Dr. Hotniar Siringoringo HOTNIAR SIRINGORINGO HOTNIAR SIRINGORINGO LEMBAGA PENELITIAN LEMBAGA PENELITIAN KAMPUS D GD 4 LT. 1 KAMPUS D GD 4 LT. 1 JL. MARGONDA RAYA NO. 100 DEPOK JL. MARGONDA RAYA NO. 100 DEPOK [email protected][email protected][email protected][email protected][email protected][email protected]http:// http:// staffsite.gunadarma.ac.id/hotniars staffsite.gunadarma.ac.id/hotniars EXPERIMENT DESIGN
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Dr. Hotniar Siringoringo
��HOTNIAR SIRINGORINGOHOTNIAR SIRINGORINGO
��LEMBAGA PENELITIANLEMBAGA PENELITIAN
��KAMPUS D GD 4 LT. 1KAMPUS D GD 4 LT. 1
��JL. MARGONDA RAYA NO. 100 DEPOKJL. MARGONDA RAYA NO. 100 DEPOK
��Controls, standards, checks, or other Controls, standards, checks, or other
item that may be used in points of item that may be used in points of
reference in an experiment or an reference in an experiment or an
investagationinvestagation
��Discrete level of factors or variables Discrete level of factors or variables
(qualitative factors). E.g. types of (qualitative factors). E.g. types of
machine, number of times ofmachine, number of times of…….., date .., date
ofof……....
��Continuous level of factors or variables Continuous level of factors or variables
(quantitative factors), e.g. temperature, (quantitative factors), e.g. temperature,
humidity, height, etc.humidity, height, etc.
A factor might be called a set of random A factor might be called a set of random
effects if the levels of that factor are a effects if the levels of that factor are a
random sample from a population of such random sample from a population of such
levels.levels.
•• A factor is called a set of fixed effects if the A factor is called a set of fixed effects if the
levels of that factor are selected by some levels of that factor are selected by some
nonrandom process. nonrandom process.
Dr. Hotniar Siringoringo
�� Mixtures of k of v factors with the Mixtures of k of v factors with the proportion of each factor being proportion of each factor being specified by experimenter or by specified by experimenter or by the nature of the phenomenon the nature of the phenomenon under study and with there being under study and with there being one level for each factor in many one level for each factor in many cases.cases.
�� Combination of two or more of the Combination of two or more of the type of treatments above.type of treatments above.
Dr. Hotniar Siringoringo
TYPES OF MODELS
• Fixed effects model: A model is called a fixed effects model if all of the factors in the model are fixed effects and it involves only one variance component.
• Random effects model: A model is called a random effects model if all of the factors in the model are random effects.
• Mixed effects model: A model is called a mixed effects model if some of the factors in the model are fixed effects and some are random effects or if all of the factors in the model are fixed effects and there is more than one variance component in the model.
Dr. Hotniar Siringoringo
�Note: Most designs are mixed! Only a few designs; completely randomized designs: e.g. one-way, factorials, response surface) might be considered fixed.
�Design issue : Should take sources of variation into consideration as fixed, random or residual effects!
Dr. Hotniar Siringoringo
Most designs involve 2 or more factors.
Generally two types of factors in an experiment:
1. Treatment structure: consists of
those factors that the experimenter has selected to study; e.g. diets, drugs, gender
2. Design structure: consists of grouping of the experimental
units into homogeneous groups or blocks; e.g. pens, litters, days (of assay), animals (repeated measures)
Dr. Hotniar Siringoringo
Experimental design: FactorialExperimental design: FactorialExperiments Experiments –– 1. Single factor1. Single factor
��Experimental designExperimental design
–– Multiple Multiple ““treatmentstreatments”” or or ““variablesvariables””
–– Multiple replicates of each treatmentMultiple replicates of each treatment
�� Statistical AnalysisStatistical Analysis
–– OneOne--way ANOVA way ANOVA –– are any are any treatments different?treatments different?
–– BonferroniBonferroni tt--tests tests –– identify which identify which treatments are differenttreatments are different
1. The elements of the design structure 1. The elements of the design structure are random effects.are random effects.
2. There is no interaction among 2. There is no interaction among elements of the design structure and elements of the design structure and elements of the treatment structure.elements of the treatment structure.
