13. 2

3. 2

- -k F

acto

rial E

xperim

entsk

Facto

rial E

xperim

ents

2

Pu

rpose

Describ

e th

e o

verall

concepts

of 2

-k F

actorials

Create

standard

ord

er d

esigns

Desig

n and

Analy

ze 2

-k F

actorials

using

Anov

a

using

Effects Plots

Graph

s and

Resid

ual

PlotsU

se C

enter P

oints in

your d

esigns

After

this sectio

n y

ou

will

und

erstand

ho

w to

:

3Ad

vantag

es of

2-k

Facto

rials

Req

uire relativ

ely few

runs p

er facto

r studied

Can

be th

e b

asis fo

r m

ore

com

plex d

esigns

Good

for early

investig

ations -

can lo

ok at

a larg

e n

um

ber

of

factors

with

relatively

few ru

ns

Lend

them

selves

well

to seq

uential

studiesA

naly

sis is

fairly easy

2kfa

ctorials

refer to

k fa

ctors

, ea

ch w

ith 2

levels

. A

22

facto

rial is

a

2x2

facto

rial. This

desig

n h

as tw

o fa

ctors

with

two lev

els a

nd ca

n b

e

don

e in

2x2

or 4

run

s. Lik

ewise

a 2

3facto

rial h

as in

cludes

3 fa

ctors

,

each

with

two lev

els. This

experim

ent ca

n b

e d

on

e in

2x2

x2 o

r 8

run

s.

4

Stand

ard

Ord

er of

2kD

esign

s

The d

esign

matrix

for 2k

facto

rials a

re u

sually

show

n is

stand

ard

ord

er. Th

e lo

w lev

el of

a fa

ctor is

desig

ned

with

a

-or -1

and

the

high lev

el is

desig

nated

with

a +

or 1

. A

n ex

am

ple of

a d

esign

matrix

for a

22

F

acto

rial w

ould

look

like this:

A 2

3Facto

rial L

ook

s lik

e this:

Tem

pC

onc

Catalyst

-1-1

-11

-1-1

-11

-11

1-1

-1-1

11

-11

-11

11

11

Tem

pC

onc

-1-1

1-1

-11

11

5Exercise

Create

a 2

4Facto

rial D

esign M

atrixW

hat

are th

e m

inimu

m n

um

ber

of ru

ns n

eeded?

6

Answ

er

ab

cd

-1-1

-1-1

1-1

-1-1

-11

-1-1

11

-1-1

-1-1

1-1

1-1

1-1

-11

1-1

11

1-1

-1-1

-11

1-1

-11

-11

-11

11

-11

-1-1

11

1-1

11

-11

11

11

11

2x2

De

sign

2x2

x2

De

sign

2x2

x2x2

De

sign

7Let

s u

se M

initab to

Gen

erate th

e M

atrixG

o to

Stat>DOE

>Facto

rial >C

reateF

actorial

Desig

n

-Defin

e th

e n

um

ber of

Facto

rs

-Click

on

the D

esign

s b

utton

1.

2.

8

Desig

n M

atrix

3. Click

on

Full

Facto

rial

Optio

n; Hit

OK

.

9Desig

n M

atrix

4. Click

on

Facto

rsB

utton

5. N

am

e F

actors;

Defin

e le

vels.

6. W

hen

you

hit O

K

the D

esign

matrix

will

be o

utput

into th

e D

ata

Wind

ow

.

10

Exam

ple of

a 2

3Facto

rialThis

exam

ple relates

two q

uantitativ

e Inp

ut V

ariables (T

emperatu

re and

Concentratio

n) and

one q

ualitativ

e Inp

ut (C

atalyst)

to Y

ield.

The facto

rs and

levels:

Tem

p: 160

o C

(-1)

, 180

oC (1)

Concentratio

n (%):

20

(-1)

, 40

(1)C

atalyst:

B

rand A

(-1)

, B

rand B

(1)Th

e D

esign M

atrix w

ith results

look

s lik

e:T

em

pC

onc

Catalyst

Yield-1

-1-1

601

-1-1

72-1

1-1

541

1-1

68-1

-11

521

-11

83-1

11

451

11

80

This is

an

exa

mple

of a

Full

Facto

rial E

xperim

ent

with

only

on

e

obse

rvation

per T

reatment

Co

mbin

ation

(Cell)

.

11

Calculating

EffectsW

e w

ill n

ow

calculate

the effects

of th

e e

xpe

riment

. First

well

look

at T

em

peratu

re. W

e sim

ple add

the yield

s asso

ciated w

ith (

-1) a

nd

the Yield

s asso

ciated w

ith (1)

and

calculate

the a

verag

e (S

um

/4).

()

()

23=

52.75

-75

.72=

445

5254

604

80+

83+

68+

72

= Effect

e

Tem

peratu

r+

++

This ca

n b

e inte

rpreted

as th

e yield

going

up by

an

ave

rage

of 23

points

as te

mp

eratu

re m

oves

from

Lo

w to

High

Tem

pC

on

cC

atalystYield

-1-1

-160

1-1

-172

-11

-154

11

-168

-1-1

152

1-1

183

-11

145

11

180

Total

-

-211T

otal +

303S

um

92M

ea

n Eff

23

12

Concentratio

n Effects

No

w w

e calc

ulate th

e C

on

centratio

n Effect

the sa

me w

ay

()

()

54

8352

7260

480

+45

+68

+54

= Effect

ion

Co

ncentrat

=

++

+

This indicates

that

, as

the C

on

centratio

n m

oves

from

20% to

40%,th

e

yield g

oes

do

wn

by a

n a

verag

e of

5 p

oints

Te

mp

Co

ncC

atalystYield

-1-1

-160

1-1

-172

-11

-154

11

-168

-1-1

152

1-1

183

-11

145

11

180

Total

-

-211-267

Total

+303

247S

um

92-20

Me

an

Eff23

-5

13

Cataly

st Effect

No

w yo

u calc

ulate th

e Effect

for C

atalyst a

nd Inte

rpret

Tem

pC

onc

Catalyst

Yield-1

-1-1

601

-1-1

72-1

1-1

541

1-1

68-1

-11

521

-11

83-1

11

451

11

80

Total

-

-211-267

Total

+303

247S

um

92-20

Mea

n Eff

23-5

()

_

4_

__

_

4_)

__

(_

=Effect

Cataly

st

=

++

+

++

+

14

Cataly

st Effect

Calculatio

n

Tem

pC

onc

Catalyst

Yield-1

-1-1

601

-1-1

72-1

1-1

541

1-1

68-1

-11

521

-11

83-1

11

451

11

80

Total

-

-211-267

-254T

otal +

303247

260S

um

92-20

6M

ean Eff

23-5

1.5

By g

oing fro

m C

atalyst A

to C

atalyst B

, w

e im

pro

ve

ou

r yield

by 1

.5 p

oints.

