Comparison of parameter estimators for cause-specific hazards in competing risks data with missing cause of failure

Compa ison o pa ame e es ima o s o cause-specific haza ds in compe ing isks da a wi h missing cause o ailu e Jimin Lee1and Kedai Cheng2 1Depa men o Ma hema ics and S a is ics Uni e si y o No h Ca olina Ashe ille, Ashe ille, NC 28804, USA Email: [email p o ec ed] 2Depa men o Ma hema ics and S a is ics Uni e si y o No h Ca olina Ashe ille, Ashe ille, NC 28804, USA Email: [email p o ec ed] ABSTRACT We conside he semipa ame ic p opo ional haza ds model o he cause-specific haz- a d unc ion in analysis o compe ing isks da a wi h missing cause o ailu e. The in e se p obabili y weigh ing equa ion is well used o es ima ing he eg ession pa ame e s unde he missing a andom (MAR), p o ided he model o he nonmissing cause case is co - ec ly specified. In his s udy, we conside a pa ame ic logis ic eg ession, disc iminan analysis, and neu al ne wo k o he model o missingness p obabili y. Simula ion s udies demons a e ha he in e se p obabili y weigh ed (IPW) es ima o s by a co ec ly specified logis ic eg ession model and by disc iminan analysis pe o m well and a e app op ia e o p ac ical use. The IPW es ima o by neu al ne wo k wo ks conside ably well o a su ficien la ge sample. The applica ion o he p oposed me hods a e illus a ed wi h da a om a bone ma ow ansplan s udy and Women’s In e agency HIV s udy. Keywo ds: Missing cause o ailu e, In e se p obabili y weigh ed es ima o , Logis ic eg es- sion, Neu al ne wo k, Disc iminan analysis. Ma hema ics Subjec Classifica ion: 62D10, 62F12, 62N01, 62N02 1 In oduc ion In medical s udies, he esponses o a ea men can be classified in e ms o ailu e om disease o in e es o om o he causes. Hence, in he compe ing isks amewo k, each indi idual is exposed o Kdis inc ypes o isks and he e en ual ailu e is classified o p ecisely one o he isks. Le T∗deno e he ime o ailu e, Δ∗ he cause o ailu e, and Zap-dimensional ec o o possibly ime-dependen co a ia es. Then, an es imable quan i y in compe ing isks In e na ional Jou nal o Applied Ma hema ics and S a is ics, In . J. Appl. Ma h. S a .; Vol. 64; Yea 2025, Issue:2 (Published on 03/10/2025 ) ISSN 0973-1377 (P in ), ISSN 0973-7545 (Online) Copy igh © 2025 by In e na ional Jou nal o Applied Ma hema ics and S a is ics www.cese .in/cese p da a is he cause-specific haza d unc ion o cause k∈{1,...,K}defined by λ∗ k( |Z) = lim h→0 1 hP( ≤T∗< +h, Δ∗=k|T∗≥ , Z), which is he ins an aneous a e o expe iencing he e en o ype ka ime , ha ing no expe- ienced any o he Kcompe ing e en s un il ime . Wi hou loss o gene ali y in ou s udy, we conside only wo causes o ailu e, he cause o in e es as cause 1 and he o he as cause 2 (i.e. Δ∗=1o 2). In many applica ions in ol ing ollow-up s udies, howe e , indi iduals may be subjec o censo ing. Le Cbe a censo ing ime and T∗=min(T∗ 1,T∗ 2), whe e T∗ 1and T∗ 2deno e he la en ailu e imes om causes 1 and 2, espec i ely. Then he obse ed da a consis o obse a ions o (T,Δ,Z), whe e T=min(T∗,C)and Δ=Δ ∗I(T∗≤C). I he ailu e ime T∗is obse ed, Δis he cause o ailu e and Δ=0o he wise. The obse able cause-specific haza d unc ion o cause kin he p esence o censo ing is gi en by λk( |Z) = lim h→0 1 hP( ≤T< +h, Δ=k|T≥ , Z),k=1,2. Th