Transcript FACTORIAL DESIGNS - Gadjah Mada University

FACTORIAL DESIGNS
(Treatment Design)
Erlina Ambarwati
Parts of Experimental Design
1. Set of experimental units.
2. Set of treatments.
3. Rules by which treatments are assigned to
experimental units.
4. Measurements made on experimental
units following application of treatment.
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Experimental Units (e.g.)
 Patients with heart disease in a drug study.
 Volunteers in a marketing study.
 Corn seeds in an agricultural study.
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Types of Treatment Structures
 One-Way Treatment Structure
 Factorial Arrangement Treatment Structure
 Fractional Factorial Arrangement Treatment
Structures
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Assignment Rules
 Completely Randomized Design
 Randomized Complete Block Design
 Latin Squares Design
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Measurements (e.g.)
 Mortality in a health outcomes study.
 Survey score in marketing study.
 Plant size at time x for agricultural
study.
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Definition of Factorial Design
 An experiment in which the effects of
multiple factors are investigated
simultaneously.
 The treatments consist of all combinations
that can be formed from the different factors.
 e.g. an experiment with 5 2-level factors
would result in 32 treatments.
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Definition of Factorial Design
 A set of factorial teratments consists of all
combinations of all levels of two or more
factors.
 Each treatment combination must contain
one level of every factor.
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Definition of Factorial Design
 The treatments are assigned randomly to
the pool of experimental units with an equal
number of units in each treatment.
 The number of experimental units assigned
to each treatment is referred to as the
number of replications.
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Problem of Factorial Experiments
 The uniformity of experimental material in
large number of treatment
 Factors A, B, C and D having levels a, b, c and
d, there are t = abcd different treatments.
 With many factors and/or many levels, the
number of treatments can get prohibitively
large.
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2 Factor Model Specification
Yi = B0 + B1X1i + B2X2i + B3X1iX2i + ei
Yi – Outcome for ith unit
B0 – Intercept coefficient
B1 – Effect 1 coefficient
B2 – Effect 2 coefficient
B3 – Interaction coefficient
X1i – Level of factor 1 for ith unit
X2i – Level of factor 2 for ith unit
ei – Error term for ith unit
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Analysis of Factorial Design
 Main Effects – effects of each factor
independent of the remaining factors.
 Interaction Effects – 2- to n-way interaction
effects between all combinations of factors.
 Design provides a lot more information
than a single factor experiment with
potentially not much more work.
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Example
Experimental units – 100 patients with
depression.
2. Set of factors – drug therapy (y/n) and
psychotherapy (y/n)
3. Rules - Randomly assign 25 patients to each of
the possible combinations in (2).
4. Measurement – Beck Depression Scale
1.
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Two-Way Factorial Design
Column Treatment
.
.
Row
Treatment
Cells
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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Purchase of Fashion Clothing By
Income and Education
Low Income
High Income
Purchase
High
Low
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High
Low
122 (61%)
78 (39%)
200 (100%)
171 (57%)
129 (43%)
300 (100%)
Erlina Ambarwati
Education
Education
High
Purchase
Low
High
241 (80%)
59 (20%)
300
Low
151 (76%)
49 (24%)
200
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Factorial Design
Amount of Store
Information
Low
Medium
High
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No
Humor
A
D
G
Amount of Humor
Medium
Humor
B
E
H
High
Humor
C
F
I
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The 2 x 2 Factorial Experiments
Block IV Aa Ba Ab Bb
Block III Bb Aa Ba Ab
Block II Ba Bb Ab Aa
Block I Ab Aa Ba Bb
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Kombinasi Perlakuan
N1: 25 kg
N2: 50 kg
N3: 75 kg
N1
P1: 25 kg
P2: 40 kg
P3: 60 kg
N2
N3
P1 N1P1
N2P2
N3P3
P2 N1P2
N2P2
N3P2
P3 N1P3
N2P3
N3P3
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Contoh: pembuatan plat elektroda dengan disepuh
menggunakan dua arus listrik berbeda dan dua
temperatur larutan. Masing-masing kombinasi ada 6
buah.
Temperatur (B)
Rendah
Rendah
Tinggi
19.8
23.4
X =3.3
X = 3.9
28.2
18.6
X = 4.7
X = 3.1
48.0
42.0
Total
43.2
Amper (A)
Tinggi
Total
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46.8
90
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Temperatur (B)
Amper (A)
Rendah
Tinggi
Total
Rendah
19.8 3.3
23.4 3.9
43.2 3.6
Tinggi
28.2 4.7
18.6 3.1
46.8 3.9
Total
48.0 4.0
42.0 3.5
90
Pengaruh sederhana faktor
A pada level rendah
dari faktor B
c1  Y 21 .  Y11 .  4 , 7  3 ,3  1, 4
Pengaruh sederhana faktor
A pada level tinggi dari
faktor B
c 2  Y 22 .  Y12 .  3 ,1  3 ,9   0 ,8
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Temperatur (B)
Amper (A)
Pengaruh utama faktor A
Rendah
Tinggi
Total
Rendah
19.8 3.3
23.4 3.9
43.2 3.6
Tinggi
28.2 4.7
18.6 3.1
46.8 3.9
Total
48.0 4.0
42.0 3.5
90
 4 , 7  3 ,1   3 , 3  3 , 9 
c 3  Y 2 ..  Y1 ..  


