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Factorial Experiments -Blocking,

-Confounding, and -Fractional Factorial

Designs.

Emanuel Msemo

Wednesday, July 30, 2014 430pm 630 pm 1020

Torgersen Hall

ABOUT THE INSTRUCTOR

Graduate student in Virginia Tech Department of

Statistics

- B.A. ECONOMICS AND STATISTICS
- (UDSM,TANZANIA)

- MSc. STATISTICS (VT,USA)

- LEAD/ASSOCIATE COLLABORATOR IN LISA

If your experiment needs a statistician, you

need a better experiment. Ernest Rutherford

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various other locations Collaboration meetings

typically held in Sandy 312 Who? Graduate

students and faculty members in VT statistics

department

HOW TO SUBMIT A COLLABORATION REQUEST

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possible to schedule a meeting

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Statistics

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personalized statistical advice and assistance

with Experimental Design Data Analysis

Interpreting Results Grant Proposals Software

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collaborators aim to explain concepts in ways

useful for your research. Great advice right now

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questions requiring lt30 mins

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assist with researchnot class projects or

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COURSE CONTENTS

1. INTRODUCTION TO DESIGN AND ANALYSIS OF

EXPERIMENTS

1.1 Introduction

1.2 Basic Principles

1.3 Some standard experimental designs designs

2. INTRODUCTION TO FACTORIAL DESIGNS

2.1 Basic Definitions and Principles

2.2 The advantage of factorials

2.3 The two-Factor factorial designs

2.4 The general factorial designs

2.5 Blocking in a factorial designs

3. THE 2K FACTORIAL DESIGNS

3.1 Introduction

3.2 The 22 and 23 designs and the General 2k

designs

3.3 A single replicate of the 2k designs

4. BLOCKING AND CONFOUNDING IN THE 2K FACTORIAL

DESIGNS

4.1 Introduction

4.2 Blocking a replicated 2k factorial design.

4.3 Confounding in the 2k factorial designs.

5. TWO LEVEL FRACTIONAL FACTORIAL DESIGNS

5.1 Why do we need fractional factorial designs?

5.2 The one-half Fraction of the 2k factorial

design

5.3The one-quarter Fraction of the 2k factorial

design

INTRODUCTION TO DESIGN AND ANALYSIS OF

EXPERIMENTS

Questions

What is the main purpose of running an experiment

?

What do one hope to be able to show?

Typically, an experiment may be run for one or

more of the following reasons

1. To determine the principal causes of variation

in a measured response

2. To find conditions that give rise to a maximum

or minimum response

3. To compare the response achieved at different

settings of controllable variables

4. To obtain a mathematical model in order to

predict future responses

- An Experiment involves the manipulation of one
- or more variables by an experimenter in order

to - determine the effects of this manipulation
- on another variable.

- Much research departs from this pattern in that

nature rather than - the experimenter manipulates the variables.

Such research is - referred to as Observational studies

- This course is concerned with COMPARATIVE
- EXPERIMENTS
- These allows conclusions to be drawn about

cause - and effect (Causal relationships)

Sources of Variation

- A source of variation is anything that could
- cause an observation to be different from
- another observation

Independent Variables

- The variable that is under the control of the
- experimenter.
- The terms independent variables, treatments,
- experimental conditions, controllable

variables - can be used interchangeably

Dependent variable

- The dependent variable (response) reflects
- any effects associated with manipulation
- of the independent variable

Now

Sources of Variation are of two types

- Those that can be controlled and are of
- interest are called treatments or treatment
- factors
- Those that are not of interest but are
- difficult to control are nuisance factors

Uncontrollable factors

Z1

Z2

ZP

.

PROCESS

INPUTS

OUTPUT (Response)

.

X1

X2

XP

Controllable factors

Adapted from Montgomery (2013)

The primary goal of an experiment is to determine

the amount of variation caused by the treatment

factors in the presence of other sources of

variation

The objective of the experiment may include the

following

- Determine which conditions are most influential

on the response

- Determine where to set the influential conditions

so that the - response is always near the desired nominal

value

- Determine where to set the influential conditions

so that variability - in the response is small

- Determine where to set the influential conditions

so that the effects - of the uncontrollable Variables are minimized

EXAMPLE

Researchers were interested to see the food

consumption of albino rats when exposed to

microwave radiation

If albino rats are subjected to microwave

radiation, then their food consumption will

decrease

TRY!

