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MER301: Engineering Reliability
LECTURE 13 Chapter 6:6.3-6.4 Multiple Linear Regression Models
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Summary of Topics
Multiple Regression Analysis Multiple Regression Equation Precision and Significance of a
Regression Model Confidence Limits
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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 T-test on Coefficients Analysis of Variance
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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 T-test on Coefficients Analysis of Variance
MER301: Engineering ReliabilityLecture 12
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ERT SSSSSS
)(/ 000
set )(/ 110
set
xxxy SS /1̂ xy 10
ˆˆ
iii xxxyy 101ˆˆˆˆ
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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
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Multiple Linear Regression Models
Multiple Regression Models are used when the dependent variable Y is a function of more than one independent variable
Consistent with the physics, the model may include non-linear terms such as
Use as few terms as possible, consistent with the physics..
).....,( ,21 ixxxfnY
etcexxxxxxx jxijiji
kii ,,ln,,,2
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General Form of Regression Equation
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Forms of Multiple Regression Equations…
22110 xxY
21 71050 xxY
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Forms of Multiple Regression Equations…
Interaction terms…
213 xx 22110 xxY
21 71050 xxY
215 xx
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Forms of Multiple Regression Equations…
Non-linear terms…
225
214 xx
21322110 xxxxY
2121 4710800 xxxxY 22
21 55.8 xx
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General Form of Regression Equation
i
k
jijji xy
10
ˆˆ
The general form of the multiple regression equation for n data points and k independent variables is
ni ,........2,1
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Matrix Version of Multi-Linear Regression
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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
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Example 13.1(page 2) Pull Strength is to be
modeled as a function of Wire Length and Die Height
Minitab is used to analyze the data set to get values of the
Wire Bond dataObservation Pull Strength Wire Length Die Height
1 9.95 2 502 24.45 8 1103 31.75 11 1204 35 10 5505 25.02 8 2956 16.86 4 2007 14.38 2 3758 9.6 2 529 24.35 9 100
10 27.5 8 30011 17.08 4 41212 37 11 40013 41.95 12 50014 11.66 2 36015 21.65 4 20516 17.89 4 40017 69 20 60018 10.3 1 58519 34.93 10 54020 46.59 15 25021 44.88 15 29022 54.12 16 51023 56.63 17 59024 22.13 6 10025 21.15 5 400
22110 xxY
1x2x
Y
s'
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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 StatisticsMultiple R 0.990512593R Square 0.981115197Adjusted R Square 0.979398397Standard Error 2.289367725Observations 25
ANOVAdf SS MS F Significance F
Regression 2 5990.476035 2995.238 571.478936 1.08952E-19Residual 22 115.3065007 5.241205Total 24 6105.782536
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 2.261049258 1.060678216 2.131701 0.04444576 0.061337283 4.460761 0.06133728 4.46076123Wire Length 2.744011123 0.093577836 29.3233 3.9636E-19 2.54994257 2.93808 2.54994257 2.93807968Die Height 0.012538881 0.002800034 4.478117 0.00018764 0.006731965 0.018346 0.00673196 0.0183458
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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
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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 StatisticsMultiple R 0.990512593R Square 0.981115197Adjusted R Square 0.979398397Standard Error 2.289367725Observations 25
ANOVAdf SS MS F Significance F
Regression 2 5990.476035 2995.238 571.478936 1.08952E-19Residual 22 115.3065007 5.241205Total 24 6105.782536
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 2.261049258 1.060678216 2.131701 0.04444576 0.061337283 4.460761 0.06133728 4.46076123Wire Length 2.744011123 0.093577836 29.3233 3.9636E-19 2.54994257 2.93808 2.54994257 2.93807968Die Height 0.012538881 0.002800034 4.478117 0.00018764 0.006731965 0.018346 0.00673196 0.0183458
(6-46)
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Confidence Interval on Mean Response
0 10 20
0
10
20
30
40
50
60
70
Wire Length
Pul
l Str
eng
t
Y = 5.11452 + 2.90270X
R-Sq = 96.4 %
Regression
95% CI
Regression Plot
(6-52)
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Precision and Significance of the Regression…
And now the Significance…. Hypothesis Testing ANOVA
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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
Regression StatisticsMultiple R 0.990512593R Square 0.981115197Adjusted R Square 0.979398397Standard Error 2.289367725Observations 25
ANOVAdf SS MS F Significance F
Regression 2 5990.476035 2995.238 571.478936 1.08952E-19Residual 22 115.3065007 5.241205Total 24 6105.782536
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 2.261049258 1.060678216 2.131701 0.04444576 0.061337283 4.460761 0.06133728 4.46076123Wire Length 2.744011123 0.093577836 29.3233 3.9636E-19 2.54994257 2.93808 2.54994257 2.93807968Die Height 0.012538881 0.002800034 4.478117 0.00018764 0.006731965 0.018346 0.00673196 0.0183458
(6-48)
(6-49)
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Analysis of Variance(ANOVA)
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SUMMARY OUTPUT
Regression StatisticsMultiple R 0.990512593R Square 0.981115197Adjusted R Square 0.979398397Standard Error 2.289367725Observations 25
ANOVAdf SS MS F Significance F
Regression 2 5990.476035 2995.238 571.478936 1.08952E-19Residual 22 115.3065007 5.241205Total 24 6105.782536
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 2.261049258 1.060678216 2.131701 0.04444576 0.061337283 4.460761 0.06133728 4.46076123Wire Length 2.744011123 0.093577836 29.3233 3.9636E-19 2.54994257 2.93808 2.54994257 2.93807968Die Height 0.012538881 0.002800034 4.478117 0.00018764 0.006731965 0.018346 0.00673196 0.0183458
(6-47)
(6-45)
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Summary of Topics
Multiple Regression Analysis Multiple Regression Equation Precision and Significance of a
Regression Model Confidence Limits