experiment design -...
Transcript of experiment design -...
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
��[email protected]@staff.gunadarma.ac.id
��[email protected]@yahoo.com
��[email protected]@gmail.com
��http://http://staffsite.gunadarma.ac.id/hotniarsstaffsite.gunadarma.ac.id/hotniars
EXPERIMENT DESIGN
Dr. Hotniar Siringoringo
Siklus Percobaan
Dr. Hotniar Siringoringo
TYPE OF EFFECTSTYPE OF EFFECTS
Type of treatments:Type of treatments:
��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
Typical modeling Typical modeling asumptionasumption::
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
∑∑= =
−=a
i
n
jijT N
yySS
1 1
2..2
N
y
n
ySS
a
i
itreatments
2..
1
2. −= ∑
=
SSE = SST - SStreatments
NN--11SSSSTTTotalTotal
NN--aaSSSSEEError Error (within (within treatments)treatments)
aa--11SSSStreatmentstreatmentsBetween Between treatmentstreatments
FF00Mean Mean squaresquare
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
a
SS streatement
( )aNSSE
− E
streatement
MS
MS
Dr. Hotniar Siringoringo
Contoh ANOVA satu arahFaktor : temperatur
Variabel random : kecepatan peleburan (menit)
303047476161757555
313150506161737366
292949496363737344
282848486262707033
292949496060727222
303050506060757511
1250125010001000750750500500
PengamatPengamatanan
TemperaturTemperatur((00C)C)
12751275
TotalTotal
177177293293367367438438YY i..i..
303047476161757555
313150506161737366
292949496363737344
282848486262707033
292949496060727222
303050506060757511
1250125010001000750750500500
PengaPenga--matanmatan
TemperaturTemperatur((00C)C)
Dr. Hotniar Siringoringo
Tabel analisis sidik ragam
∑∑= =
−=a
i
n
jijT N
yySS
1 1
2..2
N
y
n
ySS
a
i
itreatments
2..
1
2. −=∑
=
2424--1=231=236254.625TotalTotal
2424--4=204=2037.33337.333KesalahanKesalahan
44--1=31=36217.2926217.292TemperaturTemperatur
FF00Mean Mean squaresquare
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
=(75)2+(72)2+(70)2+(73)2+(72)2+(73)2+(60)2+(60)2+(62)2+…+(31)2 -(1275)2/24 = 73989-67734.375 = 6254.625
= ((438438)2+(367)2+(293)2+(177)2 )/6-(1275)2/24
= 73951.67-67734.375 = 6217.292
SSE = SST – SStreatments = 37.333
86665.120333.37
=
431.20723
292.6217
=
241.111086665.1
431.2072
=
Bandingkan F0 dengan Ftabel untuk taraf nyata 5% atau 10%
Dr. Hotniar Siringoringo
Contoh
5.145.143.943.944.084.08134.854.854.264.263.523.5244
6.386.384.084.084.444.44125.485.484.384.384.074.0733
98.5798.5780.080.022
73.9673.96TotalTotal
4.904.904.484.483.883.88186.256.255.175.174.524.5299
4.854.854.864.864.174.17175.885.884.704.704.404.4088
16
15
14
11
10
UlanganUlangan
4.494.494.084.084.044.046.216.215.225.224.804.8077
5.145.144.534.534.304.305.585.584.374.373.683.6866
4.494.494.084.083.653.655.675.674.264.264.204.2055
6.036.034.534.534.964.965.235.234.194.193.363.3622
6.206.204.754.754.444.445.805.804.144.143.453.4511
332211332211
UlanganUlangan
Dr. Hotniar Siringoringo
THE RANDOM EFFECTS MODEL�� hasilhasilpercobaanpercobaanberlakuberlakuumumumumuntukuntukpopulasipopulasi
NN--11SSSSTTTotalTotal
σσ22NN--aaSSSSEEErrorError
MSMStreatmenttreatment
ss/MS/MSEE
σσ22 + n+ nσσττ22aa--11SSSStreatmentstreatmentsBetween Between
treatmentstreatments
FF00Mean Mean squaresquare