Independent variable?

.

Dependant variable?

.

Nuisance factor (s)?

.

BASIC PRINCIPLES

The three basic principles of experimental

designs are

- Randomization
- The allocation of experimental material and the

order in - which the individual runs of the experiment

are to be - performed are randomly determined

- Replication
- Independent repeat run of each factor combination
- Number of Experimental Units to which a treatment
- is assigned

Blocking

- A block is a set of experimental units sharing a
- common characteristics thought to affect the
- response, and to which a separate random
- assignment is made

- Blocking is used to reduce or eliminate the
- variability transmitted from a nuisance factor

SOME STANDARD EXPERIMENTAL DESIGNS

The term experimental design refers to a plan of

assigning experimental conditions to subjects and

the statistical analysis associated with the

plan. OR An experimental design is a rule that

determines the assignment of the experimental

units to the treatments.

- Some standard designs that are used frequently

includes

Completely Randomized design

A completely randomized design (CRD) refer to a

design in which the experimenter assigns the

EUs to the treatments completely at random,

subject only to the number of observations to

be taken on each treatment.

The model is of the form

Response constant effect of a treatment

error

Block designs

This is a design in which experimenter partitions

the EUs in blocks, determines the allocation

of treatments to blocks, and assigns the EUs

within each block to the treatments completely

at random

The model is of the form

Response Constant effect of a block

effect of treatment error

Designs with two or more blocking factors

These involves two major sources of variation

that have been designated as blocking factors.

The model is of the form

Response Constant effect of row block

effect of column block

effect of treatment error

INTRODUCTION TO FACTORIAL DESIGNS

- Experiments often involves several factors, and

usually - the objective of the experimenter is to

determine the - influence these factors have on the response.

- Several approaches can be employed to deal when
- faced with more than one treatments

Best guess Approach

Experimenter select an arbitrary combinations of

treatments, test them and see what happens

One - Factor - at - a - time (OFAT)

- Consists of selecting a starting point, or

baseline set of - levels, for each factor, and then

successively varying - each factor over its range with the other

factors held - constant at the baseline level.

- The valuable approach to dealing with
- several factors is to conduct a
- FACTORIAL EXPERIMENT

- This is an experimental strategy in which
- factors are varied together, instead of one
- at a time

- In a factorial design, in each complete trial
- or replicate of the experiment, all possible
- combination of the levels of the factors
- are investigated.

e.g.

If there are a levels of factor A and b levels of

factor B, each replicate contains all ab

treatment combinations

The model is of the form

Response Constant Effect of factor A

Effect of factor B

Interaction effect Error term

Consider the following example (adapted from

Montgomery, 2013) of a two-factors (A and B)

factorial experiment with both design factors at

two levels (High and Low)

B High A High

B High A Low

52

30

B Low A Low

B Low A High

20

40

Main effect Change in response produced by a

change in the level of a

factor

Factor A

_

Main Effect 40 52

20 30

2

2

21

,Increasing factor A from low level to high

level, causes an average response increase of 21

units

Factor B

?

Main Effect

Interaction

A High B High

A Low B High

12

40

50

A Low B Low

20

A High B Low

At low level of factor B

The A effect

50 20

30

At high level of factor B

The A effect

12 - 40

-28

The effect of A depends on the level chosen for

factor B

If the difference in response between the levels

of one factor is not the same at all levels

of the other factors then we say there is an

interaction between the factors (Montgomery

2013)

The magnitude of the interaction effect is the

average difference in the two factor A effects

AB

(-28 30)

2

-29

A effect

1

In this case, factor A has an effect, but it

depends on the level of factor B be chosen

Interaction Graphically

B High

B High

B Low

Response

Response

B Low

Low

High

Low

High

Factor A

Factor A

- A factorial experiment
- without interaction

- A factorial experiment with
- interaction

Factorial designs has several advantages

- They are more efficient than One Factor at a Time

- A factorial design is necessary when interactions
- may be present to avoid misleading conclusions

- Factorial designs allow the effect of a factor to

be - estimated at a several levels of the other

factors, - yielding conclusions that are valid over a

range - of experimental conditions

The two factor Factorial Design

- The simplest types of factorial design involves
- only two factors.
- There are a levels of factor A and b levels of
- factor B, and these are arranged in a

factorial - design.

