1. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 1 MER301: Engineering Reliability LECTURE 13 Chapter 6:6.3-6.4 Multiple Linear Regression Models

2. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 2 Summary of Topics  Multiple Regression Analysis Multiple Regression Equation Precision and Significance of a Regression Model Confidence Limits

3. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 12 3 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 R 2 - Correlation Coefficient Confidence Limits  Significance T-test on Coefficients Analysis of Variance

4. L Berkley Davis Copyright 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 R 2 - Correlation Coefficient Confidence Limits  Significance T-test on Coefficients Analysis of Variance MER301: Engineering Reliability Lecture 12 4

5. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 5 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

6. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 6 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..

7. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 7 General Form of Regression Equation

8. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 8 Forms of Multiple Regression Equations…

9. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 9 Forms of Multiple Regression Equations…  Interaction terms…

10. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 10 Forms of Multiple Regression Equations…  Non-linear terms…

11. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 11 General Form of Regression Equation  The general form of the multiple regression equation for n data points and k independent variables is

12. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 12 Matrix Version of Multi-Linear Regression

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

14. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 14 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

15. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 15 Example 13.1(page 3) Regression Analysis The regression equation is Pull Strength = 2.26 + 2.74 Wire Length + 0.0125 Die Height

16. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 16 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

17. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 17 Example 13.1(page 4) Regression Analysis The regression equation is Pull Strength = 2.26 + 2.74 Wire Length + 0.0125 Die Height (6-46)

18. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 18 Confidence Interval on Mean Response (6-52)

19. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 19 Precision and Significance of the Regression…  And now the Significance…. Hypothesis Testing ANOVA

20. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 20 Example 13.1(page 5) Regression Analysis The regression equation is Pull Strength = 2.26 + 2.74 Wire Length + 0.0125 Die Height (6-48) (6-49)

21. L Berkley Davis Copyright 2009 Analysis of Variance(ANOVA) MER301: Engineering Reliability Lecture 13 21 (6-47) (6-45)

22. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 13 22 Summary of Topics  Multiple Regression Analysis Multiple Regression Equation Precision and Significance of a Regression Model Confidence Limits