MER301: Engineering
Reliability
LECTURE 14:
Chapter 7:
Design of Engineering Experiments
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
1
Summary of Topics
 Design of Engineering Experiments
 DOE and Engineering Design
 Coded Variables
 Optimization
 Factorial Experiments
 Main Effects
 Interactions
 Statistical Analysis
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
2
Design of Experiments and
Engineering Design
 Applications of Designed Experiments
 Evaluation and comparison of design
configurations
 Establish Production Process Parameters
 Evaluation of mechanical properties of
materials/comparison of different materials
 Selection of ranges of values of independent
variables in a design (Robust Design)
 Determination of Vital x’s (Significant Few
versus the Trivial Many…..)
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
3
Design of Engineering Experiments



Objectives of engineering
experiments include
acquiring data that can be
used to generate an
analytical model for Y in
terms of the dependent
variables Xi
The model may be linear or
non-linear in the Xi’s and it
defines a Response Surface
of Y as a function of the Xi’s
The model can be used to
generate statistical
parameters(means, std dev)
for use in product design, as
in DFSS
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
4
Factorial Experiments
 Factorial Experiments are used to establish Main
Effects and Interactions
Y  fn( x1 , x2 ,...xk , x1  x2 ,....)
 Levels of each factor are chosen to bound the
expected range of each Xi
xi , LB  xi  xi ,UB
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
5
DOE Glossary



Model- quantitative relationship Y  fn( xi ) ; also called
Transfer Function
DOE- systematic variation of Xi’s to acquire data to
generate Transfer Function Y  fn( x )
i
Factorial Experiment-all possible combinations of Xi’s are
tested



Response Surface- surface of Y generated by the Transfer
Function


Main Effects – change in Y due to change in Xi .
Interactions – joint effects of two or more Xi’s
Replicates and Center Points
Optimum Response- local max/min of Y
Partial Factorial Experiments- can run fewer points if can
neglect higher order interactions
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
6
Coded Variables

Coded Variables..


Each x typically has some (dimensional) range in which it is
expected to vary…. xi , LB  xi  xi ,UB
In Designed Experiments the lower value of each x is often
assigned a value of –1 and the upper value of x a value of +1
 1  xi  1

Coded Variables have two advantages



First, discrete variables( eg, “yes/no”, Operator A/ Operator B)
can be included in the experiment
Second, the magnitude of the regression coefficients is a
direct measure of the importance of each x variable
Coded Variables have the disadvantage that an equation
that can be directly used for engineering design is not
specifically produced
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MER301: Engineering Reliability
Lecture 14
7
DOE Process Map
L Berkley Davis
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8
Example 14.1Text Example 7-1
 Two level factorial design applied to a
process for integrated circuit
manufacturing
 Y= epitaxial growth layer thickness
 A= deposition time(
),levels are
1
long(+1) or short(-1)
 B= arsenic flow rate( x 2 ), levels are
59%(+1) or 55%(-1)
 Experiment run with 4 replicates at each
combination of A and B
x
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
9
Example 14.1(Text Example 7-1)
 Two level factorial design applied to a
process for integrated circuit
manufacturing
Epitaxy is
a kind oflayer
interface
between a thin film and a substrate. The term epitaxy
 Y= epitaxial
growth
thickness
(Greek; epi "above" and taxis "in ordered manner") describes an ordered crystalline
 A= deposition
time(
),levels
are
growth on a monocrystalline
substrate
1
long(+1) or short(-1)

Epitaxial films may be grown from gaseous or liquid precursors. Because the substrate acts as a seed crystal, the
x 2 structure
 B= arsenic
flow
rate(
), levels
are identical to those of the substrate. This is different from
deposited
film takes
on a lattice
and orientation
other thin-film deposition methods which deposit polycrystalline or amorphous films, even on single-crystal
59%(+1)substrates.
or 55%(-1)
If a film is deposited on a substrate of the same composition, the process is called homoepitaxy;
otherwise it is called heteroepitaxy.

