Lecture13

# Lecture13 - Simple Linear Regression With Thanks to My...

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With Thanks to My Students in AMS572 – Data Analysis Simple Linear Regression 1

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1. Introduction Response (out come or dependent) variable ( Y ): height of the wife Predictor (explanatory or independent) variable ( X ): height of the husband Example: George Bush :1.81m Laura Bush: ? David Beckham: 1.83m Victoria Beckham: 1.68m Brad Pitt: 1.83m Angelina Jolie: 1.70m To predict height of the wife in a couple, based on the husband’s height 2
Regression analysis: The earliest form of linear regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809. The method was extended by Francis Galton in the 19th century to describe a biological phenomenon. This work was extended by Karl Pearson and Udny Yule to a more general statistical context around 20th century. ●1 regression analysis is a statistical methodology to estimate the relationship of a response variable to a set of predictor variable. History: ●1 when there is just one predictor variable, we will use simple linear regression . When there are two or more predictor variables, we use multiple linear regression . when it is not clear which variable represents a response and which is a predictor, correlation analysis is used to study the strength of the relationship 3

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A probabilistic model We denote the n observed values of the predictor variable x as n x x x ..., , , 2 1 We denote the corresponding observed values of the response variable Y as n y y y ..., , , 2 1 4
Notations of the simple linear Regression - Observed value of the random variable Y i depends on x i i y 0 1 ( 1, 2, ..., ) i i i Y x i n β β ε = + + = i ε 2 ) ( 0 ) ( σ ε ε = = i i Var E 0 1 ( ) i i i E Y x μ β β = = + - random error with unknown mean of Y i Unknown Slope True Regression Line Unknown Intercept 5

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i Y 2 σ Linear function of the predictor variable Have a common variance, Same for all values of x . i ε Normally distributed 4 BASIC ASSUMPTIONS – for statistical inference Independent 7

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Comments: 1. Linear not in x But in the parameters and 2. Predictor variable is not set as predetermined fixed values, is random along with Y. The model can be considered as a conditional model x Y E log ) ( 1 0 β β + = 1 0 β β Example: x x X Y E 1 0 ) | ( β β + = = linear, logx = x* Example: Height and Weight of the children. Height (X) – given Weight (Y) – predict Conditional expectation of Y given X = x 8
2. Fitting the Simple Linear Regression Model 2.1 Least Squares (LS) Fit 9

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Example 10.1 (Tires Tread Wear vs. Mileage: Scatter Plot. From: Statistics and Data Analysis; Tamhane and Dunlop; Prentice Hall. ) 10
0 1 0 1 ( ) ( 1,2, ..... ) i i y x y x i n β β β β = + - + = 2 0 1 1 [ ( )] n i i i Q y x β β = = - + 11

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The “best” fitting straight line in the sense of minimizing Q: LS estimate One way to find the LS estimate and Setting these partial derivatives equal to zero and simplifying, we get 1 β 0 1 1 1 2 0 1 1 1 1 n n i i i i n n n i i i i i i i n x y x x x y β β β β = = = = = + = + = 0 1 1 0 0 1 1 1 2 [ ( )] 2 [ ( )] n i i i n i i i i Q y x Q x y x β β β β β β = = = - - + = - - + 0 β 12
Solve the equations and we get 2 1 1 1 1 0 2 2 1 1 1 1 1 1 2 2 1 1 ( )( ) ( )( ) ( ) ( )( ) ( ) n n n

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