oughou he pape , we assume ha Zis an ex e nal co a ia e p ocess (Kalbfleish and P en ice, 2002) and ha he censo ing ime Cis condi ionally independen o (T∗,Δ∗)gi en Z. Unde his assump ion, i was shown ha λ∗ k( |Z)=λk( |Z)i he dis ibu ion o Cis con inuous a (Lu and Tsia is, 2001; Gao and Tsia is, 2005; Lu and Liang, 2008). A numbe o s a is ical models o he ela ionship be ween he cause-specific haza d unc- ion o in e es and eg ession co a ia es ha e been s udied, among o he s, by (P en ice e al., 1978; Benichou and Gail, 1990; Cheng e al., 1998; Shen and Cheng, 1999; Scheike and Zhang, 2003). In his a icle we s udy he p opo ional haza ds model o desc ibing he ela ionship λ∗ 1( |Z)=λ0( )eβT 0Z( ),(1.1) whe e λ0(·)is a nonnega i e bu o he wise unspecified baseline haza d unc ion and β0is a p- dimensional ec o o eg ession pa ame e s. The pa ame e β0can be consis en ly es ima ed by ea ing all he ailu e imes wi h Δ=1as censo ed obse a ions and using he pa ial likelihood sco e equa ion p oposed by (Cox, 1972; Cox, 1975). The es ima o will be called he ull-case es ima o , deno ed by  βFin he pape . In p ac ice, howe e , he in o ma ion needed o he cause o ailu e may be los , o i may be di ficul o de e mine he cause o dea h o some indi iduals (Ande sen e al., 1996). When we ha e missing causes in da a, a nai e me hod o es ima ing he eg ession pa ame e β0is o simply igno e he missing da a and use he pa ial likelihood sco e equa ion o he comple e da a only. The so-called comple e-case analysis and i s es ima o , deno ed by  βC, is clea ly ine ficien and can lead o se ious bias. Thus, analysis o compe ing isks da a wi h missing cause o ailu e has ecei ed conside able a en ion and a numbe o me hods ha e been p oposed. (Lu and Tsia is, 2001) p oposed a pa ame ic model o model he p obabili y ha he missing cause is he cause o in e es while allowing he inclusion o addi ional auxilia y co a ia es and hen es ima ed he eg ession pa ame e s by using a mul iple impu a ion me hod. (Gao and Tsia is, 2005) conside ed linea ans o ma ion models and (Lu and Liang, 2008) conside ed he addi i e haza ds model o he ela ionship in analysis o compe ing isks da a wi h missing cause o ailu e. In e na ional Jou nal o Applied Ma hema ics and S a is ics 2 In his s udy, we de i e an es ima o o he eg ession pa ame e s in model (1.1), namely he in e se p obabili y weigh ed es ima o and es ablish i s heo e ical p ope ies. The app oach, ollowing he idea o (Ho i z and Thompson, 2012), uses he in e se p obabili y weigh ed comple e-case echnique o es ima e he eg ession pa ame e . The undamen al idea o IPW is o weigh obse ed da a poin s whe e he cause is known by he in e se o he p obabili y ha hei cause o ailu e would be obse ed. This app oach uses only he comple e cases and elies on he missing a andom (MAR) assump ion, and he model o he missing cause case mus be co ec ly specified o he IPW es ima o o be alid (Scha s ein e al., 1999; Lu and Tsia is, 2001; Gao and Tsia is, 2005; Lu and Liang, 2008). Thus, we conside se e al models o he p obabili y o missing causes; logis ic eg ession model, disc iminan analysis, and neu al ne wo k o he model. The pape is s uc u ed as ollows. In Sec ion 2, he in e se p obabili y weigh ed es ima o is de eloped. The asymp o ic p