MER301: Engineering
Reliability
LECTURE 13
Chapter 6:6.3-6.4
Multiple Linear Regression Models
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
1
Summary of Topics
 Multiple Regression Analysis
 Multiple Regression Equation
 Precision and Significance of a
Regression Model
 Confidence Limits
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
2
Summary of Topics
 Linear Regression Analysis
 Simple Regression Model
 Least Squares Estimate of the Coefficients
 Standard Error of the Coefficients
 Precision and Significance of a Regression Model
 Precision



Standard Error of the Coefficients
R2 - Correlation Coefficient
Confidence Limits
 Significance


L Berkley Davis
Copyright 2009
T-test on Coefficients
Analysis of Variance
MER301: Engineering Reliability
Lecture 12
3
SST  SS R  SS E
 Linear Regression Analysis
yˆi  y  ˆ1  xi  x   ˆ0  ˆ1  xi
 Least Squares Estimate of the Coefficientsˆ1  S xy / S xx
 Standard Error of the Coefficients
ˆ0  y  ˆ1  x
 Precision and Significance of a Regression Model
 Precision
 Simple Regression Model



Standard Error of the Coefficients
R2 - Correlation Coefficient
Confidence Limits
 Significance


L Berkley Davis
Copyright 2009
T-test on Coefficients
Analysis of Variance
MER301: Engineering Reliability
Lecture 12