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
SuatuSuatuperusahaanperusahaantekstiltekstil memproduksimemproduksibenangbenangdalamdalamgulungangulunganbesarbesar. . DiinginkanDiinginkangulungangulunganbenangbenanghomogenhomogensehinggasehinggadiperolehdiperolehdidapatkandidapatkanbenangbenangdengandengankekuatankekuatanseragamseragam. . ManajerManajerproduksiproduksimendugamenduga, , selainselainvariasivariasiyang yang umumumumdidi antaraantarasampelsampeldaridari gulungangulunganyang yang samasama, , ditemukanditemukanjugajugavariasivariasikekuatankekuatanantaraantaragulungangulunganbenangbenang. . UntukUntuk mengetahuinyamengetahuinya, , manajermanajerproduksiproduksimemilihmemilih empatempatgulungangulunganbenangbenangsecarasecaraacakacak. . DilakukanDilakukanpengukuranpengukurankekuatankekuatansebanyaksebanyakempatempatulanganulangandaridari setiapsetiapgulungannyagulungannya. Data . Data kekuatankekuatanyang yang diukurdiukur ditunjukkanditunjukkanTabelTabelberikutberikut::
contoh
Dr. Hotniar Siringoringo
388388989899999696959544
383383959597979595969633
366366929293939090919122
390390969699999797989811
TotalTotal44332211GulunganGulungan
PengamatanPengamatan
Tabel kekuatan benang
1515111.94111.94TotalTotal
1.901.90121222.7522.75ErrorError
15.6815.6829.7329.733389.1989.19GulunganGulunganbenangbenang
FF00Mean Mean squaresquare
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
Analisis Sidik Ragam
Signifikan pada taraf nyata 5%
Dr. Hotniar Siringoringo
RANDOMIZED BLOCK DESIGN
bb--11BlocksBlocks
NN--11TotalTotal
(a(a--1)(b1)(b--1)1)ErrorError
aa--11treatmentstreatments
Degrees of Degrees of freedomfreedom
Sum of squareSum of squareSource of Source of variationvariation
( )∑ ∑= =
−a
i
b
jij yy
1 1
2
N
y
b
ya
i
i
2..
1
2. −∑
=
Ny
a
yb
j
j2..
1
2. −∑
=
blockstreatmentsT SSSSSS −−
�� SeorangSeorangmahasiswamahasiswateknikteknik industriindustrimembuatmembuatpercobaanpercobaanlama lama fokusfokusmatamata. . DiaDia tertariktertarikakanakanpengaruhpengaruhjarakjarakdaridari matamataterhadapterhadaplama lama fokusfokus. . EmoatEmoatcaracaraberbedaberbedadipilihdipilih, , yaituyaitu 4, 6, 8, 4, 6, 8, dandan10 10 meter. meter. DigunakanDigunakanlima lima orangorangsebagaisebagaipercobaanpercobaan. Lama . Lama waktuwaktufokusfokusmatamataadalahadalah::
33224444661010
552233335588
661166667766
66666666101044
5544332211
subjeksubjekJarakJarak
Dr. Hotniar Siringoringo
�� PenyelesaianPenyelesaian::
FF00
1.2751.275
9.0759.075
10.98310.983
MSEMSE
4436.336.3Blocks Blocks ((subjeksubjek))
191984.5584.55TotalTotal
121215.315.3ErrorError
3332.9532.95JarakJarak
Degrees of Degrees of freedomfreedom
Sum of Sum of squaresquare
Source of Source of variationvariation
( ) ( ) ( ) ( ) ( )95.3245.4704.503
2097
519
518
526
534 22222
=−=−+++
( ) ( ) ( ) ( ) ( ) ( )3.3645.47075.506
2097
420
411
419
419
428 222222
=−=−++++
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) 55.8485.4385.4285.44
85.4485.4685.4585.42
85.4385.4385.4585.4685.4185.4685.46
85.4785.4685.4685.4685.4685.410
222
2222
2222222
222222
=−+−+−+
−+−+−+−
+−+−+−+−+−+−+−
+−+−+−+−+−+−
979720201111191919192828yy.j.j
1919
1818
2626
3434
yyii..
