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Anonymous

this class is design and analysis experiment

see the files attached

only the highlighted questions

and i attached slides for chapter4

2017_09_27_photo_00000576.jpg

2017_09_27_photo_00000575.jpg

2017_09_27_photo_00000574.jpg

Design of Engineering Experiments
– The Blocking Principle
• Text Reference, Chapter 4
• Blocking and nuisance factors
• The randomized complete block design or
the RCBD
• Extension of the ANOVA to the RCBD
• Other blocking scenarios…Latin square
designs
Chapter 4
1
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’s 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?)
Chapter 4
2
The Hardness Testing Example
• Text reference, pg 139, 140
• We wish to determine whether 4 different tips produce
different (mean) hardness reading on a Rockwell hardness
tester
• Gauge & measurement systems capability studies are
frequent areas for applying DOE
• Assignment of the tips to an experimental unit; that is, a
test coupon
• Structure of a completely randomized experiment
• The test coupons are a source of nuisance variability
• Alternatively, the experimenter may want to test the tips
across coupons of various hardness levels
• The need for blocking
Chapter 4
3
The Hardness Testing Example
• Suppose that we use b = 4 blocks:
• Notice the two-way structure of the experiment
• 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)
Chapter 4
4
Extension of the ANOVA to the RCBD
• Suppose that there are a treatments (factor levels)
and b blocks
• 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
Chapter 4
a where i (1/ b) j 1 ( i j ) i
b
5
Extension of the ANOVA to the RCBD
ANOVA partitioning of total variability:
a
b
( y
i 1 j 1
ij
a
b
y.. ) [( yi. y.. ) ( y. j y.. )
2
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
Chapter 4
6
Extension of the ANOVA to the RCBD
The degrees of freedom for the sums of squares in
SST SSTreatments SSBlocks SSE
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
Chapter 4
7
ANOVA Display for the RCBD
Manual computing (ugh!)…see Equations (4-9) –
(4-12), page 124
Design-Expert analyzes the RCBD
Chapter 4
8
Manual computing:
Chapter 4
9
Chapter 4
10
Vascular Graft Example (pg. 126)
• 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
Chapter 4
11
Chapter 4
12
Vascular Graft Example
Design-Expert Output
Chapter 4
13
Residual Analysis for the
Vascular Graft Example
Chapter 4
14
Residual Analysis for the
Vascular Graft Example
Chapter 4
15
Residual Analysis for the
Vascular Graft Example
• 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
Chapter 4
16
Multiple Comparisons for the Vascular Graft
Example – Which Pressure is Different?
Also see Figure 4.2
Chapter 4
17
Other Aspects of the RCBD
See Text, Section 4.1.3, pg. 132
• 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?
• Sample sizing in the RCBD? The OC curve
approach can be used to determine the number of
blocks to run..see page 133
Chapter 4
18
Random Blocks and/or
Treatments
σ2β
Chapter 4
Design & Analysis of Experiments
8E 2012 Montgomery
19
Chapter 4
Design & Analysis of Experiments
8E 2012 Montgomery
20
Chapter 4
Design & Analysis of Experiments
8E 2012 Montgomery
21
The Latin Square Design
• Text reference, Section 4.2, pg. 158
• 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
Chapter 4
22
The Rocket Propellant Problem –
A Latin Square Design
•
•
•
•
This is a 5 5 Latin square design
Page 159 shows some other Latin squares
Table 4-12 (page 162) contains properties of Latin squares
Statistical analysis?
Chapter 4
23
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.
• See the ANOVA table, page 160 (Table 4.10)
• The analysis for the rocket propellant example
follows
Chapter 4
24
Chapter 4
25
Chapter 4
26
Graeco-Latin Squares
=
=

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