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R2 - n p 1 n-1-1 R 2-p n-1 n-1 n p 1 ˆ σ 2 = 1-adjR 2 Syy...

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STAT5044: Regression and Anova Inyoung Kim
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Outline 1 More on R 2
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R 2 R 2 : the coefficient of multiple determination R 2 is not an estimate of a population quantity unless the data is multivariate normal R 2 can be drametically changed by how the x ’s are selected. R 2 does not capture non-linear relationship R 2 usually can be made larger by including a larger number of predictor variables
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Adjusted R 2 Let p variables in model Definition of adj R 2 = 1 - SSE / ( n - ( p + 1 )) SSTO / ( n - 1 ) adj R 2 is found as adj R 2 = ( R 2 - p n - 1 )( n - 1 n - ( p + 1 ) ) What is the rationale for adj R 2 ? If we use p terms that are unrelated to y , the average R 2 is p n - 1 To correct for this, our first adjustment is R 2 - p n - 1 However this reduction R 2 too much. If we have p X’s that perfectly predict y , this adjustment will only give 1 - p n - 1 = n - ( p + 1 ) n - 1 , but in this case we should get a value of 1
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Unformatted text preview: [ n-( p + 1 ) n-1 ]-1 , ( R 2-p n-1 )( n-1 n-( p + 1 ) ) ˆ σ 2 = [ 1-adjR 2 ] Syy n-1 , ˆ σ 2 = RSS ( n-( p + 1 )) = MSE As adj R 2 increases ˆ σ 2 decrease. Comments Let p variables in model R 2 can viewed as a coefficient of simple determination between the response Y i and the fitted values ˆ Y i A large value of R 2 does not necessarily imply that the fitted model is useful one. ⇒ Despite a high R 2 , the fitted model may not be useful if most predictions require extrapolations outside the region of observations. Again even though R 2 is large, MSE may still be too large for inferences to be useful when high precision is required....
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