assumptions: in addition to the assumptions that we already talked about this design assumes: 1)two...
TRANSCRIPT
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)