Is the mean square error s 2 ² n x i 1 y i ˆ y i 2

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is the mean square error s 2 ² = n X i =1 ( y i - ˆ y i ) 2 / ( n - 2) = SSE n - 2 , s ² = r SSE n - 2 ( S ² is called the residual standard deviation) 11.6 Partitioning the variability y i - ¯ y = y i - ˆ y i + ˆ y i - ¯ y -→ & $ % n X i =1 ( y i - ¯ y ) 2 = n X i =1 ( y i - ˆ y i ) 2 + n X i =1 y i - ¯ y ) 2 SSTotal = SSError + SSREG (Note: SS stands for ”Sum of Squares”). 3
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A useful measure of model fit is the Coefficient of determination ( R 2 ). R 2 = SSREG SSTotal , 0 R 2 1 The larger the R 2 , the closer the fit. The Sample correlation coefficient between X and Y is r X,Y = S XY S XX · S Y Y , - 1 r X,Y 1 . It measures the linear relationship between X and Y . r X,Y = +1 ⇐⇒ Y = a + bx, b > 0 , r X,Y = - 1 ⇐⇒ Y = a - bx, b > 0 . For the simple linear regression ˆ y = ˆ β 0 + ˆ β 1 x ; we have (prove!) r 2 X,Y = r 2 Y, ˆ Y and r 2 X,Y = R 2 11.7 Distributions of the estimated model parameters In order to construct the CI’s for the unknown parameters β 0 and β 1 , or to do hypothesis test such as H 0 : β 0 = - v.s. H 1 : β 1 6 = 0 . We need to know the distributions of ˆ β 0 and ˆ β 1 . To do this, we assume the distribution of the random error ² to be normal, i.e. ² N (0 , σ 2 ). Under this normality assumption, T 1 = ˆ β 1 - β 1 s ² / S XX t n - 2 ; T 0 = ˆ β 0 - β 0 s ² · r X 2 i n · S XX t n - 2 . Under H 0 : β 1 = 0, T 1 = ˆ β 1 - 0 S ² / S XX . 4
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( T 1 ) 2 = ( ˆ β 1 ) 2 · S XX s 2 ² = SSREG s 2 ² F 1 ,n - 2 . 11.8 Checking the model assumptions The constant variance assumption can be checked via a scatter plot of the residuals ( y i - ˆ y i ) versus x i (or ˆ y i ). This plot is often called the residual plot.
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