Homoepitaxy
is a kind
epitaxy performed with
one material. In homoepitaxy, a crystalline film is grown on a
 Experiment
run with
4ofreplicates
atonly
each
substrate or film of the same material. This technology is applied to growing a more purified film than the substrate
and fabricating layers with different doping levels.
combination
of A and B

Heteroepitaxy is a kind of epitaxy performed with materials that are different from each other. In heteroepitaxy, a
x
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




L Berkley Davis
Copyright 2009
crystalline film grows on a crystalline substrate or film of another material. This technology is often applied to
growing crystalline films of materials of which single crystals cannot be obtained and to fabricating integrated
crystalline layers of different materials. Examples include gallium nitride (GaN) on sapphire or aluminium gallium
MER301: Engineering Reliability
8
indium phosphide
(AlGaInP) on gallium arsenide (GaAs).
Lecture 14
Heterotopotaxy is a process similar to heteroepitaxy except for the fact that thin film growth is not limited to two
dimensional growth. Here the substrate is similar only in structure to the thin film material.
Epitaxy is used in silicon-based manufacturing processes for BJTs and modern CMOS, but it is particularly important
for compound semiconductors such as gallium arsenide. Manufacturing issues include control of the amount and
uniformity of the deposition's resistivity and thickness, the cleanliness and purity of the surface and the chamber
atmosphere, the prevention of the typically much more highly doped substrate wafer's diffusion of dopant to the new
layers, imperfections of the growth process, and protecting the surfaces during the manufacture and handling
Doping
An epitaxial layer can be doped during deposition by adding impurities to the source gas, such as arsine, phosphine
or diborane. The concentration of impurity in the gas phase determines its concentration in the deposited film. As in
CVD, impurities change the deposition rate.
Additionally, the high temperatures at which CVD is performed may allow dopants to diffuse into the growing layer
from other layers in the wafer ("autodoping"). Conversely, dopants in the source gas may diffuse into the substrate.
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Lecture 14
10
Example 14.1( con’t)

What are the questions we
need to answer?
 What is the quantitative
effect of changes in A on
the value of Y, ie the
response of Y?
 What is the response of
Y to changes in B?
 What is the interaction
effect on Y when both A
and B are changing, if
any?
 Are there values of A and
B such that Y is at an
optimum level?
L Berkley Davis
Copyright 2009
Thickness
14.037
14.165
13.972
13.907
14.821
14.757
14.843
14.878
13.88
13.86
14.032
13.914
14.888
14.921
14.415
14.932
MER301: Engineering Reliability
Lecture 14
time
30
30
30
30
60
60
60
60
30
30
30
30
60
60
60
60
Arsenic
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
11
Example 14.1(con’t)
Two level factorial design applied to
integrated circuit manufacturing

Y= epitaxial growth layer
thickness

A= deposition time( X1),levels are
long(+1) or short(-1)

B= arsenic flow rate( X2), levels
are 59%(+1) or 55%(-1)
Experiment run with 4 replicates at each
combination of A and B
How do we answer these
Questions? What is the
form of Experimental
Design ?




L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
What is the quantitative
effect of changes in A on the
value of Y?
What is the effect of
changes in B on Y?
What is the interaction
effect on Y when both A and
B are changing, if any?
Are there values of A and B
such that Y is at an optimum
level?
12
2

k Factorial Design
2 k factorial design is
used when each factor
has two levels



Establish both main
effects/interactions
Assumes linearity of
response
Smallest number of
runs to test all
combinations of x’s
Factors A B
2k
 Factor(Xi) levels often
described as “+ or –”

L Berkley Davis
Copyright 2009
Called geometric or
coded notation
MER301: Engineering Reliability
Lecture 14
7-8
13
Response of Y to A and B:
Interaction/No Interaction
 In Fig 7-1, response of
Y to change in A is
independent of B level;
there is No Interaction
between A and B
 In Fig 7-2, the response
of Y to change in A is
shown with a different
slope to illustrate an
interaction between A
and B
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
14
Interaction/ No Interaction
Interactions change the
shape of the response
surface significantly
Experimental design must
identify interactions
and allow their impact
to be quantified
7-3
7-4
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MER301: Engineering Reliability
Lecture 14
15
Main Effects
ab
b
Main Effects
B
(1)
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a
A
MER301: Engineering Reliability
Lecture 14
 Main Effect term captures the change in
the response variable due to change in
level of a specific factor
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
16
ab
b
AB Interactions
B
(1)
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a
A
MER301: Engineering Reliability
Lecture 14
 Interaction terms show the effects of
changes in one variable at different
levels of the other variables
 For Eq 7-3, this would give the effects of A at
different levels of B(or vice versa)
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
17
Example 14.1: Data Set


