TWO-­‐FACTOR  FACTORIAL  DESIGN

PREPARED  BY:  SITI  AISYAH  BT  NAWAWI

Basic  Definition  and  Principles

▪ Factorial  designs    ➢  most  efficient  in  experiments  that  involve  the  study  of  the  effects  of  two  or  more  factors.  ➢ 2k  means  there  are  k  factors  in  the  experiment  and  each  factor  has  two  levels  ➢ Factor  levels:  

❖ Quantitative  ❖ All  combinations  of  factor  levels  will  be  investigated  ❖ Number  of  treatment  combinations  =  2k  !

E.g.:  a  levels  of  factor  A,  b  levels  of  factor  B;  each  replicate  contains  all  ab  treatment  combinations.  !!!!!!!!!

!!!

32  =2  factors  with  each  factor  has  3  

levels  

THE  ADVANTAGE  OF  FACTORIALS

!• The  factorial  designs  can  be  easily  illustrated.  • More  efficient  than  one-­‐factor-­‐at-­‐a-­‐time  experiments.  • A  factorial  design  is  necessary  when  interactions  may  be  present  to  avoid  misleading  conclusions.  

• Factorial  designs  allow  the  effects  of  a  factor  to  be  estimated  at  several  levels  of  the  other  factors,  yielding  conclusions  that  are  valid  over  a  range  of  experimental  conditions.

0

9

18

26

35

no alch. one alch

no barb.one barb.

Interaction  exist

THE  TWO-­‐FACTOR  FACTORIAL  DESIGN

!

• The  effects  model

General  arrangement  for  a  Two-­‐Factor  Factorial  Design

!

• General  arrangement  for  a  Two-­‐Factor  Factorial  Design  table     it  comes  from                        yijk    where  i=1,2,3,…,a      (level  of  factor  A)                    j=1,2,3,…,b      (level  of  factor  B)                    k=1,2,3,…,n      (replication)

See  table  on  next  page…

General  arrangement  for  a  Two-­‐Factor  Factorial  Design

!

!!                                

yijk

ANOVA  Table

Source  of  Variation

Sum  of  Squares Degrees  of  Freedom

Mean  Square F

A  treatments SS a  –  1

B  treatments SS b  –  1

Interaction SS (a  –  1)  (b  –  1)

Error SS ab  (n  –  1)  

Total SS abn  –  1    

1−=aSSMS A

AE

A

MSMSF =0

1−=bSSMS B

BE

B

MSMSF =0

( )( )11 −−=

baSSMS AB

ABE

AB

MSMSF =0

)1( −=

nabSSMS E

E

Cont..

∑∑∑− = =

⋅⋅⋅−=a

i

b

j

n

kijkTotal abn

yySS

1 1 1

22

abny

ybn

SSa

iiA

2

1

2..

1 ⋅⋅⋅

=

−= ∑

abny

yan

SSb

jjB

2

1

2..

1 ⋅⋅⋅

=

−= ∑

BAtotalAB SSSSSSSS −−=

ABBATotalE SSSSSSSSSS −−−=

Example

Cont..

cont..  

• 2)  ANOVA

Now,  calculate  ANOVA  table  using  formula  given  in  

previous  slide

Cont..

Source  of  Variation Sum  of  SquaresDegrees  of  Freedom

Mean  Square F

Material  types 10683.72 2 5,341.86 7.91

Temperature 39118.72 2 19,559.36 28.97

Interaction  (Material*Temperature)

9613.78 4 2,403.44 3.56

Error 18230.75 27 675.21  

Total 77646.97 35    

cont..  

Cont..

Do  you  get  the  same  answer?  If  YES,  lets  continue..

Cont..

!

• Conclusion  :  This  analysis  indicates  that  at  the  temperature  level  700F,  the  mean  battery  life  is  the  same  for  material    types  2  and  3