These assumptions aid in constructing an These assumptions aid in constructing an appropriate model.appropriate model.
Dr. Hotniar Siringoringo
ONE WAY ANOVA
��The observed response from each The observed response from each treatments : random variable.treatments : random variable.
��Model:Model:
{ ainjijiijy ,...,2,1
,...,2,1==++= ετµ
Yij= observasi ke ij
µ=parameter umum utk semua perlakuan(rata-rata umum)
τi=pengaruh perlakuan
εij=random error componen
Dr. Hotniar Siringoringo
�COMPLETELY RANDOMIZED DESIGN
��THE FIXED EFFECT MODELTHE FIXED EFFECT MODEL
�� PerlakuanPerlakuanditentukanditentukanoleholehpenelitipeneliti τi adalah deviasi dari rata-rata keseluruhan. Hasil penelitian tidak berlaku umum
n
yyy i
n
jiji
.i.
1. y ; == ∑
=
01
=∑=
a
iiτ
N
yyyy
a
i
n
jij
....
1 1.. , ==∑∑
= =
Dr. Hotniar Siringoringo
�� HH00 : : τ1= τ2 = τ3 = …= τa = 0�H1 : τi ≠ 0 untuk paling tidak satu I
Sum of squareSum of squareSource of Source of variationvariation
Ny
p
yp
j
j2..
1
2.. −∑
=
Ny
y ijk
2..2 −∑ ∑∑
Ny
pyp
k
k2..
1
2.. −∑
=
( )( )11 −− ppSS E
( )1−p
SStreatments
E
treatments
MSMS
1−p
SS rows
N
y
p
yp
i
i2..
1
2.. −∑
=
1−p
SS columns
Contoh :
Pengaruh lima katalis berbeda (A, B, C, D, dan E) pada waktu reaksi proseskimia sedang dipelajari. Setiap batch bahan baru hanya cukup untuk lima kali percobaan. Setiap percobaan butuh waktu 90 menit, sehingga hanyalima percobaan dalam satu hari yang bisa dilakukan. Peneliti memutuskanmelakukan percobaan sebagai latin square, sehingga hari dan batch dapatdikontrol secara sistematis. Data hasil percobaan ditunjukkan tabel berikut:
Dr. Hotniar Siringoringo
C=8C=8A=8A=8B=3B=3D=2D=2E=4E=455
A=10A=10B=6B=6E=6E=6C=8C=8D=6D=644
D=5D=5E=1E=1C=10C=10A=9A=9B=4B=433
B=8B=8D=3D=3A=7A=7E=2E=2C=11C=1122
E=3E=3C=7C=7D=1D=1B=7B=7A=8A=811
5544332211
HariHariBatchBatch
14714734342525272728283333yy..k..k
2525
3636
2929
3131
2626
yyii ....
C=8C=8A=8A=8B=3B=3D=2D=2E=4E=455
A=10A=10B=6B=6E=6E=6C=8C=8D=6D=644
D=5D=5E=1E=1C=10C=10A=9A=9B=4B=433
B=8B=8D=3D=3A=7A=7E=2E=2C=11C=1122
E=3E=3C=7C=7D=1D=1B=7B=7A=8A=811
5544332211
HariHariBatchBatch
( ) ( ) ( ) ( ) ( ) ( ) ( )25
1478481138
2222222 −+++++= LLLTSS
Penyelesaian:
64.20836.8641073 =−=
Total perlakuan:
A = 42; B = 28; C = 44; D = 17; E = 16
Dr. Hotniar Siringoringo
( ) ( ) ( ) ( ) ( ) ( )
44.14136.8648.100525
1475
165
175
445
285
42 222222
=−=
−++++=catalystSS
( ) ( ) ( ) ( ) ( ) ( )
24.1236.8646.87625
1475
345
255
275
285
33 222222
=−=
−++++=hariSS
52.3924.1244.1544.14164.208 =−−−=ESS
( ) ( ) ( ) ( ) ( ) ( )
44.1536.8648.87925
1475
255
365
295
315
26 222222
=−=
−++++=batchSS
3.06412.24hari
10.74
3.86415.44batch
2424208.64TotalTotal
3.293(3)(4)=12(3)(4)=1239.52ErrorError
35.364141.44catalyst catalyst
FF00MSMSDfDfSSSSSource of Source of variationvariation
Dr. Hotniar Siringoringo
The Graeco-Latin Square Design
p-1Rows
p-1Columns
p-1Greek letter treatments
pp22 --11TotalTotal
(p(p--3)(p3)(p--1)1)SST – SSLatin letter treatments –SSGreek letter treatments-SSRows -SScolumns
ErrorError
pp--11Latin letter Latin letter treatments treatments
Degrees of Degrees of freedomfreedom
Sum of squareSum of squareSource of Source of variationvariation
Ny
p
ySS
p
j
jL
2..