Two level factorial design applied to integrated circuit
manufacturing
 Y= epitaxial growth layer thickness
 A= deposition time(x1),levels are long(+1) or short(-1)
 B= arsenic flow rate(x2), levels are 59%(+1) or 55%(-1)
Experiment run with 4 replicates at each combination of A
and B
in Effects
ngineering
ab
b
B
(1)
a
A
MER301: Engineering Reliability
Lecture 14
17
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
18
Example 14.1: Effect Values
Main Effects
b
ab
B
(1)
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L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
a
A
MER301: Engineering Reliability
Lecture 14
19
Coded Least Squares Model
 The Regression Equation is of the Form
Yˆ   0  1  x1   2  x2   3  x1  x2
 The Coded Least Squares Model is of the Form
Yˆ  Y0  ( A / 2)  x1  ( B / 2)  x2  ( AB / 2)  x1  x2
X1 and X2 are coded variables and range from –1 to +1


 0  Y0  Y is the average of all observations and the
coefficients are 1  A / 2,  2  B / 2, and 3  AB / 2
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
20
ˆ0  Y  ˆ1  x  Y
Relationship between Regression
Coefficients and the DOE Effects
Main Effects
 The Regression Equation is
b
 The Coded Least Squares Model is
B
x 0
Yˆ   0  1  x1   2  x2   3  x1  x2
Yˆ  Y0  ( A / 2)  x1  ( B / 2)  x2  ( AB / 2)  x1  x2
2k
S xy
ˆ
1 

S xx
n
 ( y
i 1 j 1
2
k
ij
 y )  ( xij  x )
n
 ( x
i 1 j 1
2k
ij

 x)
2
n
 ( y
i 1 j 1
2
k
2k
 y )  xij
n
 x
i 1 j 1
2
ij

xij  1
a
(1)
A
n
 y
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ij
ab
i 1 j 1
k
ij
MER301: Engineering Reliability
Lecture 14
 xij
2 n
y1,ab  ...  yn,a  y1,b  ...  yn,(1) A
1
ˆ
so that 1  

k 1
2
2 n
2
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MER301: Engineering Reliability
Lecture 14
21



V (0 )  V (1 )   2 / 4  n..because...x  0
Relationship between Regression
Coefficients and the DOE Effects
 The Regression Equation is of the Form
Yˆ   0  1  x1   2  x2   3  x1  x2
 The Coded Least Squares Model is of the Form
x 0
xij  1
Yˆ  Y0  ( A / 2)  x1  ( B / 2)  x2  ( AB / 2)  x1  x2
For k=2
1 y1,ab  ...  yn ,a  y1,b  ...  yn ,(1) A
ˆ
1  

2
2n
2
2
ˆ
1

V ( A)
2
2
2
2
2
ˆ
V ( 1 )  (
)  (n   ab  n   a  n   b  n   (1) ) 

4n
4n
4
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
22
Example 14.1(con’t)
Regression Analysis
Thickness
14.037
14.165
13.972
13.907
14.821
14.757
14.843
14.878
13.88
13.86
14.032
13.914
14.888
14.921
14.415
14.932
time=A Arsenic=B interaction=AB
30
0.55
16.5
30
0.55
16.5
30
0.55
16.5
30
0.55
16.5
60
0.55
33
60
0.55
33
60
0.55
33
60
0.55
33
30
0.59
17.7
30
0.59
17.7
30
0.59
17.7
30
0.59
17.7
60
0.59
35.4
60
0.59
35.4
60
0.59
35.4
60
0.59
35.4
L Berkley Davis
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coded time
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
MER301: Engineering Reliability
Lecture 14
coded arsenic
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
interaction
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
23
Example 14.1 Excel Regression Analysis
Regression Statistics
Multiple R
0.958467416
R Square
0.918659787
Adjusted R Square 0.898324733
Standard Error
0.144187523
Observations
16
ANOVA
Regression
Residual
Total
df
3
12
15
SS
2.81764325
0.2494805
3.06712375
MS
F
Significance F
0.939214 45.17617 8.19196E-07
0.02079
Intercept
time
% arsenic
interaction
Coefficients
14.388875
0.418
-0.033625
0.01575
Standard Error
0.036046881
0.036046881
0.036046881
0.036046881
t Stat
P-value
Lower 95%
399.1712 4.11E-26 14.3103356
11.59601 7.08E-08 0.339460595
-0.93281 0.369306 -0.112164405
0.436931 0.66992 -0.062789405
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MER301: Engineering Reliability
Lecture 14
24
Example 14.1:Regression Equation
Yˆ  Y0  ( A / 2)  x1  ( B / 2)  x 2  ( AB / 2)  x1  x 2
Yˆ  Y0  (0.836 / 2)  x1  (0.067 / 2)  x 2  (0.032 / 2)  x1  x 2
Yˆ  14.3889  0.418  x  0.0336  x  0.01575  x  x
1
2
1
2
 Effects are calculated as A=0.836, B=-0.067, and
AB=0.032


