Great Scott

Iron Duke

K. of K.

Kerr's Pink

Nithsdale

Tinwald Perfection

Up-to-Date

Whole Plots

1 13 32

8 20 34

24 25 35

5 7 19

10 12 27

4 6 18

9 11 31

2 14 16

22 28 29

3 15 17

26 33 36

21 23 30 Postgraduate Certificate in Statistics


Lecture 4.1 63


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



Plot 1 mean = 2.86 Plot 13 mean= 2.93 Plot 32 mean=4.23

Lecture 4.1 64


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 4.1 65


Analysis of Split Plots design

• Varieties vary between whole plots,

– variety effects evaluated with reference to

chance variation between whole plots

• Fertilisers vary between subplots

– fertiliser effects evaluated with reference to

chance variation between subplots

• Two sources of chance variation

• Implications for Analysis of Variance



Lecture 4.1 66



Lecture 4.1


2. Introduction to









Lecture 4.1 67


Another illustration

• Testing water resistance of four wood stains

• Stains applied to boards

• Boards are pretreated with one of two treatments.

• Ideal:

• standard two-factor design

– eight Stain / Pretreatment combinations

– one applied to each of 8 boards



Lecture 4.1 68



• Problem:

– Pretreatments can only be applied to whole board

– single replication requires 8 boards

• Solution:

– Apply Pretreatments to whole boards

– Cut pretreated boards into 4 panels

– Apply stains to panels

– Replicate 3 times



Lecture 4.1 69





Lecture 4.1 70





Lecture 4.1 71


Results

Stain 1

Stain 2

Stain 3

Stain 4

43.0 51.8 40.8 45.5 Pretreatment 1 57.4 60.9 51.1 55.3

52.8 59.2 51.7 55.3

46.6 53.5 35.4 32.5 Pretreatment 2 52.2 48.3 45.9 44.6

32.1 34.4 32.2 30.1



Lecture 4.1 72


Unit structure

24 panels (subunits)

grouped in 6 boards

(whole units)

Treatment structure

2 pretreatments assigned at random to boards

4 stains assigned at random to panels within boards

less variation

more variation



Lecture 4.1 73


Analysis of Split Plots design

• Pretreatments vary between boards,

– pretreatment effects evaluated with reference

to chance variation between boards

• Stains vary between panels

– stain effects evaluated with reference to

chance variation between panels

• Two sources of chance variation

• Implications for Analysis of Variance



Lecture 4.1 74



Lecture 4.1


2. Introduction to









Lecture 4.1 75


Cambridge Grassland Experiment

(1931)

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 4.1 76


Recall typical agricultural experimental

layouts

Broadbalk, Rothamsted

Rothamsted

Research



Lecture 4.1 77



layout 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



Lecture 4.1 78



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 4.1 79



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 4.1 80



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


Sub Plot 1

Sub Plot 2

Sub Plot 3

Sub Plot 4



Lecture 4.1 81



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


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 4.1 82



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


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 4.1 83


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.


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


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 4.1 84


Plot structure

72 subplots

grouped in 18 whole plots

grouped in 6 blocks

Three layers of plot structure:

subplots

in

whole plots

in

blocks

least variation

in between variation

most variation



Lecture 4.1 85


Treatment structure

3 grassland treatments randomly assigned

to whole plots within blocks

4 fertilisers randomly assigned

to subplots within whole plots

Treatments referred to

chance variation between whole plots

Fertilisers referred to

chance variation between subplots

What about Blocks?

Implications for Analysis of Variance



Lecture 4.1 86


Reasons for using split units

• 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



Lecture 4.1 87


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 4.1 88


Reading

EM §5.7, §7.4 for fractional factorial designs and

blocking

Lecture Notes: Split Units Design and Analysis,

pages 1-6, 10-11



Documents

Design and Analysis of Experiments Lecture 4 · Design and Analysis of Experiments Lecture 4.1 1. ... • General pattern of increasing rates from Block 1 to ... in a 24 design run