OnApproximateLikelihood Inf erenceinthePoissonMixedModel

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On Approximate Likelihood Inference in the Poisson Mixed Mod el

by

Zh"u-Dc' Q1I

A IllI'sis sulnuittedtotill' SchoolofGraduateStIU!iI'S

illpar t.iulIulfihucntoftho

requirementforth edegreeof 1Iaslc'r uf Sdl'un'illSt.nt,is ti<..-:;

Department.ofMathomatlcsandStatistics Mcruori ulUuivr-rsitv ofNewfoundlan d

Juunarv1095

St..Jullll \~ Newfou ndland

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1+1

NalklnalLibrary

0'''''''''''

AcqOOitionsand BibliographicseoeesBranch 395w.1ingtwlSllNl

~on:llfio

:.~uenaliOnale Oireclion des 8CQUi$itiOnS cl des seMcesbibIiograpliQues

".~- '*-(Or'In1 KIA QN4

The author has granted an Irrevocable non-exclus lvelicence allowin g the National Ubraryof

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Abst ract

TIlt' i1pl'lirilti"u

IJr

1I11' l'ujssCJ lI IlJix,·dtUOfldI.ash(~ '11hamp eredbythedifficultyofcom pu- tntiouin"VIIIIIHl ill ,ll:II II'mar,ll: illilll ikd iJII}(Jl! of lIw pa ramet ersinvolved.},[illl'yapp roxhunte lIJ'I'I"'i1d ws 11i'I'C'1"l'l'('11tlyI)("~'llIII"OI'''SCII(firil1r"n~l w('ahemthegClle1'nliZCdlinearmixed 1l,<Hl"llI'hid,!"I,r!'rs10 IIll'Puis';<J1IllIixl'dmod,'l,ISnspt>c'ia)(<lSC,forexamp le,thepenalized qn.e-i-fil...lihood(I'q r.)upproarhofBreslow(111()Clilyloll(JH!Ja),andthegt'IK'l'alizl.'(! esti- Illi,lilllJ;fllll , ' t i,ul((:E [<')HI1IJroa di nfWiwlilWiwnudLiallg(/!m:I).We show intill'thesistrial 1,"111(,III'I'QI.;'1111(:1':10'1'1'0<111 ('('itlf"olJsis\l'ul.inferenc eforIIwvariance("011l1101ICn l illthe l'" jSSlJllIIlix" ')lII"d,'1.'I'll!'l.Ilf~ isIIII'uI' WpllS('Siltwo-s tepflPlWOxirtwlt,likdilioodappro'lcll (,\ 1.)fl U'1.111'"slililiitiollurrJ II'(,,'I.I'IJ('Sofl'ilr,'IIWrl'rS(fix('(1{,m.,,:!.l'ilril llll' !.CfS,randomeffects IHI, Itln-it-mJ'iilll('(' ('Ol llllOlr('nt, )illlllf'I'ois,~(ltlmixedmodel.[IIthefirstste p,nnapproximate likdilllH,drllHdi ollor ('n U1I1,d'll.,)i,sronsu'urtr-d10 esthn atctil£'fixedeffect parmncterslind 1.lw\'ilriann ','Olllfl0Jl('111.fl,vIll'Plyi llKn I'olljngill,('Hny esiantlu-ormn. Inthesecondstep.the rnudomI'lr,,\'1,1O an-l'sliumtod bylllillitlli~illgthe irapproximateposter iormean sq ua re error.

Our"s lillln1!'10<II'(',ll w ll,\'S('nnsi sll'lI!lorhotb(I n~!iXN!dfeclpurmnctcrs endthe vari uucc I'tllllp01U'Ut., \Vhl'lI(,11f'acl.unl vlIria n('!'componrutisncarzero .our estimatesarc almost

tlplilllillfurI-Ill'rlludo1l1('lfl'CI.S,Wlwn lhenct.nalvarianc ecompououtisawayfromzero,our l'sli mil lt-s ,m'nlwlIysaS~'l1lpl\lr.inlllyuubiascd forthefixed('ffee!,parameters,whereas our ,oslill Wlc' is~IS,\'III I'lol,kllllyHC'/!;lll.i\'(,hiasl'dforthr-\,ariillll'I' l'OllIIlOllCIIL,Another<lc1Oirllblc uu-r-i! isllral.Illtli k('III<'('xi slitlp;,lp[Jl'oad l<'s1H('lItiolll'(]IlIJO\'{'.our('s1.ima1.cs[01'bo t h the

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fixod ('If('{"1 pnranu-n-rsandlilt' van.mo-romponcutonl.\·.1" I)I'lI, 1 011tlu-dislriIHlli,,"ofrnn rlom df('('tsrat .lu-r thuu lilt'l'slimah'Sof random drl",ts .'\11important,lil1dill~is llial(Ill' ilS,nll11tol,iccovnrlnuc.or 0111' ,'SIi1ll1l11'Sfor Itil' tixr-c! r-m'dparanx-n-rswilllll','\J1II.'smilll.'rill W' lll'rillilS1,1ll'\"ill"i ill\ ("('n1l11pOIIt'1I1_<111index,I[Lin-il ll nl ·d ll,~I ,'1'ilsStJ(·j;l l,i' lll, ill<T<'ilS"S,;11I,1 rnn beao t in 'a blyreducedb,\'ilssign illg1.11('ralll('s{IfHit,fix"tldf..t-l,·Il\·ill·iil l,'sliS,lilJ"!'l'lltilS W1s"ihl"lll tloll g<li/fer ('1l1.oll,' I'I"I'aliOIl,'ill1111,1"!'I11.,I,']".11011" ' \',-1',iflh,'lix,'cl,' If,'d"ll\'ilriill,-lias

variuuo-oflilt'..stiIl1111,1'fo rt.ln-('olTI'Spmulinglix,~ldfr-d pnrtum-l.r'r m,l,\"illn'-<ls,'<ISlilt' vur-iaurt-romponeru 1l."1.sIm·!!.,'1".Thisfc'al,un'IIwyIH'IIs,.futillIll'si~l lill~II\'il]id,' :.qH'rillll'lIl

orS,111I1)lingfo r,,]11'Poissonll1ix (,, 11l1o l]" 1.!llI]I'SS Itil'V<lriillWI'rtuupunout.issrunll,1lu-Ii.w ,l dre("!. rovatintcsshould1)(',k,.~iglll'dtohn\'\'\'I1III1'SH,~difl('n'l1l.il.~I'"s sil.l,·1Il1l1l1l ~,liIf,'n'lIl ohscrvarlcnsillilll,\'r-lustcr.lt is furt herSIUIW11.1 1Iwllg hslnmlatlon,tilt,PI'l II }l,s, ~ 1~l l llll't l;}t' 11

iii

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Acknowledgements

Firsl"fHII,Iwouldlik«101.111111kmySII ]I(!l'vbory('()Jl1 rni w'(~:Pro fcssol'sHrajcndr eSutradhar, Ildillli1BalasouriYiIHIlt!IioyBlIl'h·ld 1.rO!"1.1.1'i1'~Ilidll llr('andsupport,illconductingthis

1':dp;.II'(:,,.,,Iain '.lim n '\VIII-sOli, HsJudy^J,('{,.1.1)1'hlllp)l' Leafandm1l11Yotherprofessors ll lUI~ril ll liHll 'sl,lu l"ulsnl.ti,l'1)" lml"1.I1Il'lllof~'I;ll.lll'll li1li('sandS1.ill,jstit' s for1,l leirhospitality

As\\,,·11 . I11111Arll1.d"1I1 toIIll'SI"i,OO!

o r

(irmhlll!.('S1.udk'sand theDepartment ofMath-

"llIa l.ks uu.l SI.HI,jHI,k s1'01providilll;111(' withfiuilncia lsupport illtheformof aGra<luat r:

SI.IIlI"1I1Sdlolal'shii',11111(;"1\<llIa1.,' J\ssi s1.illllsllip 10makeIllystay att.he MCl110dalUniver- sityofN"\\'I'JlllldhlllCl]lossihh-,

Fiuully, I1"UltI,1lik,'10Il,'dienl(' this 1,lwliis10Mr.Yu-ChuZhu,mydearestteacher at hi/!;hsdlllol.

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a.r

LikelihoodApproxituntlon :t i Two·Ste pAp pm,Ic11.

:.1.:1 Compututioue lAH]lI'cl s . :\.-[ Remarks011 ASylllpl,ol.i("Theory

:lA.1 When(11 i:-lI\II0W ll

III III 12

III

21

~!l

:U.2 Whl'1l11~islJlIk1l0 WII • . • • . .•• • • • • • • • • . . •• .

4 Two RecentApproximateMethods of Estimation

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1.1 1',·w,li...1Qllasi·Likdil,o"..1~l...h,,<1. . . .. .• . . . • .

!i Silllll lnt ioll Stndy

;

;.1 Sjlll..lat i"III ",.i~lI .

fl Cuncluslo n snur]50meSug gest ion s

56

.')j

70

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List of Table s

gu-ssiouK~ l.i II l1\l ,('suudV~ ll'i ;II I ('<'('U11111011('1l1.SofHuudomEII{·l't.~1"'1'SI'I,'i"lc,,1 Vallie'sofrr~: ~.

=

[1(1;IIi=·1(i=I.. . .!'):Tnn-\f:dlll'sof I,IH'lI"j!;I"{'Ssi,,"

Pal'Hltwl pl's:rJI=Vi./Jl =-l.tJ./~1

=

I.U;11111fi,==OJi:;llIunSil lllll,l!,j"l1s. fil '-'.2 Comparison of Sill1l1lal ('c!l\!l'illlVahU's and Sl.arnlal"!Errurs (SE)"I'till'HI"

gn'ssi o ll Es l i l1t1\l,\, s"1111 VOIril111("<'('1Il11IlO l lI' ll 1.s"I1l;11Ullllll[·:If,·('1.sfllT'S,·lt·d, -,!

ValuesorI1l:k

=

r,O:II;=Ii(i==1•.•.t'):Tn I!'Var lit'S,,["1.111'[lq -\I'l'ssiull IJilnll1lf'I(' r~:Iii

=

2.:j,til

=- r. n.

li:1==[.11iliid11.,

=

fiJi:.')(J(J[JSill1l1l;,liI/ li S. fi:!

:i:l ComparisonofShuulutr-d1\1"1111V;IIII(':oIandSI,mlllHrd Error.s(Sr,;)ufr,11l'HI"

gl'l'ssiollr·;sl,iIIWl,'.s111111VI,l'iHllf' f'(~fJlI 1IJfJllf ·l l l.sofIli lUflo l1l1·:l r, ·d s1. /1'SI'),~ · l.I ,tl

V,l l ll{'SoffT1: ~.

=

lOll;IIi=,I(i=I.•••l·);TI"Il<' VIIIII''S

"r

1.1",1I"I',l'<'ss illll

iii

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;,.1 C"lIIl'ilri sOIlofSirHllllll ,...1~],'1111Vlfhlf"s:U111Sl illlrlllr,1Erro r!! (SE)oftheHe- .l\1"<',;si.J111·~..lilll;,I,,!,1111,1Vllrillllf""('''II'I" III''llls or HIIIl,IOlllEfff'CL'lfor S.·lec:h 'tl

Viiiu,',;"rtIl:L·

=

^IlKI;n,=I;Ii=I•.• . •1:):Trill'Vllh,,'11ortltl'Ih..'j!;fC:'siotL

1':1t1l1l1d , ·I1':.tI.=1."'.il l=-1.11.,J"

=

1.00<1111/1_1

=

^lJ.!^i:!'ilXMJ Simullltions. . 6j

;,.;1 {'''IIlI'Hr iSflllor'1'01<11 :\1<0111ISII'llln' Errorsor IIII'Ih llflolll

I·:rr,·d

1)l"l'(lir l iolls r"rS!·I,...-te...1v:,rlll'lillr"l:L'=;KlilllIlllIlI:"i=.I.fi (i=1, •••• L·);Tn lf·V alut'li

!'iHllllSiI1lIlIHliu lli<•.

