Part 13- Simple linear regression - The University of Jordan

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University of Jordan Agricultural Statistic (605150)Faculty of AgricultureDr. Amer SalmanDept. of Agri. Econ. & Agribusiness22222 2( Y − Y ) = ( Yˆ− Y ) + ( Y − Yˆ) ,[ y = yê ]∑ i ∑ i i ∑ i i ∑ i ∑ i+ ∑Total variation = Explained variation + Residual variationIn Y (or total in Y (or regression in Y (or errorSum of squares) sum of squares) sum of squares)TSS = RSS + ESSDividing both sides by TSS:RSS ESS1 = +TSS TSS2The coefficient of determination or R is then defined as the proportion of the totalvariation in Y explained by the regression of Y on X.iR2RSS= = 1−TSSESSTSS2R Can be calculated by:R2∑∑∑∑22= yîei= 1−2y y2ii∑∑Where: y ˆ = ( − ) 2iYiYi22∑ei= ∑ ( Y i− Yî)22∑ yi= ∑ ( Yi− Y )2 ˆ2R Ranges in value from 0 (zero when the estimated regression equation explains nonof the variation in Y) to 1 (when all points lie on the regression line).The Correlation Coefficient:rThe correlation coefficient is given by:∑ x2 iyi∑== ˆxiyiRb12 2x y yi=2∑i∑i∑- 14 -
University of Jordan Agricultural Statistic (605150)Faculty of AgricultureDr. Amer SalmanDept. of Agri. Econ. & Agribusinessr ranges in value from -1 (for perfect negative linear correlation) to +1 (for perfectpositive linear correlation) and dose not imply causality or dependence.Example (13 – 7):The coefficient of determination for the corn – fertilizer example can be found from:R2=∑∑e2i1−2yi47.31≅ 1−= 1−0.0290 = 0.9710 ≅ 97.10%1634Thus the regression equation about 97 % of the total variation in corn output, theremaining 3 % is attributed to factors included in error term. Then:2r = R = 0.971 = 0.9854 Or 98.54 % and is positive because ˆb 1is positive.Example (13 – 8):nXiYixiX i− XyiY i− Yxiy i2xiŶi2ei( Y ) 2i− Yîy22Xii1 1 1 -2 -1 2 4 0.6 0.16 1 12 2 1 -1 -1 1 1 1.3 0.09 4 13 3 2 0 0 0 0 2 0 9 04 4 2 1 0 0 1 2.7 0.49 16 05 5 4 2 2 4 4 3.4 0.36 25 4n=5 ∑X =15iX =3∑ iY =2Y =10∑ ix =0ˆxiyi7β1= = 0.7 ,2x 10iˆ β ô= Y − β X = 2 − (0.7)(3) =Y ˆ = −0.1+ 0. 7x=∑∑2S ˆ β =S ˆ β =oo∑1−2ei×n − k n∑∑x2i2xi0.4033 = 0.6351.1= ×5 − 20.1∑ i555(10)y =0∑ =7 ∑ 2 ix i y i60.5= = 0.4033150x =10∑ 2 ie =1.1∑ 2 iX =55( Y Y ) 2i−∑ 2 iy =6- 15 -
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Part 13- Simple linear regression - The University of Jordan

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