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Materi : DOE Minggu IV
1. Introduction 2. Simple Comparative Experiments 3. Experiments with a Single Factor 4. The Randomized Complete Block Design 5. The Latin Square Design 6. Factorial Design 7. The 2k Factorial Design 8. Two-Level Fractional Factorial Design 9. Nested or Hierarchial Design10. Response Surface Methods
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Chapter 4. Randomized Blocks …
ProcessInput Output (Y)
Z1, Z2, …, Zq
X1, X2, …, Xq
Uncontrollable Factors
Controllable Factors
A Nuisance Factors
A design factor that probably has an effect on the response, but we are not interested in that effect
unknown known
Randomization
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Chapter 4. Randomized Blocks … (continued)
ProcessInput Output (Y)
Z1, Z2, …, Zq
X1, X2, …, Xq
Uncontrollable Factors
Controllable Factors
A Nuisance Factors
A design factor that probably has an effect on the response, but we are not interested in that effect
unknown known
blocking
unknown known
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The Randomized Complete Block Design (RCBD)
When the nuisance source of variability is known and controllable,
blocking can be used to systematically eliminate its effect on the statistical comparisons among treatments
Situations for which the RCBD is appropriate: Units of test equipment or machinery (often
different in their operating characteristics and would be a typical bloking factor)
Batches of raw material, people, and time (common nuisane sources of variability in an experiment)
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Desain penggunaan 4 merk mesin … (Is it right?)
Hari Kerja atau Operator
1 2 3 4
Mesin yang digunakan (A,B,C,D)
A B C D
A B C D
A B C D
A B C D
Tidak dapat dipisahkan antara rata-rata produktifitas mesin dari rata-
rata produktifitas hari ataupun operator
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Desain penggunaan 4 merk mesin … (Right design)
Hari Kerja atau Operator -> BLOK
1 2 3 4
Mesin yang digunakan (A,B,C,D)
A B C D
B C D A
C D A B
D A B C
Randomisasi secara lengkap dilakukan dalam BLOK yang sama, sehingga rata-rata produktifitas mesin dapat dipisahkan dari
rata-rata produktifitas hari ataupun operator
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The Randomized Complete Block Design (RCBD)
Y11
Y21
Y31
…
Ya1
Y12
Y22
Y32
…
Ya2
Y13
Y23
Y33
…
Ya3
Y1b
Y2b
Y3b
…
Yab
…
Block 1
Block 2
Block 3
Block b
There is one observation per treatment (1, 2, …, a) in each block, and the order in which the treatments are run within each block is
determined randomly.
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The ANOVA: Structure Data and Model RCBD
Treatment (level i)
Block (j) Total Yi.
Average Yi.1 2 … b
1 Y11 Y12 … Y1n Y1. Y1.
2 Y21 Y22 … Y2n Y2. Y2.
... … … … … … …
a Ya1 Ya2 … Yan Ya. Ya.
Total Y.j Y.1 Y.2 … Y.b Y.. Y..
Statistical model for the RCBD: yij = ij + ij, , i = 1, 2, …, a; j = 1, 2, …, b atau yij = + i + j + ij
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The ANOVA Table for a RCBD Model
Source of Variation
Sum of Squares DF
Mean Square F
Treatments SST a 1 MST FT = MST / MSE
Blocks SSB b – 1 MSB FB = MSB / MSE
Error SSE (a-1)(b-1) MSE
Total SSTotal N 1
BTTotalE
BT
Total
SSSSSSSS
SS ; SS
SS
N
YY
aN
YY
b
N
YY
b
jj
a
ii
a
i
b
jij
2..
1
2.
2..
1
2.
2..
1 1
2
11
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Chapter 4. Randomized Blocks …
ProcessInput Output (Y)
Z1, Z2, …, Zq
X1, X2, …, Xq
Uncontrollable Factors
Controllable Factors
Metal coupon
Type of Tip
1, 2, 3, 4
People, Machine, and
other control-able factors are inthe SAME conditions
The hardness testing machine
We wish to determine whether or not four different tips produce different readings on a hardness testing machine ?
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Data from the Hardness Testing Experiment …
Type of TipTest Coupon (Block)
1 2 3 4
1 9.3 9.4 9.6 10.0
2 9.4 9.3 9.8 9.9
3 9.2 9.4 9.5 9.7
4 9.7 9.6 10.0 10.2
The metal coupon differ slightly in their hardness, as might happen if they are taken from ingots that are product in different heats, the experiment units (the coupon) will contribute to the variabiliy observed in
the hardness data.
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Graphical Analysis: Box-Plot Data …
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Graphical Analysis: Main Effect Plot …
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ANOVA of RCBD: MINITAB output …
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Wrong ANOVA : MINITAB output …
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Data, Fits and Residual: MINITAB output …
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Graphical comparison of means …
9.4 9.6 9.8 10.0
Tip 3
Tip 1 Tip 2
Tip 4
This plot indicates that tip 1, 2, and 3 probably
produce identical average hardness
measurements but that tip 4 produces a much higher mean hardness.
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Model Adequacy Checking: Normality test …
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Model Adequacy Checking: Equality variance …
Plot of residuals by tip type (treatment) and by coupon (block)
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MINITAB command for RCBD Analysis