. em

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Chapter 1 Introduction

of1,11('r-Lhdl1l'k r lUlll~.istill'lutal111l1ll1ll' 1'ofdllsl.c' l"s. Le-l,rJ<I"Hul!'a/,x I\','d." r ofunknownfix('c1"fr"f"!,pi'I'iltl lf'll'rsm;sod"l c',]withl.ln-ul,sI'l'V<',1vcr-turs,f il•..,'/"'""f1111 ' lixC'd ,·rr"I'I.('UVill'iIl1l-S.i1l1dli, l f' IIIJI ' - lll li mr i illc·rilllllull l,·lf,·rb.(:iV"lli,.I-IIl'II ,.,l,s,·nilli'III S .IIij(j".I...•I,i)withintill'ithdllst","un-assllllll',lloIll'illdl'lH'II.I I'III ,mlflto f"IIU1I'1.11l' l'oissolldistrihutiou.yickliug

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wlu-n-J(.IIkfi),1t'1I0(C'Srhol'Undit.iul\ill)ll'OlwbililJ',1'~lIsil.y^flf.l/i

=

(1101,.. ,H"" f r,,,'IIf',iwtl ii,iun l

(I.:.!)

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{

Ffl I"(!!ij

I"(;)

^""^Jlij irj=/

('fII '(!!' j .!!i j '

I

"ti)

=

(J ifj¥j'.

IU)

Fonu:

i\!" n u \'l' I'. I1'I· !Il" ,Jd

(lAI

11.5)

~' 11<1 a~~III1WthaIl'ilI " r" lIl(,rr,'d~Ii(i= I, ...II,)1It'1'id'!llt.iclllly,iu<l{'pcrt d('llllyandnormally

"I iLiJl.N(O,O"~)

II'lwr,-r<:is1I"l1all,\'11111.1101\'1).amlis "/lII" ,I1.lwI'HrilinCI'compone nt .Now because

il 111<"1lf..lloll's11ml

1:'(I'XPli i)) a'

('xp {T)

f'Xp(:!U1)_1'x p(111j (LI)

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C'fJl1( Yij, IIi/ )

e;

[('xp(,r~d

+

)j11

.'xp(.rG,d+

~).

^j⁼^I...

".\, [t-xp(·l'0i:l

+

)ill

+

1'1t,\.ll'xl'( ,r~ d

+ 1,11 ('xP(J'~i:l + ~) +

l'Xp(1J,;J,Jllc'xl.(:.!,.,.l)-

,.I'I'(""~)I.

j

=

1••• •11••

/~'(f/ij!Ii.i')- /';(l/ij)/~'(!lij' )

(I.~)

(I.!l)

Thecorrelationof!lijand!Ii)'(j

"il l

inc reasc'swhenf1~lH'fl Ul1l'slilrl!,!'!"il.S follows:

(',/,·r(,l/jj·Hij' ) = ⁽:m'IYij,lJi)')

V

ll fl/'l!Jij)lIfll'(!IV ) {

0

ir f1~

⁼^U

I lff1~...'XI.

u.u )

Therefore ."~tuay1)('considered ns1,lwindr-x

or

till'iuLr il -d us1.c'l·aSSU(~i II LiC>1ll'ill'lIllw1./·rof tile observationsin11duster.

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'1'1",II!."\'<'11111.1.,) IA;,I,, "J1,with1.'1111111IJ iisIh.'so('IIIt" 11PoissonlIlixe<llIIod d ,The ll..'!>i",h,;,I,;witll II...illlllrul"f'lI ,'!>li lllill;Hll lllf'l lrfJ,l,;(firIllrS1'" is.'i<J1lIllixt'Y1mod dpilfllmclc l"lI

'I'll"lIl/ili''11if"pl1'",d,,,,)ill iul!,1111'I'uis.o;c mmixcducdcl,based011themaximum like- Ii"" " ,]''l"lilll illi u lIurlixl'lldrl"!'''';ul.1 varianre-11I111])()IH'1I1s,audtheempirica lDaye;iilll('5- lillliltiulI ..rti' Il . I"lIIdr,"!'ls,is sllltislin,ll,\',h'Sirn J.lIol\'('\·('t,;twouldinvolveallintegral wh id l,1"•.,.lI"tI'''SI'<'SS,III1, " a l~'l i,'s..luti"" .;\1"11,\'apptoild lf"l1hill'<'111"'11proposed illorder lulI\'oi,lllli s dill k lllly,liS,1,'S,'rir.,"!1ill,1l'lItilillIIII'II1'XI. ('hllflh-'t,

Il"""l1l1y.Wiu,lillI'ill'willLiill1~(I !HI:!)lIs(',1I,hePoisson mixed1l10lIdto analyze 11 count ,Ialus1'l. orill"lllin',1111I1111111(',1('lid"lIl')'syudrotue(A IDS)('/1M's, :'I'lon.'specifically,they ,"Irlll llllilll"\lII,;ly''S1.ill1;ll l'«(IIII' AIDSillr i,I"110' grow l hrate1I('to';'''I,12slt illaindexed by seven riskAr<IIlPSill..lsixll."lI~nl"hifn'j!;iulls.Ili\S1'(1011llll'umuhr-rorA/UScaSC'!i colk-ctedover

s

(II.==.",)"<1111'<"'111h'l' 'l"llt le·tl)'IiiIff'illl " n'llls(luting1I11'p(·riotl(rom,Jallua ry198::::!10March I!J.-':I.'1'11<')' IllsfI ,'St illlalc"!l1i (i ".I...£-) whichre/ltf":;('ll h' dthe sLtallllll-:>pN:ilicAIJ)l)

~rHlvlh1';,10...f1\'I'r11",1"ho l'I' 111I"a\'I'til g('';nJw l llriltf':i.whichisdecidedbythefixed effects.

IIIfi"'~.WadawiwmillI.i;(n~(1!l!J:I).le'I'dop, ,,1IIlhtt'l'-s\<' I) iterativel':Slimalionprocedure fflrIIll',...lill1i11 iflll

"r

IILI'I'I'1)'111"11

fir

I'at illlll'h'rs,ri.1;and Ol;llthegeucralieedlineermixed

""HI,-\.whi rl.i1t"t'f1IlIl1I,,,I..u-d 1111'l'uisSlllIlIIix("(llI1ot li"1II~I\.spf'CiKIr ase .Thisthree- ato p ik l'lll;\"'l'tfll"-llun' I'HlIIH, .Il'SC'I'il...r] IIs fflllflws:

I.I\ SSllllliligallillili~I I Ii,,('(1v..[ne'IorfT'~.III('fixl'(l dfl'{"iflitrl1tt1cl er sparcupdated II)' llsill~Ill<'~1-lIl'l'lllizl'(II'~1iUlillilrg l'lIUillioliapproacho(Zt'gcrctnl,(19&1),Note that

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this111~lfl'dl'I'I'dm'snotpn'SHI111'SllC'f ilk\·"l lll'Sof1111'r;llll(Ul11"If,'d sIHII'llll.\·1111' kllowlt'd w' 11H11rnndoml'lf('(' lsh,m'a t:ll11ssinll dislributiou.lIS illLti.

'},\ssll111ilLgt.hn!(1~andtl.11""lixcd,tln-SII·in.l,q w ('sl imal urs fur llll'randomt'll'p,'ls)i (i=I..• .J..)nrt-d('\'I' lol)('(lwil htln-lmrodurt.iuu()fl"'l i11lill ill ~Iuurtions.

(1'lisupdated h.\'1J,~i llg"1ll()[1l1'1l1,nu-thodumk-r1.11l'Hssllmpliu llHlll l!.Ill'df,·,!.sof,.

"I'oss'llrod l1{'1tcrru."11"1'11"1~1il;ihll',J)"l llilsalmut.IhpvllH,lilyofl<U..t1asslIll1lll,illllS,II"', 1I0\\"'\"·f.II',!.kuowu./\ sshewnill(~111lpl,·r .1.1111'ir,,,,I,illlil1.I'of(1'1is1101...,usislP lll.

Theabove threes!,'psofHIl'i1.l'l'1Il h',· pl'(l("l'd illl'dl'sni l,,'ill'Olllpll'1J' fydl ',Nol!'1,llill.willt

<1111'\1'updated\',11 11(,for(1'lIromtlu-I,hinlsklJ. 1Il101,I"'f full,'yd!'is IIl'olllp li·d.<111111111'1'1"0' reduro rontlnuosillitcircularrOlslliol11I11t.i1 l'OIl\',·rg" Il t'l'ill(11(J1'Oil"oftill'0l.111'1'pm·illll,,1."fs is nr-hievcd.Hilt\\"1[('1111'1'sud.arcnvcrgono-\\,(uIIIIIll'il<'l li "I'( ~.1isunknown.

Tit"p;"I[(,l"ali;.wdlillt'arl11ixl'dlI)(u!l,I,Sill1illlrto t.lloS('Ill'\V,wlltwiw ,,11< 1 l.ia.lltl;(1!J!l:I).Wil S

also,IIUIIF ('dhyHreslowandCla,\'l oli (I!m:!).lIo \\"!\'I'1'.unlikeWndill viwand Liml/!;(I!)!):~J,

Hreslow1111dClavton(I!J!):!) didnotilSSl1l1ll's[,,'(~ili rd,·n~i1.rfunet.lonsfor .rli11,11"'11./"wIIN ,' 'YiisII \'I'd,orofmultivarintcnonual(!isl,riIJl1!,('(! 1'lIl11 loHI dfl'd s, ItI~I,.'ad,1,111',vi1:1~um,'.1 thntIor

, I

given'Yi'1.lwlirstnmls",:utldrondltionnlmonu-ntsofIii"xistn l,;11111l.]u- s"""1I'1 roudltionul11101111'111\VII S 11~lJed ri('11[unctjonof1.11(·IiI'S! '·lJ111 li1.i uII;II IIIIIl1I.' t1l..Ih~'sl. ,wnurl Clayt on(l!m:l)lirsl ns,'(l l.he11I't1i\1i1l1'c1quusl-likclihood"sl,imilLiotl appnmdlI,ll('stimal<'fl

;l111!'Y;,Tit.·)'theng,~ t1'~ n\1.I'(1IImodiliodJlmrill~(llllI:li.likdihoOlI fll1ll"l,iliJ[Forinr,·t'l:Ill""fill 171, Sixprnct icnlproblemswerediscussed toillltstrll!'~tll(~wid erimg" ofilil pl inlt.iultsof

(19)

thelr apl'"wwl.,l-or cxumpb-,BI"l~I"willlll Clayton (IJl!J:I) llsedthePoisson mixer!model10 1Iflaly;wi,,",!!luI.,llIla sdof Sl'iZlll"l:S[rum.'j!Jl'pil"1ltirswhowore randomizedto 11 newdrug 'Ir.1 pl' ln ·l",as nn Ildj IlViIllI.1.olh,~s1.iulI[;lrIlr11l~lIlotl ler" I'Ydurillgthe 1.11'0 weeks before each urfour dillkvisit s.IIIanother"X1llllplf',they appliedthePoisson mixedmodelto analyze '11101,11" ""Olllll ,.1;11,11 Sf'lofIJn'1Is1,',1111""["raIl'Sill Icelandaccording10ye a rofbirth illII mli"l'tsI"WlilIXIm-HH!JlnI!HO-I!H!Jandi1W~ill 1:1gt'OllpS from20-2·1yearsto SO-oS"

yr-nrs,lIuwl'\'f'r,tln-ir~df'rivnti(Jll"ortill'p"111l1izl'dqunsl-likclihoodaudthe modified profile '1II<I,~i· lir'I'lil"'l H tilll'(Jlvf"!SI'I'f'rallul llor'lIljllst.lrwlll,s11lidapproximations forwhich¹¹⁰formal jus l.i[il-ill.i" rrWil Sp,i\'f'II,AsSl lll W tlinClrilpLf'l'.J,thisf.os1.im<lL...isalso notconsisten tfor(J~,

