Web-appendix 1: macro to calculate the range of ( ρ, for which R is positive definite

Transcription

1 Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm Wb-appdix : macro o calcula h rag of ( ρ, r) for which R is posiiv dfii * am: oh Prissr; * wha: chck p.d. of doubly sd corrlaio srucur; * This program plos h rgio of (r,rho) for a giv; * _s ad _ ha givs a posiiv dfii R; * This program usd SAS/IML's EIGVAL fucio; * If smalls igvalu > 0, h R is p.d.; * Thr plos ar providd i his xampl; * No: h prcisio of h boudary rgio ca b improvd; * by sig 'ic' o smallr valu; %macro pd(,s,daou); proc iml; s = &s; = &; = s#; BD = I(s)@(,,); *pri BD; ic = 0.0; /* choic should lav o rmaidr as divisor of 2 */ dim = (2/ic + )**2; EVS = (dim,7,0); j=0; psilo = 0**(-2); Do rho = - o by ic; if (abs(rho) < psilo) h rho=0; do r = - o by ic; if (abs(r) < psilo) h r=0; Rmarix = rho # (,,) + (r - rho)#bd + (-r)#i(); val = EIGVAL(Rmarix); mival = mi(val); if mival > 0 h pd=; ls pd=0; val= val[,]; /* hr ar o of hs */ val2= val[2,]; /* hr ar s(-) of hs */ val3= val[,]; /* hr ar s- of hs */ * pri rho r val val2 val3 mival pd; j=j+; EVS[j,] = rho r val val2 val3 mival pd; *if pd= h pri rho r pd; d; d; C = {rho, r, val, val2, val3, mival, pd}; cra &daou from EVS [colam=c]; appd from EVS; %md; *Exampl usd i h Scio Applicaio ; %pd(5,3,valou); proc pri daa=valou(obs=40); whr pd=; ru; * shows boudaris oly; proc plo;

2 Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm il 'Plo shows boudaris of p.d. rgio for _s=5,_=3'; whr abs(mival) < 0.005; /* may d o wak his umbr, ry ic/2 */ plo r*rho='.'; ru; /* proc plo; plo r*rho=pd; ru; */ daa posi; s valou; if pd=; ru; * shows ir p.d. rgio; proc plo; il 'Plo shows shadd rgio ha givs p.d. Corr Marix, _s=5,_=3'; plo r*rho='.'; ru; *shows p.d. rgio ha corrspods o axs i Figur 3; proc plo; il 'Plo shows shadd p.d. rgio for subrgio of (r, rho) for _s=5,_=3'; whr (0 < r < 0.8) ad (0 < rho < 0.4); plo r*rho='.'; ru;

3 Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm Parial oupu: h rag for which h hr-lvl xchag corrlaio marix of h Scio Applicaio is posiiv dfii Plo shows shadd rgio ha givs p.d. Corr Marix, _s=5,_=3 Plo of R*RHO. Symbol usd is '.'. R.05 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ RHO

4 Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm Wb-appdix 2: drivaio of Droppig h subscrip i i block diagoal marix wih Ri s R ad i R i ad oig ha R = A + B whr A = BDiag{ } is h blocks of marix lm A s A = r ρ ) + ( r) I, ad ( apply h followig marix ivrs rsul (Hdrso ad Sarl, 98): ( + B) = A A B( I + A B) A A (A) Noig ha 2 =, h followig is usd rpadly blow: x s s B = ρ s, a b a + b ( ai + b ) = I (A2) Dfi γ = + ( ) r ρ ad ϕ γ + ρ = + ( ) r + ( ) ρ, = s s { } ad apply (A2) wic o obai A = [ /( r) ] BDiag I [( r ρ ) / γ ] [ + A B] = s I I ( ρ / φ) s (A) givs a xprssio ha rducs o. Subsiuig hs io, A B = ( ρ / γ ), ad s R = I r s ρ φ s BDiag I r ρ γ (A3) BDiag =, h lar ca also b wri as (). Usig ha { } s s Fially, oig ha, h sum of h lms of = s R is ρ r ρ = I s s BDiag I r φ γ = sρ ( r ρ) s = r φ. γ φ R Hdrso, H.V. ad Sarl, S.R. (98). O drivig h ivrs of a sum of marics. Siam Rviw 23,

5 Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm Wb-Appdix 3: dails of h simulaio sudy Th mhod of Qaqish (2003) is a gral mhod for graig a T-vcor of Broulli varias Y wih possibly uqual mas ad arbiary corrlaio marix. For = 2,..., T, dfi Z = Y, K, Y ), ϑ = E( Z ), G = cov( Z ), s = cov( Z R, Y ). No ha G ad ( s ar drmid from V. For a giv (μ, V ), a ( ) vcor b is dfid as b = G s ( = 2,..., T). Th codiioal liar family of Qaqish (2003) is dfid by T ν = ν ( z, μ, V ) : = P( Y = Z = z ) = μ + b ( z ϑ ) = μ + b ( y μ ) ( = 2,..., T). () Th simulaio algorihm procds as follows. Firs, simula as Broulli Y wih ma μ, h for = 2,..., T, simula Y as Broulli wih codiioal ma ν giv by h quaio abov. I h follows ha E(Y ) = μ ad for < T, cov(z, Y ) = cov(z, T b j= j j Z ) = G b = s. Th vcor Y hus obaid has h rquird ma, μ, ad covariac, V. Thr ar som rsricios o allowabl μ ad V as discussd by Qaqish. Ths iclud, bu ar o limid o, rag rsricios placd o h lms of R as drmid by μ (i.., s quaio (8) of Qaqish) ad bouds sablishd by h rquirm ha R is posiiv dfii. Th full joi disribuio of Y, whos xplici spcificaio is o rquird, ca b compud via (). For ay valid (μ, V ) ha is rproducibl by h codiioal liar family, hr is a corrspodig uiqu valu of h vcor of joi probabiliis. 2 s For applicaio o h hr lvl clusr rial dsig, T = s ad Y is a vcor of Broulli varias wih qual mas ad covariac marix V dfid by R(r, ρ). Closd-form formula for G aalogous o quaio () may b implmd o spd compuaio. A dscripio of simulaig corrlad biary logiudial daa wih h Qaqish algorihm is providd i h Appdix of Prissr, Lohma ad Rahouz (2002). j