ope ies o he co esponding es ima o is es ablished in Sec ion 2. We e iew logis ic eg ession, disc iminan analysis, and neu al ne wo k o es ima e he p obabili y o missing causes in Sec ion 3. In Sec ion 4, we in es iga e and compa e he fini e sample p ope ies o he p oposed es ima o s h ough simula ions. A bone ma ow ansplan da a se and Women’s In e agency HIV s udy da a se a e analyzed in Sec ion 4. Some con- clusions and discussions a e gi en in Sec ion 5. Technical no a ions a e de ailed in Appendix. 2 Es ima ing equa ion and asymp o ic p ope ies Since he cause o ailu e may no be obse ed o some indi iduals, we define he misssing- ness indica o Ras ollows. I an indi idual’s ailu e is obse ed, hen R=1when he cause o ailu e in o ma ion Δis obse ed and R=0o he wise. I an indi idual is censo ed, we always define R=1. We also in oduce auxilia y co a ia es Awhich a e no o in e es o modeling he cause-specific haza d unc ion bu may be used o desc ibe he misssingness mechanism. The u iliza ion o auxilia y in o ma ion has been conside ed by (Lu and Tsia is, 2001; Gao and Tsia is, 2005; Lu and Liang, 2008; Gilbe e al., 2008), among o he s. Then he obse ed da a will consis o Oi={Ti,Z i,A i,Δi,R i}i R1=1, and Oi={Ti,Z i,A i,R i}i R1=0 o i=1,...,n. We assume ha {Oi,i=1,...,n}a e independen iden ically dis ibu ed. We also assume ha he cause o ailu e is missing a andom (Rubin, 1976); ha is, he p obabili y ha he cause o ailu e is non-missing gi en Δ>0and W=(T,Z,A)depends only on he obse ed W, bu no on he unobse ed Δ, P(R=1|Δ,Δ>0,W)=P(R=1|Δ>0,W).(2.1) The assump ion implies ha P(R=1|Δ>0,W)=P(R=1|Δ=1,Δ>0,W)=P(R=1|Δ=2,Δ>0,W). See (Lu and Tsia is, 2001; Gao and Tsia is, 2005; Lu and Liang, 2008). In e na ional Jou nal o Applied Ma hema ics and S a is ics 3 2.1 In e se p obabili y weigh ed es ima o Following he in e se selec ion p obabili y idea o (Ho i z and Thompson, 2012), he me hod o in e sely weigh ing he p obabili y o comple e-case has been commonly used in missing da a p oblems. To do ha , we need o es ima e he p obabili y o a comple e case, π(Q)≡ P(R=1|Q), whe e Q=(W, Δ). By he MAR assump ion and R=1when Δ=0,weha e π(Q)=P(R=1|W, Δ) =P(R=1|W, Δ,Δ>0) ·I(Δ >0) + P(R=1|W, Δ,Δ=0)·I(Δ = 0) =P(R=1|W, Δ,Δ>0) ·I(Δ >0) + 1 ·I(Δ = 0) =P(R=1|W, Δ>0) ·I(Δ >0) + I(Δ = 0) = (W)·I(Δ >0) + I(Δ = 0),(2.2) whe e (W)=P(R=1|W, Δ>0). The p obabili y o comple e-case (Wi)may be specified by a pa ame ic model (Wi,ψ 0), in e ms o a ew unknown pa ame e s ψ0. Acco dingly, le π(Qi,ψ 0)= (Wi,ψ 0)I(Δ >0) + I(Δ = 0). By (2.1) and (2.2), he likelihood L ega ding o π(Q, ψ0)is L(π)= n  i=1 π(Qi)I(Ri=1)(1 −π(Qi))I(Ri=0) = n  i=1 π(Qi)I(Ri=1)I(Δi>0) ·π(Qi)I(Ri=1)I(Δi=0) ·(1 −π(Qi))I(Ri=0)I(Δi>0) = n  i=1 (Wi)I(Ri=1)I(Δi>0) ·1·(1 − (Wi))I(Ri=0)I(Δi>0), because π(Qi)= (Wi)when Δi>0,and π(Qi)=1when Δi=0 = n  i=1 (Wi)RiI(Δi>0) ·(1 − (Wi))(1−Ri)I(Δi>0) . This implies ha he maximum likelihood es ima o  ψo ψcan be es ima ed by maximizing he likelihood based on only uncenso ed da a n  i=1{ (Wi,ψ)}RiI(Δi>0){1− (Wi,ψ)}(1−Ri)I(Δi>0). We define he coun ing p ocess Ni( )=I(Δi=1)I(Ti≤ )and a - isk p ocess Yi( )=I(Ti≥ ). Le a⊗0=1,a⊗1=a, and a⊗2=aaT o a ec o a. Le  S(m)( , β, ψ)= 1 n n  i=1 Ri π(Qi,ψ)Yi( )eβTZi( )Zi( )⊗m,m=0,1,2,  Z( , β, ψ)=  S(1)( , β, ψ)  S(0)( , β, ψ),  