InSllllltll<U"Y.1,ltI,11/l l' lklll.ioli"I'III,'Poisson mixedmorlelhnsbeen hamperedbythe lack

"I"till'illlillyl,kIortufor1,111'illl,('Anllo flllf'joint deusltyfunctionofclusteredcorrelatedcount ,lnlH.11111rllll. loill<'Iff'l,tswil l!1"I'~ Jlf'("I.1.()\.1]('random ('{rc('Lsinevaluatingthe ruarginalIikeli- huwl.Milll,\"Hppl'Uxi lllilLI'methods IUlVl'1't'1·I'U t.ly])('('nproposed ,forexam ple, the I'ClHtlil'.cd qllllsi·likf'liluulll HlJllI'lli\chIll'Iln'Sln\\'RlI ll<:lily1.01l (ImJ:I).111111l.h~generalizedes t ima t ing Inurf.lonilp p l'oiwll01"\Vilrlilwiwand URng(lll!);J).But LhcselIle t!to.l s arcfoun d 1,0produce iu("ullsis ll'l11.inf,'n 'lwl'forth('vnrinncc (Ifrandom ctleets . Thisinconsisten test ima te of the varium-r-"OIllIIl1l1l'1I111111.1' fllrtr ll'1'1[I'Kn llll'I,ll(' I'sl,illlatiollofothe r pararnekcrasuchusfixed dl" dpa1'aHl"1."rsilllilnuulmudrf'l'ls.Onl.1lf'other 11I111d.boththe penalisedqueai-Hkclihood Olne!,e;"Ill'I'uliy,,' ,1,'s l,illl a l.illt!;furwt.iunn1l'lhOl!.~Il('('(ltill:ir.I'l'a1.iollarnongtile three typCliof lili ril l l l<'ll' r.~ .illill111111'1IIS!!;ll l,\";lll'oll'"<llargl'lo a dofeompu uu lon,

IIIIIu-ll lf'l'Ii,~.^11"I'jJl"Ill'U_l'il~\\"o.sll 'pappro xlm at e likelihoodapproachto estimatethe fixed

(20)

erfed parameters.random I,rfrd sali t!thclr\"l l"i'l llt""cOlll ptm t 'tl lilltil('II"i llllO lll11ixc'llll\t"II,r,

hllst'(l onawell-groundedfndt.hat1111'rO~Mi [rllnofH~,l lllll,al",lIUl1l111\',II'j "IIII'is11(',lrl,l"

uorutailydistribut('IIwhenilsnll'i llll('I'is Ill'M;;'(' 1'0 .nndisuton-1"'lI kcxlIlrtl lllll ii1s1'0'111"1"

l.hulll, hc dell,~j lJ'ofa1I01'I11alr-urve-wilhthe1I11111l'l1IP,U1'1Ill i\'arimIH'wln-nlrs\'ilri~IIIl"I'is111\'11,\' fmlll xc ro(1I arl,ll' tt,lUll l\elU];ll l !!I·lr;),111111C'1irsl,slc-p,tlu' C'llllj11W111' Illl,\,psilllll.lll'o l"l,tlli s

ill)p li(' d10t'OlllIt l'1lctan'l p proxiI11i11.('likl,lihoo!lIuuej.ionorC"llllll,I'I"l'(ll'{lrn,I,ltc'llfUllll tIlillil,Vi (i

=

I, ",~,)ill orc!l'r [0<,sl.i lllillt'IIII'IiIIn,1 (11,TIll' resul till~Illlll ru:-:i ll lal,psr'un- flll ll't.i"l\s fOI"tilt'fixedeffe('tparauw-u-rsnrcsllrprisi ugly1.I1l'SHIll<' i1Stil .. IIIHl'p;illl d t',<tili li l l.illll;fllll d .ill llS usedillth eGEr.ASiIresult,if(1t\\'('1"1'known.thisapprtmdl\1'011111yi('111till'S'lll\t'l'sl,itilill.t's ferthefixederred parametersas1.111:emF,\-\'111'11".1 ilSI,lf1I('l '(ls1,0Ill'c'sl.illlal.c'tlil sillIlslIlll (,ilSI'S, thi sapproach producestheap jl nlXill l1l11'likpl i h omlhil~t'll l'IllI,~isll'lIt,l'Stillml,,'Sfil l' 1,111.11 thefixedeffectpa nnnotcrsandthe'I'ariall c('c" lI 11P0 I1I'III"audallYilCTlIr;lt'yor 1111'pslimatt'S canhI'nchicvodbyiucrt-asiugt.llt' numlx-rof1',111,[( 1111)'st' II'ft,'ddll,;1r'I':' ill pl"ilwi l' l ".Fill"

smal lqt,curestimatesI'11"C<llmos l,ellldeut(illtheSt' IISI'tllillL1l1~yklllitIl ,)(' plli,'il'llt.ilS theq~gOt'Sto zero] (orboththefixeddfcdPHI'Hllldl'l 'Sandtil('va ri llll t"l't'CUllplllll'llL I'ill' [ilr g('(1t.0111'c'stillliltesarcilsJ'll1p l ol,k l\lIy11llhi' lSt'd1'01"lilt'[ixI'l1t'lrl'f~lIJ1l1"iIlIll'lI'TlI, WI II~Tt'lIS

OUI'est ;l11al,I'isas.Yll1plol.it'aU)'Iw g llliwhillsl'l lfor-1,lw\'1lr i ' lIu ,t, t:oll1polll'n1..IIIthr-St"'lnll l

stell.l1siugtl w (..~tillmtl'softJnud"'jfro mtilt'lirs1.SII' II,WI:l'slilllilti'~I,(i=I" , ,k)I,y l1linim iziltgtheirapproximateposterior 1J11'HIISII IllU'" error-bas''l l Oilt!...I'mpi r;"illBny, 'sill1l pro ced u re.Theresult in g estima tesarc allilostcpt.hnal(illtil"St~IIS"thatthey1,'~lIti1..,Ill ' optimal,ISthe(1'tgoa; 1.0zcru )for~I;(i

=

^I^'.^"t')wlwlI"~is slll1ll1, Alu!tlwrfll:sirlll,lf~11l1·ril

(21)

is11ml,"ll lik"I I",IJrI'\'icollsill'J!rfJild ",s,UHr1'Sl illlilll'Sfo rhot l,thefixeddfectpa rametersand tlw~·;,ri;...rr-'·coUlI....'!II·" t1I11ly ,1"IK'lId 1111lIlf'Ilisl rilHltiolloftill'rand om effec tsrather th/lll tl... ,,,,ti lll "u.,."ftill'1.1I..luIII,-'f,'("t s , Furt!lt:nnon:,t.heproposedapPro1lchisdemonst ra ted tll;otllw,"sYlIll,t.fJt i.·fu ....'rilIllN:"TtIll't'Stilllillt'Sfo rfi\yilll)(~!Illli',l1crillgcncrll.la.~(1'1•

•111i",l,-x"fll,,-iut.r..·dusll-ras.';(I(;il,tioll_gl'l!ilarge r,alldCAn helIignificantly reducedby IlloOill~th.-\,.,1",,,, uTL1,,'fix...1.,If,·(·ll'uvari a !c'".rijasl1ilf,."n·ulillipoeibk-i1nlongdiffe rent

snun-"I'.,lll~lloOl"'lIwl....dll,'!!nllluug,liffer('ulobservationsill11Ilycluster,th easymp totic Vill'iall"" of111l-'-stilllillt-forth..'·fll·lt'.~pOlldin~fixeddfedparame t erlIIay increase as112 p,dslilrJ.\f'I','l'his1"',11111'1-lIWJ'Ill'usefulilld('.~igll illgav/llidcxpcrirncut orsnmpling forthe I'"iss"ulIIix,~llllocl,'I. 1l1l1,'!iSHu-III'I.U/ll"1 issmall,thefixedrlTf'Ct covariat esshould1)('

Tin''II",\~'n'SlI llsfil l'1.11('I,n'I'OSl'l1IIppr(lftr!larcpresentedilldl!l ailinChaplN:J.Chap- lo...:lillln"lul'l 'lillll'hisin ril'lll l",,'k,q olllldoTUIl'Pois sonmixl'<IIllC<Icl,In('1,apt l!r1,\1'('

:<1','l1l l1llIIIC'''lltilllitliullforllrulill'uftheGEl-'1I1l11tllC'I'QI..,fOf'the Poissonmixed model, 1111,1"Isc,Shllll'lllil l_IllI'''' ' lwuntl'lllIJ(l~I'rVtllll1'illl'u'lsi~ l('ntt.':<t i1l1llt io llfor thevar ianceCOlli- I'UIN'Ul.TI"' lwrfu l'II"' I]f"(' ufIIII'l,mp":'I'11two..stepprocedurei~Iurthcrcomp aredwithtile

<:1';1"illloltl,,·I'QI.Ihmn ,;11n"illlllllltiullstudy,illChlllller5.Theproposedapproachlip-

l'QI..fu rsumllill<\1'1'11 aslilr~,'rr'~,Ourilpl,ro<l fhcanII('IIsl~dinthe clusteredcount.datil

"lIU!i,'li,wllil'lIlIs,,"lll.1'h;I\'pIIlal'p;,'numberofr1lls l.f'rshut.relat ively asrualluumber of cluster

(22)

l>i)(l'li.providodthr-assumpt ions\lrIlll'l'ui:<'''Illillli:W l;IIlII> , I.,I IITl'vnlid,('hal'lt ' rIi~i\",,:<tl...

cnncluslouaU11~11I1'llllAAI'Slio llSrnrIurrhc rTl'Sl·arrh.

(23)

Chapter 2 Historical Background of the Pois son Mixed Model

2 .1 Po isso n P r oces s

C"lIsid"r "IknlllUUiprocvs s(k(ill<~1over 1111intcrve!of time(o r~pacc)sothatpisthe pl~lh;dli lit~·thill. ,111 "I'('Ut.ruayOC("Il'duriJigthet.imc inte r val.Ifth etime intervalis allowe d I,u''' 'COIIIC'"hurt,'r1111<1~11(Jrl('fsullwt1,lwprnlmbility,p,of alleve ntocclIITing intheinterval 1-\,'1,1slIwll"fHUdI,hl'IllllTllwr(Iftr-ials,II.iurrcnsesillsuchafashion1I1i1tn71remain sconstant, l.I11'lII.lu-"XIll'l'!,I'i11l11111 Iwruforvurreuccsinany lollliti meinte rval remainsthe sallie. It s-an1,,-SIiUlI'l1l.hnl.,IS/Iw'1.s lar w'1I11d11gl'1,sSl1ll1llso thatup remains aconstan t, II,the hilIUllli..[,lisl ri h llj,inn apPHlildwstil<'PoissondistributiongiI'C1l by

.r(!I;ll ) = ~I'XP(-Jl }

^:r^{= O,l ,} ^{.. ill}^>^O.

