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REGRESSION/LS FORMULAS Simplest case. S (  ) =  (Y i -  )2 dS/d  = -2  (Y i -  )

REGRESSION/LS FORMULAS Simplest case. S (  ) =  (Y i -  )2 dS/d  = -2  (Y i -  ) Normal equation . A  1 =  Y i A = Y-bar S(A) =  (Y i - A) 2 =  E i 2 Anova identity.  Y i 2 =  (Y-bar + Y i - Y-bar) 2

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REGRESSION/LS FORMULAS Simplest case. S (  ) =  (Y i -  )2 dS/d  = -2  (Y i -  )

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  1. REGRESSION/LS FORMULAS Simplest case. S() =  (Yi - )2 dS/d = -2(Yi - ) Normal equation. A  1 =  Yi A = Y-bar S(A) =  (Yi - A)2 =  Ei2 Anova identity.  Yi2 =  (Y-bar + Yi - Y-bar)2 =  (Y-bar2) + (Yi - Y-bar)2 + (Y-bar)(Yi - Y-bar) = nY-bar2 + (Yi - Y-bar)2

  2. Simple regression. (,) = (Yi -  -  Xi)2 S/ = (-2)(Yi -  -  Xi) S/ = (-2)(Yi -  -  Xi)Xi Normal equations. An + B  Xi =  Yi (1) A Xi + B  Xi2 =  XiYi (2) A = Y-bar - B X-bar B =  (Yi - Y-bar)(Xi - X-bar)/(Xi - X-bar)2 Residuals. Ei = Yi - A - BXi From (1)  Ei = 0 From (2) Xi(Yi - A - BXi) = 0 =  Xi Ei

  3. Fitted values. Yi-hat = A + B Xi  Ei Yi-hat = 0 Anova identity. (Yi - Y-bar)2 = (Yi-hat - Y-bar)2 + (Yi - Yi-hat)2 TSS = RegSS + RSS Yi - Y-bar = Yi-hat - Y-bar + Yi - Yi-hat (Yi-hat - Y-bar)(Yi - Yi-hat) = (Yi-hat - Y-bar)Ei = 0 S(A,B) = Ei2

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