V( , β, ψ)=  S(2)( , β, ψ)  S(0)( , β, ψ)− Z( , β, ψ)⊗2. Then we conside he ollowing in e se p obabili y weigh ed es ima ing equa ion o β0, UI(β,  ψ)= n  i=1 τ 0 Ri π(Qi, ψ)Zi( )− Z( , β,  ψ)dNi( ),(2.3) In e na ional Jou nal o Applied Ma hema ics and S a is ics 4 whe e τ>0is he end o ollow-up ime. The in e se p obabili y weigh ed es ima o o βsol es he abo e equa ion and is deno ed by  βI. When he e is no missing cause, he equa ion (2.3) consequen ly becomes he pa ial likelihood sco e equa ion p oposed by (Cox, 1972). The cumula i e baseline haza d unc ion Λ0( )= 0λ0(u)du can be es ima ed by  ΛI0( )= n  i=1  0 Ri π(Qi, ψ) dNi(u) n S(0)(u,  βI, ψ). 2.2 Asymp o ic esul s Condi ions C. (C.1) λ0( )is con inuous on [0,τ]. The dis ibu ion o Cis con inuous on [0,τ]and P(C>τ)> 0. The co a ia e p ocesses Zi( )ha e pa hs ha a e le con inuous and o bounded a ia ion, and sa is y he momen condi ion EZi( )4exp(2MZi( ))<∞, whe e M is a cons an such ha β∈[−M,M]pand A=max k,l |akl| o a ma ix A=(akl). (C.2) Each componen o s(j)( , β),j =0,1,2is con inuous on [0,τ]×[−M,M]p o some M>0,s(0)( , β)>0on [0,τ]×[−M,M]p, and sup ∈[0,τ],β∈[−M,M]p S(j)( , β)−s(j)( , β)=Op(n−1/2). (C.3) The ma ix Σ=τ 0 ( , β0)λ0( )s(0)( , β0)d is posi i e defini e. (C.4) The e is a σ>0such ha (Wi)≥σ o all iwi h Δi>0. (Wi,ψ)is wice con inuously di e en iable wi h espec o ψ. The e exis s ψ∗sa is ying he equa ion ES∗ ψi =0, whe e S∗ ψi is he co esponding sco e unc ions o (Wi,ψ)gi en in (5.2) in he Appendix. The in o ma ion ma ix I∗ ψalso gi en in (5.2) in he Appendix is posi i e defini e. We le ψ0be he ue alue o ψsuch ha (Wi)= (Wi,ψ 0). In gene al, unde Condi ion (C.4), he e exis s ψ∗such ha  ψcon e ges in p obabili y o ψ∗and  ψconsis en ly es ima es ψ0(Habe man, 1974; Habe man, 1977; Gou ie oux and Mon o , 1981; Whi e, 1982). Thus, i he model o (Wi)is co ec ly specified, we ha e ψ∗=ψ0and  ψcon e ges in p obabili y o ψ0. Theo em 2.1. Assume Condi ions C a e sa isfied. I (Wi,ψ)is co ec ly specified o (Wi), hen  βIcon e ges in p obabili y o β0and √n βI−β0con e ges in dis ibu ion o a ze o- mean Gaussian andom ec o wi h co a iance ma ix Σ−1Eω1ωT 1Σ−1, whe e Σ=τ 0 ( , β0)λ0( )s(0)( , β0)d , ωi=τ 0{Zi( )−z( , β0)}Ri π(Qi,ψ 0)dMi( )−VψI−1 ψSψi, Mi( )=Ni( )− 0Yi(u)eβT 0Zi(u)dΛ0(u),Vψis gi en in (5.1), Iψand Sψi a e gi en in (5.3) in he Appendix. In e na ional Jou nal o Applied Ma hema ics and S a is ics 5 The asymp o ic co a iance ma ix Σ−1Eω1ωT 1Σ−1can be consis en ly es ima ed by  Σ−1n−1 n  i=1 ωiωT i Σ−1, whe e  Σ= 1 n n  i=1 τ 0 Ri π(Qi, ψ) V( ,  βI, ψ)dNi( ), ωi=τ 0 Ri π(Qi, ψ){Zi( )− Z( ,  βI, ψ)}d Mi( )− Vψ I−1 ψ Sψi, and  Mi( )=Ni( )− 0Yi(u)e  βT IZi(u)d ΛI0(u).He e  Vψ, Iψand  Sψi a e ob ained by eplacing wi h hei espec i e sample es ima o s and subs i u ing ( βI, ψ) o (β0,ψ 0)in Vψ,Iψ, and Sψi. See (Hyun e al., 2012) o he p oo o Theo em 2.1. 