P total variation ym0y p m0 i iii 1i the m matrix for

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p Total variation = = yM0y . p M0 = I – i(i’i)-1i’ = the M matrix for X = a column of ones. n 2 i i=1 (y - y) ™    6/33
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Part 5: Regression Algebra and Fit Decomposing the Variation = = = = - = - = + i i i i i N N N 2 2 2 i i i 1 i 1 i 1 y e y y e         =  ( ) + e (y y) [( ) ] e (Sum of cross products is zero.) Total variation =  regression variation +                         residual variation b x x b i i i i x b + x b - x b + x - x -              Recall the decomposition:      Var[y]  =  Var [E[y|x]]  +  E[Var [ y | x ]]                 =  Variation of the conditional mean around the overall mean                     +  Variation around the conditional mean function.  ™    7/33
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Part 5: Regression Algebra and Fit Decomposing the Variation of Vector y Decomposition: (This all assumes the model contains a constant term.  one of the columns in  X  is  i .)        y  =  Xb  +  e  so        M0y  =  M0Xb  +  M0e  =  M0Xb  +  e .         (Deviations from means. Why is  M0e  =  e ? )        y¢ M0y  =  b ¢ ( X’ M0 )( M0X ) b  +  e¢ e                  =  b¢ X¢ M0Xb  +  e¢ e.         ( M0  is idempotent and  e’ M0X  =  e X  =  0 .) Total sum of squares = Regression Sum of Squares (SSR)+                                       Residual Sum of Squares      (SSE) ™    8/33
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Part 5: Regression Algebra and Fit A Fit Measure R2 =  b¢ X¢ M 0 Xb / y¢ M0y        (Very Important Result.)   R 2 is bounded by zero and one only if: (a) There is a constant term in X and  (b) The line is computed by linear least squares.   = = - = - N 2 i i 1 Regression Variation  1 Total Variation (y y) e'e ™    9/33
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Part 5: Regression Algebra and Fit Adding Variables p R2 never falls when a z is added to the regression. p A useful general result 2 2 2 2 *2 R with both and variable equals R with only plus the increase in fit due to after is accounted for: R R (1 R )r = + - Xz X X yz|X X z X z X 2 ™    10/33
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Part 5: Regression Algebra and Fit Adding Variables to a Model What is the effect of adding PN, PD, PS, ™    11/33
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Part 5: Regression Algebra and Fit A Useful Result Squared partial correlation of an x in X with y is We will define the 't-ratio' and 'degrees of freedom' later. Note how it enters: squared  t - ratio squared  t - ratio  +  degrees of freedom ( 29 ( 29 2 2 2 * 2 2* 2 2 2 Xz X X yz yz Xz X X R R (1 R )r r R R / 1 R = + - = = - - ™    12/33
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Part 5: Regression Algebra and Fit Partial Correlation     Partial correlation is a difference in R2s.
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