15

Interactions

We h

ave ju

st calculated

the M

ain Effects

for this

experim

ent. In

other

word

s, w

ev

e o

nly in

vestig

ated th

e sing

ular Effects

of

Tem

peratu

re, C

oncentratio

n and

Cataly

st.

We are

also interested

in th

e co

mbin

ed effects

of th

ese th

ree. Th

e

questio

n to

be an

swered

is, Is

there

aparticular

com

binatio

n of

Input

settings th

at im

pro

ve yield

s o

ver

and ab

ove th

e sing

ular

effects?

We w

ill b

ack-up

to th

e 2

x2 facto

rial and

learn h

ow

the interactio

n

terms are

represented

statistically. Th

en w

e w

ill co

me b

ack to

our

exam

ple.

16

Interaction Effects

Tem

pC

onc

TxC

-1-1

11

-1-1

-11

-11

11

The Inte

raction

Effect is

represe

nted by

multiplying

the

colum

ns to

be rep

resented

. F

or th

e 2

x2 e

xam

ple, th

e T

em

peratu

rex

Co

nce

ntration

interactio

n co

ntrast is

created

by m

ultiplying th

eT

em

peratu

re C

ontrast

by th

e C

on

centratio

n C

ontrast

.

Tem

pC

onc

-1-1

1-1

-11

11

Main

Effects D

esign

Interactio

n Effects

Desig

n

TxC

= T

em

p *

Co

nc

17

Interactions fo

r th

e 3

-way

Desig

n

To

simplify

, let

s say

we h

ave

Facto

rs A

, B

and

C. Th

e

interactio

ns w

e ca

n test

will

be A

*B, A

*C, B

*C a

nd A

*B*C

.

Tem

p(T)C

onc

(C)C

at (K)

T*C

T*K

C*K

T*C

*KYield

-1-1

-160

1-1

-172

-11

-154

11

-168

-1-1

152

1-1

183

-11

145

11

180

Calc

ulate th

e Inte

raction

Co

ntrasts fo

r this

desig

n.

18

Interactions

Total

-

-211-267

Total

+303

247S

um

92-20

Mean Eff

23-5

Calc

ulate th

e Inte

raction

effects

Ente

r th

ese C

ontrasts

into M

initab a

nd C

orrelate

them

.

Tem

p(T)C

on

c(C)C

at(K)

T*C

T*K

C*K

T*C

*KYield

-1-1

-11

11

-160

1-1

-1-1

-11

172

-11

-1-1

1-1

154

11

-11

-1-1

-168

-1-1

11

-1-1

152

1-1

1-1

1-1

-183

-11

1-1

-11

-145

11

11

11

180

19

InteractionsTe

mp(T)

Conc

(C)C

at (K)

T*C

T*K

C*K

T*C

*KYield

-1-1

-11

11

-160

1-1

-1-1

-11

172

-11

-1-1

1-1

154

11

-11

-1-1

-168

-1-1

11

-1-1

152

1-1

1-1

1-1

-183

-11

1-1

-11

-145

11

11

11

180

Total

-

-211-267

Total

+303

247S

um

92-20

Mean Eff

23-5

1.5

1.5

100

0.5

Listed b

elow

are

the fin

al effects

:

No

w th

e ch

alleng

e is

, w

hich effects

are

imp

orta

nt (sig

nificant)

.

Let

s n

ow

mo

ve to

Minitab

and

wo

rk w

ith th

e d

ata. W

e w

ill follo

w

the sa

me p

roced

ure

as b

efore

, b

ut w

ell

look

at diffe

rent

ways

to

an

alyze

the

data

.

So

, n

ow

ente

r th

e d

ata into

a M

initab file

. Y

ou

only

need

to e

nter

the M

ain d

esign

matrix

(not

all th

e inte

raction

colum

ns)

and

yield.

20

Minitab

Pro

cedu

res

Stat>AN

OV

A>O

new

ayStack

ed allo

ws y

ou to

do m

ultiple C

om

pariso

ns

Can

do b

alanced

or u

nbalan

ced D

esigns

Stat>AN

OV

A>O

new

ay(Unstack

ed)P

ermits

data

from

each g

roup

to b

e in

a different

colum

n

No m

ultiple co

mpariso

ns

Stat>AN

OV

A>B

alanced

AN

OV

AA

dditive, full

or any

mod

el sp

ecified, b

alanced

desig

n

only

Mix

ed m

odels

(Fixed

and R

andom

Facto

rs) p

ermitted

Stat>AN

OV

A>G

LMA

NO

A plu

s u

nbalan

ced o

r n

estedTh

e m

ost

pow

erful A

NO

VA

com

mand

-tak

es m

ore

com

puting

time

21

Minitab

Pro

cedu

res C

ontin

ued

Stat>DO

E>A

naly

ze F

actorial

Desig

ns (o

r A

naly

ze C

usto

m

Desig

ns)

Used

for 2

-k, 2

-k w

ith C

enterpoints

, 2

-k w

ith Blo

ckingU

sed fo

r 2

-k F

ractional

Facto

rialsN

otation is

different th

an A

NO

VA

pro

cedures

22

No

w let

s g

o th

rough

ou

r ex

ample:

Here

s th

e D

ata ag

ain

Te

mp

Co

nc

Catalyst

Yield-1

-1-1

601

-1-1

72

-11

-154

11

-168

-1-1

152

1-1

183

-11

145

11

180

23

Create

the D

ata M

atrixG

o to

Stat>DOE

>Create

Facto

rial D

esign

Data

Matrix

1.

