### The Randomized Complete Block Design

```Design and Analysis of
Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
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Experiments with
Blocking Factors
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
2/33
Outline
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The Randomized Complete Block Design
The Latin Square Design
The Graeco-Latin Square Design
Balanced Incomplete Block Design
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The Randomized Complete
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
4
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
5
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?)
6
The Randomized Complete
Block Design-example
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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.
7
The Randomized Complete
Block Design-example
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To reduce the error from testing coupon,
randomize complete block design(RCBD) is
used
8
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
9
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)
10
The Randomized Complete
Block Design– Extension from ANOVA
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Suppose that there are a treatments (factor
levels) and b blocks
11
The Randomized Complete
Block Design– Extension from ANOVA
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Suppose that there are a treatments (factor
levels) and b blocks
 i  1, 2,..., a
yij     i   j   ij 
 j  1, 2,..., b
H 0 : 1  2 
 a where i  (1/ b) j 1 (    i   j )    i
b
12
The Randomized Complete
Block Design– Extension from ANOVA
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A statistical model (effects model) for the
RCBD is
 i  1, 2,..., a
yij     i   j   ij 
 j  1, 2,..., b

The relevant (fixed effects) hypotheses are
H 0 : 1  2 
 a where i  (1/ b) j 1 (    i   j )    i
b
H1 : at least one i   j
13
The Randomized Complete
Block Design– Extension from ANOVA
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Or
H 0 : 1   2     a  0
H1 : at least one i  0
14
The Randomized Complete
Block Design– Extension from ANOVA
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Partitioning the total variability
a
b
a
b
2
(
y

y
)
 ij ..   [( yi.  y.. )  ( y. j  y.. )
i 1 j 1
i 1 j 1
( yij  yi.  y. j  y.. )]2
a
b
i 1
j 1
 b ( yi.  y.. ) 2  a  ( y. j  y.. ) 2
a
b
  ( yij  yi.  y. j  y.. ) 2
i 1 j 1
SST  SSTreatments  SS Blocks  SS E
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The Randomized Complete
Block Design– Extension from ANOVA
The degrees of freedom for the sums of squares in
SST  SSTreatments  SS Blocks  SS E
are as follows:
ab  1  a  1  b  1  (a  1)(b  1)
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
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The Randomized Complete
Block Design– Extension from ANOVA
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Mean squares
a
E MStreatment    2 
b i2
i 1
a 1
a
E MSBlock    2 
E MSE    2
a   j2
i 1
b 1
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The Randomized Complete
Block Design– Extension from ANOVA
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F-test with (a-1), (a-1)(b-1) degree of
freedom
MSTreatments
F0 
MSE
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Reject the null hypothesis if
F0>F α,a-1,(a-1)(b-1)
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The Randomized Complete
Block Design– Extension from ANOVA
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ANOVA Table
19
The Randomized Complete Block
Design– Extension from ANOVA
Manual computing:
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The Randomized Complete Block
Design– Extension from ANOVA
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Meaning of F0=MSBlocks/MSE?
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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
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Meaning of F0=MSBlocks/MSE?
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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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The Randomized Complete Block
Design– Example
23
The Randomized Complete
Block Design– Example
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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
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Vascular-Graft.MTW
The Randomized Complete
Block Design– Example—Minitab
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StatANOVATwo-way
Two-way ANOVA: Yield versus Pressure, Batch
Source DF
SS
MS
F P
Pressure 3 178.171 59.3904 8.11 0.002
Batch
5 192.252 38.4504 5.25 0.006
Error 15 109.886 7.3258
Total 23 480.310
S = 2.707 R-Sq = 77.12% R-Sq(adj) = 64.92%
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Vascular-Graft.MTW
The Randomized Complete
Block Design– Example—Minitab
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The Randomized Complete
Block Design– Example —Residual Analysis
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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)
the constant variance assumption, possible
outliers
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Vascular-Graft.MTW
The Randomized Complete
Block Design– Example—Minitab
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Vascular-Graft.MTW
The Randomized Complete
Block Design– Example —No Blocking
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StatANOVAOne-way
One-way ANOVA: Yield versus Pressure
Source DF SS MS F P
Pressure 3 178.2 59.4 3.93 0.023
Error
20 302.1 15.1
Total 23 480.3
S = 3.887 R-Sq = 37.10% R-Sq(adj) = 27.66%
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Vascular-Graft.MTW
The Randomized Complete
Block Design– Example—No Blocking-Residual
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The Randomized Complete
Block Design– Other Example

1
2
3
4
y. j
1
1.3
2.2
1.8
3.9
9.2
y. j
2.30
2
1.6
2.4
1.7
4.4
10.1

3
0.5
0.4
0.6
2.0
3.5
4
1.2
2.0
1.5
4.1
8.8
5
1.1
1.8
1.3
3.4
7.6
yi.
