Estimating Variance Components
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1 Estimting Vince Components Why e we inteested in them? -fo thei own ske - to get b.l.u.e. of n estimble function A. Blnced Dt (using ANOVA method) This method equtes SS (o MS) in ANOVA tble to its expected vlue yields equtions tht e line in the vince components. Conside RCB expeiment: Bke lye ckes t diffeent tempetues nd mesue thei heights. y ij height of cke fom j th btch of dough bked t i th tempetue This is modelled by y ij i j e ij, i 1,, 3, ; j 1,, 3, 4, 5 whee i epesent fixed effects nd j epesent ndom effects nd E j 0j v j j cov j, j E j j 0j j 3 tetments, b 5 blocks Suppose we obtin the following esults: y ij y i y i Dt: y j y y j y 1
2 ESSE E b i1 j1 y ij y i y j y 1b 1 e 14 fo ou exmple ESSC E b y j y j1 b 1 e 30 fo ou exmple b y ij y i y j y e i1 j1 1b 1 SSE 1b1 MSE 1 b1 b y j y e j1 Theefoe we hve e 14 4 e which gives nd Now 3 1 e 14 8 e ij y ij Ey ij j
3 since we hve mixed model (If we hd fixed effects model we would hve witten e ij y ij Ey ij ) Theefoe Ee ij 0 ve ij Ee ij Ee ij Ee ij e i,j cove ij, e ij Ee ij Ee ij e ij Ee ij Ee ij e iji i,j j cov j,e ij Ee ij Ee ij j E j Ee ij j 0i,j,j Ey ij j E i j e ij j i j Ee ij j i j Ey ij E j Ey ij j E j i j i E j j i yij v i j e ij v j e ij v j ve ij cov j, e ij e, which e clled vince components Conside the cse of genel mixed model y X Zue 3
4 whee X e fixed effects nd Zu e ndom effects nd y NX,V Then e yey u Ey u X Zu Ey X V vy vzue covu, e 0 ve e I V ZvuZ e I Now ptition u into subvectos so tht u u 1 u u with ech u i hving s elements the effects coesponding to ll levels in the dt of single ndom effects fcto. e.g. u 1 would be the s coesponding to the btches of dough in ou cke-bking exmple. (If we hd ndom effect fo (tetment x block ) intection in ou model, tht would be epesented by u. To be confomble with u, Z must be ptitioned similly s Z Z 1 Z Z i Z y X Z i u i e i1 V vy v Z i u i e i1 But 4
5 Eu i 0 i vu i i I qi, covu i,u i 0 i i i so covu i,e 0 V Z i Z i i e I i1 Note: We cn define u 0 e 0 e Z 0 I y X Z i u i i0 In fixed effects model whee with using lest sques V Z i i i0 y X e e N0, I noml equtions 0 XX OLSE Xy (These cn still be used fo mixed models to get estimtos but they my not be b.l.u.e. becuse they e obtined without egd fo the vince components.) To get b.l.u.e. s, we must wite ou estimto s line combintion of the elements of i.e.. 5
6 b.l.u.e.x XXV 1 X XV 1 y This wy, the vince components get tken into ccount in estimting estimble functionsx of the fixed effects. vb.l.u.e.x XXV 1 X X Note: we need V vy nd vince of ny obsevtion involves the vince components so ) how e the s estimted? b) how do we use these estimtes fo estimtingx? Given set of estimted vince components, we hve b.l.u.e. VX X XV 1 X XV 1 y This is n unbised estimto ofx fo bod clss of estimtos. Fo blnced dt OLSEX XXX Xy XXV 1 X XV 1 y b.l.u.e.x Meits: 1. Bod pplicbility - thee e no distibutionl ssumptions e the dt othe thn vince nd covince ssumptions in the bsic model.. Unbisedness - the estimtos e lwys unbised (best qudtic unbised) - of ll qudtic functions of the obsevtions tht e unbised fo ech vince component, the ANOVA estimtos hve minimum vince. Assuming nomlity of the eo tems nd ndom effects, the estimtos e best unbised i e. of ll unbised estimtos, the ANOVA estitos hve minimum vince. 3. Smpling vinces- Unde nomlity, smpling vinces of the estimtos e edily vilble. The MS in the ANOVA tble e independent nd e elted to by constnt of popotionlity. 6
7 k i M i i whee M i MS with degees of feedom nd M i EM i nd M i s e independent. Then v v k i M i i i k i EM i Note: This is function of unknown vince components though EM i. To use it on dt, one needs to estimte it. We do this unbisedly s follows: Nomlity ssumptions give but we know so vm i EM i vm i EM i EM i EM i EM i This suggests tht is n unbised estimto of M i EM i Theefoe 7
8 v k i M i i Fo ou cke-bking exmple, e MSE 14 8 MSB MSE 3 1 EMSE e EMSB e v EMSB b1 e b1 EMSE 1b1 e 1b1 v MSB b 1 MSE 1b 1 so fo ou exmple, v Demeits: 1. Negtive estimtes: If ou exmple hd just ows nd columns e.g. 8
9 Dt: ow ow 6 then 9 e 49 e e Distibutionl popeties: Even unde nomlity ssumptions, the distibution of ANOVA estimtos is unknown since they e not simply sums ondependent s. They e line combintions of s with some hving negtive coefficients hence we hve poblems with foming confidence intevl estimtes. 9
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