33224444661010
552233335588
661166667766
66666666101044
5544332211
subjeksubjekJarakJarak
85.42097 ==y
6141.8275.1983.10
=
Dr. Hotniar Siringoringo
The Latin Square Design
p-1Columns
p-1Rows
pp22 --11TotalTotal
(p(p--2)(p2)(p--1)1)SST – SStreatments –SSrows-SScolumns
ErrorError
pp--11Treatments Treatments
FF00Mean Mean squaresquare
Degrees Degrees of of freedomfreedom
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
Dr. Hotniar Siringoringo
�� YoudenYoudenSquares : incomplete Squares : incomplete latinlatin square square design (design (columnscolumns≠≠rowsrows))
�� 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)
� Interaction = Difference1 – Difference2 = Difference3 – Difference4
Dr. Hotniar Siringoringo
1.1. TwoTwo--way classification analysis of varianceway classification analysis of variancea.a. Fixed Effects ModelFixed Effects Model
(a(a--1)(b1)(b--1)1)interaction
FF00MSEMSE
bb--11B treatmentsB treatments
abnabn--11TotalTotal
ab(n-1)ErrorError
aa--11A treatmentsA treatments
dfdfSSSSSVSV
abny
bny
SSa
i
iA
2...
1
2.. −= ∑
=
abny
ySSa
i
b
j
n
kijkTotal
2...
1 1 1
2 −= ∑ ∑ ∑= = =
BAsubtotalsAB SSSSSSSS −−=abny
n
ySS
a
i
b
j
ijsubtotals
2...
1 1
2. −= ∑ ∑
= =
BAABtotalE SSSSSSSSSS −−−=
abny
an
ySS
b
i
jB
2...
1
2.. −= ∑
=
Dr. Hotniar Siringoringo
ContohContoh::
VoltaseVoltaseoutput output maksimummaksimumtipetipe bateraibateraitertentutertentudipengaruhidipengaruhioleholehmaterial material pembentukpembentukbateraibateraidandansuhusuhuruanganruangandimanadimanabateraibateraidigunakandigunakan. . EmpatEmpatulanganulangandiujicobakandiujicobakandidi dalamdalamlaboratoriumlaboratoriumdengandengan3 level 3 level material material dandan3 level 3 level suhusuhu. . VoltaseVoltasebateraibateraidiukurdiukur padapadasetiapsetiapkombinasikombinasiperlakuanperlakuandandanulanganulangan, , sepertisepertiyang yang ditunjukkanditunjukkantabeltabelberikutberikut::
60608282139139150150160160168168
1041049696120120174174110110130130
33
45455858115115106106126126159159
70702525122122136136188188150150
22
58588282757580801801807474
7070202040403434155155130130
11
808065655050
SuhuSuhu((00F)F)TipeTipematerialmaterial
Dr. Hotniar Siringoringo
379937997707701291129117381738yy.j.j..
60608282139139150150160160168168 15011501
1041049696120120174174110110130130
33
45455858115115106106126126159159 13001300
70702525122122136136188188150150
22
58588282757580801801807474 998998
7070202040403434155155130130
11
yyii……808065655050
SuhuSuhu((00F)F)TipeTipematerialmaterial
( ) ( ) ( ) ( )96.77646
363799
601551302
222 =−+++= LTotalSS
( ) ( ) ( ) ( )72.10683
363799
4315011300998 2222
=−×
++=materialSS
Penyelesaian
( ) ( ) ( ) ( )72.39118
363799
4377012911738 2222
=−×
++=suhuSS
( ) ( ) ( ) ( )77.961372.3911872.10683
363799
4342229539 2222
int =−−−+++= L
erakSS
75.1823077.961372.3911872.1068396.77646 =−−−=ESS
Dr. Hotniar Siringoringo
28.91
3.56
7.91
2403.4449613.77interaksi
FF00
675.21
19558.36
5341.86MSEMSE
239118.72suhusuhu
3577646.96TotalTotal
2718230.75GalatGalat
210683.72MaterialMaterialdfdfSSSSSVSV
Kesimpulan: tolak H0suhu, H0material, H0interaksi. Ada pengaruh suhu, material, dan interaksisuhu dan material terhadap voltase baterai.
