Faculty Research Interest Seminar Department of Biostatistics, GSPH University of Pittsburgh. Gong Tang Feb. 18, 2005

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1 Faculty Research Iterest Semar Departmet of Bostatstcs, GSPH Uversty of Pttsburgh Gog ag Feb. 8, 25

2 Itroducto Joed the departmet 2. each two courses: Elemets of Stochastc Processes (Bostat 24). Aalyss of Icomplete Data (Bostat 265). Some research terests: Statstcal Aalyss of Mssg Data. Aalyss of Correlated Outcomes. Sem-Parametrc Statstcs. Statstca of the Bostatstcal Ceter of NSABP ( Work o cacer clcal trals. Recetly volved studes to vestgate the assocato betwee geomc data ad clcal outcomes (wth Drs. Joh Bryat ad Joe Costato).

3 A Iterestg Pheomeo Pseudolkelhood method Lkelhood fucto of : L( ; α) = L( α). : parameter of terest; α: ussace parameter. ˆ α s a cosstet estmator of α; α s the true value. Cosder ˆ = arg max L( α ; ) ad = arg max L( ; ˆ α) <pseudolkelhood estmate>. I some crcumstaces, var( ) < var( ˆ ). Demostrate through a mssg-data problem. Questo: how to mprove the effcecy whe α s actually kow, from other source?

4 Bvarate data wth outcome-depedet orespose Model assumptos: ( a) X f( x; α), [ Y X] g( y x, ). ( b) Pr[ R= X, Y ] = w( y; ψ ). A codtoal lkelhood method m L( ; F) = p( x y ;, F) (for x R, gve y ) m g( y x, ), m=# of c.c. g( y x, ) df( x) where F( x) = F( x; α ) represets the true CDF of X. ˆ = arg max L( ; F) = arg max L( ; F( x; ˆ α)), where ˆ α = arg max f( x; α). α X Y R = R =

5 Full lkelhood for 2-patter data Model assumptos: ( a) X f( x α), [ Y X] g( y x, ). ( b) Pr[ R= X, Y ] = w( y ψ ). Full lkelhood: m L ( αψ,, ) = f( x α) g( y x, ) w( y ψ) Full m+ f( x α) g( y x, ){ w( y ψ )} dy m m+ = { f ( x α) } [{ g( y x, ) w( y ψ)} g( y x, ){ w( y ψ)} dy] = L( α) L( ψ, ) X Y R = R =

6 Asymptotc Varace of ˆ Let l( ) = log L( ; α), from ˆ = arg max L( ; α ), the = l ( ˆ α ; ) l ( ; α) + l ( ; α)( ˆ ) ˆ herefore, l ( ; α ) l ( ; α) E( l,) l, ( ; α). ( ) { } Var( ( ˆ )) E( l ) Var( l ( ; α )) E( l ),,, = El ( ) El ( l ) El ( ),,,,, (A sadwch-type estmator).

7 Asymptotc Varace of (I) Let S( α ) = log f( x; α), ˆ α = arg max f( x; α) s MLE of α, ( ˆ α α ) ( ) (, ( ) ). ESαα Sα, N ESαα From = arg max ( ; ˆ) arg max ( ; ˆ L α = l α), = l ( ; ˆ α) l ( ; ˆ α) + l ( ; ˆ α) ( ), the ( ) { } α l ( ; ˆ α) l ( ; ˆ α) l ( ˆ ; α) + lα ( α α) E( l,) El l El (,) {, ( α,) ES ( αα ) Sα, }.

8 Asymptotc Varace of (II) From ( ) ( ) { ( ) ( ) El, l, Elα, ESαα Sα, Var( ( )) El ( ) { El ( l ),,, El ( S ) E( S ) E( l },, α, αα α, El l l ( α,) ES ( αα ) ES ( α, l, ) El ( α,) ES ( αα ) E( α, ) } E(,). O the other had, = E( l ) = l,, = l, ( α, ) f( x; α) g( y x; ) p( r x, y ; ψ) dx dy dr ( α, ) L( α) L(, ψ) dµ So, = El (, ) = El ( α, ) + S l,, L( ) L(, ) d α α ψ µ α = El ( ) + ES ( l ) = El ( ) + ES ( l ). herefore, α, α,, α, α,, Var( ( )) E( l ) { E( l l ) El ( ) E( S ) El ( ) } El ( ),,, α, αα α,, )

9 Aother Pseudolkelhood Estmator Whe the fuctoal form of = arg max L( ; F ) m m j= F( x) g( y x, ) = arg m ax, g( y x, ) df ( x) j s ukow, let = arg max g( y x, ), ( PL2 ) g( y x, ) where, F ( x) = I( x x) s the emprcal dstrbuto of X. Smulato studes suggest that s eve more effcet eve though t has o assumpto o F( x). X Y R = R =

10 Auxlary Iformato If F( x) = F( x; α ) s kow, for example, some survey studes, ca we get more effcet estmator? Aswer: yes, wth emprcal lkelhood.

11 Emprcal Lkelhood " Emprcal lkelhood s a oparametrc method of ferece based o a data-drve lkelhood rato fucto" (Art Owe). Suppose { x} s a radom sample of X F( x), estmator for F( x) s F( x) = I{ x x}, the the o-parametrc or p s maxmzed subject to p =, where p = pr{ X = x}. Wth auxlary formato Ewx ( ( )) = avalable, the emprcal lkelhood estmator maxmzes p subject to costrats p = ad pw( x ) =.

12 Icorporate Auxlary Iformato For example, EX ( ) = µ s kow. he PL2 estmator solves the estmatg equatos New estmator: = l ( ; F ) = l ( ; F )., () Maxmzes p subject to costrats p = ad p ( x µ ) =. Get {p ˆ }. () Let solves = p ˆ l ( ; F ), the, ( ) = ( ) { ( ( ; )) ( ) ( ) ( )}( ). Var El Var l F E l x Var x E xl El X Y R = R =

13 X Y A smulato study Complete data: () [ x] N(,), (2) [ y x] N( + x,). Mssg-data mechasm: pr[ R = x, y] =Φ( y -). Compare the performace of the PL ad EL estmates for the regresso model (2), where β β σ = 2 =(,, ) (,,). Note: sample sze =3, average # of c.c.=5.

14 Smulato results able: emprcal bases ad stadard devatos of two estmators over replcates Methods β β σ 2 PL.2 (.33).9 (.5).5 (.97) EL.2 (.29).7 (.).9 (.86)

15 Referece ag, Lttle ad Raghuatha. Aalyss of multvarate mssg data wth ogorable orespose. Bometrka, vol. 9, pp , 23. ag, Lttle ad Raghuatha. Aalyss of multvarate mootoe mssg data by a pseudolkelhood method. Proceedgs of the 2 d Seattle Symposum Bostatstcs: Aalyss of Correlated Data. Lecture Notes Statstcs, Vol. 79. Edtor, D. L ad P.J. Heagerty. 25. A upublshed mauscrpt.

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