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)

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

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

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

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

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

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