chapter11 - Chapter 11 Linear Regression and Correlation...

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Unformatted text preview: Chapter 11 Linear Regression and Correlation Linear Regression and Correlation Explanatory and Response Variables are Numeric Relationship between the mean of the response variable and the level of the explanatory variable assumed to be approximately linear (straight line) Model: ) , ( ~ 2 1 e N x Y + + = 1 > 0 Positive Association 1 < 0 Negative Association 1 = 0 No Association Least Squares Estimation of , 1 2200 Mean response when x =0 ( y-intercept) 2200 1 Change in mean response when x increases by 1 unit (slope) , 1 are unknown parameters (like ) + 1 x Mean response when explanatory variable takes on the value x Goal: Choose values (estimates) that minimize the sum of squared errors ( SSE ) of observed values to the straight-line: 2 1 1 ^ ^ 1 2 ^ 1 ^ ^ ^ = = +- = - = + = n i i i n i i i x y y y SSE x y Example - Pharmacodynamics of LSD Score (y) LSD Conc (x) 78.93 1.17 58.20 2.97 67.47 3.26 37.47 4.69 45.65 5.83 32.92 6.00 29.97 6.41 Response ( y ) - Math score (mean among 5 volunteers) Predictor ( x ) - LSD tissue concentration (mean of 5 volunteers) Raw Data and scatterplot of Score vs LSD concentration: LSD_CONC 7 6 5 4 3 2 1 SCORE 80 70 60 50 40 30 20 Source: Wagner, et al (1968) Least Squares Computations ( 29 ( 29 ( 29 ( 29 ( 29 xx xy yy yy xy xx S S S SSE y y S y y x x S x x S 2 2 2- =- =-- =- = ( 29 ( 29 ( 29 2 2 1 2 ^ 2 1 ^ ^ 2 1 ^- =- - =- = =--- = = n SSE n y y s x y S S x x y y x x n i i i xx xy e Summary Calculations Parameter Estimates Example - Pharmacodynamics of LSD 72 . 50 01 . 9 10 . 89 10 . 89 ) 33 . 4 )( 01 . 9 ( 09 . 50 01 . 9 4749 . 22 4872 . 202 333 . 4 7 33 . 30 087 . 50 7 61 . 350 2 ^ 1 ^ ^ 1 ^ =- = =-- =- =- =- = = = = = e s x y x y x y Score (y) LSD Conc (x) x-xbar y-ybar Sxx Sxy Syy 78.93 1.17 -3.163 28.843 10.004569 -91.230409 831.918649 58.20 2.97 -1.363 8.113 1.857769 -11.058019 65.820769 67.47 3.26 -1.073 17.383 1.151329 -18.651959 302.168689 37.47 4.69 0.357 -12.617 0.127449 -4.504269 159.18868937....
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This note was uploaded on 06/04/2011 for the course STA 6166 taught by Professor Staff during the Fall '08 term at University of Florida.

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chapter11 - Chapter 11 Linear Regression and Correlation...

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