3 Se e al models o (W) Logis ic eg ession Logis ic eg ession is o en used o model he p obabili y ha an expe imen al uni alls in o one o he wo g oups based on p edic o a iables measu ed on he expe imen al uni . Since Ris bina y, one can posi a pa ame ic logis ic model, logi { (Wi,ψ)}=WT iψwhich was o en used in his s udy a ea (Lu and Tsia is, 2001; Gao and Tsia is, 2005; Lu and Liang, 2008; Hyun e al., 2012), among o he s. Howe e , model misspecifica ion occu s, leading o biased pa ame e es ima es and inaccu a e in e ences. Disc iminan analysis Disc iminan analysis is a me hod used o classi y obse a ions in o nono e lapping g oups based on p edic o a iables. Fo example, a doc o migh used disc iminan analysis o iden- i y pa ien s a low o high isk o a disease based on ac o s like choles e ol le els, sex, blood p essu e, and o he a iables, The e a e di e en ypes o disc iminan analysis, including lin- ea disc iminan analysis which is a common app oach ha assumes a linea ela ionship be- ween he p edic o a iables and he g oups, and quad a ic disc iminan analysis ha allows non-linea ela ionship be ween he p edic o a iables and he g oups. Disc iminan analysis elies on ce ain assump ions, including independence o obse a ions, no mali y o p edic- o a iables, and equal a iance-co a iance ma ices ac oss g oups o linea disc iminan analysis. The only di e ence om quad a ic disc iminan analysis is ha we do no assume ha he a iance-co a iance ma ix is iden ical o di e en g oups. We conside IPW es i- ma o unde linea and quad a ic disc iminan analysis o he p obabili y o non-missingness, (W)=P(R=1|W, Δ>0). In e na ional Jou nal o Applied Ma hema ics and S a is ics 6 Neu al ne wo k A (a ificial) neu al ne wo k is a compu a ional model inspi ed by he s uc u e o biological neu al ne wo ks. A neu al ne wo k ypically consis s o an inpu laye , one o mo e hidden laye s, and an ou pu laye . Each node in he hidden laye ecei es inpu , p ocesses i , and send an ou pu o o he connec ed nodes. See Figu e 1 o neu al ne wo k diag am. Neu al ne wo ks a e being used in he a ea o p edic ion and classifica ion o ou comes in medicine a eas whe e eg ession models ha e adi ionally been used. Howe e , logis ic eg ession assumes a linea ela ionship be ween inpu a iables and he p obabili y o he ou come, bu neu al ne wo ks can cap u e complex, non-linea ela ionship be ween inpu s and he ou pu . We conside IPW es ima o unde neu al ne wo k model o he p obabili y o non-missingness, (W)=P(R=1|W, Δ>0). Figu e 1: A neu al ne wo k is an in e connec ed g oup o nodes. Each ci cula node ep esen s an a ificial neu on and an a ow ep esen s a connec ion om he ou pu o one neu on o he inpu o ano he . x0 x1 x2 x3 Inpu laye h(1) 1 h(1) 2 h(1) 3 h(1) 4 Hidden laye 1 h(2) 1 h(2) 2 h(2) 3 Hidden laye 2 ˆy1 Ou pu laye 4 Nume ical esul s In his s udy, we es ima e he pa ame e β0unde hese me hods and e alua e he es ima es based on Mean Squa ed E o (MSE) and Mean Absolu e E o (MAE), whe e MSE calcula es he a e age o he squa ed di e ences be ween IPW es ima es and β0, and MAE is he a e age o he absolu e di e ences be ween IPW es ima es and β0. 4.1 Simula ion s udies We p esen simula ion s udies conduc ed o e alua e he pe o mance o he p oposed me h- ods. We conside a uni a ia e co a ia e Z, whe e Z ollows a no mal dis ibu ion N(0.5,1/12). Gi en Z, he la en ailu e ime T∗ 1o in e es is gene a ed om he p opo ional haza ds model λ∗ 1( |Z)=λeβ0Z, whe e λ=1and β0=−0.5. The o he la en ailu e ime T∗ 2is gene a ed om a Gompe z dis ibu ion wi h a haza d unc ion λ∗ 2( |Z)=eθ+ν , whe e θ=−0.5and ν=0.2. The censo ing ime Cis gene a ed om an exponen ial dis ibu ion which yields abou 25% censo ing le el. We