2
2

 

 3 ,9  3 , 6  0 ,3
atau
c3 
1

1
2
2

1
2
Y
21 .
Y
21 .
 Y 22 .  
1
 Y11 .  
1
2
2
Y
11 .
Y
22 .
 Y12 . 
 Y12 . 
1
c1  c 2   1, 4    0 . 8   0 ,3
2
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Pengaruh sederhana faktor B pada
A rendah c 4  Y12 .  Y11 .  3,9  3,3  0 , 6
A tinggi c 5  Y 22 .  Y 21 .  3,1  4 , 7   1, 6
Pengaruh utama faktor B:
c 6  Y. 2 .  Y. 1 .  3 , 5  4 , 0   0 , 5
atau
c6 
1

1
2
2

Y
12 .
Y
12 .
 Y 22 .
  1 Y
 Y11 .  
c4  c5
11 .
2
Y
1
2
 Y 21 . 
Temperatur (B)
Rendah
Tinggi
Total
Rendah
19.8 3.3
23.4 3.9
43.2 3.6
Tinggi
28.2 4.7
18.6 3.1
46.8 3.9
Total
48.0 4.0
42.0 3.5
90
Amper (A)
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22 .
 Y 21 . 
Erlina Ambarwati
Kemungkinan dalam kombinasi perlakuan
Ada interaksi
Ada interaksi
Arus lemah
Arus lemah
Tidak ada interaksi
Arus lemah
ketebalan
Arus tinggi
Arus tinggi
Arus tinggi
Temperatur
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Perbandingan Kontras Ortogonal
Untuk mengetahui efek utama dan interaksinya
Temperatur (B)
Rendah
Tinggi
Total
Rendah
19.8 3.3
23.4 3.9
43.2 3.6
Tinggi
28.2 4.7
18.6 3.1
46.8 3.9
Total
48.0 4.0
42.0 3.5
90
Amper (A)
c 3   Y11 .  Y12 .  Y 21 .  Y 22 .
c A    19 ,8  28 , 2  23 , 4  18 , 6  6  0 , 6
c 6   Y11 .  Y12 .  Y 21 .  Y 22 .
c B    19 ,8  28 , 2  23 , 4  18 , 6  6   1, 0
c 7   Y11 .  Y12 .  Y 21 .  Y 22 .
c A * B    19 ,8  28 , 2  23 , 4  18 , 6  6   2 , 2
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
0 , 6 
JK

 0 , 54 
A

4

6

2

1, 0 
JK

 1, 5
 9 ,3
B
4

6


  2 , 2 2
JK 
 7 , 26 
I
4

6

2
JK

plk

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
 19


, 8 2  28 , 2 2  23 , 4 2  18 , 6 2 

6
2080 , 8
6

90 2
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 337 , 5  9 , 3
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Formulas for Computing
a Two-Way ANOVA
R
SSR  nC 
i 1
C
SSC  nR 
R
( X i X )
( X j X )
(X
SST 
(X
c 1 r 1 a 1
M SR 
M SC 
M SI 
M SE 
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ijk
j 1 k 1
R
n
ijk
SSR
R 1
SSC
C 1
SSI
 R  1 C  1
R
 R 1
2
df
 ( X ij  X i  X j  X )
i 1 j 1
R
C
n
i 1
C
df
j 1
C
SSI  n 
SSE 
2
 X ij )
X)
C
2
df
I
w h ere :
 C 1
n = n u m b er o f o b serv atio n s p er cell
  R  1 C  1
C = n u m b er o f co lu m n treatm en ts
R = n u m b er o f ro w treatm en ts
2
df
E
 R C  n  1
i = ro w treatm en t lev el
j = co lu m n treatm en t le v el
2
df
F
F
F
T
R
C
I
 N 1