- There are n replicates, and each replicate of the
- experiment contains all the ab combination.

Example

An engineer is designing a battery for use in a

device that will be subjected to some extreme

variations in temperature. The only design

parameter that he can select is the plate

material for the battery. For the purpose of

testing temperature can be controlled in the

product development laboratory (Montgomery, 2013)

Life (in hours) Data

Temperature

Material Type

15

70

125

130 74 150 159 138 168

155 180 188 126 110 160

34 80 136 106 174 150

40 75 122 115 120 139

20 82 25 58 96 82

70 58 70 45 104 60

1

2

3

- The design has two factors each at three levels

and is - then regarded as 32 factorial design.

- The design is a completely Randomized Design

The engineer wants to answer the following

questions

1. What effects do material type and temperature

have on the life of the battery?

2 .Is there a choice of material that would give

uniformly long life regardless of temperature?

- Both factors are assumed to be fixed,
- hence we have a fixed effect model

Analysis of Variance for Battery life (in

hours) Source DF

Seq SS Adj SS Adj MS F

P-value Material Type 2

10683.7 10683.7 5341.9 7.91

0.002 Temperature 2

39118.7 39118.7 19559.4 28.97

0.000 Material TypeTemperature 4 9613.8

9613.8 2403.4 3.56 0.019 Error

27

18230.7 18230.7 675.2 Total

35 77647.0

We have a significant interaction between

temperature and material type.

Interaction plot

Significant interaction is indicated by the lack

of parallelism of the lines,Longer life is

attained at low temperature, regardless Of

material type

The General Factorial Design

- The results for the two factor factorial
- design may be extended to the general
- case where there are a levels of factor A,
- b levels of factor B, c levels of factor C,
- and so on, arranged in a factorial
- experiment.

- Sometimes, it is not feasible or practical
- to completely randomize all of the runs
- in a factorial.

- The presence of a nuisance factor may
- require that experiment be run in blocks.

The model is of the form

Response Constant Effect of factor A Effect

of factor B interaction

effect Block Effect Error term

The 2K Factorial designs

- This is a case of a factorial design with K

factors, each - at only two levels.

- These levels may be quantitative or qualitative.

- A complete replicate of this design requires
- 2K observation and is called 2K factorial

design.

Assumptions

1. The factors are fixed.

2. The designs are completely randomized.

3. The usual normality assumptions are satisfied.

- The design with only two factors each at two

levels is - called 22 factorial design

- The levels of the factors may be arbitrarily

called - Low and High

Factor

A

B

Treatment Combination

- -

- -

A Low, B Low A High, B Low A Low, B High A

High, B High

(1) a b ab

The order in which the runs are made is a

completely randomized experiment

- The four treatment combination in the design can

be - represented by lower case letters

- The high level factor in any treatment

combination is - denoted by the corresponding lower case letter

- The low level of a factor in a treatment

combination is - represented by the absence of the

corresponding letter

- The average effect of a factor is the change in

the - response produced by a change in the level of

that - factor averaged over the levels of the other

factor

- The symbols (1), a, b, ab represents the total
- of the observation at all n replicates
- taken at a treatment combination

A main effect 1/2nab a b (1)

B main effect 1/2nab b - a (1)

AB effect 1/2nab (1) a b

- In experiments involving 2K designs, it is
- always important to examine the magnitude
- and direction of the factor effect to

determine - which factors are likely to be important

- Effect Magnitude and direction should always
- be considered along with ANOVA, because the
- ANOVA alone does not convey this information

We define

Contrast A ab a b (1) Total effect of A

- We can write the treatment combination in the

order - (1), a, b, ab. Also called the standard order

(or Yates order)

Factorial Effect

Treatment Combination

I

A

B

AB

(1) a b ab

- -

- -

- -

The above is also called the table of plus and

minus signs

- Suppose that three factors, A ,B and C, each at

two levels - are of interest. The design is referred as 23

factorial design

Treatment Combination

Factorial Effects

I

A

B

AB

C

AC

BC

ABC

- - - -

(1) a b ab c ac bc abc

- - - -

- - - -

- - - -

- - - -

- - - -

- - - -

A contrast ab a ac abc (1) b c - bc

B contrast ?