10

(2.1)

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Thomenuand\'ariallf('ofthePoi sso ll t1istrihul ioll.1,,·110111I' ,

ThePOiS';OIIdildriIJl1tioll llOl"C'S~'llti,l'additlvr- 1)n,!It'fl yIhatI Ill' slimuftwoil1tl"III'1I'/"l1t Poissonrandomvarillbll'Swit hI'Clril UIt'l I'l'!I1' 1alit!1'1i1'laI'ois,,,,tlll"'lit/o m\-aria"I,·willI I)a.r il ll lf'l cr/':::/'1+"2'

,\1'0isSOlJIJr·Ol "f'!>....fo r a('OlllillllOI~~lilll!'!Will i'('il UIII'dl'Ji'It~11lI1.,It~tlllSIt.IIBI'n "" 11Ii prO!1'!>....onaIliscrl"'j(' lillll':>1·.../1·.'l'l,f'I)ois sun pmn'.'lSIT·rl·...to 1111'... ·I·lIrl1·lIn·uf1'\'I'l lhillulI~

1\C':lJlllillllOlllilillll'" lorIO(,fll-iou)";('a ll', Forill1('l npirif"111l...·k,l;fl'III ultukc-rallll" m1'11'1111'1 suchasdisillLl'grationsofpllrlkl('s,illt"OI ll i llp;11·11·l'hUl IC''·Hlls.'lll. ldlfl'IIIII:;tJl1It·1....·;lki l)l,,'S Hild erharmful in m li'lLilllL, All Ol'l'Urn'II('('S ItH·"S.~l1l1ll'rlt.oIll'orI.II!' Si ll lU'kind,1l11d\\'1' 111'('fUllt' f'l"Il ('(1withIlwlot..d1I1111l111'rorO('('UTn'II('f'Sinan1ll'IJi1.tHrytiull'irlkr \'illurII'n~111I.

gllr hllt't'Ul'rI'lll'I'isl'l'\lIT'S('IIIN!byIIpointouthr-tinll'llsi s,,11111111',lt'I' \\1'ilrl'n',llI yt'lll1fl'r11I 'l1 withccrl a i llraudo llll,lill'cmcnls ofI'ointson IIlim-.f'lu-1I11l 11'flyill~\lhy"in, 1a"-~lIml'liuni~

lha ltil("fon."l'll andi..nll ('IH.· !.'l'I,;o\'(' r nil1';tl1l'IIO"'!''';'' "'1lmi ll('lIll!tll<lIlillllIllll lIlt'l.ruh..l.ilily orilUYIla rlk lll"reventistlWltill11('foralltillN'iuter valecf(lllraliul lI. illlilisiUII"I"'l1lll '1ll" r l]u'IMsI Ill' \'Clop m l'ntoftl,l'Ilmn'SS.InIllil l!II'1lla l inlkrillSt1,ht111"il.1I"t1wtti" ,111't1l"'!ISis

I.Till'"probahilityoffill{'\'l'IlLillall}'sltortilll<' rvalIlul

+

61itll,6lC!lrtlplIr tiullllIItl IIIC~1('111;111uftheilll l"'r\'lll)forallWII Ill'llofl.Thispro!ll'rlyisknowu;L~:;1.I,ticmOltity.

:I.'l'he1I111l1he rof(·...entHilllillyiub'rvil!Orliuwi"illd(~pl~IIl/'~lltIIftIll'111111I111'1' lJf••....·nll'!

"

(25)

in,IllY«tln-r11(jIl·UV('r hlp pill~ill lc~rvaluflime .

'1'1...pr"halo ilil y11Ii'~~fllllr t iclIIlIflll'~numlx-rof('wills11;11 limeffora POis.·KmpnK'Cll!lis

~iVl'1I1Iy

[(!Ji/ll)==

(J~r " XI~-,Il)

U==O.I •...;l>O; p>0_ (2.2) wll<'Il' !(y;/d j i>lUII'IlrHlmhilityIIf!/.~VI~II!.l<illlilll(,t,

2 .2 Mode ls for C lus tered Co unt Data

(111.1"ri,Il'll l i ~,l', 1"XI"' I'imf'lll,a lC'(lI11 lit.iOlI~W1WllSIlCCL'l4sivceVI'II1sOCCIlI'independently1111<1II~

Lln-";'llll' Tilt." tll!'ll[Ill'illf"f('lIn' furPoissouronutda~aisr{·lal.i\'dY (,il.~y.ilurlthelradi li llllal

IliAlilll'lIr IlImll·jnudma"i lllllilllik{·li IIOlllj l'!itima lioncallbeusedforthispurpose.Howe ver,

ilr "l..I..· l·:-il>l...-u ...rrur....·vl·r lll1l·/I.'<l)JIJ!I. Forexa mple,inhdulVio llrlllstudiesinvolvin g pri- lIliIlI'llll r ol l...rm,illlill>l.iudcll'IlI-llIl>lllllllyoccurinspllruorc111s1 ~.ThenetelTcct. isthai IIll' 1Il11111,,'f"fll.'n>nl,"(I('n'll~is 1110re variablethan the simplePoissonmodel wouldsuggC31,

""Il'.1l1lll'llSthe-n-i>ls!rollgr-vitll·III·.'tothe-1"Ull tril.ry, WI'avoidtheassump t i onofPoisson

";'rialillllilll t!as/'UIn' ·llll'aplH'MiUll'I'ofm'('rclisJlI'niiollin1)0;S5011countdata .Inbiomcdi- I'ali1111,li"illi1111SiLis "Isoture-lylilt'('as('lhillVrlr (y )=gCIJ)asi!impliedhythePolsscn ilS>lllltlpt.inli.TYl'knlly.I,11l'vlIriulll"l'C'X('l'l't1sl,hemeau(Ul'(-slow,I!JS,j).This ove r-dispe rsion ('lUIlu-,' xpln ill(~1h,l'aSSllll1il1P;thi\1.then-isnatura l heterogeneityamong the ex peele dre-

12

(26)

marg ina l disl rihl1li oll oftlu-countsisIIII'1l('.u:a 1iw hinumi a ldist .ribut.ion.SIl,'cifir arr,\'. this distr ibutionarisesfromthe As su mpt ion s l,hat

I.rondltlouelallIii.tln-rcsllullsP\'l1rinhh'IIi)hnsaI'nisJ'();\diJ'l rihllli llllwithllll'i\1II' ,.

TIll'JI,till.'marginaldistribntionorlliiisIll'gi\1.i\"t' hiuotuiulwith

'1'111"uscofUK'nl~gil ti VI~binomialmodeldateshackill(1" lS1.totill'workIlr(:I'l'''llwoOlI,'1111 Yule (1920) who modelled eve r-dispe rsed al'l:idl'ul,(,OIln ts.Hrcsl ow(IBS'I),llr illilll!;\'r(I !Jl'ili) , La wless(HJ8711,h)andMCCllllilgli and~I'hl,'r(I!IS!J, S"I',(;.~)disl'll,sstill' ,milly,sisofnlllllL datuwhenextra-l'oissollvarintlon isprrsout,[Lisdl'siral ,l"1.0(lSI'iI1111l1Mthaillllmv:>fur th ('possihilityofextra -Poissonvnnntlonir wI'IlrI'illL" f('sl l'iI !,rilllilrilyill illf"l'I'llrI' nlll l'C'l'lliug

!'egressio nparam ete rsandifthr-~itllationi~onl'illwhichoVI'rd ispI·r.lioll mllti rll·l)'"'TnrN.

Recently,])1.·a11 and Lawlos s(illS!)develop!.I'sls fordekdi l1 ~l' xl,I'1l-I·ois.'iuli vnl'illt.io llill illlillyxing':OUII1.dala.])£'1111(l!)!J~)lunher,levf'1olJsII1I11ifyill~1111't1r\ldfor()[,I.ilillin~1.I'SI.S forovcrdispcrulcuwithrespectto11naturalexponentialIamilywhir-hrd( 'I':>1.0I.JII·r-xf.ru- Poissonvariut.ion asa s[ll't:ia lCI1SI\

Thesimp lestextens ion ofthenegativeblnotnialmodclis10/lss mUI!1.I111ttill!11,,1~!IJl'lI r1 lJlI cov ariat e!>'1';through S()Il1C]HH1tl11d l'icfUlII:l ioll. Till'lll\ls1.l:OrllfllO!li:>HI(' IU/l,-lilw;lr1110,[,,1 forwhich

(27)

lI~illV; llll~lo V;·lill"i1l'm,,,ldtohllilIY~I~ in, h~ pelldellLcou ntdatawit h overdiapersion is also Ili~I'nsSl~IIJY(:Inyl.<muudKuldor(1!/..'17) ilSwclIasbyA.JcCuUagllandNeide r(J!!S9). Actually, CI;,,Vt l>llmid]{;'[dor(HJIl7)lJHI'd lllf:log·lillca rmodeltoaJl;ll~'zeohservedand eXIlCcled 1IIII1lIJPf S lI(Iijll'aJwl' rnISI'Sint.Ilf~.'i(if:(JlwliCHorSm1.1iwdwithfl.viewtowardproducinga m'lI'1.11"1.\\'oul,1disp layl1'gillllalvariill.i(JII"~in n"wel' incidenceye tnvohlthe presentationor 11IIs',all l" f11I,('H [url.ln 'slnilll,'r{"Ollllti,-s.

()n" illlpurt;1II1lim i1.i1li"northislog-lilil-iII'modelfo r applicati o ntoclustered da laisthat tl",PXI,l;i1I1l I.OI'yvari..I,I" sinI,he rrgrcssiouabovedonotvntywithindusters,Itis unlikely thaI.1,11l'r1l1st('n~1('o lltlldalaill·I'illdcpf'llllt'lIl..Theresp onseswi t hin aclusterar cgeneral ly

11'11"\1l.'-~l.in/!;1.1l!'uvvrnlldli.:j('IHT

or

itlI'-Wdrug.IIIotherstud ies,thedepende ncyisLhe IIlllillr","us,fil l"('Xn llll, I,'.\\'III'1.I1£' 1"adisl'flSl'runsinIamilicsorhowadiseasctendsto progress.

'l'ln-l.r;llli1ioll ll lf'-grf's sillllllS...;tl lllp1.iolls llmlHLl"cosponseaareslatistically independent with l'UllS!.;lIl1. l'ari<lhili1.yil,holl(,th" i!'"XIWCI,('f1\llJlICSarc IIOtsatisfied. Asii,l"esu1t,thc classical

~1,all(l imlll"~l"f'S~i'J1 I 1lll'l.lrod~auchastilt'logJiucnrmodelruuygivcinco nsistent amiinvalid in(I'f"('lwl'S.E:\lI'II~ioltsflftill'log lilleilr U!OI!t'1 whichac cca»t fordepen dencearcnecessary illonl,'rtoohl;lillndillillr"l'pt u· ('S.