3.

4.

2.

24

Desig

n M

atrix

StdOrd

er

Ru

nOrd

erB

locks

T

em

p C

on

cC

at

1

11

-1-1

-1

22

11

-1-1

33

1-1

1-1

44

11

1-1

55

1-1

-11

66

11

-11

77

1-1

11

88

11

11

25

Set

up th

e D

ata M

atrix

Add th

e resp

on

se

variable

StdOrd

er

Ru

nOrd

erB

locks

T

em

p C

on

cC

at Yield

11

1-1

-1-1

602

21

1-1

-172

33

1-1

1-1

54

44

11

1-1

685

51

-1-1

152

66

11

-11

837

71

-11

145

88

11

11

80

26

Set

up th

e D

ata M

atrixAdd

the resp

on

se

variable

27

Analy

ze th

e D

ata

This exp

eriment

only

has

one ob

servatio

n p

er treatm

ent co

mbin

ation.

Therefo

re w

e can

t an

alyze

the full

factorial

using

the A

nova

pro

cedures

we learn

ed b

efore

. (W

ell, w

e really

can, b

ut w

e h

ave to

learn so

me

tricks)

.

In a

situatio

n w

here

there

is o

nly o

ne ob

servatio

n p

er treatm

entco

mbin

ation, w

e can

use

the n

orm

al p

robability

plot tech

nique to

plot th

e

effects w

e calculated

befo

re.

If th

ere is

no effect

at all

(The n

ull hyp

othesis

is tru

e fo

r ev

ery M

ain

Effect and

Interaction)

we w

ould

expect

to see

these

effects b

e n

orm

ally

distributed

around

a m

ean of

zero.

Any

outlying

effect is

consid

ered im

portant

or sig

nificant.

Choose

Stat>D

oe>

Facto

rial>A

nalyze

Facto

rial D

esign

and co

mplete

the

Dialog

Box.

28

Analy

zing a

DO

EG

o to

Stat>Do

e>F

actorial>A

naly

ze F

actorial

Desig

ns

1. E

nter R

espo

nse

2.

4.

3.

29

5.

6.

7.

Analy

zing a

DO

E

30

An

alysis

Fractio

nal

Facto

rial Fit

Estim

ated Effe

cts a

nd C

oefficients

for Yield

Te

rm Effect

C

oef

Con

stant

64

.250

Te

mp

23

.000 11

.500

Con

c-5

.000 -2

.500

Catalyst

1

.500 0

.750

Te

mp

*Conc

1.500

0

.750

Te

mp

*Catalyst

10

.000 5

.000

Con

c*C

atalyst 0

.000 0

.000

Te

mp

*Conc

*Catalyst

0

.500 0

.250

An

alysis of

Va

rian

ce fo

r Yield

So

urce

DF

S

eqSS

AdjSS

AdjM

S F

P

Main

Effects 3

1112

.50 1112

.50 370

.833

**

2-W

ay Inte

ractio

ns 3

204

.50 204

.50 68

.167 *

*

3-W

ay Inte

ractio

ns 1

0

.50 0

.50 0

.500

**

Resid

ual

Erro

r 0

0

.00 0

.00 0

.000T

otal 7

1317

.50

These

are

the

contrasts

you

pre

viou

sly c

alculated

Notice

the

re a

re n

o F

-

tests b

ecau

se th

ere

is

only

on

e sco

re in

each

cell

31

An

alysis

We

see h

ere

that

the

Effects asso

ciated w

ith

A(Te

mp)

and

the

A*C

(Te

mp

eratu

re *

Catalyst)

Interactio

n a

re im

po

rtant

. S

o w

e

will

evalu

ate th

e high

est o

rde

r inte

raction

and

not

wo

rry ab

out

the

Main

Effect.

32

Pareto

of Effects

This ch

art

pareto

sth

e effects

and

uses

a p

>0.10

as a

cutoff

. Y

ou

can

see th

atnth

e A

and

A*C

interactio

ns a

re id

entified

33

Looking

at Interactio

ns

Go

to Stat>DO

E>F

actorial>F

actorialPlots

1. C

heck

Interactio

n

Blo

ck

4.

3. E

nter R

espo

nse

4. S

elect F

actors

34

Interaction Plot

We ca

n u

se th

e inte

raction

plot fo

r a

naly

zing th

e

Tem

peratu

re by

Catalyst

Interactio

n.

35

Cub

e Plots

Let

s lo

ok at

a C

ube plot

for this

exp

erim

ent

. Y

ou

ll find

the

Cub

e Plot

option

in Stat>D

oe>F

actorial>F

actorialPlots

36

Math

ematical

Mod

el

Yield =

64.250

+ 11

.500(Tem

p) -2

.500(Co

nc)

+ 0

.750(Cat)

+

0.750(T

*C) +

5

.000(T*K)

+ 0

.000(C*K)

+

0

.250(T*C

*K)

We ca

n u

se th

e C

oefficie

nts fro

m th

e a

nalysis

to d

erive

the

follow

ing m

athem

atical m

odel:

Wh

at is

the m

odel

wh

en

eve

rything is

set to

zero?

W

hat

do

es

that

value rep

resent?

Estim

ate Yield

wh

en

all co

efficients

are

at (+1)

.

37

Red

uced

Mod

elW

e ca

n u

se th

e sa

me d

ata to

run

a red

uced

mod

el. W

e fo

und

that

the

Tem

p a

nd T

em

p*C

atalyst effects

were

imp

orta

nt, so

no

w w

e ca

n u

seth

e A

no

vap

roced

ure

to ru

n o

nly th

ose

term

s. C

ho

ose

Stat>An

ova

>Gen

eral

linea

r m

odel

.