1
2.. −= ∑
=
N
yy
i j k lijkl
2....2 −∑ ∑ ∑ ∑
N
y
p
ySS
p
i
iRows
2..
1
2... −=∑
=
Ny
py
SSp
l
lColumns
2....
1
2... −=∑
=
N
y
p
ySS
p
k
kG
2..
1
2... −=∑
=
Seorang teknik industri melakukan percobaan untuk mengetahui pengaruhempat metode perakitan (A, B, C, dan D) pada waktu perakitan komponentelevisi. Empat operator dipilih untuk melakukan perakitan. Dia mengetahuibahwa setiap metode perakitan menghasilkan kelelahan, sehingga waktuperakitan periode akhir mungkin lebih besar dibandingkan dengan periode awal, sehingga dianggap ada tren kenaikan waktu perakitan. Disamping itu, dia jugamenduga bahwa tempat kerja yang digunakan juga memberikan sumberkeragaman lainnya. Fakor keempat, tempat kerja disimbolkan dengan α, β, γ, dan δ. Waktu perakitan terukur adalah sbb:
Dr. Hotniar Siringoringo
BBδ=6=6CCα=18=18AAβ=8=8DDγ=9=944
CCγ=15=15BBβ=7=7DDα=11=11AAδ=9=933
DDβ=12=12AAγ=10=10CCδ=12=12BBα=8=822
AAα=8=8DDδ=14=14BBγ=10=10CCβ=11=1111
44332211
OperatorOperatorUrutanUrutanperakitanperakitan
1681684141494941413737yy……ll
4141
4242
4242
4343
yyii……
BBδ=6=6CCα=18=18AAβ=8=8DDγ=9=944
CCγ=15=15BBβ=7=7DDα=11=11AAδ=9=933
DDβ=12=12AAγ=10=10CCδ=12=12BBα=8=822
AAα=8=8DDδ=14=14BBγ=10=10CCβ=11=1111
44332211
OperatorOperatorUrutanUrutanperakitanperakitan
Penyelesaiaan
( ) ( ) ( ) ( ) ( )5.95
16168
446563135 222222
..
1
2.. =−+++=−= ∑
= Ny
p
ySS
p
j
jL
( ) ( ) ( ) ( ) ( )5.7
16168
441443845 222222
..
1
2... =−+++=−=∑
= Ny
py
SSp
k
kG
y.k.: α=45; β=38; γ=44; δ=41
y..j. : A=35; B=31; C=56; D=46
Dr. Hotniar Siringoringo
F0
9.17
31.83
MS
30.5Rows
319Columns
37.5Greek letter treatments
1515150TotalTotal
3327.5ErrorError
3395.5Latin letter Latin letter treatments treatments
dfdfSSSSSVSV
( )150
16168
61410112
22222....2 =−++++=−∑∑∑∑ KN
yy
i j k lijkl
( ) ( ) ( ) ( ) ( )5.0
16168
441424243 222222
..
1
2... =−+++=−=∑
= Ny
py
SSp
i
iRows
( ) ( ) ( ) ( ) ( )19
16168
441494137 222222
....
1
2... =−+++=−=∑
= Ny
py
SSp
l
lColumns
Dr. Hotniar Siringoringo
INCOMPLETE BLOCK DESIGNS
FF00MSEMSE
bb--11BlocksBlocks
NN--11TotalTotal
(a(a--1)(b1)(b--1)1)ErrorError
aa--11treatmentstreatments
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
a
Qk i
λ∑ 2
Ny
k
y j2..