IIIAI'II,'raJ.II\('Hll ;ll.v~i~ofdisrrct.r-forrl'lillt'ddataiiidillicult.partlybecaus etheirjoi nt llist,rilllllinliis hal',lIy~p,·,'iri('d11'1-11. Itisneuailyroasoueblc1.0 assumetheclusteredrc- Spll WWSfromlli~L;Il(' 1dusl,'rs<H I'indcpcndcnt,hut withiua cer-tainc1ustcr,thecluster ed

Tl -SI1llIW ' I1:11_11ill~'(·ut'l"(·lah-',1.Thi~distingulsh cs Ih,!e!lls lcl'I,(jdata [romot hertyp esofmo re

, . ,

(28)

2.3 Mixed Effect s Model for Clu ste re d Co u nt Data

inrecentyearst1lM 1.11("1«'11l()(ldshave,l! lrar l,'duuu-hattentionilllilt'sl"l lisl.jl"alr<'S" ;IIT h lltcraturc.Thesimph'l\tmal welldevelopedmixed modr-lswithilSSUl l1l'drout.iuuous(:illissiall rcs pon s os arc till'lillearmixedmodel, illwh ic htil,'rc'Slll Jlli'll'isllSSIlIH "t lIIIIll'illhu-ar [unci,jonof cxplauatorv \'ilrinlJl.·swit hr('Arl'll.~i()llru d lk il '1I1,sl1wtV;'l')'Irorn(JIll'il),lh'itillill

for ill fallLgrowthwbcrethecod fid"lll,sn'Ilrt'S( 'll~blrth\\·,'igh1.umlAlllwt.h1",11.1'.(:llildl"l'l l obvious lynrc born nldilrl'lt'litwI'i ghtsHlllihill'('dHf('l"{'111-I-tI'll\\'t.hI'ill. 'sdlll'til W'IH'tio-'lIlt l cuvlronmenta l factor swhich111'l'dilliclll t orilllp()ssjh l(~to Il1IlUlti fy .Amixeddr.'I'ls1II11l1t·!

is11l'C'lls(JllllblC'dcs cript.ion iftill':owl.ofc" M'lfi d f'U1.SIromi~pOIJIl ]; llioliorrhildn-nrnuI",

linear mixed('rredsmode lfur therilS~lllllf'Stha tllw ohsI'I'vatillllS011rhildrr-urill' !.lliItIamily lire iudcpendent. The.' correlat ionatnougclirrc'l"I'nL"lisPI'vatiolls ,u"i.-;C'SI" 'C'III IM'WI'nlllllOl I ob.~f'r\'clllf'underlayingfamilyeffect,lha lis,\.1 11~truel'C~Kff"';si(1Ilc:o dlidf'll ls,IJllt 111lv c'lJllly imperfectmeasurementsorw('iglll 011r-arhillfa ll1.,

Auni!iN! HlJpnJllC'!Tto liUillglhc'lilll'al'lIlixc'c1 l1lodpl.Ims('<1 em II"o llll,illill,ilJllIJ[till' empir-icalBay ('sinllnndthe max imum likdillooc l c'\;liULali"l l uf1110 , 1..11'1I1'illlll'l,"I"1<;'lI rlIlSjll ~

(29)

(I!/!J:tlpn·sf'rrl".1ilImlillll:flw'ril/l,1'1IJ'tIll'Iillf"iHmixl'llmodel.

TI.i,~id/'i' f'xll'HIlsllal.llrldl.l'torq!;l'l~s i(J1JlllCldelsfo rdiscreteand non-Gaussiancontinuous ff'.~J"JJlSf'S,II isilsslllm~1I,h"l,1.11l'dutaforiIsubjectnrc independent observationsfollowing

iI/l,1'lll'rllli;;,,',1liuc-m-mode-l.hilI,lh..lIll/'I"l'gl"l~si()l1coefficients1;<111 varyfromperson 10 p"l"slfnilITO"IliIiAl.u ..r1isl!"i hlllioli.F,To illustrate,consider a log linearmodelstudiedhy WII,.[I,wi\\,il'l,1l,i"IlA(I!m:l)fnt'tllf'l'l'nl,al, ililyfirtil "uumher-of theAIDSincidence across S"\','r;,1W't,/.!,rill' hk1"'l\iolis.Wf'mighli1SSI1IllI'thatthe AIDSincidencegrowth rate Viides

;",,·.,ssA''t'j!,l"iIIJ I,if'l"!'l\iOlIS_ l"f'f1I't,tillp;tlu-ir diff" l'cn t eulutrcs, livinghl'lhitsandunmeasured illll'lf'IWI'Soff'I.,'!"'"lllll'lll alInrtm-s.This Sill1l'll'sLmuddwoul dassumeLhat C\'CI)'geographic n'/.!,iunIIiISits O\\'1[AIDSinri ,ll,tIt'I'growlh1'1111'htltlitec/fect of the....vcrnge 11l1111WJincome linIll isjll'"IJll bilil,\' i,'ttlrf'SHl1l1'fUI'I'\wy W'tJ/{l"ilphkregion. Thismodel rakestheform

(2.·1)

\\'11t'1"f'IIi)IlIlll%')rt-prr'st-utIh,'11l1l1l111'rof1\lllS('''!;l'Sandthe averageannualincomeillthe i1.hp,"' Ip,I"i1phkr<',l!;i"nIII1111' JUIvear, andIirepresents thegeographicregion-specificrandom

"trl'I'I,All h"llp,h11,,1 \'f'r,1"n-asonebh-,\-\"a <'1 all'll1'and Liang(1!)9:l)assume thatgivenIi,th(' n'p"i1I,'dOhs" ITa l,iullsIii)(j= J", • •lIi)furtl[(,hhgeogra phicregionMeindependent

or

^one

nmutn-r.Fillil l l.\"till'mOII"1n'f1uin'l-;nn ussutnptionnbout thedist r ibution ofthe 1';IICI'O~S ).\1"'P,nlll lli<'1"'p,iUIIill1111'IIOI,ulilt ioll. 'I'YI,il'lIIIY,1IPilfilll ld l'ic modelsuch i1Stho Gaussiall wil h1111',111;;,f'rllnmlunknowu\'ill'iiIUl'f',f1~,i,'t Ilst'd,Thisvariance representsthedegr eeof 11<'1,'n ').\I'Ill'iIY ,U'1"IlSS).\f'll).\rilp h irn'p,iull sillrlu-1\IDS iucidcuccgrowthrate,notaunbnt ablo

I(i

(30)

1.0,I' ij .

Thegeucrnl specificati on of lilt'gencrnlizedlilll'ill'mixed1ll1ll!t·1isas follows:

I.Gi\,(' I1, ;. the'['('spOUS('S!Iii••.•lIi",<In'll1l1t11al l.l'lndl'IWl1dl'ut,andfoll l)\\' nP;1'1l1'l'ilI1:f.C'11

nndq,uri'1I11kll O\\'Ilpurunn-tcrs, Midt-,'1I1l1I'un-kuowuflllll'liolls ,'I'll<'c""diti ,,nu l 11l011lCIlI.S,I'ij::::1~'(lflj11';):= rj,l(f )ij)umlI'ij

=

^"^{lIr e lfij}¹^1';)

=

^4·^1I(f"j h'l.silli.~r.\·

h(f l ;'; )'=;I'Vj

+

^If;/'linndl',j:=I'(/Iij)rf> Whl'l'I 'h111111I'art'knownlinkaJlll\'Ilr1a lln' Iuuct icns.n'll[ll'('li\'d y.J1is1111unknown]lill'ilHW1!'1'\,pc·l,m·.111111Iii)illa:-1I1",l'I,llf.rip

'} The1'<11,,10111dfeds"i,i:=I••••,t·.un-11I1Il,UOIII)'iudc'lll'lulelll,with11l'U11l1l10111111111' 1"

lyil1gmultivarlntodistributiou,F.

'l'hcmodelthatist.1ll'fO<'llsof1,11l~mnaiuderofthisl,h('sisistlu-I'Oi!l."o linrlx«!llLodc,l wuh11l1iVil1' iill.l'1'nlldolllc'lfi'cls asfollows;

,}Gi\,el11'i.theresponses1111" .,IIi ",I1I'C~illdqwlulc'l11. Poissonvariuhlcs with1111'.111";( IIIJ1

:1.the·iiill'l'illdt' p(·udclll.rellli:f.iLtioliSfnnnanormal,lislr il>l1 l.iollwith1Il1'1111~,r'I'O.1IIrl

\'i1,l' iIlIH 'f'(J~.

11ITOSSiudivldualsin\.hdl' n'IJ,I'('ssion('(/I'Hil'ic'nlsaudLhutthis11I ~ll ~ r(Jl!,l~rll~i lyn,"Itc·n'pn"

(31)

~"llt" ,ll>y" dlJ~I" I'

"rl',,,'"

wlli.-11IHIHiIprol.'lh ilitydistribution.UorrclutiouamongobH('r\'(l- l.i"lIsk» 1>11"dll~l..rilri~,'HFmmun-ir~ltilrillguucbscrvablovariables,I; 'Inthe mixed model.

II...1"U11.li1.ioIiHI I'ro],' I]'ilitydi,~lri'llitiflIlHoftil"r('S1'01l5(~5atglvou difforoutsubjec thhelong tu'I~:ihgl.,f,lIl1ily,lmL Ill!'1"1111(10111r'Ilcctsvary,ICI"OSSsubjects ,with a ronunon distribution .w1Ill'fir,~I,two,'OIIlIlIfJlIIII11lll"lI ls, sllI'l'ifi,'dut1.111' secondS{.IIW),Therefore, theyapparently

"" I1"' ·l,11l'l<'roW'IIl'il..viln'lSSJ1;rtlll llH illl ]u'r! 'p;l"l'SHi(}l 1nll'ffie ic' III.S. il li dasscclnj.lcnwithinthe

S,OIII",l!,roUIlillOil'"I>S"fViII,iwIS.SlidlIIlL",'dlumldshavese\'{~l'illdesirablefeatures.There

isIIlJJ'I" I'lin 'IIlPll t fHt'hal,IlI"''(!Ilil~ilill e1i/ferl'uL/.(nmps,They allowexplicitmodellingand :t!lill.rsi,~ur ),1'1.\\','('n-illIIlwitbi».gnlllpJ"I'SIWIIH('S.'I'lu-1'111111011)

"rr"Cl5

panunotcrs havea llHI,III'HIinl"I'jlrl'l,i1tioliwltkhlsrn " pwntlyrek-vantto the'p;oalsofstndi('s,lI11d theirestimates nUl1lC'UHl" Ifor"XI' I<lwl or,\'11I1111,rsis .TI\l'S!'llltlfidsl11HOIacilitau-thestudyoffixe d effects

TIll'IlI; XPt[l,ff, 'r1,Smodelismos!. Ils"rulwhenLI!(' objective istomakelnferonceabout illtliv;,lnalsral.llf'r1111111tln-1'OIIllI'II,iollavc'ragl',IIItIl t)IIIJOv('AIDSincidencegrowth rate

":I;illIlll!<',Lln-l1Ii ~,',1('I[l','(,slH\lIlt ,1wouldpl'I'mi t.inferencenbont theAIDSincidencegrowt h 1'111,'[01'aI'Hrlinl!lIl").!;''tl.u;nlpltkn'.u;illll ,TIll'n'gn o.ssioll(t)t'lfiti c llts,11,n!pn's('IlLth" effects ..I'011','xl"I11111101'.1"\'iI1'illhks 011 anindivirinulchild's chance orinfection. Thisiliill cont rast l,u!.Ill'1Il'lr.u;illlllIIlmld!'Oc'flidt'liiswhirl ldf'Scrilwtill' cllccl.ofexplanatoryvariables011the 1"'lurJiI,j'lIl<1""1'11/:;",

11'\

(32)

2.4 Met h od s for Estimating the Poisson Mixed Model

theilllAlysisof sortaldichotomousft'S!>OIISI'S1II'o\'l ,!t'd11,\'111111111'1ofst.lltlyIla rl idi lilll ls.1,;.,dL sllllj"d'si'wri llll'C'spoIlS('Swereussnuu-d10ilri~'fromill()~islklilll'lI1"mmll",hill.with1""I-\l"( 'S' slonrodlicll'lltsthat.rill'YIw (,\'J('{'l1sllhj''('I,S.TIll'logisl it'I'Pg l'(' ssi oliPil l'ml l"le'f s\\','f!';V'~lI nll'( '

bilSI .. [onthemnxhnumIikclihoodcstiurat.ionof[i)((',[('lfl'1'Isnud\'i\ rii llll'P('0Il1111111t·I\I,s.;11111

('mpirici\11Iar<'silllll'StillHlti01Iofrandomdl;'('I.sIVIIS used.TIll',\'found111011,'-X;I<'I,SUhliioilS

\\'('1'('allil ly l.icilHrHIIlIrompuf.atiouullvillfeil Sihll',and1,11I 1sPl'tJl'0Sl'd<IIIilpp rnxilwl1,iullhasl..]