M SR
M SE
M SC
M SE
M SI
M SE
k = cell m em b er
X
X
X
X
ijk
ij
i
j
= in d iv id u a l o b serv atio n
= cell m ean
= ro w m ean
= co lu m n m ean
X = gran d m ean
SSE
R C  n  1
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The Linear Model for a Two-Factor
Factorial
X ijk     i   j  ( ) ij   k  e ijk
i  1, 2 ,...., a
j  1, 2 ,...., b
k  1, 2 ,...., r
a

 
b
r

i 1
j 1 k 1
X ijk
 X ...
a .b .r
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a
a

 


i 1
b .r

 
a

i 1
a .r
a

 
i 1
j 1
r
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 X i ..  X ...
b
r
X ijk
j 1 k 1
 X . j .  X ...
a .b .r
b

j 1 k 1

X . j.
j 1
X ijk
a .b .r
b

r

X i ..
i 1
b
b
a

X ij .


X i ..
i 1
b .r

a
a .r
r

X . j.
j 1
b

i 1
Xijk
j 1 k 1
a .b .r
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
 


X .. k
k 1


 ijk 
i 1

 ijk 
b
Erlina Ambarwati
i 1
j 1
r
a
r

X ij .

r

X .. k
k 1
b

i 1
a .b
X ijk
j 1 k 1
a .b .r
r

i 1
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X ijk 
j 1 k 1
a
b

r

 X .. k  X ...
a .b .r
a
b
X ijk
j 1 k 1
i 1
a .b
a
r
b
a
r
X ijk  X ij .  X .. k  X ...
j 1 k 1
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a

JKtotal 
b
(

i 1
2
X ijk 
r
X ijk )
2
j 1 k 1
a .b .r
db total = abr-1
a
a

X i j.
i 1
Jkperlk 
(
2

i 1
b

a
JKA 
i 1
b .r
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(
2

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2
dbperlk = ab-1
a .b .r
a
X i ..
X ijk )
j 1 k 1
r

r
i 1
b
r

j 1 k 1
a .b .r
X ijk )
2
db A = a-1
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b
a

JKB 
X . j.
(
2
j 1
i 1

b

b
i 1
2
dbB= b-1
a .b .r
b
a

JKAxB 
X ijk )
j 1 k 1
a .r
a
r
X ij .

2
j 1

X i ..
i 1
r
a

2

X . j.
(
2
j 1
b .r

i 1
b
r

X ijk )
j 1 k 1
a .r
a .b .r
db AxB = (a-1)(b-1)
a
a
JKerror 

r

i 1
31
b
b
2
X ijk 
i 1
j 1 k 1
db error = (r-1)(ab-1)
j 1
r
a
r
X ij .

2

X .. k
k 1
(
2

a .b
i 1
b
r

X ijk )
2
j 1 k 1
a .b .r
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2
ANOVA OF FACTORIAL
Degrees of
freedoma
Sums of
square
s
(SSQ)
Blocks (B)
b-1
SSQB
SSQB/(b-1)
MSB/MSE
First factor (F1)
f-1
SSQF1
SSQF1/(f-1)
MSF1/MSE
Second factor (F2)
s-1
SSQF2
SSQF2/(s-1)
MSF2/MSE
(f-1)*(s-1)
SSQFxS
SSQFxS/((f-1)*(s-1))
MSFxS/MSE
Error (E)
(f*s-1)*(b-1)
SSQE
SSQE/((f*s-1)*(b-1))
Total (Tot)
f*s*b-1
SSQTot
Souce of variation
First X Second (FxS)
Mean
square (MS)
F
awhere
f=number of treatments in the first factor. s=number of treatments in the second
factor and b=number of blocks or replications.
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Another example
Factor B
Factor A
A1
A2
A3
Total B.j.
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Total Ai..
B1
B2
B3
B4
11
12 32
9
8
10 28
10
12
10 35
13
9
11 30
10
125
13
11 38
14
14
10 34
10
8
12 30
10
9
9 26
8
128
9
9
9
27
10
8 29
11
11
11 31
9
7
11 24
6
111
97
91
96
80
364
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continued
CF  364
2
36  3680 , 44

