Post on 15-Dec-2015
Design and Analysis of Experiments
Dr. Tai-Yue Wang Department of Industrial and Information Management
National Cheng Kung UniversityTainan, TAIWAN, ROC
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Experiments with Blocking Factors
Dr. Tai-Yue Wang Department of Industrial and Information Management
National Cheng Kung UniversityTainan, TAIWAN, ROC
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Outline
The Randomized Complete Block Design The Latin Square Design The Graeco-Latin Square Design Balanced Incomplete Block Design
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In some experiment, the variability may arise from factors that we are not interested in.
A nuisance factor ( 擾亂因子 )is a factor that probably has some effect on the response, but it’s of no interest to the experimenter … however, the variability it transmits to the response needs to be minimized
These nuisance factor could be unknown and uncontrolled use randomization
The Randomized Complete Block Design
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If the nuisance factor are known but uncontrollable use the analysis of covariance.
If the nuisance factor are known but controllable use the blocking technique
Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units
The Randomized Complete Block Design
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Many industrial experiments involve blocking (or should)
Failure to block is a common flaw in designing an experiment (consequences?)
The Randomized Complete Block Design
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We wish determine whether or not four different tips produce different readings on a hardness testing machine.
One factor to be consider tip type Completely Randomized Design could be
used with one potential problem the testing block could be different
The experiment error could include both the random and coupon errors.
The Randomized Complete Block Design-example
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To reduce the error from testing coupon, randomize complete block design(RCBD) is used
The Randomized Complete Block Design-example
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Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips
“complete” indicates each testing coupon (BLOCK) contains all treatments
Variability between blocks can be large, variability within a block should be relatively small
In general, a block is a specific level of the nuisance factor
The Randomized Complete Block Design-example
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A complete replicate of the basic experiment is conducted in each block
A block represents a restriction on randomization
All runs within a block are randomized Once again, we are interested in testing the
equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)
The Randomized Complete Block Design-example
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Suppose that there are a treatments (factor levels) and b blocks
The Randomized Complete Block Design– Extension from ANOVA
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Suppose that there are a treatments (factor levels) and b blocks
The Randomized Complete Block Design– Extension from ANOVA
1,2,...,
1, 2,...,ij i j ij
i ay
j b
0 1 2 1: where (1/ ) ( )
b
a i i j ijH b
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A statistical model (effects model) for the RCBD is
The relevant (fixed effects) hypotheses are
1,2,...,
1, 2,...,ij i j ij
i ay
j b
0 1 2 1: where (1/ ) ( )
b
a i i j ijH b
The Randomized Complete Block Design– Extension from ANOVA
jiH oneleast at :1
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Partitioning the total variability
The Randomized Complete Block Design– Extension from ANOVA
2.. . .. . ..
1 1 1 1
2. . ..
2 2. .. . ..
1 1
2. . ..
1 1
( ) [( ) ( )
( )]
( ) ( )
( )
a b a b
ij i ji j i j
ij i j
a b
i ji j
a b
ij i ji j
T Treatments Blocks E
y y y y y y
y y y y
b y y a y y
y y y y
SS SS SS SS
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The degrees of freedom for the sums of squares in
are as follows:
Therefore, 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 that can be used to test the hypothesis of equal treatment means
T Treatments Blocks ESS SS SS SS
1 1 1 ( 1)( 1)ab a b a b
The Randomized Complete Block Design– Extension from ANOVA
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Mean squares
The Randomized Complete Block Design– Extension from ANOVA
2
1
2
2
1
2
2
1
1
E
a
ij
Block
a
ii
treatment
MSE
b
aMSE
a
bMSE
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F-test with (a-1), (a-1)(b-1) degree of freedom
Reject the null hypothesis if
F0>F α,a-1,(a-1)(b-1)
The Randomized Complete Block Design– Extension from ANOVA
E
Treatments
MS
MSF 0
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The Randomized Complete Block Design– Extension from ANOVA
Meaning of F0=MSBlocks/MSE? The randomization in RBCD is applied only to
treatment within blocks The Block represents a restriction on
randomization Two kinds of controversial theories
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The Randomized Complete Block Design– Extension from ANOVA
Meaning of F0=MSBlocks/MSE? General practice, the block factor has a large effect
and the noise reduction obtained by blocking was probably helpful in improving the precision of the comparison of treatment means if the ration is large
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To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin
Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures
The Randomized Complete Block Design– Example
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The Randomized Complete Block Design– Example—Minitab
StatANOVATwo-way
Vascular-Graft.MTW
Two-way ANOVA: Yield versus Pressure, Batch
Source DF SS MS F PPressure 3 178.171 59.3904 8.11 0.002Batch 5 192.252 38.4504 5.25 0.006Error 15 109.886 7.3258Total 23 480.310
S = 2.707 R-Sq = 77.12% R-Sq(adj) = 64.92%
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Basic residual plots indicate that normality, constant variance assumptions are satisfied
No obvious problems with randomization No patterns in the residuals vs. block Can also plot residuals versus the pressure
(residuals by factor) These plots provide more information about
the constant variance assumption, possible outliers
The Randomized Complete Block Design– Example —Residual Analysis
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The Randomized Complete Block Design– Example —No Blocking
StatANOVAOne-way
Vascular-Graft.MTW
One-way ANOVA: Yield versus Pressure
Source DF SS MS F PPressure 3 178.2 59.4 3.93 0.023Error 20 302.1 15.1Total 23 480.3
S = 3.887 R-Sq = 37.10% R-Sq(adj) = 27.66%
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化學品 樣品
類 別 1 2 3 4 5 yi . yi . 1 1.3 1.6 0.5 1.2 1.1 5.7 1.14 2 2.2 2.4 0.4 2.0 1.8 8.8 1.76 3 1.8 1.7 0.6 1.5 1.3 6.9 1.38 4 3.9 4.4 2.0 4.1 3.4 17.8 3.56 y j. 9.2 10.1 3.5 8.8 7.6 39.2 1.96
y j. 2.30 2.53 0.88 2.20 1.90 y.. y..
