RCBDblog.ub.ac.id/emiliawidyasari/files/.../2012-kul-7-RCBDRAK-rancob.pdfRCBD (Randomized Complete...
Transcript of RCBDblog.ub.ac.id/emiliawidyasari/files/.../2012-kul-7-RCBDRAK-rancob.pdfRCBD (Randomized Complete...
RCBD
(Randomized Complete Block Design)
Randomized Block DesignRancangan Acak Kelompok
(RAK)
EPI809/Spring 2008 2
Types of Experimental Designs
ExperimentalDesigns
One-Way Anova
Completely Randomized
Randomized Block
Two-Way Anova
Factorial
Kondisi Percobaan Yang sesungguhnya:-Ada nuisance factor (pengganggu), homogenitas materi terganggu :
-(Data HETEROGEN)
Misalnya: Pengaruh ransum terhadap ADG (kg)
Umur juga berpengaruh terhadap ADG
sehingga : umur mrpk faktor pengganggu
Pilihan:1. Umur juga diteliti : RAL Pola Faktorial,
umur sebagai faktor perlakuan juga
2. menggunakan umur untuk pengelompokan (sebagai BLOK):
Mengeluarkan variasi yang bersumber pada umur dari variasi error
percob.
Asumsi TIDAK ADA interaksi antar perlakuan
Catatan: jika ragu-ragu dengan Asumsi . Sebaiknya
faktor pengganggu dijadikan perlakuan , gunakan
RAL Faktorial.
EPI809/Spring 2008
4
Graphs of Interaction
Effects of Gender (Jantan-Betina) & dietary group (Rendah,Sedang,Tinggi) energi terhadap pertumbuhan
Interaction No Interaction
AverageResponse
RDH SDG TINGGI
male
female
AverageResponse
RDH SDG TINGGI
male
female
Occurs When Effects of One Factor Vary According
to Levels of Other Factor
Detected : In Graph , Lines Cross
Persyaratan RAK :
Keuntungan;
Kerugian;
EPI809/Spring 2008 7
Randomized Block Design
1.Experimental Units (Subjects) Are Assigned Randomly within Blocks
– Blocks are Assumed Homogeneous
2.One Factor or Independent Variable of Interest
– 2 or More Treatment Levels or Classifications
3. One Blocking Factor
The Blocking Principle
• Blocking is a technique for dealing with nuisance factors
• A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized
• Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units
• Many industrial experiments involve blocking (or should)
• Failure to block is a common flaw in designing an experiment (consequences?)
The Blocking Principle
• If the nuisance variable is known and controllable, we use blocking
• If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to statistically remove the effect of the nuisance factor from the analysis
• If the nuisance factor is unknown and uncontrollable , we hope that randomization balances out its impact across the experiment
• Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable
Randomized Complete Block Design
• An experimental design in which there is one independent
variable, and a second variable known as a blocking
variable, that is used to control for confounding or
concomitant variables.
• It is used when the experimental unit or material are
heterogeneous
• There is a way to block the experimental units or materials
to keep the variability among within a block as small as
possible and to maximize differences among block
• The block (group) should consists units or materials which
are as uniform as possible
A Randomized Block Design
Individual
observations
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Single Independent Variable
Blocking
Variable
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MSE
MST
EPI809/Spring 2008 12
Randomized Block Design
Factor Levels:
(Treatments) A, B, C, D
Experimental Units
Treatments are randomly
assigned within blocks
Block 1 A C D B
Block 2 C D B A
Block 3 B A D C.
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Block ... D C A B
EPI809/Spring 2008 13
Randomized Block F-Test Hypotheses
H0: 1 = 2 = ... = p
– All Population Means are Equal
– No Treatment Effect
Ha: Not All j Are Equal– At Least 1 Pop. Mean is
Different
– Treatment Effect
– 1 2 ... p Is wrong
X
f(X)
1 = 2 = 3
X
f(X)
1 = 2 3
Randomized Block Design
100 Subjects
New Medication- 25 subjects
Old Medication-25 subjects
Compare level of pain relief as reported by subjects
Ran
do
m A
ssig
nm
ent
50 Women
50 Men
New Medication- 25 subjects
Old Medication-25 subjects
Compare level of pain relief as reported by subjects
Ran
do
m A
ssig
nm
entB
lock
by
Gen
der
EPI809/Spring 2008 15
Randomized Block F-Test Test Statistic
• 1. Test Statistic
– F = MST / MSE
• MST Is Mean Square for Treatment
• MSE Is Mean Square for Error
• 2. Degrees of Freedom
– 1 = p -1
– 2 = n – b – p +1
• p = # Treatments, b = # Blocks, n = Total Sample Size
Partitioning the Total Sum of Squares
in the Randomized Block Design
SStotal
(total sum of squares)
SST
(treatment
sum of squares)
SSE
(error sum of squares)
SSB
(sum of squares
blocks)
SSE’
(sum of squares
error)
ANOVA Table for a
Randomized Block Design
Source of Sum of Degrees of Mean
Variation Squares Freedom Squares F
Treatments SST t – 1 SST/t-1 MST/MSE
Blocks SSB r - 1
Error SSE (t - 1)(r - 1) SSE/(t-1)(r-1)
Total SSTot tr - 1
Contoh:Percobaan mengetahui efek Level lemak (L1.L2.L3) terhadap pertambahan BB
Bloking dilakukan terhadap BB sbb
Perlakuan
Blok
1 2 3 4 5 6
L1 89 89 87 92 92 85
L2 96 94 96 98 94 100
L3 96 97 99 101 102 103
281 280 282 291 288 298
SSY = 356,44
SSP =248,44
SSB = 82.444
SSE = 25.556
Sumber variasi
df SS MS F-stat F Tabel
Perlakuan 2 248.444 122.222 48.60 0.001,2,10=7.56
Blok 5 82.444 16.489
Error 10 25,556 2.556
Total 17 356,444
Tabel ANOVA:F- stat lebih
besar dari F-Tab.
Kesimpulan:
terdapat
perbedaan efek
Lemak (P<0.,01)
Extension of the ANOVA to the RCBD
ANOVA partitioning of total variability:
t
1i
r
1j
2
...ji.ij
r
1j
2
...j
t
1i
2
..i.
t
1i
r
1j
2
...ji.ij...j..i.
t
1i
r
1j
2
..ij
)yyy(y)yy(t)yy(r
)yyy(y)yy()yy()y(y
EBlocksTreatmentsT SSSSSSSS
Extension of the ANOVA to the RCBD
The degrees of freedom for the sums of squares in
are as follows:
• Ratios of sums of squares to their degrees of freedom result in mean squares, and
• The ratio of the mean square for treatments to the error mean square is an F statistic used to test the hypothesis of equal treatment means
EBlocksTreatmentsT SSSSSSSS
)]1)(1[( )1( )1( 1 rtrttr
ANOVA Procedure
• The ANOVA procedure for the randomized block design requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error.
• The formula for this partitioning is
SSTot = SST + SSB + SSE
• The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments,
b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.
Example: Eastern Oil Co.
Automobile Type of Gasoline (Treatment) Blocks
(Block) Blend X Blend Y Blend Z Means
1 31 30 30 30.333
2 30 29 29 29.333
3 29 29 28 28.667
4 33 31 29 31.000
5 26 25 26 25.667
Treatment
Means 29.8 28.8 28.4