'Yield' =

Tem

p C

atalyst T

em

p* C

atalyst;

Anal

ysi

s of

Vari

ance f

or Yi

eld, usi

ng Adj

usted

SS

for Tests

Source DF S

eq

SS Adj

SS Adj

MS F P

Temp 1

1058

.00 1058

.001058

.0076

.95 0

.001

Catal

yst 1

4

.50 4

.504.50

0.33

0

.598

Temp*Catal

yst 1

200

.00 200

.00200

.0014

.55 0

.019

Error 4

55

.00 55

.0013

.75

Total

7

1317

.50

S = 3

.70810

R-S

q = 95

.83% R-Sq(

adj) = 92

.69%38

Diag

nostics

39

2kF

actorial

Steps

1. C

reate th

e d

ata set

in M

INITA

B su

ch th

at all

of th

e v

alues

for th

e resp

onse

variable

are in

one colu

mn. E

ach inp

ut v

ariable, o

r facto

r, is

assigned

to a

colum

n, w

hich d

esignates

the v

arious lev

els of

that

factor.

2. R

un th

e D

OE

pro

cedure

specifying

the d

esign

-If

there

is o

nly o

ne ob

servatio

n p

er exp

erimental

run, u

se th

eEffects

Plot optio

n.

-G

enerally

, w

ith m

ore

than

3 facto

rs, ru

n th

e m

odel

show

ing o

nly 3

-

way

and 2

-way

interactions.

3. (O

ptional

at this

point)

Perfo

rm diag

nostic

run o

n resid

uals

using

the

Resid

ual

Plot in

the D

oe

section of

Minitab

4. Interp

ret th

e T

-test fo

r th

e high

est o

rder

interaction first

, o

r, if

using

the

Effects Plot

, id

entify th

e o

utlying effects

and an

alyze

.

-W

ith d

esigns u

sing C

enter P

oints, in

spect

the F

-test fo

r C

urv

ature

. If

this is

large, th

en y

ou m

ay h

ave a

curv

ature

effect to

analy

ze.

40

2kF

actorial

Steps -C

ontin

ued

5. U

se th

e Interactio

n Plot

feature

of M

initab fo

r th

e 2

-way

interactions.

6. If

none of

the interactio

ns are

significant

, ex

amin

e th

e m

ain effects

.

Interpret

these

in th

e sam

e m

anner

as a

one-w

ay A

NO

VA

. U

se th

e M

ain

Effects plot

to in

vestig

ate g

raphically.

7. B

ased o

n th

e ab

ove results

, reru

n th

e red

uced

mod

el w

ith o

nly th

e

significant

effects and

rerun and

exam

ine th

e resid

uals

.

8. C

alculate Ep

silon

-Squ

ares fo

r each

significant

effect to

test fo

r p

ractical sig

nificance

.

-Do

this o

nly if

the M

ean Sq

uare

Erro

r (M

SE) is

greater

than

0.7

-Yo

u w

ill h

ave to

use

the A

no

va

pro

cedu

re to

get

the S

um

-of-Sq

uares

for each

effect.

9. F

orm

ulate co

nclu

sion

s and

recom

mend

ation

s

10. Plan

the n

ext exp

eriment

or

11. R

eplicate O

ptimu

m setup

or in

stitutionalize

the ch

ange.

41

Create

Data

Set(File

: E

xercise\D

OE

2K\C

on

vers

.mtw)

StdOrd

er

Ru

nOrd

er

Blo

cksC

at-Ch

arg

Te

mp

Press

Co

nc

Co

nve

rs

11

1-1

-1-1

-171

22

11

-1-1

-161

33

1-1

1-1

-190

44

11

1-1

-182

55

1-1

-11

-168

66

11

-11

-161

77

1-1

11

-187

88

11

11

-180

99

1-1

-1-1

161

1010

11

-1-1

150

1111

1-1

1-1

189

1212

11

1-1

183

1313

1-1

-11

159

1414

11

-11

151

1515

1-1

11

185

1616

11

11

178

42

Effects Plot

2010

0

10-1

Effect

Normal Score

A

D

BD

B

Norm

al P

robability Plot

of the

Effects(re

spo

nse

is C

on

vers

, Alph

a =

.10)

A:

Cat

-Cha

rB

:T

em

pC

:P

ress

D:C

on

c

2010

0

BADBDCBCABBCDABCACABD

ABCDACD

CDAD

Pareto

Chart

of th

e Effe

cts(response

is C

onvers

, Alpha

=

.10)

A:C

at-Char

B:

Tem

pC

:Press

D:C

onc

43

Dotplot

of Effects

To

create

a d

otplotof

the effects

, g

o to

Stat>DOE

>An

alyze

Facto

rial D

esgn

>Storag

eand

check

the Effects

bo

x.

Go

to G

raph >C

haracte

r G

raphs>D

otplotsand

do

uble-click

on

the colu

mn

titled EFFE1

.

MTB > DotPl

ot

'EFFE1

'.

Character Dotpl

ot

:

. . . .:::. .

.

-----+---------+---------+---------+---------+---------+-EFFE1

-6.0

0

.0 6

.0 12

.0 18

.0 24

DA

BD

B

44

Which

Graph

s?

Set

up sim

ple tables

listing m

ain effects

, 2

-way

interactio

ns,

3-w

ay inte

raction

s, etc

.

Main

Effects

2-W

ay Inte

raction

s

Cro

ss o

ut th

e M

ain Effects

that

are

involved

with

higher o

rder

interactio

ns.

45

Which

Graph

s?

Set

up sim

ple tables

listing m

ain effects

, 2

-way

interactio

ns, 3

-way

interactio

ns, etc

.

Main

Effects

2-W

ay Inte

raction

s

We w

ill d

raw

the B

D inte

raction

plot a

nd th

e A

main

effects plot

.

AB

DBD

46

Interpret

L

arge Effects

(High

est O

rder

Interactions First)

We see

the T

em

peratu

re x

Co

nce

ntration

interactio

n is

imp

orta

nt

(BD)

, so

we g

o to

Stats>DO

E>F

actorial

Plotsand

create

the

interactio

n g

raph:

-1 1-1 1

1 1-1-1

85756555

Co

nc

Temp

Mean

Interactio

n Plot

- M

eans

for C

onve

rt

47

Investig

ate A

pprop

riate G

raphs

Conc

Pressure

Tem

pC

at-Chrg

8478726660

Convert

Main

Effects

Plot - M

eans

for C

onve

rt

Main

Effects Plots

are

very

handy

for in

vestigating

all m

ain effects

very

quickly

. N

otice th

e la

rge T

em

p effect

. T

em

peratu

re w

as

involved

in a

n im

po

rtant

interactio

n a

nd th

e inte

raction

plot

dem

on

strated a

simila

r res

ult. Th

e effect

for C

atalyst C

on

centratio

n

was

also im

po

rtant

and

you

can

insp

ect th

at g

raph b

elow

.