0111.111'11I0dl'

or

till'[losl'('riordLst.rilmlio ll

or

111('randomp,lriltlld ,t'ts, illllllt'llIl'lIh' ,1 II)'IlW;llrs oftheE:"Inlgoritluu,'I'Ill'maindilli-ultyhr-n-e'lwollulf'rI'llwil.hI'il ll\> r nlaxitlrrutrlikdiJre'u,1 01·empirk-al Bayl'sill1l apprOiwlresistlw!.t.hc-dOSf,d'(Ofll l"Xflr,·.ssions fur11l'l"l'SSitl"Y;ul.".u;r;tls ,10 1101.exist..Tllis ('(llllpUl.itliClllilldifficullyirflpl'al'.sill ol.lU'rgf'lll'ril li~I'llllli.'(I'II'llOllf'ls.sudl as1.11('Poissonmixedmodel,and ifhasJlf~'Ollll'II l·lIITI'll!.slilUsl.inl!nos!'ill..-lltupic· will.11 'righ1('\'1,1ofilll.f'U'S!..

hl'gl'tandI\lIrilll (11J!J1)nlst tIlt' gl'J1I-'I'aJizl',ll illf'iLtmixedHlod,'I.sillitfullyBHy,'sia u Irnmework and\ISI·1!tIreGihhs saillplillgt.1',·huiquf'toflV,· rc·Ul1lf'Un' larkofr1"s, ·,j·fOI·1lIex- pressio nsfor Iwn'SSllryintegrals.Compan-dwit h"ady usednuun.riealillt.f'p;r;r1.iollIlldho ds thathas11lou /!;hisl ory(101' l'xlllllple ,Uml/iwinPH!l:CrUllf"lr;\1111Sl'i" /!,l'IIll;lll1!I!lU),1.1ll' sampliug-basod nppruac hesnn-nlll{'(~ptllallysimpl,'Hill! "1Isy1.0iIllIJlf' IIII'lI1.for II.S"t s wi1.11

il\'llilllblermuputiugl"I'S OIll TC Shutwithoutnumericalilllllly1.kill l.",~rHliollI'l'l,,·rth;I'.l'o1..' H-

HI

(33)

till!,Ir••w!l",·bine-l..,I" 1,11f"illlf·t1 ~i \"f'nmlp ul" l iollSHilli(l llf'li lim l ~aboruwhen1111'sampling I'I'"n ':<.'IIII's iw /tjf' w'l l l'lillili l.rilllll(Hillierif/lllKlrklanrl1!1..IO),sudtilerequircrncmtha t con- ,Iili" "if ' III' plilll ,I'·II.'lil y,lisl r illlll ioll S fnri11lfiX!'l1.mel randomeffects.aswellas va riance nllJljlll,,,,,,bsl,olll,1Ill'"1I11j''l·liw·l.r(Imt lIIi1.\"Il<lnotpropocrl.v)aSl,IIIHet!(G el fa ndandSmit h I!I!KI).1.' ')\''1'all,l Karim11!J!Il ) ":0<'11111" "'\;1 IIo11iufo rlll ali \'1'prierlorvariancecom pc ncuts . 111,,1n[lat",iu r rorli~,,,,',·' f''''·l s.TIll'validityorsuehi1S.'1UllIllli o IlS isillneedofci\rcrllljus-

may11I'1"llln'diff ..n-utr<'lilills .'l'ln-n-Iore,strictlyslwak illg,hull.ofthoinlin'appro ximat e in r,'n'III'"11111'1"11;1,.] 1"'1.

Iin 'sl' l\\'1111,1(~ ] II )' I"II(I!I!J:I)nll, lW••dllwiwaud!.iilllg(1m):!)propo sedtwo dilfc relllbut n·llIlI..1 HPlll"Uxilllillt'lIp p rt li ll'h"stot'liLillmli' Ih"gcucrnlixcdmixcdmodel inorder\0 avoid tI..."tllll lll1lilti w lill .lillicllh it'lf.110\\"1'1"1'1'. as it is '11101\"11 ill Chapter,I,holh methodsproduce illnm sis lo'lit,o:<l illlalo'furII...\',lfi ,IIlf"l' ('(}IlIIKll,,' ul iUllu'Poisson mixedmodelwith univariat e r;11l,lullll'lr' 'l'Il' .

111

(34)

Chapter 3 The Proposed Two-Step Approach

Thischapterpr l-'SI' U!,K1111approxiuratclikc'lihoodnpproarhfor Hit'I'lliS.""!11lIIixc'd111l1l1d, Imsl'don!.lit'filettl111lthe logarit hm ofitgam111arandom\'il ri al;[,·islll'ar 1yIlUl'll1id l.vdis- tributedWIIl'11it svHria u('I'is s1l11111,nnd111011'peakedaroundit so-ntorI,ha nIIll'Ilt'IISit.yof ilnormalcurvewith11]('salll!' 111('il 11;\IId\'ilriil1ll'('whmlii,,,\'a~ii'IL"" isIIl!'W'(11'11"1 11'\.1.alul I\l·ndallI[Hli).'I'hlsapproximat elikelihoodHPllm m:1iwliHis t.softwoNtl'I's.[IItill'lirs t.SLl'p, theconj ugate lJayt'sil1l1 theoremisapl,til'd10yieldtill'itPi'rmd11li~I,I'likc'nlwotlfortill'Ii.\(,',]

elfl'l"lpe rnmetersnud thevariall('(~C"UllIP OIlI'IILIIIllll:!iI'rOlIIISlll!;I',W!'d"fllll'!'ll l"aplJrox - ill1!1kempiricalB'lyf'Slllll('S1.ill lHt io llforr,h~'l'ill ldo ll1 f·m ,d.sby1IIilliI11i;du /!,tlll';IPIU"fIX!III,II ,!' posteriormeanSfl ' lilll'f' ITO"

o r

theramloruf'lrf'rl s"

(35)

3.1 Likelihood Approxim a tion

"1t,11<1ll~11IIIlYI'stJJl~JfI~1lI<:it/IJII~ilpp:it:r110rombincany priordistributionwit hanylike- jilt" ",I,itisj"Ull vl' ll it'n l toIIS/'nmjllA"llteprio rsfOl'the unknownparametersbecause these 11·;ttlto Sil ll!llf'1lIlSWl'I'l'.Fo r'f'xmnpll',aPoissonlikelihoodundgemma conjugateprior canhe

illtq~ml,\\Ilif'n'asPoisson1i11l,Hllo()f1andnormalprior rnnnot .Billt)wapplicationofsuch

iInllljllll;al.('l'rior1I1~'llsfllrdllljllstiliratiollinoachcase.

CUlljllll;ilkpr iurshave-!J1'( '11widely usediutimr-sl)l'ieslindregressionproblems(for exam- ph-,W"f^l,HarrisonandMi~(lliI!J~;)).llnrvcy111)(1Ft'r1l1111dcs (IH89)applied this approach to l<lnwllll"alronntdill,a11l0\JI'lswhk-hdl'lirrilll'onlynilecorrelate d seriesorcount data ,Clayton 0II1r ll\1I1,lul'(I!lHi) ill' lllil',1it 10 illHlll'S('iIUI"[)I'II.IClllrrnml,,la tawithcverdlspcrsion.

Hwtill'l'\li~~IJIlmix('(1FIlmldwilhnnivari.ucrandomeffects,the gamma distributionfor '.Ill'1'.~p(jrlJ'lIl,i'llfllrtr'l.il)jJofrcrU.IOItldfl'l'l.~wouldhe11conjug atepriordistribution. Duthe

01.11<'1"han.I,tln-llll':Iune-tionuri\l':i l ll1 ll1l lrnudomvnrlublo isfound to he nearly normally

di~',rilm t.l'dfol'1.111'slIlall\'ill'iillll'l'.andtuI,...more peakedarounditscentertha nthedensity

of1I1ll'IIlidIlislrih lllinll fo rtill '11IrW'vartauce. 'l'hoso interestingpropertiesarcusedhe re to

f'UI1Slr ll \'l I IIf'i1Pl'l"tIxil1w\('likl'lilwoll fordllsl,erl 'dcountdate. inthePoissonmixed model.

Furtil<'[m'sl'u l,111001,,1.1111'likl'li!lomlFntu-tkm forIl audO'~hastheform

(:1.1) Il'hl'rl'f(f/,!1di~Ill!'t'<lIldil,ioIl1l1Poisson (!('llsily lISilll.d. Iliswei!kuownthattheintegral ill,o\'I'(llI('~nut hm'('au unnlyficsolution, HencethelikelihoodInferencerequiresnumerical

(36)

cvaluei.ion,which is notonlydillirl1lt to Wit'.hutalsoyi,'ld" ;lppruxil1l;lll'illf.,rt'IW", :\sn remedy.we11011' propose1.0 rcnsf.ruct.annpproxlmatclik,'liholldIum-tiou<ISfollows,

HI'\\'rill' theconditional Poissonmodel1..1;11 tlu-furlll

whereI()i

=

,'xp(-Yj).ThenI,llt' likelihood Iunrt.ionin:1,1 iscquivnk-ntIII

(:\.:1)

wlwn'.f/lw;)is t,hc'prohabil ityd"llSil,yor

It" =

,'XII(1 ;) ,Ingr-m-rnl,Y(It'j)i:<nu!.kuowu.

IWl";IU SCthellisl.dllll1.iollof')jis1101known. H')iisilSSIIlJ1l'dtoIll'nlll'lll11l,whh-hist.lu-

ell""inourPoisson11li);f,d 111011,,1:1.1,tlll'lI,1/(wj)tunyIll'routputcd,whichhy:1.:1yil'llls

nil'

exact likelihoodIuuctiou.Bill"ilSit.was lllt'lll,inlll'd,'adj"r ,t.lu-ill1.l'I;rl1lill:1,:1due'snothave' thell llnlyl ksolutiuu.Toovercmnet.hisinll',(!;1'il1prohh-ru,IV"slI!\.W~lI' WlIlllI l;]'wor killJ.(' dist l'ihnl.iollfor

"'j,

;\ [1ll'1'spl'('ififa lly,WI'liSt'

A"

,Il(W;)=f(;0l1>i'-1";<p(- >'III; )

HsI,lli''working'prohahllitydonsityofWj,whureI,lli'pllrilllldl'l1Inand>.ill """\'i1IWII.,..1by eq ml!.ill!\.lIlt' firsttwomome-nts or lhis'working'dislrilllll.iollIIItill'r!'Sp"I:tiV('IlIIJlIU'II(,Suf

un:

,"1I1TI'c ldistri butionofWi=('xp(, d .Thatis.

am i

(Hi)

(37)

I n =(' Xl'(11 1 )- ]'

,\= I 1 •

('XIl( T Hm:p(17¹)-I] (3.7)

N,,!.,.1.l1Il1.e"m~'~I'(JlI(liJlp;totill' 'Wl'rkillg'd(~lIsily:1.'1orWi,1;=logW;has theprobability

I/I~(I) = J;;~~~) ,

undtln-'"ll llllI l illl1. I-\I'Ill' ra ti l ll-\fll ll d iol1

k..,,(I)=]og lll.,,(1)=logl'(o

+

L)-I.log>.-clcgI'[c].

(3.8)

'l'hust.ln-Iir:!![ou rind"xI's of tIll' shi1Jl{' or(.Ill',list ri llllt ioll(Ifl i-nK'ill l,varlance,skewness

~'(o)- Iog.\.

,111,1

FII/"(l i ) 1"(0), 1"'(0) 1'/«(1 )3/2'

(3,1))

~"(n) = iJl{)~ I'(Il)

⁼^_(^_

1.. + f- ._ "_.