t0  0 / se(0 ) t0  1 / se(1 )
4
Regression Analysis
 For those cases where there is not a
Mechanistic Model of an engineering
process, data are used to generate an
Empirical Model. A powerful technique for
creating such a model doing is called
Regression Analysis
 In Simple Linear Regression, the Dependent Variable
Y is a function of one Independent Variable X
 Multiple Linear Regression is used when Y is a
function of more than one X
 The form of regression models is based on
the underlying physics as much as possible
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
5
Multiple Linear Regression Models
 Multiple Regression Models are used when
the dependent variable Y is a function of
more than one independent variable
Y  fn( x1 , x 2, .....xi )
 Consistent with the physics, the model may
include non-linear terms such as
xi2 , xik , xi  x j , xi  ln x j , xi  e  x j , etc
 Use as few terms as possible, consistent
with the physics..
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
6
General Form of Regression
Equation
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
7
Forms of Multiple Regression Equations…
Y   0   1  x1   2  x2
Y  50  10  x1  7  x2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
8
Forms of Multiple Regression Equations…
 Interaction terms…
Y   0  1  x1   2  x2
 3  x1  x2
Y  50  10  x1  7  x2
 5  x1  x2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
9
Forms of Multiple Regression Equations…
 Non-linear terms…
Y   0  1  x1   2  x2   3  x1  x2
  4  x12   5  x22
Y  800  10  x1  7  x2  4  x1  x2
 8.5  x12  5  x22
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
10
General Form of Regression Equation
 The general form of the multiple regression equation
for n data points and k independent variables is
k
yi  ˆ 0   ˆ j  xij   i
i  1,2,........n
j 1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
11
Matrix Version of Multi-Linear
Regression
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
12
Example 13.1
 The pull strength of a wire bond in
a semiconductor product is an
important characteristic.
 We want to investigate the
suitability of using a multiple
regression model to predict pull
strength (Y) as a function of wire
length (x1) and die height (x2).
 Excel file Example13.1.xls
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
13
Example 13.1(page 2)
Pull Strength Y is to be
modeled as a function of
Wire Length
1and Die
Height
x2
x
Y   0   1  x1   2  x2
Minitab is used to analyze
the data set to get values
of the  ' s
L Berkley Davis
Copyright 2009
Observation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
MER301: Engineering Reliability
Lecture 13
Wire Bond data
Pull Strength
Wire Length
9.95
2
24.45
8
31.75
11
35
10
25.02
8
16.86
4
14.38
2
9.6
2
24.35
9
27.5
8
17.08
4
37
11
41.95
12
11.66
2
21.65
4
17.89
4
69
20
10.3
1
34.93
10
46.59
15
44.88
15
54.12
16
56.63
17
22.13
6
21.15
5
Die Height
50
110
120
550
295
200
375
52
100
300
412
400
500
360
205
400
600
585
540
250
290
510
590
100
400
14
Example 13.1(page 3)
Regression Analysis
The regression equation is
Pull Strength = 2.26 + 2.74 Wire Length + 0.0125 Die Height
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.990512593
0.981115197
0.979398397
2.289367725
25
ANOVA
df
Regression
Residual
Total
Intercept
Wire Length
Die Height
L Berkley Davis
Copyright 2009
2
22
24
SS
MS
5990.476035 2995.238
115.3065007 5.241205
6105.782536
Coefficients
2.261049258
2.744011123
0.012538881
Standard Error
t Stat
1.060678216 2.131701
0.093577836 29.3233
0.002800034 4.478117
F
Significance F
571.478936
1.08952E-19
P-value
0.04444576
3.9636E-19
0.00018764
MER301: Engineering Reliability
Lecture 13
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.061337283 4.460761 0.06133728 4.46076123
2.54994257 2.93808 2.54994257 2.93807968
0.006731965 0.018346 0.00673196
0.0183458
15
Precision and Significance of
the Regression…
 Dealing with the
Precision first….
 Standard Error of
the Coefficients
 Coefficient of
Determination
 Confidence
Interval on the
Mean Response
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
16
Example 13.1(page 4)
Regression Analysis
The regression equation is
Pull Strength = 2.26 + 2.74 Wire Length + 0.0125 Die Height
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.990512593
0.981115197
0.979398397
2.289367725
25
ANOVA
df
Regression
Residual
Total
Intercept
Wire Length
Die Height
L Berkley Davis
Copyright 2009
2
22
24
SS
MS
5990.476035 2995.238
115.3065007 5.241205
6105.782536
Coefficients
2.261049258
2.744011123
0.012538881
Standard Error
t Stat
1.060678216 2.131701
0.093577836 29.3233
0.002800034 4.478117
F
Significance F
571.478936
1.08952E-19
P-value
0.04444576
3.9636E-19
0.00018764
MER301: Engineering Reliability
Lecture 13
(6-46)
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.061337283 4.460761 0.06133728 4.46076123
2.54994257 2.93808 2.54994257 2.93807968
0.006731965 0.018346 0.00673196
0.0183458
17
Confidence Interval on Mean
Response
(6-52)
Regression Plot
Y = 5.11452 + 2.90270X
R-Sq = 96.4 %
70
60
Pull Strengt
50
40
30
20
Regression
10
95% CI
0
0
10
20
Wire Length
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
18
Precision and Significance of
the Regression…
 And now the
Significance….
 Hypothesis Testing
 ANOVA
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
19
Example 13.1(page 5)
Regression Analysis
The regression equation is
Pull Strength = 2.26 + 2.74 Wire Length + 0.0125 Die Height
SUMMARY OUTPUT
(6-48)
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.990512593
0.981115197
0.979398397
2.289367725
25
ANOVA
df
Regression
Residual
Total
Intercept
Wire Length
Die Height
L Berkley Davis
Copyright 2009
2
22
24
SS
MS
5990.476035 2995.238
115.3065007 5.241205
6105.782536
Coefficients
2.261049258
2.744011123
0.012538881
Standard Error
t Stat
1.060678216 2.131701
0.093577836 29.3233
0.002800034 4.478117
F
Significance F
571.478936
1.08952E-19
P-value
0.04444576
3.9636E-19
0.00018764
MER301: Engineering Reliability
Lecture 13
(6-49)
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.061337283 4.460761 0.06133728 4.46076123
2.54994257 2.93808 2.54994257 2.93807968
0.006731965 0.018346 0.00673196
0.0183458
20
Analysis of Variance(ANOVA)
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.990512593
R Square
0.981115197
Adjusted R Square
0.979398397
Standard Error
2.289367725
Observations
25
(6-47)
ANOVA
df
Regression
Residual
Total
Intercept
Wire Length
Die Height
L Berkley Davis
Copyright 2009
2
22
24
SS
MS
5990.476035 2995.238
115.3065007 5.241205
6105.782536
Coefficients
2.261049258
2.744011123
0.012538881
Standard Error
t Stat
1.060678216 2.131701
0.093577836 29.3233
0.002800034 4.478117
F
Significance F
571.478936
1.08952E-19
P-value
0.04444576
3.9636E-19
0.00018764
MER301: Engineering Reliability
Lecture 13
Lower 95% Upper 95% Lower 95.0% Upper 95.0%
0.061337283 4.460761 0.06133728 4.46076123
2.54994257 2.93808 2.54994257 2.93807968
(6-45)
0.006731965 0.018346 0.00673196
0.0183458
21
Summary of Topics
 Multiple Regression Analysis
 Multiple Regression Equation
 Precision and Significance of a
Regression Model
 Confidence Limits
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 13
22