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 +   ^ ^

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 ^

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 ^^ ^

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. ^ ^

x y

x y y = x x y Residual = 7 Residual = -13 Residual = -5 Residual = 11 ^

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. ) ^ ^ ^

Total deviation ( y - y ) Unexplained deviation ( y - y ) Explained deviation ( y - y ) (5, 32) (5, 25) (5, 17) y = x ^ y = 17 ^ ^ y x y = 25 y = 32 ^

( 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

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

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

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)

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

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