Large effect of deposition rate A
Small effect of arsenic level B and interaction AB
ˆ
2 , pooled data set)=0.02079
 Sample Variance(
 Sample Mean( Y0  Y   0 ) = 14.3889
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
25
Example 14.1(con’t)
Two level factorial design applied to
integrated circuit manufacturing

Y= epitaxial growth layer
thickness

A= deposition time( X1),levels are
long(+1) or short(-1)

B= arsenic flow rate( X2), levels
are 59%(+1) or 55%(-1)
Experiment run with 4 replicates at each
combination of A and B
How do we answer these
Questions? What is the
form of Experimental
Design ?




L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
What is the quantitative
effect of changes in A on the
value of Y?
What is the effect of
changes in B on Y?
What is the interaction
effect on Y when both A and
B are changing, if any?
Are there values of A and B
such that Y is at an optimum
level?
26
One Factor at a Time
Optimization
7-5
7-6
 The One Factor at a Time method of
conducting experiments is intuitively
appealing to many engineers
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
27
One Factor at a Time
Optimization
 One Factor at a Time
will frequently fail to
identify effects of
interactions


Learning to use DOE
factorial experiments
often difficult for new
engineers to accept
DOE’s however are
the most efficient and
reliable method of
experimentation
7-7
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MER301: Engineering Reliability
Lecture 14
28
Optimization
 Screening tests
establish factors (vital
i ’s) that affect Y
x
 Range of x i’s is a
critical choice in the
experimental design
Xi
 Optimization will
require multiple
experiments
7-7a
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MER301: Engineering Reliability
Lecture 14
29
Example 14.1:Regression Equation
Optimization
Yˆ  Y0  ( A / 2)  x1  ( B / 2)  x 2  ( AB / 2)  x1  x 2
Yˆ  Y0  (0.836 / 2)  x1  (0.067 / 2)  x 2  (0.032 / 2)  x1  x 2
Yˆ  14.3889  0.418  x  0.0336  x  0.01575  x  x
1
2
1
2
 Take partial derivatives wrt X1 and X2
 Set equal to zero and solve for X1 and X2
 X1~2 and X2~-26
No Optimum of Y in Range of X’s
-1<=X,=1
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Lecture 14
30
Example 14-1:Statistical Analysis of the
Regression Model
There are three ways of conducting a statistical analysis
of the regression model- and all will lead to the same
conclusion




Standard Error of the Effects calculated from Sample Data
Sum of Squares based on Mean and Interaction Effects
ANOVA/Significance Analysis of the Regression Equation
All of these methods are based on analysis of the single
data set generated in the DOE.
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
31
Example 14-1:Statistical Analysis of the
Regression Model
There are three ways of conducting a statistical analysis
of the regression model- and all will lead to the same
conclusion
 Standard Error of the Effects calculated from Sample
Data



Sum of Squares based on Mean and Interaction Effects
ANOVA/Significance Analysis of the Regression Equation
All of these methods are based on analysis of the single
data set generated in the DOE.
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
32
SS E 
2k
n
i 1
j 1
  ( yij  yˆ i. ) 2

yi.  yi.  either...(1), a, b, ab
Standard Error of the Effects
calculated from Sample Data




2 
The magnitude/importance of each effect can be judged
by comparing each effect to its Estimated Standard Error.
The first step in the analysis is to calculate the means
and variances at each of the i factorial run conditions
using data from the n replicates. For the variances
The second step is to calculate an overall(pooled)
variance estimate for the k factorial run conditions
2
SSE
2k  (n  1)
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MER301: Engineering Reliability
Lecture 14
33
Calculation of Variance

