assumptions: in addition to the assumptions that we already talked about this design assumes:

Assumptions:

In addition to the assumptions that we already talked about this design assumes:

1) Two or more factors, each factor having two or more levels.

2) All levels of each factor are investigated in combination with all levels of every other factor. If there are a (= 3) levels of factor A and b (= 3) levels of factor B then the experiment contains a x b (= 3 x 3 = 9) combinations. (the treatment levels are completely crossed).

3) Random assignment of experimental units to treatment combinations. Each experimental unit must be assigned to only one combination.

Completely Randomized Factorial DesignWith Two Factors

Assignment of Experimental Units:

Assume we have 3 factors. Factor A has three levels a1 , a2 and a3 and factor B has three levels b1, b2, and b3 then the layout of the completely randomized design is as follows:

a1b1 a1b2 a1b3 a2b1 a2b2 a2b3 a3b1 a3b2 a3b3

y111

y112

y113

…

y11n

y121

y122

y123

…

y12n

y131

y132

y133

…

y13n

y211

y212

y213

…

y21n

y221

y222

y223

…

y22n

y231

y232

y233

…

y23n

y311

y312

y313

…

y31n

y321

y322

y323

…

y32n

y331

y332

y333

…

y33n

Total sample is nab = n(3)(3) randomly assigned to the different combinations, with a minimum n = 1 (in this case we have to assume no interaction between the different factor levels).

Completely Randomized Factorial DesignWith Two Factors

Linear Model

Completely Randomized Factorial DesignWith Two Factors

1,2,..., 1, 2,3

1,2,..., 1, 2,3

1,2,...,

ijk i j ijkijy

i a

j b

k n

Completely Randomized Factorial DesignWith Two Factors

yijk Response of the kth experimental unit in the ij factor combination.

The grand mean of all factor combinations’ population-means.

i Factor effect for population i, and should obey the condition:

j Factor effect for population i, and should obey the condition:

ij Joint effect of factor levels i and j,

and should obey both:

ijk The error effect associated with Yijk and is equal to:

.i i

1

0a

ii

.i i

1

0b

jj

. .ij i jij

1 1

0 & 0a b

ij iji j

ijk ijk i j ijY

Completely Randomized Factorial DesignWith Two Factors

A\B b1 b2 b3 Grand Means

a1 11 12 13 1.

a2 21 22 23 2.

a3 31 32 33 3.

Grand means .1 .2 .3

Means

Completely Randomized Factorial DesignWith Two Factors

Hypotheses:

'

'

' ' ' '

' ' ' '

1 1. 2. .

'1 . .

2 .1 .2 .

'2 . .

3

3

: ...

: ,

: ...

: ,

: 0 ,

: 0 ,

o a

a i i

o b

a j j

o ij i j ij i j

a ij i j ij i j

H

H for some i i

H

H for some j j

H for all i j

H for some i j

Completely Randomized Factorial DesignWith Two Factors

A\B b1 b2 b3 Grand Means

a1

a2

a3

Grand means

Means

11.y 12.y 13.y 1..y

21.y 22.y 23.y 2..y

31.y 32.y 33.y 3..y

.1.y .2.y .3.y ...y

Completely Randomized Factorial DesignWith Two Factors

What are we comparing?

A/B b1 b2 b3 Grand Means

a1 11= + 1+ 1 + ()11

12= + 1+ 2 + ()12

12= + 1+ 3 + ()13

1.= + 1

a2 23= + 2+ 1 + ()21

23= + 2+ 2 + ()22

23= + 2+ 3 + ()23

2.= + 2

a3 33= + 3+ 1 + ()31

33= + 3+ 2 + ()32

33= + 3+ 3 + ()33

3.= + 3

Grand means .1= + 1 .2= + 2 .3 + 3

Completely Randomized Factorial DesignWith Two Factors

Hypotheses:

1

1

2

2

3

3

: 0

: 0

: 0

: 0

: 0 ,

: 0 ,

o i

a i

o j

a j

o ij

a ij

H for all i

H for some i

H for all j

H for some j

H for all i j

H for some i j

Completely Randomized Factorial DesignWith Two Factors

A\B b1 b2 b3 Grand Means

a1

a2

a3

Grand means

Means

11... 1 1ˆy 12... 1 2

ˆˆy 13... 1 3ˆˆy ... 1

ˆy

21... 2 1ˆy 22... 2 2

ˆˆy 23... 2 3ˆˆy ... 2

ˆy

31... 3 1ˆy 32... 3 2

ˆˆy 33... 3 3ˆˆy ... 3

ˆy

... 1y ... 2ˆy ... 3

ˆy ...y

Where

.. ...ˆ

i iy y

. . ...i jy y

. .. . . ...ij ij i jy y y y

...ˆ y

Completely Randomized Factorial DesignWith Two Factors

2 2 2... .. ... . . ...

1 1 1 1 1

2 2. .. . . ... .

1 1 1 1 1

( ) ( ) ( )

( ) ( )

a b n a b

ijk i ji j k i j

a b a b n

ij i j ijk iji j i j k

y y bn y y an y y

n y y y y y y

( )ijk i j ij ijkY

breakdown:

1 1 1 ( 1)( 1) ( 1)

T A B AB ESS SS SS SS SS

df

abn a b a b ab n

Completely Randomized Factorial DesignWith Two Factors (Fixed Effects)

2

2 1

2

12

2

1 12

2

1

1

1 1

a

ii

A

b

jj

B

a b

iji j

AB

E

bnE MS

a

an

E MSb

n

E MSa b

E MS

Completely Randomized Factorial DesignWith Two Factors (Fixed Effects)