In General

- The design with K factors each at two levels is
- called a 2K factorial design

- The treatment combination are written in
- standard order using notation introduced
- in a 22 and 23 designs

A single replicate of the 2K Designs

- For even a moderate number of factors, the total

number of - treatment combinations in a 2K factorials

designs is large.

- 25 design has 32 treatment combinations

- 26 design has 64 treatment combinations

- Resources are usually limited, and the number of

replicates that - the experimenter can employ may be restricted

- Frequently, available resources only allow a

single replicate of the - design to be run, unless the experimenter is

willing to omit some - of the original factors

- An analysis of an unreplicated factorials assume
- that certain high order interaction are

negligible - and combine their means squares to estimate

the - error

- This is an appeal to sparsity of effect

principle, - that is most systems are dominated by some

of - the main effect and low order interactions,

and - most high order interactions are negligible

- When analyzing data from unreplicated factorial
- designs, its is suggested to use normal

probability - plot of estimates of the effects

Example

A chemical product is produced in a pressure

vessel. A factorial experiment is carried out in

the pilot plant to study the factors thought to

influence filtration rate of this product. The

four factors are Temperature (A), pressure (B),

concentration of formaldehyde (C), and string

rate (D). Each factor is present at two levels.

The process engineer is interested in maximizing

the filtration rate. Current process gives

filtration rate of around 75 gal/h. The process

currently uses the factor C at high level. The

engineer would like to reduce the formaldehyde

concentration as much as possible but has been

unable to do so because it always results in

lower filtration rates (Montgomery, 2013)

The design matrix and response data obtained from

single replicate of the 24 experiment

Factors

Treatment Combination

A B C D Response - - - 48 - - - 43 -

70 - - 80 - - - 68 - - - 71 - - 60 -

- 45 96 - 86 - 65 - -

100 - - 75 - - 65 - - - - 45

- 104

(1) a b ab c ac bc abc d ad bd abd cd acd bcd ab

cd

The Normal probability plot is given below

The important effects that emerge from this

analysis are the main effects of A,C and D and

the AC and AD interactions

The main effect plot for Temperature

The plot indicate that its better to run the

Temperature at high levels

The main effect plot for Concentration of

Formaldehyde

The plot indicate that its better to run the

concentration of formaldehyde at high levels

The main effect plot for Stirring rate

The plot indicate that its better to run the

stirring rate at high levels

- However, its necessary to examine any
- interactions that are important

- The best results are obtained with low

concentration of formaldehyde - and high temperatures

- The AD interaction indicate that stirring rate D

has little effect - at low temperatures but a very positive

effects at high temperature

- Therefore best filtration rates would appear to

be obtained when A - and D are at High level and C is at low level.

This will allow - Formaldehyde to be reduced to the lower levels

Factor effect Estimates and sums of squares for

the 24 Design

Model Term Effect Estimates Sum of Squares Percent Contribution

A 21.63 1870.56 32.64

B 3.13 39.06 0.68

C 9.88 390.06 6.81

D 14.63 855.56 14.93

AB 0.13 0.063 1.091E-003

AC -18.13 1314.06 22.93

AD 16.63 1105.56 19.29

BC 2.38 22.56 0.39

BD -0.38 0.56 9.815E-003

CD -1.12 5.06 0.088

ABC 1.88 14.06 0.25

ABD 4.13 68.06 1.19

ACD -1.62 10.56 0.18

BCD -2.63 27.56 0.48

ABCD 1.38 7.56 0.13

ANOVA for A, C and D

Source DF Seq SS Adj SS

AdjMS F P A

1 1870.56 1870.56

1870.56 83.36 lt0.0001 C

1 390.06 390.06

390.06 17.38 lt0.0001 D

1 855.56

855.56 855.56 38.13 lt0.0001

AC 1 1314.06

1314.06 1314.06 58.56 lt0.0001

AD 1 1105.56

1105.56 1105.56 49.27 lt0.0001

CD 1 5.06

5.06 5.06 lt1

ACD 1 10.56

10.56 10.56 lt1

Residual Error 8 179.52 22.44

Total 15 5730.94

Blocking and Confounding in the 2K factorial

designs

- There are situations that may hinder the

experimenter to - perform all of the runs in a 2K factorial

experiment under - homogenous conditions

- A single batch of raw material might not be large

enough - to make all of the required runs

- An experimenter with a prior knowledge, may

decide to run a - pilot experiment with different batches of

raw materials

- The design technique used in this situations is

Blocking

Blocking a Replicated 2K Factorial design

- Suppose that the 2K factorial design has been
- replicated n times

- With n replicates, then each set of homogenous
- conditions defines a block, and each replicate