48

Epsilo

n Sq

uared

Go

to Stat>A

no

va>B

alan

ced A

no

vaand

run

the follo

wing

mod

el:'C

on

vert

' =

'C

at-C

hrg

' T

em

p C

on

cT

em

p* C

on

c

in th

e Dialog

Bo

x.

Set

up colu

mn

s C8

, C9

and

C10

as S

ou

rce, SS

and

E-Sq

uare

.

1. C

opy S

ou

rce Effects

from

Sessio

n

2. C

opy SS

colum

n fro

m sessio

n w

indo

w

3. U

se th

e follo

wing

Let

statem

ent:

MTB

> let

c10=

c9/2801

49

Epsilo

n Sq

uared

Source

SSE

-Square

Cat

-Chrg

2560.09140

Tem

p2304

0.82256

Conc

1210.04320

Tem

p*C

onc

810.02892

Erro

r39

0.01392

Total

28011.00000

We see

that

Tem

p is

by fa

r th

e stro

ngest

factor in

this

exp

erim

ent

.

50

Resid

ual

Analy

sis fo

r th

e R

edu

ced M

odel

21

0-1

-2

3210-1-2-3

Norm

al S

core

Residual

Norm

al Plot

of R

esiduals31

-1-3

9876543210

Residual

Frequency

Histogram of

Residuals

1510

50

50-5

Observation N

umber

Residual

I Chart

of R

esiduals

X=0

.000

UC

L=4

.699

LCL

=-4

.699

9080

7060

50

3210-1-2-3

Fit

Residual

Residuals

vs. Fits

Re

sidual

Mod

el Diag

nostics

We see

the resid

uals

are

well

beh

aved

.

51

Form

ulate C

onclu

sions

Fo

r th

e b

est co

nve

rsion

the p

rocess

sho

uld b

e ru

n at

the high

est te

mp

eratu

re a

nd th

e lo

west

catalyst ch

arg

e.

Co

nce

ntration

is n

ot im

po

rtant

as lo

ng as

the te

mp

eratu

re is

at th

e high

level

. (Th

e p

rocess

is rob

ust

to co

nce

ntration

at

high te

mp

eratu

res) P

ressu

re is

not

imp

orta

nt to

con

versio

n

(at least

not

acro

ss th

e le

vels u

sed in

the e

xperim

ent)

.

52

Multi

-Facto

r E

xperim

ent

5 facto

rs :

25

= 32

com

binatio

ns

6 facto

rs :

26

= 64

com

binatio

ns

7 facto

rs :

27

= 128

com

binatio

ns

2

2

2

2

2

2

2

On

Off

Up

Dow

nL

eftR

ightF

ront

Ba

ckIn

Out

Hi

Lo

Wh

at h

appen

if w

e d

on

t h

av

e

eno

ugh reso

urces

53

Why

do F

ractional

Facto

rial E

xperim

ents?

As th

e n

um

ber

of facto

rs in

creases, so

do th

e n

um

ber

of ru

ns.

2x2

Facto

rial =

4 ru

ns

2x2

x2 F

actorial

= 8

runs

2x2

x2x2

Facto

rial =

16 ru

ns

ets.

If th

e exp

erimenter

can assu

me high

er o

rder

interactions are

neglible

, it

is p

ossible

to d

o a

fraction of

the full

factorial

and

still g

et g

ood

estimates

of lo

w-o

rder

interactions.

Majo

r u

se of

Fractio

nal

Facto

rials is

screening: A

relatively

large n

um

ber

of F

actors

in a

relatively

small

num

ber

of ru

ns.

Screening

experim

ents u

sually

done in

the early

stages

of a

pro

cess im

pro

vem

ent p

roject.

54

Facto

rial E

xperim

ents

Successful

factorials

are b

ased o

n:

The Sp

arsityof

Effects P

rinciple

System

s are

usu

ally d

riven

by M

ain Effects

and L

ow

-

ord

er interactio

ns

The P

rojective P

roperty

Fractio

nal

Facto

rials can

represent

full-facto

rials o

nce

som

e effects

dem

onstrate

weak

ness

Seq

uential

Exp

erimentatio

n

Fractio

nal

Facto

rials can

be co

mbin

ed into

more

pow

erful d

esigns

Half

-Fractio

ns can

be fold

ed o

ver

into

a full

factorial

By elim

inating

uninteresting

Input

Variables

,

fractions can

beco

me full

factorials

.

55

Half

-Fractio

n

Recall

that

this is

the e

xpand

ed rep

resentatio

n of

a 2

x2x2

Facto

rial.

Supp

ose

we w

anted

to in

vestigate

fou

r Inp

ut V

ariables

. Sin

ce all

the

contrasts

are

indep

end

ent

(orth

ogo

nal)

we ca

n select

any

interactio

n as

the co

ntrast to

represe

nt th

e fo

urth

variable

. U

su

ally w

e select

the

highest

ord

er inte

raction

, in

this case

the A

xBxC

Interactio

n.

AB

CAXB

AXCBXC

AXBXC-1

-1-1

11

1-1

1-1

-1-1

-11

1-1

1-1

-11

-11

11

-11

-1-1

-1-1

-11

1-1

-11

1-1

1-1

1-1

-1-1

11

-1-1

1-1

11

11

11

1

Fa

ctor D

In this

case, w

hen

we replace

the A

xBxC

Interactio

n w

ith F

actor D

, w

e

say th

e A

BC w

as aliased

with

D. O

bvio

usly

, A

BC ca

n n

o lo

nger b

eestim

ated.