The Randomized Complete Block Design– Other Example
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SS yy
abT ijji
2
1
5
1
4 2
2 2 22
13 16 3 439 2
2025 69
..
( . ) ( . ) ( . )( . )
.
SSy
b
y
abFactorsi
i
a
. ..
( . ) ( . ) ( . ) ( . ) ( . ).
2
1
2
2 2 2 2 257 8 8 6 9 17 85
39 220
18 04
SSy
a
y
abBlocksj
j
b
. ..
( . ) ( . ) ( . ) ( . ) ( . ) ( . ).
2
1
2
2 2 2 2 2 29 2 10 1 3 5 8 8 7 64
39 220
6 69
SS SS SS SSE T Blocks Factors 0 96.
The Randomized Complete Block Design– Other Example
Two-way ANOVA: 濃度 versus 化學品類別 , 樣品
Source DF SS MS F P化學品類別 3 18.044 6.01467 75.89 0.000樣品 4 6.693 1.67325 21.11 0.000Error 12 0.951 0.07925Total 19 25.688
S = 0.2815 R-Sq = 96.30% R-Sq(adj) = 94.14%
One-way ANOVA: 濃度 versus 化學品類別
Source DF SS MS F P化學品類別 3 18.044 6.015 12.59 0.000Error 16 7.644 0.478Total 19 25.688
S = 0.6912 R-Sq = 70.24% R-Sq(adj) = 64.66%
The Randomized Complete Block Design– Other Example
Blocking effect
Without blocking effect
The Randomized Complete Block Design– Other Example
Blocking effect
Without blocking effect
Source of Variation
Sum of Squares
自由度
df
變異數
MS
F0
化學品 18.04 3 6.01 75.13
集區 6.69 4 1.67
誤差 0.96 12 0.08
總和 25.69 19
Source of Variation
Sum of Squares
自由度
df
變異數
MS
F0
化學品 18.04 3 6.01 12.59
誤差 7.65 16 0.48
總和 25.69 19
The RCBD utilizes an additive model – no interaction between treatments and blocks
Treatments and/or blocks as random effects Missing values What are the consequences of not blocking if we
should have?
The Randomized Complete Block Design– Other Aspects
1,2,...,
1, 2,...,ij i j ij
i ay
j b
Sample sizing in the RCBD? The OC curve approach can be used to determine the number of blocks to run..see page 133
The Randomized Complete Block Design– Other Aspects
1,2,...,
1, 2,...,ij i j ij
i ay
j b
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The Latin Square Design
These designs are used to simultaneously control (or eliminate) two sources of nuisance variability
Those two sources of nuisance factors have exactly same levels of factor to be considered
A significant assumption is that the three factors (treatments, nuisance factors) do not interact
If this assumption is violated, the Latin square design will not produce valid results
Latin squares are not used as much as the RCBD in industrial experimentation
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The Latin Square Design
The Latin square design systematically allows blocking in two directions
In general, a Latin square for p factors is a square containing p rows and p columns.
Each cell contain one and only one of p letters that represent the treatments.
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Statistical Analysis of the Latin Square Design
The statistical (effects) model is
The statistical analysis (ANOVA) is much like the analysis for the RCBD.
1,2,...,
1, 2,...,
1, 2,...,ijk i j k ijk
i p
y j p
k p
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Other Topics
Missing values in blocked designs RCBD Latin square Estimated by
)1)(1(
2)( '...
'..
'..
'..
pp
yyyypy kji
ijk
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Other Topics
Replication of Latin Squares To increase the error degrees of freedom Three methods
1. Use the same batches and operators in each replicate
2. Use the same batches but different operators in each replicate
3. Use different batches and different operator
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Other Topics
Crossover design p treatments to be tested in p time periods using
np experiment units. Ex : 20 subjects to be assigned to two periods
First half of the subjects are assigned to period 1 (in random) and the other half are assigned to period 2.
Take turn after experiments are done.
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Graeco-Latin Square
For a pxp Latin square, one can superimpose a second pxp Latin square that treatments are denoted by Greek letters.
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Graeco-Latin Square
If the two squares have the property that each Greek letter appears once and only once with each Latin letter, the two Latin squares are to be orthogonal and this design is named as Graeco-Latin Square.
It can control three sources of extraneous variability.
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Graeco-Latin Square --Example
In the rocket propellant problem, batch of material, operators, and test assemblies are important.
If 5 of them are considered, a Graeco-Latn square can be used.