  CF  117 ,56
 12   CF
SStotal  11  12  ...  6
SS
A

2
2
 125  128  111
2
2
2
2
 3694 ,12  3608 , 4  13 , 73

SS B  97  91  96  80
2
2
2
2
 9   CF
 3700 , 67  3680 , 4  20 , 23

Komb . plk  32  28  ...  24
2
2
2
 3   CF
 3738 , 67  3680 , 4  58 , 23
SS
A* B
 SS
plk
 SS
A
 SS B
 58 , 23  13 , 73  20 ,3  24 , 27
34
SS Error  117 ,56  58 , 23  59 , 33
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ANOVA
SR
df
SS
MS
A
B
A*B
Sesatan
2
3
6
24
13.73
20.23
24.27
59.33
6.86
6.74
4.04
2.47
Total
35
117.56
Fhit
2.77
2.72
1.63
Ftab
3.42
3.03
2.51
Bagaimana jika dipecah
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Factorial 3 x 2 +1 control
replication
Control
A1B1
A1B2
A1B3
A2B1
A2B2
A2B3
1
2
2
3
3
2
2
2
3
2
4
3
4
3
3
4
Total
4
5
6
4
7
6
8
CF 1 
40
CF 2 
36
2
14
2
12
40
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
SS total  2  2  3  ...  4
SS ctrl
vs treat
2
2

A1
A2

2
Erlina Ambarwati
  CF
1
 7 .8
  6 x 4  1 x 5  1 x 6  1 x 4  1 x 7  1 x 6  1 x 8 2

2  6 1 1 1 1 1 1
2
2
2
B1
B2
B3
5
7
6
6
4
8
12
12
12
SS treat  5  6  ...  8
37
2
2
2
2
2
2
2
2

 1 . 71
15
21

2  CF 2  113  108  5
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SS
A
SS B
 15 2  21 2
 
3 .2


  CF 2  3


 12 2  12 2  12 2
 
2 .2


  CF 2  0


SS error  SS total  SS treat  SS ctrl
vs treat
 7 ,8  1, 71  5  1, 09
SS
A* B
 SS treat  SS
A
 SS B
 530  2
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ANOVA
SV
df
SS
MS
Fstat
Ftab
1.71
5
3
0
2
1.09
1.71
1
3
0
1
0.28
7.84*
4.58ns
13.7*
6.61
5.05
6.61
Error
1
6-1
2-1
3-1
(3-1)(2-1)
5
4.58
5.79
Total
3x2x2-1
7.8
Ctrl vs treat
Treatment
-A
-B
-A*B
7/20/2015
Erlina Ambarwati
39
Factorial 3 x 4 x 2
Sum of treatment based on 5 replication
B
C
A
1
2
3
4
1
1
1
1
2
3
10
9
12
6
8
11
8
7
8
7
7
10
2
2
2
1
2
3
19
14
16
16
15
14
12
15
13
18
15
18
SS treat
 10
7/20/2015
2
 6  ...  18
2
Erlina Ambarwati
2
 x ijk
x ....  288
CF   288  120  691 . 2
2
 5   691 .2  73 .2
40
Level B
1
2
3
4
Total A
(Yi…)
1
2
3
29
23
28
22
23
25
20
22
21
25
22
28
96
90
102
Y.j..
80
70
63
75
Level A
Level B
1
2
3
4
Total A
(Y..k.)
31
49
25
45
23
40
24
51
103
185
Level C
1
2
Level C
Level A
1
2
3
1
2
31
31
41
65
59
61
41

 5 . 2   CF  12
 31  65  ...  61  5 . 4   CF  60 , 3
 31  25  ...  51  5 . 3   CF  63 ,33
 96  90  102  5 . 4 . 2   CF  1,8
 809  70  63  75  5 . 3 . 2   CF  5 , 27
 103  185  5 . 3 . 4   CF  56 , 03
S AxB  29  22  ...  28
S AxC
S BxC
SS
A
SS B
SS C
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
SS AxB  S AB  SS A  SS B  12  1,8  5 , 27  4 ,93
SS AxC  S AC  SS A  SS C  60 ,3  1,8  56 , 03  2 , 47
SS BxC  S BC  SS B  SS C  63 ,33  5 , 27  56 , 03  2 , 03
SS ABC  SS treat  SS AB  SS AC  SS BC  SS A  SS B  SS C
 73 , 2  4 ,93  2 , 47  2 , 03  1,8  5 , 27  56 , 03  0 , 67
42
Erlina Ambarwati
7/20/2015
ANOVA
SV
df
SS
A
B
C
AB
AC
BC
ABC
Error
2
3
1
6
2
3
6
23
1.8
5.27
56.03
4.93
2.47
2.03
0.67
73.20
7/20/2015
Erlina Ambarwati
43