5.7
8.8
6.9
17.8
39.2
yi.
1.14
1.76
1.38
3.56
1.96
2.53
0.88
2.20
1.90
y..
y..
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The Randomized Complete
Block Design– Other Example
4
5
SST   
i 1 j 1
yij2
y..2

ab
( 39 .2 )2
 ( 1.3 )  ( 16
. ) ( 3.4 ) 
 25.69
20
2
2
2
yi2. y..2


ab
i 1 b
a
SS Factors
( 57
. )2  ( 8 .8 )2  ( 6 .9 )2  ( 17 .8 )2 ( 39 .2 )2


 18 .04
5
20
SS Blocks
y.2j
y..2


ab
j 1 a
b
( 9 .2 )2  ( 10 .1 )2  ( 3.5 )2  ( 8 .8 )2  (7 .6 )2 ( 39 .2 )2


 6 .69
4
20
SS E  SST  SS Blocks  SS Factors  0.96
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The Randomized Complete
Block Design– Other Example
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Blocking effect
Two-way ANOVA: 濃度 versus 化學品類別, 樣品
Source
DF

4
Error
12
Total
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SS
MS F P
18.044 6.01467 75.89 0.000
6.693 1.67325 21.11 0.000
0.951 0.07925
25.688
S = 0.2815 R-Sq = 96.30% R-Sq(adj) = 94.14%
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Without blocking effect
One-way ANOVA: 濃度 versus 化學品類別
Source
DF SS MS F
P

Error
16 7.644 0.478
Total
19 25.688
S = 0.6912 R-Sq = 70.24% R-Sq(adj) = 64.66%
The Randomized Complete
Block Design– Other Example
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Blocking effect
Source of
Variation
Sum of Squares

MS
6.01

18.04
df
3

6.69
4
1.67

0.96
12
0.08

25.69
19
F0
75.13
Without blocking effect
Source of

Variation Sum of Squares
df
MS
18.04
3
6.01

7.65
16

25.69
19
0.48
F0
12.59
The Randomized Complete
Block Design– Other Aspects
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The RCBD utilizes an additive model – no
interaction between treatments and blocks
 i  1, 2,..., a
yij     i   j   ij 
 j  1, 2,..., b
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
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Sample sizing in the RCBD? The OC curve
approach can be used to determine the number of
blocks to run..see page 133
 i  1, 2,..., a
yij     i   j   ij 
 j  1, 2,..., b
The Latin Square Design
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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
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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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A Latin Square Design – The Rocket
Propellant
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This is a 5  5 Latin square design
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Statistical Analysis of the
Latin Square Design
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The statistical (effects) model is
 i  1, 2,..., p

yijk     i   j   k   ijk  j  1, 2,..., p
k  1, 2,..., p
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The statistical analysis (ANOVA) is much like the
analysis for the RCBD.
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Statistical Analysis of the
Latin Square Design
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Statistical Analysis of the
Latin Square Design
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The Standard Latin Square
Design
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A square with first row and column in
alphabetical order.
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Other Topics
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Missing values in blocked designs
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RCBD
Latin square
Estimated by
yijk 
p( yi'..  y.' j.  y..' k )  2 y...'
( p  1)( p  1)
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Other Topics
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Replication of Latin Squares
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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
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Replication of Latin Squares
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ANOVA in Case 1
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Other Topics
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Replication of Latin Squares
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ANOVA n Case 2
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Other Topics
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Replication of Latin Squares
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ANOVA n Case 3
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Other Topics
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Crossover design
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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.
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Take turn after experiments are done.
Other Topics
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Crossover design
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ANOVA
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Graeco-Latin Square
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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
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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
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ANOVA
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Graeco-Latin Square --Example
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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.
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Graeco-Latin Square --Example
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Graeco-Latin Square --Example
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