H0material : Tidak ada pengaruh material terhadapkekuatan voltase yang dihasilkanbaterai.
H0suhu : Tidak ada pengaruh suhu terhadapkekuatan voltase yang dihasilkanbaterai.
suhu H0interaksi: Tidak ada pengaruh interaksimaterial dan suhu terhadap kekuatan
voltase yang dihasilkan baterai.
Dr. Hotniar Siringoringo
RANDOM EFFECT MODEL
HH00 : : σστβτβ = 0= 0
(a(a--1)(b1)(b--1)1)interaction
FF00MSEMSE
bb--11B treatmentsB treatments
abnabn--11TotalTotal
ab(n-1)ErrorError
aa--11A treatmentsA treatments
dfdfSSSSSVSV
E
AB
MS
MS
AB
A
MSMS
AB
B
MS
MS
Contoh:
8.13
3.56
2.22
2403.4449613.77interaksi
FF00
675.21
19558.36
5341.86MSEMSE
239118.72suhusuhu
3577646.96TotalTotal
2718230.75GalatGalat
210683.72MaterialMaterialdfdfSSSSSVSV
Dr. Hotniar Siringoringo
Mixed Model
H0 : τI = 0 (fixed effect)
H0 : σβ2 =0 (random effect)
H0 : στβ2 = 0 (random effect, interaction)
(a(a--1)(b1)(b--1)1)interaction
FF00MSEMSE
bb--11B treatmentsB treatments
abnabn--11TotalTotal
ab(n-1)ErrorError
aa--11A treatmentsA treatments
dfdfSSSSSVSV
E
AB
MS
MS
AB
A
MSMS
E
B
MS
MS
Dr. Hotniar Siringoringo
Contoh
PercobaanPercobaandilaksanakandilaksanakanuntukuntukmempelajarimempelajaripengaruhpengaruhsuhusuhuoperasioperasidandan3 3 tipetipe gelasgelaspermukaanpermukaandalamdalammenghasilkanmenghasilkansinarsinar. . SuhuSuhuoperasioperasidipilihdipilihsecarasecaraacakacakdandantipetipe gelasgelasadalahadalahfixed. fixed. KesimpulanKesimpulanapaapayang yang bisabisaditarikditarik daridari percobaanpercobaantersebuttersebut??