⁼^¢J(o

+

1)-

1..,

(a.l0 )

()(1 f\ j=l)( O + J) 0

t·l

(38)

in whichl=O.[iii~l. ",Eu[('r'i'Icoustant,1..'1~"(ll ),l!'" (ll)11I1I11/''''(n)n'l'n':'l'utt.ln- li....I,

~',1(o) = iN'(nl

_;hl

= f: _' _._. = t" '(Il+I) +~,

^P,ll)

):,, (/ 1+JF ••~

and

,,''" (n)

= ij4I~~I'i'l )

_do"·

= l ii:

_j",u

-

_{n_+J

' -. -

_).⁼"''''(11 +

I ) + ~.

^(:l.!:I)

III

TlIl'S('l"l,:,nlts ('01111)1'fOlIII']ill1111,)'~11\11(larcll.l'xlhotlki'l11"],illl.Jll !JIISlIll;\11.11\\,1,1,(I!li l l, pilg.,sl!m- I!IS) IIIIl!Villidcr1,111111;11111TC' 11 1111C'(1!1 ~1,IHIW'i'I11 i.I ~!I).TIll',\"shuw 1.hlll1.l,..

'working'density Iunrt.ion:1.1'1 of"'Iiusually follows lIilf"l"l''' !,clillt!'illllliull Inun till' lI11rll1ll1(JIll' ufrt- ••1 random dr."d,s .IIUW("""f,ilisiUll'f('!<l illgtollloS'"r\','lImlwllf'1Il1w1II"l1lalVnt"illlU"""J i"~m'nrzero,Ull" 111".... II"",'Tnylllr's""'ril'li"XI><Ill11illllto;' l' l'rtlXilllil!.l'111..I'Tul.al,ilil,\",1"lIsh )' ur :J.g hranormald. 'n silywith111l'IllC'i\l\ aeroall.1vOlrilllW(',,1,whidlillLllI'I'W'III"11:'1till'

h.,approximated hy

Froma,a,WC'hav('

"

-:\:::::"xlJ(lJj=I,

t.hnt ii'l,

,, ::::::.\.

(39)

I

I)

=;;.

FillilHy,W"hilv,'

11(1 ;)

~ -\'·';~:;~~.\l " >:I)(_~J

IX

I'Xpr-~1

whichis1,111'1101"111111d"lIsil,ywil,1111lf'11IlzeroIIlidvntiancef11,andthe sumo111'1thedensity of

1;ill1,Ii,'I'11lI~.fur«mul](11,1,111'Jl;lllllll Hl'working"^,!cIIsily

a. s

fo,"'l'ireducesalmosttothe l,rIll',]jsl rilmti tHI of 1;,ilIlll\\~'I'illlllXp('C1111IlLourlikelihoodinferenceba.<;Cl'1011the'working' .ll'llsily :J..lurIl'iWIIlIl.1 1M" lIlmns l (,md"111(ill 111('SCIISl'that theytendLobeefficient"...the n',,1,,~ ~UI'!lI..1A·ru).Oiltil<' oth"f hand,it follows(rom:1.9 that,when the act ual002is not lilllilll.L1II'kllrltll'ill uftil l''wur kitlp;'ll.·nsity:1.8of)",islargerthan3.thekurtosisofnormal

in:I.S\\'11111,1I..,mut<' 1>I·ak...1 anulI1,1il...t'I'lllc'rthaI!thedCII~il)'

or

thenorm alcurvewith II... !I;lII11'IIIt'lInllllilnlfimw l' .Thc'l'4'IlruIM-rlil'Ssulllricntlyjll~lirythe uscorthe'working' c1I'II~H.\·:1.-1urWiillrt>l1Iplltiu,l!;llll' J1Pllroxilll<l1f'likC'iihoodfunctionforthePoisson mixed

Nul\',hyn~i n,l!;1/(11',1rl'U1I1:\..1ill:1.:1allliinl.t-gnlti ngoutUti.weobtaintht·upproxhunte

:!li

(40)

log likelihood Iunetionforr:llUIlIt1~ il~

I _, _,

(11,"2)

=

L{o log'\-IIl& l'(nl-E lnr;!I;j!+L Yjj.r~I~

j*1 J,,"I J ..I

+J;·nlog.\. (:1.1·11

Tile abov elikel ihood functi oni~exploill'l:\ill LIll'lIl'xL~uhst"('LitmInnhLa;1l Lilt' t'!iLilllilt.e'!<uf

3.2 Two- S t ep Appro ach

I. TIll' lil1'l,sll 'p

ap proxilUlIh'likelihoodIuuctiougi~'I'1I;11:1.1·1.In~11'PI.till'S!'('Kl iIlIll LI'liitto·1I1 11l1ill.t~ 1 bysolvingLltl'scorr-('(IIIBli o lis

and

17

(41)

(3.16)

!1; = n+ L .'Iij,

j=l

IIi=.\

+ I:e

^Xp(.l}jf.l J,

.i=1

n'la ' ) 11/1,1

;),\ :11'xp(a' ) -1

iJ;;i

= -

2CXJl( ~ )jP.Xp{t72)_1)2'

Nuli-I.I'ill,IIII'SC'OIVfHIICliOJJ:1.1:,forlJi.~the SIIJIle astheestimat ing functionfor1Iin WildawiwnudLi;lII~(I!J!l:l)which lI'illlll'furthereli-'"I1SSl-'(]illthe next chapter.'I'hus furknownn~.ho UIthepr(' !if'lIlapproachan dtheestilll llli llgfu ncti onapproachyield t.ll1'Sall1l!ill1l""C'IlI"1'forfl.But.in pfadin',(7'isrnrnly known.1"'01' unknownu~,the t.WUi1pl'rwwl ll's .,·idddirTI'l"I'ntillferc-'un'Sfor{iandfT' ,Unlike Waclawiwand Liang (I Dn:!).till'pn-scut.nppronrh provkk-stlu-[ointapp roximatelikelihoodestim at es for

fJ

lnxtr-p2.\\'1'.ll'illwithtltl'pre,lirlioll oft.llt' randomdfl'Clll'Y;(i=I,. .,~').tel

v;

1)(' flu-ruiniuuun mean~111li1r('I'rrorpredictionofIi.Itt.lrcn follows that11"" B("Y;lu;).

Now.IlyI'xpluili tll{1.I1l'{'(Imlil,iollill ll" ltsily 1.'1of!Iiforagh'{'11"(i,andthe'working'

(42)

probabiliwdl'lls it:v

a,s

of1;, nut-Oht.l1i ll,~

flY'

1, ,1 "(0;)

J.rUI; I ,;)h('); )r!·y;

l'xp{(n*

+

E;'':-I!li jli i-I,\'

+

L:~';" 1o:p(,r jj,d')]t' xp()jl}

fl'Xp{((\ *

+

L:j;"1!/Ulii

p-+

Ej';"I('x p(,r iJd')jt'Xp() i H, h i' Therefore.

ism-ar\(t'1"O,this('St.1111al.f'isalmostolJlilllal (illlilt,S"IlSl'Ih.lt,it.tends t.oI" ,"1'I i11l.i1

3.3 Co m p u tationa l A spect s

TIll'traditionalNewlonHaphsonitx-rulionpt'on',llIl't'lllll)"1'1111lutu"UII\,,'rl-\"II('"1,n ,I, II'IIISill solving the score1'l]Il11tiollS:I.Iii 1111(1a,ll i,~il1lll1t.i111 '~l1l sly, 1';\'1'11if(11isknown,llll' Sl'Ul't' fuuct .iou:J. li)maylondtonlocalmaximum01'mininuuu. WlwlI(11ii'\Ill1knUWII, asill1.111' generalruse,litesolution hecotucs11l0 W("ollllllifu!('(1h( '(";1I1SI'IIft.11I'nosl.1"idi ullnrIlIIlIl ll!lIry

l.Iwl,l"Il I'maxinuunoft:ursIlI'HI'nit' bcnuduryof1,11l'l';l ri lll ll' 1.l'r1'111111'1'fTl

=

o.Tuilvuid

follo ws:

: w

(43)

I, ",

L L{f/'J-" xp(,rl;tJJ/,rij=cf), i=lj=1

'l'llI'l'd"n·.11', 'ilppruxittl;ll,,·lyhavr-

~, ".

E

L:[l<>gf/ii-,,.~tJJ,rij~O.

;=Ii =1

<Iud

k !o, l: ".

!~("J::=(E L ,rir"0)-1L L,r i,;Jllg,llij

i=I,, = 1 i=Ii= 1

tT~(lIl

^::=

Lt.,I IJ~~

^{wJulF •}

(:1.18)

CUl)

(:1.21)

(:].22 )

,

'·XI,(..,.1{llJ_ I ) ' I

;1II,1,\,The«illil ial,'!;linl1l ll'~111<1)'he'ill1HTll ra l l'nndwillbeimprovedintheIollowiug

:1()

(44)

:1.TIll'modifiedNI' \I'lo l lBllph"oll itt-rutire';ll~orilhmd.

(ill Solviugth"S('()I'C'function:I.If! 11.\'IlSill ~1111'lirsl-ordr-r'I'ar 1or expansion.We' 11111'<'.

(;1.:.!.1)

wit h

Il

^UI⁼

f: ,riie'''11(.rl~i1IlJl ).

1"' 1

'ill)T;::

^{f :}

,rD('XII(,rl~i1lU) ),

.ie l

!I;(II)

=

nIU)+L!Ji.i'

)",1

I S') = f>'xp(.rf~I:I((I)),r·ij.rf~,

.i=l

11;(lJ)

= ~(II) + f>xrl(.r!~/~(II))

^.

.i"'.

:11

(45)

is,

(:).26)

III this 1\"i'Y.tlrl~dHIIIJt'·Sil!/'I1ff1(U, IJl"«)II"'!IshorterAIIIIshorteruntil thelog likf~ihIH...l/lplll.".llull>'lJ1l11J. ".lllll).Thi sc~ngllllralll('l'thatour estimate

J1i

¹1 'IfflllI" "t1o·;o.l lutl"' lII..xhumulikelihoodOlin'\.

{"plil t1UlI:I, l l ij;,rtJIIll'l'xpluit iuJtIII('J;l"lll'rillIonnufI,IU'EMalgor it hmoff)l'mp,~ll'r.

Lilird 111111Hllllili(!!Iii).

hilSfllllyslal i,milrypoiut,11ll'1Itl1l' El\1algorith m,'!<I illl<ltiollcc uvcrgcsto theunique tuaxirmuulik,·liIKlIKII'!<liuml... '1'11('sohuionisIlUi111lt·11I~redueto':If"1\1.·II-kIlOWII

('m'f·lli'<1!1,1f.it"( ~ull'H"I.(l !~".!. P<1~I'!<~J{).;J().))pointl'dI1Ut ,all iuipc rtan tat!\'lIl1lag eof tIll'1-:711al~"rilhlllisIllillIhl'ilf'TlIlioJI~willillwa~':'1n'mainill 1·llepaTa;'lCICrspare,since itisIH'rr" rllli liA11K'IW1Xi lll llIlIlik..lihOtxl""tillliitiollforthecom ple le dilla.Mor~\'{'T.

till"EMlllAurillllll1llumllySiUll' lilil'!<I,IIP diTl'l'1("lI1rulatioll llfthC" maximumlikclihccd ''!fliIl11l1 1ulI.

Fut1u\\"i ll!~lln- i,I,·,] IIIStil"a lt'lli,l.a il'lland\V1ll'l' (llIS·I),wealsothink oftheill- {·UllIp!t·!l',lOll,]liSh"ill~th,' nhs ,'r\"l'(I,Ia l a!Jjjalulllll!romplotedatn lhe unobservable rundom''If''c"l s 1..11111tlu-npp lil'lll iullof IJwEM algorithm lu'r ei~slight lydifferent

:t!

(46)

Fromt.hoirsillthel>!'!lS"thnr.tlu- gcucrnlIcnnral lwrthansiulplt, ulll,.1fI':"lalp;lIril lllH is used.

FOl'litepresent

.