56

Half

Fractio

n

AB

CD

-1-1

-1-1

1-1

-11

-11

-11

11

-1-1

-1-1

11

1-1

1-1

-11

1-1

11

11

The n

ew

desig

n m

atrix lo

oks like

this:

This is

a H

alf-fractio

n of

a 2

4 d

esign

. In

stead of

16

run

s, w

e o

nly n

eed 8

run

s. This

is a

Resolutio

n IV

desig

n.

57

Half

Fractio

n

We w

ould

call this

a h

alf-fractio

n sin

ce a

full 2

x2x2

x2

Facto

rial w

ould

take 16

run

s to

com

plete. H

ere

we ca

n

estimate

4 facto

rs in

8 ru

ns. B

ut th

ere

is a

cost:

W

e lo

ss th

ehigh

er o

rder inte

raction

. W

hen

assessing w

hat

we h

ave

to

lose

, w

e u

se th

e co

ncept

of R

esolution

.

Resolutio

n III

Desig

ns

Resolutio

n III

Desig

ns:

No m

ain effects

are aliased

with

other

Main

Effects. M

ain Effects

aliased w

ith tw

o-facto

r interactio

ns

Resolutio

n IV

Desig

ns

Resolutio

n IV

Desig

ns

No M

ain Effect

aliased w

ith oth

er M

ain Effects

or w

ith tw

o-facto

r

interactions.

Tw

o-facto

r interactio

ns aliased

with

other

two-facto

r interatio

ns.

Resolutio

n V

Desig

ns

Resolutio

n V

Desig

ns

Main

Effects ok

ay, T

wo-facto

r interactio

ns aliased

with

3-facto

r

interations

58

Notatio

n

The g

en

eral

notatio

n to

desig

nate

a fractio

nal

factorial

desig

n is

:

2R k

p

k is

the n

um

ber of

factors

to b

e in

vestigated

2

k-pis

the n

um

ber of

run

s

R

is th

e resolutio

n

E

xam

ple: Th

e d

esign

ation

belo

w m

ean

s fo

ur facto

rs

will

be in

vestigated

in ru

ns. This

desig

n is

a

resolution

s IV

.

24

1IV

28

3=

122

22

22

25

15

51

51

==

=

59

Fractio

nal

Facto

rials and

Minitab

Let

s take

a lo

ok at

the M

initab Dialog

Bo

xes fo

r th

e

Stat>DOE

>Create

Facto

rial D

esign

>Display A

vailable

Desig

ns

pro

cedu

re:

60

Desig

ning a

Fractio

nal

Facto

rialS

uppo

se yo

u w

ant

to see

som

e d

esign

option

s fo

r a

n e

xperim

ent

with

5 facto

rs a

nd yo

u ca

nt

afford

a full

factorial

. G

o to

Stat>DOE

>Create

Facto

rial D

esign

1.

2.

61

Desig

n O

ptions

This table

sho

ws th

ree optio

ns: T

wo

fraction

al d

esign

s

and

the full

factorial

desig

n

62

Exercise

(File :

Ex

ercise\DO

E 2K

\%R

EAC

T (fractio

nal

factorial)

.mtw)

Objectiv

e: T

o d

esign and

analy

ze a

fractional

factorial

experim

ent u

sing M

initabO

utput

Variable:

%

Reacted

Inputs:F

eed R

ate (liters/m

inute)

10, 15

Cataly

st (%)

1,2

Agitatio

n R

ate (rp

m)100

, 120

Tem

peratu

re (C)

140,180

Concentratio

n (%)

3, 6

Use

Minitab

to setup

the D

esign M

atrixY

ou o

nly h

ave fu

nds to

do 16

run

s

Step 1

: N

ame th

e colu

mns fo

r th

e F

actors

Step 2

: G

o to

DO

E>C

reate F

acto

rial D

esign

63

Exercise

64

Desig

n M

atrixStdO

rde

rR

unO

rde

rBlo

cks F

eedrate

Catalyst

Agitatio

n T

em

p C

on

centrt

16

110

1100

1406

27

115

1100

1403

39

110

2100

1403

415

115

2100

1406

51

110

1120

1403

65

115

1120

1406

712

110

2120

1406

814

115

2120

1403

916

110

1100

1803

103

115

1100

1806

1113

110

2100

1806

128

115

2100

1803

1311

110

1120

1806

1410

115

1120

1803

152

110

2120

1803

164

115

2120

1806

65

Interaction Plots

1212

180180

140140

85756555

Temp

Catalyst

Mean

Interaction Plot

- M

eans for

Reacted

140180140180

6633

807060

Conc

Temp

Mean

Interaction Plot

- M

eans for

Reacted

66

Run th

e R

edu

ced M

odel

Since

the o

nly F

actors

(Inputs)

that

are

imp

orta

nt a

re

Catalyst

, T

em

peratu

re a

nd C

on

centratio

n, w

e ca

n re

run

the

desig

n w

itho

ut u

sing F

eed R

ate a

nd Agitatio

n R

ate. W

e n

ow

have

a F

ull 3

-factor d

esign

with

two

replication

s.

Ru

n this

mod

el eith

er u

singA

no

vao

r DO

E

MTB

> N

am

e c11

= 'R

ESI1' c12

= 'FITS1

'

MTB

> A

NOVA

'R

eacted' =

c2 c4

c5 c2

*c4 c4

*c5;

SUBC

> R

esidu

als 'R

ESI1';

SUBC

> Fits

'FITS1

'.

67

Anova

Results

Analysis of Variance for Reacted

Source DF SS

MS F P

Catalyst 1 1681

.00 1681

.00 239

.29 0.000

Temp 1 600

.25 600

.25 85

.44 0.000

Conc

1 156

.25 156

.25 22

.24 0.000

Catalyst*Temp 1 462

.25 462

.25 65

.80 0.000

Temp*Conc

1 361

.00 361

.00 51

.39 0.000

Error 10

70

.25 7.03

Total 15

3331

.0068

Diag

no

stics

21

0-1

-2

6543210-1-2-3-4

Norm

al S

core

Residual

Norm

al Plot

of R

esiduals64

20

-2-4

76543210

Resid

ual

Frequency

Histogram of

Residuals

1510

50

100

-10

Observatio

n N

umber

Residual

I Chart

of R

esiduals

X=0

.000

UCL=7

.225

LCL

=-7

.225

9585

7565

5545

6543210-1-2-3-4

Fit

Residual

Residuals

vs. Fits

Diagnostics

for R

educed M

odel

69

Oth

er F

ractio

nal

Facto

rial D

esign

Con

sideratio

ns

Fractio

nal

Facto

rials can

be blo

cked

and u

se center

points

,

just

as in

full facto

rialsF

ractional

Facto

rials can

be fold

ed o

ver

to

add to

the

desig

n. E

xam

ple: A

half

-fraction fold

ed o

ver

can b

ecom

e a

full facto

rial w

ith tw

o blo

cks. F

olding o

ver

is ju

st ch

ange th

e

signs of

the o

riginal

fraction and

rerunning

the exp

eriment

.