is run in - one of the blocks

- The run in each block (or replicate) will be made

- in random order

Confounding in the 2K Factorial designs

- Many situations it is impossible to perform a

complete - replicate of a factorial design in one block

- Confounding is a design technique for arranging a

- complete factorial experiment in blocks, where

the block - size is smaller than the number of treatment
- combinations in one replicate

- The technique causes information about certain
- treatment effects (usually) higher order

interactions) - to be indistinguishable from or confounded

with blocks

Confounding the 2K Factorial design in two Blocks

- Suppose we want to run a single replicate of the

22 design

- Each of the 22 4 treatment combination requires

a quantity - of raw material

- Suppose each batch of raw material is only large

enough for two - treatment combination to be tested, thus two

batches of raw - material are required

- If batches of raw materials are considered as

blocks, then we must - assign two of the four treatment combinations

to each block

Consider table of plus and minus signs for the 22

design

Treatment Combination

Factorial Effect

I

A

B

Block

AB

(1) a b ab

- -

- -

- -

1 2 2 1

Block 1

Block 2

- The order in which the treatment
- combination are run within a block
- are randomly Determined

(1) ab

a b

- The block effect and the AB interaction are

identical. That is, AB is - confounded with blocks.

- This scheme can be used to confound any 2K design
- into two blocks

- Consider a 23 design run into two blocks

- Suppose we wish to confound the ABC interaction

with - blocks

Factorial Effect

Treatment Combination

I

A

B

AB

C

AC

BC

ABC

Block

(1) a b ab c ac bc abc

- - - -

- - - -

- - - -

- - - -

- - - -

- - - -

- - - -

1 2 2 1 2 1 1 2

- Again, we assign treatment combinations that are
- minus on ABC to Block 1 and the rest to block 2