The variances for the i factorial runs are
2
 (21)  0.0121,.. a2  0.0026,.. b2  0.0059,..and .. ab
 0.0625

The overall(pooled) variance estimate for the 2k=4 factorial run
conditions is
2k
 i2
i 1
2k
ˆ  
2
L Berkley Davis
Copyright 2009

1
 (0.0121  0.0026  0.0059  0.0625)  0.0208
4
MER301: Engineering Reliability
Lecture 14
34
Standard Error of the Effects
calculated from Sample Data (con’t)…..
 Given the overall variance, the effect variance is
calculated as follows

The Effect Estimate is a difference between two
means, each of which is calculated from half of the N
measurements. Thus the Effect Variance is
ˆ 2
ˆ 2
2  ˆ 2
ˆ 2
V (effect)  V ( A)  V ( B)  V ( AB) 



N / 2 N / 2 N / 2 N / 22
where
N  n  2k
 The Standard Error of each Effect is then
se(effect)  se( A)  se( B)  se( AB)  ˆ 2 /( n  2 k  2 )
L Berkley Davis
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MER301: Engineering Reliability
Lecture 14
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Standard Error of the Effects
calculated from Sample Data (con’t)…..
 Given the overall variance, the effect variance is
calculated as follows

The Effect Estimate is a difference between two
means, each of which is calculated from half of the N
measurements. Thus the Effect Variance is
ˆ
V (effect)  V ( A)  V ( B )  V ( AB ) 
where

ˆ
 0  Y  ˆ1  x  Y
ˆ
2  ˆ
ˆ between Regression
 Relationship


N / 2 N Coefficients
/ 2 N / 2 N / and
2 2 the DOE Effects
2
2
2
2
N  n  2k
 The Regression Equation is of the Form
The Standard Error of each Effect is then
Yˆ   0   1  x1   2  x 2   3  x1  x 2
 The Coded
Least
2
k  2 Squares Model is of the Form
se(effect)  se( A)  se( B )  se( AB )  Yˆ ˆ Y /(n(A2/ 2) )x  ( B / 2)  x  ( AB / 2)  x  x
0
1
2
1
2
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 14
1
se( ˆ1 )   se( A), etc
2
For k=2
xij  1
35
1 y1,ab  ...  yn ,a  y1,b  ...  yn ,(1) A

2
2n
2
1 2
ˆ 2 V ( A)
2
2
2
2
ˆ
V ( 1 )  (
)  ( n   ab  n   a  n   b  n   (1) ) 

4n
4n
4
ˆ1  
Union College
Mechanical Engineering
L Berkley Davis
Copyright 2009
x 0
MER301: Engineering Reliability
Lecture 14
37
Standard Error of the Effects
calculated from Sample Data (con’t)…..

Because N  n  2 k is the same for all of the effects and
is used to calculate the Standard Error of any specific
Effect( ie, A, B, AB,…) the value calculated will be the
same for each one…
se( EffectA)  se( EffectB)  se( EffectAB)  ˆ 2 /( n  2 k 2 )

For the Epitaxial Process
se( Effect)  ˆ 2 /( n  2 k 2 )  0.0208 /( 4  2 22 )  0.072

The value of the Effect is twice that of the coefficient in
the Regression Equation. Similarly, the Standard Error for
the Coefficient is half that of the Effect
se(Coefficient )  se( Effect) / 2  0.036
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
37
Standard Error of the Effects
calculated from Sample Data (con’t)…..

A Hypothesis Test is carried out on each of the Main
Effects and the Interaction Effects. This is a t-test.
7-5

The A Effect is significant and the B and AB Effects are
not
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
38
Example 14.1 Excel Regression Analysis
Regression Statistics
Multiple R
0.958467416
R Square
0.918659787
Adjusted R Square 0.898324733
Standard Error
0.144187523
Observations
16
ANOVA
Regression
Residual
Total
df
3
12
15
SS
2.81764325
0.2494805
3.06712375
MS
F
Significance F
0.939214 45.17617 8.19196E-07
0.02079
Intercept
time
% arsenic
interaction
Coefficients
14.388875
0.418
-0.033625
0.01575
Standard Error
0.036046881
0.036046881
0.036046881
0.036046881
t Stat
P-value
Lower 95%
399.1712 4.11E-26 14.3103356
11.59601 7.08E-08 0.339460595
-0.93281 0.369306 -0.112164405
0.436931 0.66992 -0.062789405
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
39
Standard Error of the Effects calculated
from Sample Data (con’t)…..
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
40
Example 14-1:Statistical Analysis of the
Regression Model