Using

sequ

ential assem

bly of

fractions, effects

can b

e isolated

from

other

confo

und

ed effects

.

Red

uced

fractions can

beco

me full

factorials

with

replication.

70

Gen

eral A

dvice

M

ake su

re y

ou h

ave tied

potential

busin

ess results

to y

our

project

.

Th

e b

est tim

e to

desig

n an

experim

ent is

after th

e p

revious o

ne

is finish

ed.

D

ont

try to

answ

er all

the q

uestio

ns in

one study

. R

ely o

n a

sequen

ce of

studies.

U

se tw

o-lev

el d

esigns early

Sp

end less

than

25% of

budg

et o

n th

e first

experim

ent

Alw

ays v

erify results

in a

follow

-on study

Be ready

for ch

anges

A fin

al rep

ort

is a

must!!

71

Tru

e Effect

Tru

e Effect

Y Y

Lo

(-)

Hi

(+)

Exp

erimental

EffectE

xperim

ental Effect

Facto

r S

ettings

Wh

at h

appen

if

.

72

Adding

Center

Points

Adding

Center

Points

There

is alw

ays a

risk in

2-lev

el d

esigns of

missing

a

curvilin

ear relatio

nship

by o

nly in

cluding tw

o lev

els of

the

Input

Variable

.

The additio

n of

Center

points

is an

efficient w

ay to

test fo

r

curv

ature

with

out

adding a

large n

um

ber

of exp

erimental

runs.

Let

s lo

ok at

the follo

wing

exam

ple.

A ch

emical

engineer

wants

to im

pro

ve yield

. Th

ere are

two

inputs

of interest:

R

eaction Tim

e and

Reactio

n T

emperatu

re.

The engin

eer d

ecides

to co

nduct

the exp

eriment

using

a 2

x2

desig

n (R

eaction Tim

e x

Reactio

n T

emp)

, b

ut w

ill add

five

center p

oints to

estimate

experim

ental erro

r and

curv

ature

.

Inputs:R

eaction T

emp: 150

, 155

and 160

Reactio

n Tim

e: 30

, 35

and 40

.

73

Desig

n M

atrix

We ca

n u

se M

initab to

desig

n o

ur study

. C

ho

ose

Stat>DOE

>Create

Facto

rial D

esign

>Desig

n

74

Desig

n M

atrix

StdOrd

er

Ru

nOrd

er

Blo

cks T

em

p Tim

e

11

1150

302

21

16030

33

1150

404

41

16040

55

1155

356

61

15535

77

1155

358

81

15535

99

1155

35

Cente

r P

oints

75

Center

Points

Center

Points

The e

xperim

ent

is ca

rried o

ut a

nd th

e follo

wing

data

result:

Ch

oo

se Stat>D

oe>Fit

Facto

rial M

odel

Ru

nOrd

er

Blo

cks T

em

p Tim

eYield

11

15030

39.3

21

16030

40.0

31

15040

40.9

41

16040

41.5

51

15535

40.3

61

15535

40.5

71

15535

40.7

81

15535

40.2

91

15535

40.6

76

An

alysis

Estim

ated Effects

and

Co

efficients

for Yield

Te

rm Effect

C

oef

Std C

oef

t-valu

e P

Co

nsta

nt 40

.4444 0

.06231 649

.07 0

.000T

em

p 0

.6500 0

.3250 0

.09347 3

.48 0

.018Tim

e 1

.5500 0

.7750 0

.09347 8

.29 0

.000T

em

p*Tim

e -0

.0500 -0

.0250 0

.09347 -0

.27 0

.800

An

alysis of

Varia

nc

e fo

r Yield

So

urce

D

F S

eqSS

AdjSS

AdjM

S F

P

Main

Effects 2

2

.82500 2

.82500 1

.41250 40

.42 0

.0012-W

ay Inte

raction

s 1

0

.00250 0

.00250 0

.00250 0

.07 0

.800R

esidu

al E

rror 5

0

.17472 0

.17472 0

.03494C

urvatu

re 1

0

.00272 0

.00272 0

.002720.06

0

.814P

ure

Erro

r 4

0

.17200 0

.17200 0

.04300T

otal 8

3

.00223((( ())) )c

f

cf

cf

nn

yy

nn

+++ +

=== =

2

Cu

rvatu

reSS

Not

Imp

orta

nt

File :

Exe

rcise\DOE

2K\Re

actio

nA.m

tw

77

Cub

e Plot

41.5

40.0

40.9

39.3

40.46

Time

Tem

p

160150

4030

Cub

e Plot

- M

ea

ns fo

r Yield

Cente

rpoint

Facto

rial P

oint

78

Main

Effects Plots

Time

Te

mp

41.2

40.8

40.4

40.0

39.6

Yield

Main

Effects Plot

- M

eans for

Yield

Ce

nter P

oints

79

Center

Point

Revisited

Reru

n th

e sa

me e

xperim

ent

BUT

AD

D C

ENTER

POIN

TS

Yield2 is

exactly

the sa

me as

Yield o

nly Ive

added

2 p

oints to

the C

ente

r P

oint valu

es.