2. −∑
Ny
yij
2..2 −∑∑
1)(
−a
SS adjtreatments
1−b
SS blocks
1+−− baN
SS E
E
adjtreatments
MS
MS )(
Balance incomplete block design
FF00MSEMSE
bb--11BlocksBlocks
bkbk--11TotalTotal
bkbk--bb--a+1a+1ErrorError
aa--11Treatments Treatments ((adjadj))
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
∑=
a
iii Q
1
τ̂
bk
yy
k
b
jj
2..
1
2.
1 −∑=
bky
yij
2..2 −∑∑
1)(
−a
SS adjtreatments
1−b
SS blocks
1+−− abbk
SS E
E
adjtreatments
MS
MS )(
Partially Balance incomplete block design with 2 associate classess
�� Lattice design: a balanced incomplete Lattice design: a balanced incomplete block design with kblock design with k22 treatments arranged treatments arranged in b=k(k+1) blocks with k runs per block in b=k(k+1) blocks with k runs per block and r=k+1 replicatesand r=k+1 replicates
Dr. Hotniar Siringoringo
FACTORIAL EXPERIMENT
� Two factors A and B
� Two levels per factor
– A1, A2 (e.g. AC and without AC)
– B1, B2 (e.g. 60 db vs. 70 db)
� Four different “treatment” combinations: A1B1, A1B2, A2B1, A2B2 Main effect of A = 0.5 (Difference1+ Difference2)
� Main effect of B = 0.5 (Difference3+ Difference3)
Tolak H0 interaksi pada taraf nyata 0% dansuhu pada 10%, terima H0 tipe gelas.
Ada pengaruh interaksi suhu dan tipe gelaspada kekuatan sinar yang dihasilkan yang sangat kuat, dan pengaruh suhu padakekuatan sinar yang dihasikan. Tidak adapengaruh signifikan tipe gelas terhadapkekuatan sinar yang dihasilkan
Dr. Hotniar Siringoringo
GENERAL FACTORIAL
(a(a--1)(b1)(b--1)1)(c-1)ABC
(b(b--1)(c1)(c--1)1)BC
(a(a--1)(c1)(c--1)1)AC
(a(a--1)(b1)(b--1)1)AB
C-1C
FF00MSEMSE
bb--11BB
abcnabcn--11TotalTotal
abc(n-1)ErrorError
aa--11AA
dfdfSSSSSVSV
E
AB
MS
MS
E
A
MSMS
E
B
MS
MS
E
C
MS
MS
E
AC
MS
MS
E
BC
MS
MS
E
ABC
MS
MS
H0 : Tidak ada pengaruh faktor A pada responseTidak ada pengaruh faktor B pada responseTidak ada pengaruh faktor C pada responseTidak ada pengaruh interaksi faktor AB pada responseTidak ada pengaruh interaksi faktor AC pada responseTidak ada pengaruh interaksi faktor BC pada responseTidak ada pengaruh interaksi faktor ABC padaresponse
Dr. Hotniar Siringoringo
Contoh
PersentasePersentasekonsentrasikonsentrasihardwoodhardwood dalamdalambuburbuburkertaskertas, , tekanantekananpadapadatabungtabung, , dandanwaktuwaktupemasakanpemasakanbuburbubursedangsedangdipelajaridipelajaripengaruhnyapengaruhnyapadapadakekuatankekuatankertaskertasyang yang dihasilkandihasilkan. . TigaTiga level level masingmasing--masingmasingkonsentrasikonsentrasihardwoodhardwood dandantekanantekanan, , dandan2 level 2 level waktuwaktupemasakanpemasakandiujicobakandiujicobakan. Level . Level perlakuanperlakuanadalahadalahtetaptetap(fixed). (fixed). DilakukanDilakukan2 kali 2 kali ulanganulangan. . KekuatanKekuatankertaskertasyang yang dihasilkandihasilkanadalahadalah::
Kesimpulan: Tolak H0 pada taraf nyata 1% (konsentrasi), 0% (waktu dan tekanan), terima H0
untuk semua interaksi
Dr. Hotniar Siringoringo
Rancangan Faktorial 2k dan 3k
ababTinggi-tinggi
bbRendah-tinggi
aaTinggi-rendah
11rendah-rendah
KonvensiKonvensiKombinasiperlakuan
[ ] ( )[ ]{ } ( )[ ]121
121 −−+=−+−= baab
nabab
nA
2k factorial design: k faktor dengan 2 level perlakuan.