There are three ways of conducting a statistical analysis
of the regression model- and all will lead to the same
conclusion

Standard Error of the Effects calculated from Sample Data

Sum of Squares based on Mean and Interaction Effects

ANOVA/Significance Analysis of the Regression Equation
All of these methods are based on analysis of the single
data set generated in the DOE.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
41
Sum of Squares:Main Factors
and Interaction
 n  A2
 n  B2
 n  AB 2



The Sum of Squares for the Effects can be expressed as
SS A  n  A2 ,.SS B  n  B 2 ,.and.SS AB  n  AB 2
Sum of Squares from the Main and Interaction Effects can be
used to assess the relative importance of each term
2k
n
2
2
The Total Sum of Squares is obtained from SS 
y

4

n

y
T
ij
and the Mean Square Error from
j 1 i 1

SS E  SST  (SS A  SS B  SS AB )
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
42
2k
ˆ  
2
 i2
2
k
SS E 
 0.0208
ˆ 2
n  (k  1)

Sum of Squares:
Two Calculation Methods
i 1

0.0208
 0.2495
12
The Total Sum of Squares,Effect Sum of Squares,and Mean
Square Error are obtained from
2k
SST  
j 1
2
n
y
i 1
2
ij
2k
 4n y  
2
j 1
n
y
i 1
2
ij
 4  (14.3882) 2  3.0672
SS A  4  0.0836  2.7956, SS B  4  (0.067) 2  0.0181, SS AB  4  0.032 2  0.0040
SS E  SS T  ( SS A  SS B  SS AB )
SS E  3.0672  (2.7956  0.0181  0.0040)  0.2495
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
43
Example 14-1:Statistical Analysis of the
Regression Model

There are three ways of conducting a statistical analysis
of the regression model- and all will lead to the same
conclusion


Standard Error of the Effects calculated from Sample Data
Sum of Squares based on Mean and Interaction Effects

ANOVA/Significance Analysis of the Regression
Equation
All of these methods are based on analysis of the single
data set generated in the DOE.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
44
Example 14.1 Regression Analysis
and ANOVA (con’t)
7-6
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
45
A is Significant
B and AB are not Significant
ANOVA: for each term

The Significance of each term (A,B,AB) can be obtained from
ANOVA
SS A  4  0.0836 2  2.7956, SS B  4  (0.067) 2  0.0181, SS AB  4  0.032 2  0.0040
ˆ 2
0.0208
SS E 

 0.2495
n  (k  1)
12
2k
ˆ  
2
i 1
 i2
2
k
 0.0208
f A  (SS A / 1) ˆ 2 .... f B  (SS B / 1) ˆ 2 ... f AB  (SS AB / 1) ˆ 2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 14
46
Example 14.1 Regression Analysis
and ANOVA(con’t)
7-6
Example 14.1 Excel Regression Analysis
Regression Statistics
Multiple R
0.958467416
R Square
0.918659787
Adjusted R Square 0.898324733
Standard Error
0.144187523
Observations
16
ANOVA
L Berkley Davis
Copyright 2009
Regression
Residual
Total
df
3
12
15
SS
2.81764325
0.2494805
3.06712375
MS
F
Significance F
0.939214 45.17617 8.19196E-07
0.02079
Intercept
time
% arsenic
interaction
Coefficients
14.388875
0.418
-0.033625
0.01575
Standard Error
0.036046881
0.036046881
0.036046881
0.036046881
t Stat
P-value
Lower 95%
399.1712 4.11E-26 14.3103356
11.59601 7.08E-08 0.339460595
-0.93281 0.369306 -0.112164405
0.436931 0.66992 -0.062789405
Union College
Mechanical Engineering
MER301: Engineering Reliability
MER301: Engineering Reliability
Lecture 14
Lecture 14
47
43
Summary of Topics
 Design of Engineering Experiments
 DOE and Engineering Design
 Coded Variables
 Factorial Experiments




L Berkley Davis
Copyright 2009
Main Effects
Interactions
Optimization
Statistical Analysis
MER301: Engineering Reliability
Lecture 14
48