An

alysis of

Va

rian

ce fo

r Yield2

So

urce

DF

S

eqSS

Main

Effects 2

2

.82502

-Way

Interactio

ns

1 0

.0025R

esidu

al E

rror 5

9

.3747C

urvatu

re

1 9

.2027P

ure

Erro

r

4 0

.1720T

otal

8 12

.2022

F P

0.75

0

.5180

.00 0

.972

214.02

0

.000

80

Main

Effects Plot

Tem

pTim

e

41.2

40.8

40.4

40.0

39.6

Yield

Orignal

Center

Points

Te

mp

Time

42.4

41.8

41.2

40.6

40.0

Yield2

Revised

Center

Point

81

The U

se of

a D

esign

Center

Point

In

clusio

n of

one o

r m

ore

center p

oints in

a 2

level

factorial

desig

n allo

ws th

e estim

ation of

curv

ature

.

U

se of

a center

point

allow

s th

e interio

r sp

ace to

be

investig

ated.

Th

e center

point

may

be replicated

with

out

destroying

the

balan

ce of

the d

esign.

Th

e n

um

ber

of center

points

will

not

affect th

e estim

ation

of m

ain effects

or interactio

ns.

Th

e n

um

ber

of center

points

will

not

affect th

e p

recision of

effect estim

ates.

AB

(1)-

-

a+

-

b-

+

ab+

+

*o

o

*o

o

*o

o

A AB B

Center

Point

Ru

ns

82

Real

Data

Exercise

Objectiv

e: Th

e in

vestig

ate th

e effects

of C

oncentratio

n, R

atio B/A

and

Tem

peratu

re o

n Y

ield.

Outp

ut: Y

ield (%)

Inputs:C

oncentratio

n (L

ow

, M

ed, H

igh)R

atio B/A

(Low

, M

ed, H

igh)T

emperatu

re (L

ow

, M

ed, H

igh)D

esign: 2

x2x2

Facto

rial w

ith 3

Center

Points

Pro

cedure:

Use

Minitab

FileA

naly

ze th

e d

ata fo

r C

urv

ature

, Interactio

ns and

Main

EffectsG

raphically A

naly

ze app

ropriate

EffectsR

un D

iagnostics

Calculate

Epsilo

n-Sq

Be p

repared

to state

your results

and co

nclu

sions.

83

Effects Plot

0-2

-4-6

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

Stand

ardized

Effect

Normal Score

A

Norm

al P

robability

Plot of

the Sta

ndardized

Effects

(resp

onse

is Yield

, Alph

a =

.10)

A:

Conce

nAB

:R

atioB/AC

:T

em

p

65

43

21

0

AAB

ACCBBC

ABC

Pa

reto C

ha

rt of

the

Stand

ardized

Effects

(resp

onse

is Yield

, Alph

a =

.10)

A:C

once

nAB

:RatioB/A

C:T

em

p

84

Test

for C

urv

ature

An

alysis of

Varia

nce

for Yield

So

urce

DF

S

eqSS

AdjSS

AdjM

S F

P

Main

Effects

3 26

.6050 26

.6050 8

.86833 14

.49 0

.0272-W

ay Inte

raction

s

3 3

.0100 3

.0100 1

.00333 1

.64 0

.3473-W

ay Inte

raction

s

1 0

.4050 0

.4050 0

.40500 0

.66 0

.476R

esidu

al E

rror

3 1

.8364 1

.8364 0

.61212C

urvatu

re

1 0

.0297 0

.0297 0

.02970 0

.03 0

.873P

ure

Erro

r

2 1

.8067 1

.8067 0

.90333T

otal

10 31

.8564

85

Main

Effects Plot

Tem

pR

atioB/AC

oncenA

89.2

88.4

87.6

86.8

86.0

Yield

Main

Effects

Plot - M

eans

for Yield

Co

nclu

sion

s?

86

Basic

Steps F

or A

naly

zing D

OE

using

Minitab

1) C

reate

DOE

in M

initab (Stat>DO

E>F

acto

rial>Cre

ate F

acto

rial D

esig

n)or

Defin

e d

ata w

orksh

eet

into DO

E fo

rmat

(Stat>DOE

>Facto

rial>Defin

e

Custo

m F

acto

rial D

esig

n)

2) A

nalyze

DOE

(Stat>DOE

>Facto

rial>Analyze

Facto

rial D

esig

n)G

raph>Effect

Plot>Norm

al, P

areto

>Alpha =

0.1

(For S

creening

DOE)

Alpha =

0.05

(For N

on S

creening)

Storag

e >

Fits , R

esid

uals

3) C

heck

the R

esid

ual

Plot of

the DO

E fo

r yo

ur co

nfident

level

your a

nalysis

(Stat>Reg

ressio

n>R

esid

uals

Plots)

87

4) L

ook

up fo

r th

e infe

rential

facto

rs &

their

effects

Fro

m S

essio

n W

indow

(P-valu

e DO

E>F

acto

rial>Facto

rial Plots

>Main

Effects Plot)

or

(Stat>Anova

>Main

Effects

Plot)Inte

ractio

n Plot

(Stat>DOE

>Facto

rial>Facto

rial Plots

>

Interactio

n Plot)

or

(Stat>Anova

>Intera

ctions Plot)

Basic

Steps F

or A

naly

zing D

OE

using

Minitab

(Co

nt)

88

5) R

educe

the m

odel

with

only

the sig

nificant

effect

, ru

n A

nova

(Stat>Anova

>General

Linear M

odel)

6) C

heck

the Ep

silon-Sq

uare

s fo

r p

ractical

significa

nce

(Prio

rity setting)

Sum

-of-Sq

uare

s fo

r e

ach

effect

Total

Sum

-of-Sq

uare

s

7) S

um

marize

your finding

and

plan fo

r th

e n

ext

actio

n.

Basic

Steps F

or A

naly

zing D

OE

using

Minitab

(Co

nt)

89

Gen

eral A

dvice

Mak

e su

re y

ou h

ave tied

potential

busin

ess results

to y

our

project

.

The b

est tim

e to

desig

n an

experim

ent is

after th

e p

revious

one is

finished

.

Dont

try to

answ

er all

the q

uestio

ns in

one study

. R

ely o

n a

sequen

ce of

studies.

Use

two-lev

el d

esigns early

Spend

less th

an 25%

of b

udget

on

the first

experim

ent

Alw

ays v

erify results

in a

follow

-on study

Be ready

for ch

anges

A fin

al rep

ort

is a

must!!