- The treatment combinations within a block are run
- in a random order

Block 1

Block 2

(1) ab ac bc

a b c abc

- ABC is confounded with blocks

Alternative method for constructing the block

- The method uses the linear combination

L a1x1 a2x2 ..... akxk

This is called a defining contrast

- For the 2K ,xi 0 (low level) or xi 1 (high

level), ai 0 or 1

- Treatment combination that produces the same

value of L (mod 2) - will be placed in the same block

- The only possible values of L (mod 2) are 0 and

1, hence we - will have exactly two blocks

- If resources are sufficient to allow the

replication - of confounded designs, it is generally better

to - use a slightly different method of designing

the - blocks in each replicate

- We can confound different effects in each

replicate - so that some information on all effects is

obtained

- This approach is called partial confounding

Consider our previous example

Two modification

- The 16 treatment combination cannot all be run

using one batch of raw - material. Experimenter will use two batches

of raw material, hence two - blocks each with 8 runs

2. Introduce a block effect, by considering one

batch as of poor quality, such That all the

responses will be 20 units less in this block

Experimenter will confound the highest order

interaction ABCD

The defining contrast is

L x1 x2 x3

The two resulting blocks are

(1) ab ac bc ad bd cd abcd

a b c d abc bcd acd abd

The half Normal plot for the blocked design

Source DF Seq SS Adj SS

Adj MS F P-Value Blocks

1 1387.56 1387.56 1387.56 A

1 1870.56

1870.56 1870.56 89.76 lt0.0001 C

1 390.06 390.06

390.06 18.72 lt0.0001 D

1 855.56 855.56

855.56 41.05 lt0.0001 AC

1 1314.06 1314.06 1314.06

63.05 lt0.0001 AD 1

1105.56 1105.56 1105.56 53.05

lt0.0001 Residual Error 9 187.56

187.56 20.8403 Total 15

7110.94

- Similar methods can be used to confound the 2K

designs - to four blocks, and so on, depending on

requirement

NOTE

Blocking is a noise reduction technique. If we

dont block, then the added variability from the

nuisance variable effect ends up getting

distributed across the other design factors

Two level Fractional Factorial Designs

- As the number of factors in a 2K factorial

designs increases, the - number of runs required for a complete

replicate of the design - rapidly outgrows the resources of most

experimenters

- If the experimenter can reasonably assume that

certain high-order - interactions are negligible, information on

the main effects and - lower order interactions may be obtained by

running a fraction of a - complete factorial experiment

- Fractional factorials designs are widely used for

product and - process designs, process improvement and

industrial/business - experimentation

- Fractional factorials are used for screening
- experiments

The successfully use of Fractional factorials

designs is based on three key ideas

1. The sparsity of effect principle

2. The projection property

3. Sequential experimentation

The one half Fraction of the 2K Design

- Suppose an experimenter has two factors, each at

two - levels but cannot afford to run all 23 8

treatment - combinations

- They can however afford four runs

- This suggests a one half fraction of a 23 design

- A one half fraction of the 23 design is often

called a - 23-1 design

Recall the table of plus and minus signs for a 23

design

- Suppose we select those treatment combinations

that have a plus - in the ABC column to form 23-1 design, then

ABC is called a - generator of this particular design

- Usually a generator such as ABC is referred as a

WORD

- The identity column is always plus, so we call

I ABC , The defining relation for our design

- Now, It is impossible to differentiate between A

and BC, B and AC, - and C and AB

We say the effects are aliased

- The alias structure may be easily determined by

using a defined relation - by multiplying any column by the defining

relation

A I A ABC A2BC BC

A BC

BI B ABC AB2C AC

B AC

This half fraction with I ABC is called the

Principal fraction

Design Resolution

- A design is of resolution R if no p-factor

effect is - aliased with another effect containing less

than R-p - factors

- Roman numeral subscript are usually used to

denote - design resolutions

- Designs of resolution III, IV and V are

particularly - important

Resolution III designs

- These are designs in which no main effects are

aliased with any - other main effects, but main effects are

aliased with two factor - interactions and some two factor interactions

may be aliased with - each other

e.g. the 23-1 design with I ABC is of

resolution III

Resolution IV designs

- No main effects is aliased with any other main

effect or with any two - factor interactions, but two factor

interactions are aliased with each - other

e.g. A 24-1 design with I ABCD is a resolution

IV design

Resolution V designs

- No main effect or two factor interactions is

aliased with any other - main effect or two factor interaction, but

two factor interactions are - aliased with three factor interactions

e.g. A 25-1 design with I ABCDE is a

resolution V design

Construction of One half Fraction

- A one half fraction of the 2K design is obtained

by - writing down a basic design consisting of the

runs for - the full 2K-1 factorials and then adding the

kth factor by - identifying its plus and minus levels with the

plus and - minus signs of the highest order interactions

ABC..(K-1)

- The 23-1 resolution III design is obtained by

writing - down the full 22 factorials as the basic

design and then - equating C to the AB interactions

One half fraction of the 23 design

Full 22 Factorial (Basic Design)

Resolution III, I ABC

Run

A

B

A

B

C AB

1 2 3 4

- -

- -

- -

- -

- -

Consider the filtration rate example

- We will simulate what would happen if a half

fraction - of the 24 design had been run instead of the

full factorial

- We will use the 24-1 with I ABCD, As this will

generate - the highest resolution possible

- We will first write down the basic design, which

is 23 design

- The basic design has eight runs but with three

factors

- To find the fourth factor levels, we solve I

ABCD for D

D I D ABCD ABCD2 ABC

The resolution IV design with I ABCD

Basic Design

Treatment Combination

Run

D ABC

A

C

B

(1) ad bd ab cd ac bc abcd

1 2 3 4 5 6 7 8

- - - -

- - - -

- - - -

- - - -

Estimates of Effects

Term Effect A

19.000 B 1.500

C 14.000 D

16.500 AB

-1.000 AC -18.500 AD

19.000

A ,C and D have large effects, and so is the

interactions involving them

Thank you

Reference

Montgomery, D.C (2013). Design and analysis of

experiments. Wiley, New York.