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Published byDavid Bates Modified over 9 years ago

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Definition Regression Model Regression Equation Y i = 0 + 1 X i ^ Given a collection of paired data, the regression equation algebraically describes the relationship between the two variables Y i = 0 + 1 X i + ^ ^

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y -intercept of regression equation 0 0 Slope of regression equation 1 1 Dependent Response Variable Independent Explanatory Variable Residuals (error) Population Parameter Estimate ^ ^ YiYi XiXi YiYi ^

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Definition Regression Equation Given a collection of paired data, the regression equation Regression Line (line of best fit or least-squares line) is the graph of the regression equation algebraically describes the relationship between the two variables Y i = 0 + 1 X i ^^ ^

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Definitions Residual (error) for a sample of paired ( x,y ) data, the difference ( y - y ) between an observed sample y -value and the value of y, which is the value of y that is predicted by using the regression equation. Least-Squares Property A straight line satisfies this property if the sum of the squares of the residuals is the smallest sum possible. ^ ^

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x 1 2 4 5 y 4 24 8 32

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x 1 2 4 5 y 4 24 8 32 y = 5 + 4 x 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 12345 x y Residual = 7 Residual = -13 Residual = -5 Residual = 11 ^

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Total Deviation from the mean of the particular point ( x, y ) the vertical distance y - y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y Explained Deviation the vertical distance y - y, which is the distance between the predicted y value and the horizontal line passing through the sample mean y Unexplained Deviation the vertical distance y - y, which is the vertical distance between the point ( x, y ) and the regression line. (The distance y - y is also called a residual. ) ^ ^ ^

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Total deviation ( y - y ) 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Unexplained deviation ( y - y ) Explained deviation ( y - y ) (5, 32) (5, 25) (5, 17) y = 5 + 4 x ^ y = 17 ^ ^ y x 0123456789 y = 25 y = 32 ^

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( y - y ) = ( y - y ) + (y - y ) (total deviation) = (explained deviation) + (unexplained deviation) (total variation) = (explained variation) + (unexplained variation) Σ ( y - y ) 2 = Σ ( y - y ) 2 + Σ (y - y) 2 ^ ^ ^ ^ SST = SSR + SSE

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Q=SSE=Σ (ε) 2 =Σ (y - y) 2 ^ =Σ (y - 0 - 1 X i ) 2 ^ ^ Minimize with respect to 1 and 0 ^^

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0 = ( y) ( x 2 ) - ( x) ( xy) n( xy) - ( x) ( y) n( x 2 ) - ( x) 2 1 = n( x 2 ) - ( x) 2 ^ ^ ^ ^

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Multiple Regression Models Polynomial Model Y k = 0 + 1 X 1k ……… k X nk + k Y k = 0 + 1 X+ 2 X 2 ……… k X k + k Y k = 1 X 1k ……… k X nk + k Multiple Regression Models (no intercept)

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Y = XB + e Y is the n x 1 response vector (n x 1) X is the n x (k + 1) design matrix B is the n x 1 regression coefficients vector e is the n x 1 error (residual) vector 0 ≤ k ≤ n

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Y k = 0 + 1 X 1k + 2 X 2k + k Describe Y, X, B & e for

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