L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 1 MER301: Engineering...

L Berkley DavisCopyright 2009

MER301: Engineering ReliabilityLecture 13

1

MER301: Engineering Reliability

LECTURE 13 Chapter 6:6.3-6.4 Multiple Linear Regression Models

L Berkley DavisCopyright 2009

MER301: Engineering ReliabilityLecture 13

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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

L Berkley DavisCopyright 2009

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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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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MER301: Engineering ReliabilityLecture 13

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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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MER301: Engineering ReliabilityLecture 13

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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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MER301: Engineering ReliabilityLecture 13

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Forms of Multiple Regression Equations…

Interaction terms…

213 xx 22110 xxY

21 71050 xxY

215 xx

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MER301: Engineering ReliabilityLecture 13

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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)

L Berkley DavisCopyright 2009

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Summary of Topics

Multiple Regression Analysis Multiple Regression Equation Precision and Significance of a

Regression Model Confidence Limits