Level : rendah dan tinggi.
1 a
b ab
rendah
rendah
tinggi
tinggi
Pengaruh rata-rata faktor A pada level rendah dan tinggifaktor B adalah:
[ ] ( )[ ]{ } ( )[ ]121
121 −−+=−+−= abab
nbaab
nB
[ ] ( )[ ]{ } ( )[ ]baabn
ababn
AB −−+=−−−= 121
121
Pengaruh rata-rata faktor B pada level rendah dan tinggifaktor A adalah:
Pengaruh interaksi faktor AB sebagai perbedaan rata-rata antara pengaruh A pada level rendah dan tinggifaktor B adalah:
2 faktor, A dan B : 22
Dr. Hotniar Siringoringo
( )[ ]4
1 2
×−−+=
n
baabSSA
( )[ ]4
1 2
×−−+=
n
baabSSB
( )[ ]4
1 2
×−−+=
nbaab
SSAB∑∑∑= = =
−=2
1
2
1 1
22
4i j
n
kijkT n
yySS K
( )1−−+= baabContrast A
+1+1--11--11+1+1ABAB
+1+1+1+1--11--11BB
+1+1--11+1+1--11AA
ababbbaa(1)(1)
++++++++abab
PengaruhPengaruhfaktorialfaktorial
--++--++bb
----++++aa
++----++(1)(1)
ABABBBAAII
KombinasiKombinasiperlakuanperlakuan
Tanda aljabar untuk menghitung pengaruh padadesain 22
Dr. Hotniar Siringoringo
( )[ ] ( )[ ]bccbaacaban
bcabccacbaban
A −−−−+++=−+−+−+−= 141
141
Pengaruh rata-rata faktor A adalah:
( )[ ]accaabcbcabbn
B −−−−+++= 141
( )[ ]cacbcabcababn
AB +−−++−−= 141
Pengaruh rata-rata faktor B adalah:
Pengaruh rata-rata faktor C adalah:
Desain 23 : 3 faktor
( )[ ]abbaabcbcaccn
C −−−−+++= 141
( )[ ]abcbcaccabban
AC +−+−−+−= 141
( )[ ]abcbcaccabban
BC ++−−−−+= 141
Pengaruh rata-rata interaksi faktor AB adalah:
Pengaruh rata-rata interaksi faktor AC adalah:
Pengaruh rata-rata interaksi faktor BC adalah:
Pengaruh rata-rata interaksi faktor ABC adalah:
[ ] [ ] [ ] ( )[ ]{ }
( )[ ]141
141
−++−+−−=
−+−−−−−=
ababcacbcabcn
ababcacbcabcn
ABC
Dr. Hotniar Siringoringo
++++++++++++++++abcabc
--++--++--++--++bcbc
----++++----++++acac
++----++++----++cc
--
--
++
++
BCBC
--
++
--
++
ACAC
--
++
++
--
ABCABC
--
--
--
--
CC
++++++++abab
PengaruhPengaruhfaktorialfaktorial
--++--++bb
----++++aa
++----++(1)(1)
ABABBBAAII
KombinasiKombinasiperlakuanperlakuan
Tanda aljabar untuk menghitung pengaruh padadesain 23
Dr. Hotniar Siringoringo
Desain 2k tanpa ulangan
• Tanpa ulangan tidak memungkinkan menghitunggalat percobaan (MSE).
• Asumsikan interaksi yang lebih tinggi diabaikan, dan
karena semua E(MS) = σ2, maka semua E(MS) dapatdigunakan untuk memperkirakan galat percobaandesain ini direkomendasikan hanya untuk model paling tidak 24.