lec2_ppt2019.pdf - Stat 206 Linear Models Lecture 2 Sept 30 2019 Simple Linear Regression Model(Review n cases(trials/subjects Yi – the value of the

lec2_ppt2019.pdf - Stat 206 Linear Models Lecture 2 Sept 30...

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Stat 206: Linear Models Lecture 2 Sept. 30, 2019
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Simple Linear Regression Model (Review) n cases (trials/subjects): Y i – the value of the response variable in the ith case; X i – the value of the predictor variable in the ith case. Model equation : Y i = β 0 + β 1 X i + ε i , i = 1 , . . . , n . (1) Model assumptions : i s are uncorrelated, zero-mean, equal-variance random variables: E ( i ) = 0 , Var ( i ) = σ 2 , i = 1 , . . . , n Cov ( i , j ) = 0 , 1 i , j n . Unknown parameters : β 0 – regression intercept; β 1 – regression slope σ 2 : error variance
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Given X i s, the distributions of the responses Y i s have the following properties: The response Y i is the sum of two terms: which is which has i s have constant variance = i s are uncorrelated =
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In summary, the simple linear regression model says that the responses Y i are whose means are whose variances are Moreover, two responses Y i and Y j ( i , j ) are
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Regression Function y = β 0 + β 1 x A β 1 is the of the regression line: the change in per unit change of X . β 0 is the of the regression line: the value of E ( Y ) when We will study how to model and fit the regression function from data.
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Figure: Regression line: y = β 0 + β 1 x 0 1 2 3 4 5 0 1 2 3 4 5 6 7 x y y=beta_0+beta_1 x { 1 unit of x { beta_1 { beta_0
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Least Squares Principle For a given line: y = b 0 + b 1 x , the sum of squared vertical deviations of the observations { ( X i , Y i ) } n i = 1 from the corresponding points on the line is: ( X i , b 0 + b 1 X i ) is the point on the line with as the i th observation point ( X i , Y i ) . The least squares (LS) principle is to fit the observed data by the sum of squared vertical deviations. LS line has the sum of squared vertical deviations among all straight lines.
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Figure: Illustration of LS principle 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 3 5 7 x y Q(3,0.5)= 5.539 y=3+0.5x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 3 5 7 x y Q(2.5,1)= 3.041 y=2.5+x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 3 5 7 x y Q(2,1)= 2.984 true regression line: y=2+x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 3 5 7 x y Q(2.09,1.07)= 2.659 LS line: y=2.09+1.07x Which line has the smaller sum of squared vertical deviations, the LS line (a.k.a. the fitted regression line) or the true regression line?
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Least Squares Estimators LS estimators of β 0 , β 1 are the pair of values b 0 , b 1 that minimize the function Q ( · , · ) : ( ˆ β 0 , ˆ β 1 ) = argmin b 0 , b 1 Q ( b 0 , b 1 ) .
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