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Lecture14 - Multiple Linear Regression General Linear...

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Unformatted text preview: Multiple Linear Regression, General Linear Model, & Generalized Linear Model With Thanks to My Students in AMS 572: Data Analysis 1 Outline 1. Introduction to Multiple Linear Regression 2. Statistical Inference 3. Topics in Regression Modeling 4. Example 5. Variable Selection Methods 6. Regression Diagnostic and Strategy for Building a Model 2 1. Introduction to Multiple Linear Regression 3 Multiple Linear Regression Historical Background • Regression analysis is a statistical methodology to estimate the relationship of a response variable to a set of predictor variables • Multiple linear regression extends simple linear regression model to the case of two or more predictor variable • Francis Galton started using the term regression in his biology research • Karl Pearson and Udny Yule extended Galton’s work to the statistical context • Legendre and Gauss developed the method of least squares used in regression analysis • Ronald Fisher developed the maximum likelihood method used in the related statistical inference (test of the significance of regression etc.). Example: A multiple regression analysis might show us that the demand of a product varies directly with the change in demographic characteristics (age, income) of a market area. 4 History Hi, I am Francis Galton (1822/2/16 – 1911/1/17). You guys regard me as the founder of Biostatistics . In my research I found tall parents usual have shorter children; and vice versa. So the height of human being has the tendency to regress to its mean. Yes, It is me who first use the word “regression” for such phenomenon and problems. I am Adrien-Marie Legendre (1752/9/18 - 1833/1/10) . In 1805, I published an article named Nouvelles méthodes pour la détermination des orbites des comètes . In this article, I introduced Method of Least Squares to the world. Yes, I am the first person who published article regarding to method of least squares, which is the earliest form of regression. Hi, I am Carl Friedrich Gauss (1777/4/30 – 1855/4/23). I developed the fundamentals of the basis for least-squares analysis in 1795 at the age of eighteen. I published an article called Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum in 1809. In 1821, I published another article about least square analysis with further development, called Theoria combinationis observationum erroribus minimis obnoxiae . This article includes Gauss–Markov theorem HI, I am Karl Pearson . Hi, again. I am Ronald Aylmer Fisher . We both developed regression theory after Galton. Most content in this page comes from Wikipedia 5 Probabilistic Model 1 1 , 1,2,..., k i i i k i i Y x x x i n β β β β ε 1 2 = + + +... + + = i y is the observed value of the random variable (r.v.) i Y 1 2 , , , i i ik x x x K according to the following model: Here 1 , , , k β β β K are unknown model parameters, and n is the number of observations....
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