867867
904904
889889
10451045
10531053
10661066
546546
575575
599599
33
13281328
13121312
12991299
10701070
10351035
10001000
550550
530530
579579
22
13921392
13801380
13861386
10901090
10871087
10851085
580580
568568
570570
11
150150125125100100
SuhuSuhuTipeTipe gelasgelas
Dr. Hotniar Siringoringo
Penyelesaian
1.0381.038
198.726198.726
13.56313.563
7263772637.926.926
44290551.704290551.704interaksiinteraksi
FF00
7543275432.259.259
98516985167.2597.259
MSEMSE
22150864.519150864.519TipeTipegelasgelas
2626TotalTotal
1818ErrorError
221970334.5191970334.519SuhuSuhu
dfdfSSSSSVSV
Kesimpulan:
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::
199.8199.8197.8197.8198.4198.4198.1198.1196.2196.2196.6196.6
198.5198.5197.0197.0197.6197.6197.4197.4195.6195.6197.5197.588
199199198.0198.0198.1198.1197.6197.6196.9196.9197.2197.2
199.6199.6198.7198.7197.5197.5198.4198.4196.0196.0198.5198.544
200.9200.9200.4200.4198.6198.6199.4199.4196.0196.0196.0196.0
200.6200.6199.6199.6198.4198.4199.8199.8197.7197.7196.6196.622
650650500500400400650650500500400400
TekananTekananTekananTekanan
WaktuWaktumasakmasak4 jam4 jamWaktuWaktumasakmasak3 jam3 jam% % konsenkonsentrasitrasihardhard--woodwood
Dr. Hotniar Siringoringo
Penyelesaian
1.3760.48641.943Konsentrasi*tekanan*waktu
3.3141.170222.340Waktu*tekanan
4.5141.594446.374Konsentrasi*tekanan
3.0471.07622.152Konsentrasi*waktu
56.089
27.043
10.566
9.548219.096Tekanan
FF00
0.353
19.803
3.730
MSEMSE
119.803WaktuWaktu
361412320.470TotalTotal
186.355ErrorError
27.461KonsentrasiKonsentrasi
dfdfSSSSSVSV
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.
• Contoh:
Suatu bahan kimia dipoduksi pada tangki bertekanan. Penelitian dilakukan untuk mengetahui faktor yang mempengaruhi laju filtrasi. Empat faktor, yaitu suhu (A), tekanan (B), konsentrasi reaktan ©, dan laju pengadukan(D) dengan masing-masing 2 level digunakan. Laju filtrasitanpa ulangan ditunjukkan tabel berikut:
969610410486861001007070454575754343DD11
65656565606071718080484868684545DD00
CC11CC00CC11CC00CC11CC00CC11CC00
BB11BB00BB11BB00
AA11AA00
Dr. Hotniar Siringoringo
General 2k
ContrastAB…K=(a=(a±±1)(b1)(b±±1)1)……(k(k±±1)1)
11
22kk(n(n--1)1)
N2N2kk--11
=1 k=1 k--factors interactionfactors interaction
ABCABC……KK
ErrorError
Total Total
11
11
::11
threethree--factors factors inetractionsinetractions
ABCABC
ABDABD
::IJKIJK
11
11
::11
twotwo--factors interactionsfactors interactions
ABAB
ACAC
::JKJK
11
11
::11
k main effectsk main effects
AA
BB
::KK
DfDfSum SquareSum SquareSource of variationSource of variation
2
k
3
k
k
k
Dr. Hotniar Siringoringo
Penyelesaian
��AsumsikanAsumsikaninteraksiinteraksi3 3 faktorfaktor dandan4 4 faktorfaktordiabaikandiabaikan, , dandandapatdapatdigunakandigunakanuntukuntukmemperkirakanmemperkirakangalatgalat. .
15155730.945730.94TotalTotal
25.5725.5755127.84127.84ErrorError
< 1< 15.065.06115.065.06CDCD
< 1< 10.560.56110.560.56BDBD
< 1< 122.5622.561122.5622.56BCBC
43.2443.241105.561105.56111105.561105.56ADAD
51.3951.391314.061314.06111314.061314.06ACAC
< 1< 10.060.06110.060.06ABAB
33.4633.46
15.2515.25
1.531.53
73.1573.15
FF00
855.56855.56
390.06390.06
39.0639.06
1870.561870.56
Mean SquareMean Square
11855.56855.56DD
11390.06390.06CC
1139.0639.06BB
111870.561870.56AA
DfDfSum SquareSum SquareSVSV
Dr. Hotniar Siringoringo
Desain Faktorial 3k
Rendah
0
sedang
1
tinggi
2
Faktor A
tingg
i
2
seda
ng
1
rend
ah
0
Fak
tor
A
00 10 20
01 11 21
02 12 22
Kombinasi perlakuan desain 32
Dr. Hotniar Siringoringo
contoh�� SuatuSuatupercobaanpercobaandilakukandilakukanuntukuntukmempelajarimempelajaripengaruhpengaruhtipetipe
botolbotol (A), (A), tipetipe rakrak (B), (B), dandanoperator (C). operator (C). MasingMasing--masingmasingfaktorfaktor terdiriterdiri daridari 3 level, 3 level, dengandengan2 2 ulanganulangan. . ResponsResponsyang yang diukurdiukur adalahadalahwaktuwaktupenyimpananpenyimpanan, , dandanhasilhasilpercobaanpercobaanditunjukkanditunjukkantabeltabeldidi bawahbawah..
