RCBD (Randomized Complete Block Design)
Randomized Block Design Rancangan Acak Kelompok (RAK)
Types of Experimental Designs Experimental Designs Completely Randomized
Randomized Block
One-Way Anova
Factorial
Two-Way Anova
EPI809/Spring 2008
2
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.
Graphs of Interaction Occurs When Effects of One Factor Vary According to Levels of Other Factor Effects of Gender (Jantan-Betina) & dietary group (Rendah,Sedang,Tinggi) energi terhadap pertumbuhan
Interaction
Average Response
No Interaction
male
Average Response
male
female RDH
SDG
TINGGI
female RDH
SDG
Detected : In EPI809/Spring Graph , Lines Cross 2008
TINGGI 4
Persyaratan RAK :
Keuntungan;
Kerugian;
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
EPI809/Spring 2008
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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 Single Independent Variable
. . Blocking Variable
M ST M SE
Individual observations
.
. . .
. . .
. . .
. .
. . .
Randomized Block Design Factor Levels: (Treatments) Experimental Units Block 1 Block 2 Block 3 . . .
Block ...
A, B, C, D Treatments are randomly assigned within blocks A C D B C D B A B A D C . . .
. . .
. . .
. . .
D
C
A
B
EPI809/Spring 2008
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Randomized Block F-Test Hypotheses H0: 1 = 2 = ... = p
– All Population Means are Equal – No Treatment Effect
f(X)
1 = 2 = 3
Ha: Not All j Are Equal
– At Least 1 Pop. Mean is Different – Treatment Effect – 1 2 ... p Is wrong
X
f(X)
EPI809/Spring 2008
1 = 2 3
X 13
50 Men
Random Assignment
100 Subjects
Block by Gender
50 Women
Random Assignment
Randomized Block Design New Medication- 25 subjects
Compare level of pain relief as reported by subjects
Old Medication-25 subjects
New Medication- 25 subjects Compare level of pain relief as reported by subjects Old Medication-25 subjects
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 EPI809/Spring 2008
15
Partitioning the Total Sum of Squares in the Randomized Block Design SStotal (total sum of squares)
SSE (error sum of squares)
SST (treatment sum of squares)
SSB (sum of squares blocks)
SSE’ (sum of squares error)
ANOVA Table for a Randomized Block Design Source of Variation
Sum of Squares
Degrees of Freedom
Treatments
SST
t–1
Blocks
SSB
r-1
Error
SSE
Total
SSTot tr - 1
(t - 1)(r - 1)
Mean Squares SST/t-1
F MST/MSE
SSE/(t-1)(r-1)
Contoh: Percobaan mengetahui efek Level lemak (L1.L2.L3) terhadap pertambahan BB Bloking dilakukan terhadap BB sbb
Perla kuan
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
Tabel ANOVA: Sumber df SS variasi
MS
F-stat
F Tabel
Perlakuan
2
248.444
122.222
48.60
0.001,2,1 0=7.56
Blok
5
82.444
16.489
Error
10
25,556
2.556
Total
17
356,444
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
r
t
r
(yij y.. ) (yi. y.. ) (y.j y.. ) (yij yi. y.j y.. ) 2
i 1 j1
2
i 1 j1
t
r
t
r
r (yi. y.. ) t (y.j y.. ) (yij yi. y.j y.. ) 2 2
i 1
2
j1
i 1 j1
SST SSTreatments SS Blocks SS E
Extension of the ANOVA to the RCBD
The degrees of freedom for the sums of squares in
SST SSTreatments SS Blocks SS E are as follows:
tr 1 (t 1) (r 1) [(t 1)(r 1)] • 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
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
