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chapter11

# 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 σ ε ε β β + + = • β 1 > 0 ⇒ Positive Association • β 1 < 0 ⇒ Negative Association • β 1 = 0 ⇒ No Association Least Squares Estimation of β , β 1 2200 β ≡ Mean response when x =0 ( y-intercept) 2200 β 1 ≡ Change in mean response when x increases by 1 unit (slope) • β , β 1 are unknown parameters (like μ ) • β + β 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 β β β β 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 β β β 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 β β β 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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chapter11 - Chapter 11 Linear Regression and Correlation...

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