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MAR 5621 Advanced Statistical Techniques Summer 2003 Dr. Larry Winner

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Chapter 11 – Simple linear regression Types of Regression Models (Sec. 11-1) Linear Regression: Y X i i i 0 1 Y i - Outcome of Dependent Variable (response) for i th experimental/sampling unit X i - Level of the Independent (predictor) variable for i th experimental/sampling unit 0 1 X i - Linear (systematic) relation between Y i and X i (aka conditional mean) 0 - Mean of Y when X =0 ( Y -intercept) 1 - Change in mean of Y when X increases by 1 (slope) i - Random error term Note that 0 and 1 are unknown parameters . We estimate them by the least squares method. Polynomial (Nonlinear) Regression: Y X X i i i i 0 1 2 2 This model allows for a curvilinear (as opposed to straight line) relation. Both linear and polynomial regression are susceptible to problems when predictions of Y are made outside the range of the X values used to fit the model. This is referred to as extrapolation . Least Squares Estimation (Sec. 11-2) 1. Obtain a sample of n pairs ( X 1 ,Y 1 )…( X n ,Y n ). 2. Plot the Y values on the vertical (up/down) axis versus their corresponding X values on the horizontal (left/right) axis. 3. Choose the line Y b b X i i ^ 0 1 that minimizes the sum of squared vertical distances from observed values ( Y i ) to their fitted values ( Y i ^ ) Note: S S E Y Y i i n i ( ) ^ 1 2 4. b 0 is the Y -intercept for the estimated regression equation 5. b 1 is the slope of the estimated regression equation
Measures of Variation (Sec. 11-3) Sums of Squares Total sum of squares = Regression sum of squares + Error sum of squares Total variation = Explained variation + Unexplained variation Total sum of squares (Total Variation): S S T Y Y d f n i i n T ( ) 1 2 1 Regression sum of squares (Explained Variation): S S R Y Y d f i i n R ( ) ^ 1 2 1 Error sum of squares (Unexplained Variation): S S E Y Y d f n i i i n E ( ) ^ 1 2 2 Coefficients of Determination and Correlation Coefficient of Determination Proportion of variation in Y “explained” by the regression on X r l a i n e d i a t i o n t o t a l i a t i o n S S R S S T r 2 2 0 1 e x p v a r v a r Coefficient of Correlation Measure of the direction and strength of the linear association between Y and X r s i g n b r r ( ) 1 2 1 1 Standard Error of the Estimate (Residual Standard Deviation) Estimated standard deviation of data ( V Y V i i ( ) ( ) 2 ) S S S E n Y Y n Y X i i i n 2 2 2 1 ^

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Model Assumptions (Sec. 11-4) Normally distributed errors Heteroscedasticity (constant error variance for Y at all levels of X ) Independent errors (usually checked when data collected over time or space) Residual Analysis (Sec. 11-5) Residuals: e Y Y Y b b X i i i i i ^ ( ) 0 1 Plots (see prototype plots in book and in class): Plot of e i vs Y i ^
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