Download Design of Engineering Experiments Part 3 – The Blocking Principle

•A nuisance factor 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
•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
remove the effect of the nuisance factor from the
analysis
•If the nuisance factor is unknown and
uncontrollable, use randomization to balance out its
impact across the experiment
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The Blocking Principle
• Blocking is a technique to systematically eliminate
the effect of nuisance factors
• Failure to block is a common flaw in designing an
experiment (consequences?)
• The randomized complete block design or the RCBD
– one nuisance variable
• Extension of the ANOVA to the RCBD
• Several sources of variability can be combined in a
block, to form an aggregate variable
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The Latin Square Design
• Text reference, Section 4-2, pg. 136
• These designs are used to simultaneously control
(or eliminate) two sources of nuisance
variability
• 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 Rocket Propellant Problem – A
Latin Square Design
•
•
•
•
Five different formulations of a rocket propellant
Five different materials, and five operators
Two nuisance factors
This is a 5x5 Latin square design
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Examples of Latin squares
A B D C
B C A D
C D B A
D A C B
4X4
A D B
E C
D A C
B E
C B E
D A
B E A
C D
E C D
A B
5X5
Examples of standard Latin squares
A B C D
A B C
D E
B C D A
B A E
C D
C D A B
C D A
E B
D A B C
D E B
A C
E C D
B A
4
576
56
161,280
First row and column
consist of the letters
written in alphabetical
order
# of standard Latin squares
5
Total # of Latin squares
Statistical Analysis of the
Latin Square Design
• The statistical (effects) model is
 i  1, 2,..., p

yijk    i   j   k   ijk  j  1, 2,..., p
k  1, 2,..., p

• The statistical analysis (ANOVA) is much like the
analysis for the RCBD
• SST = SSRows + SSColumns + SSTreatments + SSE
Respective degrees of freedom
p2 – 1 = p-1 + p-1 + p-1 + (p-2)(p-1)
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• Mean squares, Fo = MSTreatments/MSE
• Fo is compared with Fp-1,(p-2)(p-1) for testing the null
hypothesis of equal treatments
• The effects of rows and columns can also be tested
using the ratios of MSRows and MSColumns to MSE,
but may be inappropriate
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Example: The Rocket Propellant Problem
• Coding (by subtracting 25 from each observation)
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• There is a significant difference in the means of
formulations
• There is also an indication that there are differences
between operators
• There is no strong evidence of a difference between
batches of raw materials
• Model adequacy can be checked by plotting residuals
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• The observations in a Latin square should be taken in
random order
• Select a Latin square from a table
• Arrange the order of the rows, columns, and letters
at random
• The Latin square design assumes that there is no
interaction between the factors
• Small Latin squares provide a relatively small number
of error degrees of freedom (p-2)(p-1), therefore,
replication is often needed to increase error DOF.
• The analysis of variance depends on the method of
replication
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The Latin square is replicated n times –
• Case 1: same levels of the row and column blocking
factors are used in each replicate (repeat the square).
Total number of observations: N = np2. ANOVA:
Table 4-13
• Case 2: same row (column) but different columns
(rows) are used in each replicate (fix the level of the
row factor, and change the level of the column
factor). N = np2. ANOVA: Table 4-14
• Case 3: different rows and columns are used in each
replicate. N = np2. ANOVA: Table 4-15
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DATA PREPARATION
 Identifying Outliers
 Calculating standardized residuals.
 Values outside 95% of confidence interval are regarded
as outliers.
 These outliers should be taken away so the model is not
affected.
d ij 
eij
MS E
LoadRate
(A)
ButtonDia.
(B)
HoldTime
(C)
PeakLoad
Residual
Std. Residual
1
-1
1
4.97
-6.4594
-4.54
Note: this example is taken from the peak load
observations of Material 01.
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 Filling Missing Points
 The missing observation x is estimated by
minimizing its contribution to the error sum of
squares.
 If there are several missing points, the
equation is iteratively used to estimate the
missing values, until convergence is obtained.
1
1
1
SS E  x 2  ( yi'.  x) 2  ( y.' j  x) 2  ( y..'  x) 2  R
b
a
ab
x
ayi'.  by.' j  y..'
(a  1)(b  1)
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 Filling Multiple Missing Points
•If several observations are missing, they may be estimated by
writing the error sum of squares as a function of the missing
values, differentiating with respect to each missing value,
equating the results to zero, and solving the resulting equations.
•Alternatively, the equation can be iterated to estimate the missing
values. Suppose that two values are missing. Arbitrarily choose
the first missing value, and then the equation is used with this
assumed value along with the real data for estimating the second.
The equation is then used to re-estimate the first missing value
using the real observations and the estimated second value.
Following this, the second is re-estimated. This process is
continued until convergence is obtained. In any missing value
problem, the error degrees of freedom are reduced by one for each
missing observation.
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