II101l c" ,Ih.,H-Sh'lloftlu-Ei\ialp;urilhlllinvolvesliullillp; 111",'Xp l' I'·

lal,io!lof{ ;IO/!;!f(w,l!1i'(7~ClIl.,ill).when-!f(U"!lIi.(T~lU],j" ll])is Illl'nlll(lil,iullillll"!lsil,,\' fll lldiollolWiwilhthep;i1ll1l111: 'wor kinp;' Ilt'llsily~i\'l'1lill:1.·1,ronditiouuiou111l' 011·

sl'l"\'(,.1dalllve-ctor.'Ii;mt!gi\·" l! till'init ia l ('Sl im1l11'.•1T~11l)1IlHili (l ).Ast,ri1i~llll't>r\\'ilI'd

....(1) ~

.

1\'{1()I-\II'il .'li.rr1(1t),tiClll

t {if'(n^llJ1

+

I:'!/jj)_IUI-\!-'CIl I

+ I:c'xp(.r~ji)(I)H,

^{(:I ,:Uq}

''''I .i=' j"'l

IIlid

'"

,.~ _~

.

_{/~{ Il',}I.IIj,,,.'lIU], /:l( I)) L' nClJl +L: j':'II/;j

~).l")+Lj':'I">:Jl(.r~/i(lll '

The romputat.ioninvolvestlu:fll11diflllIll~l' i l l )Ilndltsd,'ri\'atiV<'s.'I'lreyal""tlu l"IISy 10Ill' dlrcctly('llklllil1.t,c!fromtheirft/m lilliis:1.111,:1.11,;l.l:.!illlt! ;J,I:I, Vall dt·""mil l

(47)

;,rlt l '!'l,rlll rW(I~IK'l)lisl'~l ltill'Iollcwlugronvnnicutapproximat eformula forlog1'(0):

IfIAJ'(n )

=

(fl-fl.ij)I(JAf\ - fttO.

.'jI[Jg(2r.) + ~

-

:If)(~na ⁺

^11.li^o;,i^-

](N~lct' ⁺

⁽⁾⁽ⁿ^-^fJ⁾ ^(:1,:10)

Wfwl!nisnet. Ifoss1harr~,tlr!' formula :t:.10canIwusedto computolog l'{o)withvery

Irj~11lItTIIJ'iIl',Ywil,lroutlasttermO(n-!' ).WlrellnislCl.stltau 2hul.lil l'l~crthanI,tire snuu-Ili.!!;lra''l'UI'OIf·yranI",~lIf, I'llnl(,,'dlJrl.h(~romhinarionof^tire^above(Ol'lllH!~,:1.:\0

;rflll l,lll'rull,'lI'iuJ!,ro-nrn-uo-(,wlrrul",

lUI!:1'(11)=log I'(n

+

^I)^-loglo],

l1S' ! tJislilIw 'r ll1l111 2. 'l'1H11is,

1"v;I'(f1)

=

(n+II.!'I}log(n +I)-fl-I

+

O,.'ilog( 211' )

+

12(01

+

I)

I I I

-:l(iIJ(t~+l~liO((\+Jr'-lhSO(n t l ) ' -!(,g(fl)+O (n

+

J)-!').

Silllilarl,\',whr-nn Lsle'Ss1,11/IrrIlml,lilrW'rthan0,then

(:1.:")

(".:12)

(op;l'(nl IOI-\I'(n+2)- log(u

+

() - Iol-\(n)

(Il+1.:"I)log(o+:1)-fl- 2+0.!'Jfog(2;r)+12(fll +2)

I I I

-:!{i(/{;.+2)'1 + 12IjlJ(<l+2)5 - 1680(£1+2 )'

-1(lA(n

+

I)-I0l-\(n)

+

O(n

+

2)-~' }. ('1.":1)

~·(III.~"'( fl l .,;."(nl iludl/,III(n)n,l1IJ('computedIromthedcrivat.ivcsofthe above

;II

(48)

flislnrgcr tllllll 'l. <Jl1l'uhtlli m.

/!-1llg l'(n) ,In

-~ + lu.L\ll-

^1:.L^l

+ l'lt~o l

^-

'l!i~n' ;

^-

'l'lt~'l"

⁺^{()(. l-}^IU), ^(:U^I)

IIIorder10obtelutilt'impruve-d!'s!i1lla t.'sof1'1alut(T~.,..,.~lllFrcuutil<'lin.'I·yd..isIlSt~1

ThisisdoneIJ~'Ilsill.L\1.11('modilir-dNI'II'!.OllRuphsou iterafiouIlllu'l'd u n'!;ns illI,ll!'lirs.'",1'1'11"

'1'111'11at1.1[('Sl'COIHl

Slil~f'

^of^this^serond"rd l'.II'PIlIilXillliJW}I'!

i:

1u.L\,l/ (u 'd ,l/i'lT'!llJ,ll11ll!

'L "' I

toobtnlntheimprtll',,,1r-st.imntr-a'~t'j)[o r,,'1.Tlws" 1,II'n-s l a p;!'hils" ,1(·.I'd!' "orn,mp UI.i1I,j"nH ("Ollt,illlll' lIut.i1,'OHw rg"IU'('is,whi"I'I~ 1. '1'11<'liuul ('stiulill,'sill'l'II·illI,1^(1.¹furII'1I1l1"1

3.4 Remarks on Asymptotic Theory

3.4.1 When(11.isKnown

II)'exploitingi.lu-SOli"!'equation:t!.'i,II'"11111'"ti lt'fo!luwi rlp;Il'!;lIlt.wlll'lIn'lisknuwu.

Theor em 1IfEf"IU.;-~ )i.• pm,ilil" Ilrji llil l'111/11rrl i,.1.:11.,,1111,II""II""I,/u '(u ;IIIIII , lil.:rlilllJ(J/lls/ill!(tlr_ .~i1·oj{1II/'rI·O /l.•;"lr/lI,"" illll/l/oliI'lJII!I'1II/,ilwri ."",IJf.{f1·-fI) i,..(/,'.1/1"/'- lo/irn lf y(~'-+cc ]di.~!I" i/llllf/1 II,~"II,(lil'II";II /'11II/'/11lllwilh1111I111zr/"II1111/1: x/1/'IHm,. iflJJl"I

(49)

fllIIl l';:rf/ ;lIUI I'1/

",il/l

I'rllf,r:

t ~'jj('xp(;rJ~fI),

J=l

tl'XIJ(.r/;!~),r;j,r?;,

j=l

It

=

f:

_{j", (}

,r.J>)(p(":J~fI},

'ulIl

1';

=.\

+ ~>xll( ;l'D{I)·

j=1

'thY!!!I' C'XI'IlIlSiOli Hn,!i~lIorillg!lll'highun let'renns ,.;f.({i"-illcan!J(' approximate dby

B,l'lIsin$!, I,S,I.!I.1.111,:l.rlilll<l;l.(i .1I'l·1111\,{·

HII,lln

td:lJ"!I; j.r,) -

^(I

+ L:j'~I /';!Jijl;)

i"'lJ= I I';

t{f!<,xll( .r~iJ + (1~/2).r;jJ

^_^(l

+ E~'':''I (·xJl~,rltJ+ a'J/"l)I;1

;=(j=l I/i

j. . n

~(I; 1'XI'(I'T~/2) -"i/;)

~ 1;1,')(p(I'T~/2)-

. *1

⁼^tl

:m

(50)

k •• 01, I. , I

~

IE

j~l(.rij-; ;llt'xp(:.!<T ! )-I',I·J'« (T~l]r.tJl(.rD/:1-I-."~'il)(,l·;.,.-

i/:)

T

'" <T~ J. ,

-I-LIJ·/I(.'·~iJ +

"7))(.rij-

--:-){,t'J-

-';)T I

.;=1 - 1/, I'i

t{[l'XI)(:lI7~ )

^_^1.1"//(/)"2 )][1_

L;'~ I

^f

.r~'(JJ II )r:I,I!

i",,' 1/;

,,2 1.(T /-(F '"

+PXIl(T)[1.; -:l~ + 7;frEt'XI'('l' ~ IJlJ)

/)"~~. If !

c':'I: I'( - )~)f"-...!....!....).

:! ;",1 /Ii

l.otindl.l' nOrl1ll1! di,'itrilJllliolllI'illlIItC,',IIlzeroam IcUI'a d ml n' \·,,, II:dl

J)).I)ilf"rl'u ti ati,," "fti l<'

aud

ilP(rJ,/)" 2)

iiifl

t {(11

+

t.ll;.;)[L:~"^I(px[l(.r7jt/)2j ]L:;~I~'.~II~",~IJ).r;~]

i=-1 j;;1 [-\+L:,i:=,P:-;;jI(Jijlll ] E;;' I [·':-;;I)(·rL I1 ).I'ij."~[

,\+L:;';.,

I'X/l(J'~ lj) II

t{fl +I:.. ,ij)(!4 -!:; J

;=1 j",1 t', II;

(:1.:17)

1,·( _iJl;;/~iJ)) - tUn+

_,₌₁ _)00'1

ft,xJI(."~!J+ ;'j](~

_• _I',

^-!:;Jl

_I',

,,~ (,

u:

t':-;;p(7! )~ (I., -~)

"",.((.', (;1» .

:ri'

(:I.:IK)

(51)

1:1·'19'

TII< ~ JIl ·t1 1Iill']icld.C'~Ihlltt.ln-I\~Ylllllll)lil'('u \,Mjal1cnorLbl'ilpproxillll1lclikelihoo dest.i- 1l1/11.'S/1'11t'IWll< l~UII1.111'1'll riftllC1'ufl'arulutll,'lr('d~11'1,whidrb1I11' indexortheiut rs-cluster

~",".;c,,·illl~'11within1111' s<IIIII'dlll<ll,.illl.!l,· oIN'I"\,aliolili.a..."xllia illcdillChAllll'rI.Wllellt.hc

ca -40)

\\,,11"11 Ihl' 111"111111(r~is 11\1',1)'IromlC'I'l),audsolarge1.hlll,\ iNwrysmallcompan-dwith

" " • " I "

L{lIJII)fI; lLlI) -[(0.iI,IILlfljll; )=:j

L

[lIj (I)'(II;- Uj')(U)_uj'ITj~O (:lA2)

,,=1 ).. J" I lei -iJ';1

(52)

Proof:

t

¹^{(I/ 'i'(U}ⁱ-uj')(lli -lIi,)Tl ,i,i'='

and

t

^[Ui"j ' "JIIT -f/jllj' II,iUJ. -/I)'I)dl/III

+ "J"j'uj'lI~l

j,i'=1

.

^"

^..

^"

~L(lIi"jllr l L^II⁾^-'2L {lIjll j)L(l/JIIJ )

j : l j=l j"" )= 1

t

!rlj lly ( lIj _ IIj' )( rlj _ lIj , )Tj '2:: n,

j.j'=' whichyickl1,lu'proo fof 1,1l('k-unnn. a

followingIa cl..Ast.ho;11'\.11<11fT~prt.sInrW'I",lit!'ilsy rn pl.ol,il'rovariuuo-or;1'lI'ilIlll't"lIlIlC' stnallr-rillge-neral,1IIIII'S.~Ihr-l'OI'll' SPOI\c1 i lll!;lixI~II'lfl'('11'(lI'i1r i a l.l·lws1111'SillllC'',I'llt'"rl,\'l'I[ l l ill

10 p1'01'111('l'onJlkl.inginfl'1'I.'l1C1111'11l'1Irnmpan-rl10the1rlllli l.iOllal1IIl,t1y si s1I'1r1'1"' "~isI,holl/!,II~

10III' 11dispersion01'overdiaperaienpanlllll 'I,(' I'only.III rllf'l ,for1,11<'Ilrps,· 111.1110.11'1.n 21'11I'ylf1111