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SSTO = SSRCX , ) t SSECX , )= SSRCX , ,Xa) t SSE (Xi , XD
ssrcxzlxi) -- SSRCX , ,Xz) - SSRCX , )⇒ SSTO -
- SSRCX , ) + SSRCXAIX , )tSSE(Xp , ,Xz)T T T remainder
total variability variability residual after Xi ,in Y attributable to variability xn
after accounting haveX, for X , that is been
attributable to accountedXz for
SSTO -- SSRCX , )t SSRCXZIXDTSSRCXslxz.XDTSSECX.kz , Xs)where SSRCXSIX , ,
Xz)=SSR(Xs ,Xz , XD - SSRCX , ,Xz)
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Decomposition of SSR into extra sums of squaresGiven Xz
,SSTO = SSR ( XD t SSE ( X1 )
Given Xz as well,SSTO = SSR (XD t SSR (Xzlxse) t
SSECXI,Xa)
SSTO = SSR ( Xs,Xa) t SSE (Xs , XD
= SSR ( Xa ) t SSR (X11 Xs) t SSE ( Xe ,Xz)
Recall. .
. in ANOVA conversations previously ,we talked
about SSTO SSR SSE. . .
but that was inMSTO MSR MSE
the context of simple linear regression
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In multiple linear regression ,an ANOVA table usually
contains decompositions of the regression sum of
squares into extra sums of squaresYi -- foot §
,pnxikt Ei ,
Ei dN ( o,or)
ANOVA Table for fitting this modelsums of squares Degrade freedom mean squaresSSR ( Xs ,Xs,X3) 3 MSRCX , ,Xz ,XDSSRCX,) I MSRCX , )
Irtysh.ua#tnexfssrcx.m, IfIY£T a Msrcx.milSSRLXI ,XnXd
,model I MSRCXslx.pk/
regression SSR (Xzlxn XDsum of MSECXI ,XqX3)squares SSE ( XI ,Xr , Xs) n - 4
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Use of Extra Sums of Squares for Testing Hypotheses about pkcontext : Assume we have I , Ii , . . .
. Xp- i , we're assumingYi = Bot Xik But Ei
,EidN( 0,02)
*Testing whether a single pre-
- o
Null,Ho : pre -- O
we already know how
Alternative,Ha : put o ) to do a t - test for this
,
where we compare t*=sYbTzto quantile s of at - distribution with n- p d.f .Consider k =3
Ho corresponds to the redirect model ,Yi -- foot pi Xist for Xist §÷
,Buxiut Ei , Eii NCO ,
02 )
Fit both models or ,make an ANOVA table for the full
model and get SSR Hi , Xs , Xy, . .. and SSE ( Xi,. -
-
, Xp - i )Then we can construct f*=(ssRCX3lXzXziX4i-yXp#)÷fssECx.,X)Under Ho
, Ftt n Fi, n - p n - p
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Use of Extra Sums of Squares for Testing Hypotheses about piecontext : Assume we have I , Ii , . . .
. Xp- i , we're assumingYi = Bot Xik But Ei
,Ei d NCO
,T2)
*Testing whether a single pre-
- o
Null,Ho : fu = 0
Alternative,Ha : put 0
Com
ssrc*Hxj÷fssEnkIDecicision Rule for a level 2 - test will be :
* If F * E F ( t - d ; I,n - p) conclude Ho
* If F* > FCI- d ; I , n - p) conclude Ha! Ppk 8
p - value given by Pr ( FIFA) , where F - Fi , n -p
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Use of Extra Sums of Squares for Testing Hypotheses about piecontext : Assume we have I , Ii , . . .
. Xp- i , we're assumingYi = Bot Xik But Ei
,Eid NCO
,T2)
*Testing whether multiple pie , Be e# u are jointly equal to 0Null
,Ho : fu = Be = 0
Alternative,Ha : Puyo or Geto
com
ssrcxu.ie#mmueD:fssEnkIjxP-d/2DecicisionRule for a level 2 - test will be :
* If F * E F ( I - d ; 2 ,n - p) conclude Ho
* If F-*> F ( l- d ; 2 , n - p) conclude Ha
p - value given by Pr ( FIFA) , where F - Fa , n-p
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Use of Extra Sums of Squares for Testing Hypotheses about
context : Assume we have 4, Ii , . . .
. Xp- i , we're assumingYi = Got Bixiitfszxiztfbzxizt Ei Eid N ( O
,T2)
*Testing whether multiple paz , Bz are jointly equal to 0Null
,Ho : far -- Bss -- O
Alternative,Ha : Pato or B , 40
Com
ssrcx.it#I).fssEnHiihiH/2 4Decicision Rule for a level 2 - test will be :
* If F * E F ( l - d ; 2 ,n - 4) conclude Ho
* If F* > F ( l- Lj 2 , n - 4 , conclude Ha
p - value given by Pr ( FIFA) , where F - Fa , n- y