4.494.49
4.594.59
4.904.90
4.084.08
4.084.08
4.484.48
3.653.65
4.044.04
3.883.88
3.943.94
4.534.53
4.864.86
4.084.08
4.304.30
4.174.17
PlastikPlastik
28 mm28 mm
38 mm38 mm
33
5.885.88
6.206.20
6.386.38
4.704.70
4.654.65
4.754.75
4.404.40
4.444.44
4.394.39
5.225.22
5.155.15
5.175.17
4.804.80
4.524.52
4.964.96
PlastikPlastik
28 mm28 mm
38 mm38 mm
22
5.235.23
4.854.85
5.585.58
4.194.19
4.264.26
4.374.37
3.363.36
3.523.52
3.683.68
4.144.14
4.384.38
4.264.26
3.453.45
4.074.07
4.204.20
PlastikPlastik
28 mm28 mm
38 mm38 mm
11
PendiPendi--nginngin
PermaPerma--nennen
TipeTipebotolbotol
PendiPendi--nginngin
PermaPerma--nennen
TipeTipebotolbotol
OperaOpera--tortor
UlanganUlangan22UlanganUlangan11
Dr. Hotniar Siringoringo
Penyelesaian
1515TotalTotal
55ErrorError
0.700.70880.5580.558Operator*Operator*tipetipebotolbotol** tipetipe rakrak
0.270.27440.1090.109TipeTipe botolbotol** tipetipe rakrak
0.4100.410441.6401.640Operator*Operator*tipetipe rakrak
10.46310.463
FF00
0.270.27
8.8858.885
3.8433.843
MSMS
440.1080.108Operator*Operator*tipetipe botolbotol
2217.77017.770TipeTipe rakrak
220.4200.420TipeTipe botolbotol
227.6867.686OperatorOperator
DfDfSSSSSVSV
Dr. Hotniar Siringoringo
CONFOUNDINGBlokBlok dalamdalamdesaindesainfaktorialfaktorial
CONFOUNDING DALAM DESAIN 2CONFOUNDING DALAM DESAIN 2kk
Let k=2 and 2 blocksLet k=2 and 2 blocks
(1)
ab
Blok 1
a
b
Blok 2 Pengaruh utama A dan B:
( )[ ]121 −−+= baabA ( )[ ]1
21 −−+= ababB
[ ]baabAB −−+= )1(21
++++++++abab
PengaruhPengaruhfaktorialfaktorial
--++--++bb
----++++aa
++----++(1)(1)
ABABBBAAII
KombinasiKombinasiperlakuanperlakuan
--
++
++
--
++
--
--
++
++
++
--
--
--
--
++
++
++
--
++
--
--
++
--
++
--
--
--
--
++
++
++
++
--
--
++
++
--
--
++
++
--
++
--
++
--
++
--
++
++
++
++
++
++
++
++
++
(1)(1)
aa
bb
abab
cc
acac
bcbc
abcabc
ABCABCBCBCACACCCABABBBAAII
PengaruhPengaruhfaktorialfaktorialKombinasiKombinasiperlakuanperlakuan
Dr. Hotniar Siringoringo
IN THE CONTEXT OF AMICROARRAY EXPERIMENT
A, B, & C: 3 different treatments (experimental conditions)A, B, & C: 3 different treatments (experimental conditions)
Dr. Hotniar Siringoringo
Dr. Hotniar Siringoringo
Dr. Hotniar Siringoringo
Dr. Hotniar Siringoringo