also conside a single auxilia y co a ia e Awhich ollows he s anda d no - mal dis ibu ion N(0,1) o a Be noulli dis ibu ion wi h success p obabili y o 0.5. Fo missing In e na ional Jou nal o Applied Ma hema ics and S a is ics 7 cause o ailu e, we conside a logis ic eg ession model logi { (W, ψ)}=ψ1+ψ2Tb+ψ3Z+ψ4A, whe e Tb=Tλ−1 λ, he Box–Cox ans o ma ion wi h λ=0.25 and Ais he s anda d no mal dis ibu ion (Model 1). We also conside logi { (W, ψ)}=ψ1+ψ2T+ψ3Z+ψ4A, whe e Ais he Be noulli dis ibu ion (Model 2). Wi h ψ=(0.5,0.5,0.5,0.5), we ha e abou 32% missingness (Model 1) and abou 17% missingness (Model 2). To s udy he pe o mance o he es ima o s when (W)is misspecified, we posi wo di e en pa ame ic models o (W, ψ), whe e one is a co ec ly specified logis ic model and he o he is a misspecified model logi { (W, ψ)}=ψ1+ψ4A. The simula ion s udies consis o 1000 uns wi h he sample size n= 400 and n= 800. Neu al ne wo k consis s o 3 nodes in one hidden laye . We conside IPW es ima o s om linea and quad a ic disc iminan analysis. Model 1 has all no mally dis ibu ed p edic o s which a e equi ed o disc iminan analysis. The esul s om Tables 1 and 2 show ha as expec ed, he ull-case es ima o  βFshows he smalles biases, SE, MSE, and MAE in all he se ings, bu he comple e-case es ima o  βC shows la ge biases, SE, MSE, and MAE in almos all he se ings. When he pa ame ic logis ic model o (W)is co ec ly specified, he IPW es ima o  βIc shows small biases, bu he IPW es ima o  βIm ends o ha e la ge bias, MSE, and MAE when logi { (W, ψ)}is misspecified and  βIm is clea ly sensi i e o he misspecifica ion. Co ec ly speci ying a s a is ical model in eal-wo ld da a analysis is inc edibly di ficul in mos cases. The IPW es ima o unde linea disc iminan analysis,  βIL pe o ms qui e well and shows sim- ila esul as  βIc. Since neu al ne wo ks gene ally equi e a la ge numbe o obse a ions o achie e good pe o mance,  βIN shows poo pe o mance when n= 400 and as sample size inc ease i s pe o mance imp o es. Model 2 has ha only he auxilia y p edic o Ais no mally dis ibu ed and he o he wo p edic- o s a e no no mally dis ibu ed. Thus, i does no mee he no mally condi ion o disc iminan analysis. Howe e , he esul s om Table 2 show ha  βIL and  βIQ pe o m well and all he o he es ima o s show simila esul s and pa e ns as in Table 1. In conclusion,  βIL and  βIQ unde disc iminan analysis and he co ec ly specified IPW es i- ma o  βIc show simila pe o mance and ha e be e pe o mance han he o he es ima o s, excep he ull-case es ima o . 4.2 Bone ma ow ansplan (Sie a e al., 2002) desc ibed he cha ac e is ics and ou comes o 452 pa ien s wi h p ima y myelodysplas ic synd ome (MDS) who ecei ed ansplan s om HLA-iden ical siblings and we e egis e ed wi h he In e na ional Bone Ma ow T ansplan Regis y (IBMTR). The s udy has wo compe ing isks; ea men ela ed dea h defined as dea h in comple e emission and elapse defined as ecu ence o myelodysplasia. In his example we conside 408 pa ien s wi h comple e co a ia e in o ma ion ob ained om he ime eg R-package. Among hese 408 pa ien s, 161 pa ien s died in comple e emission, 87 pa ien s elapsed, and 160 pa ien s we e censo ed. The co a ia es conside ed in ou s udy a e age o pa ien s anda dized a mean o 35 yea s old and pla ele be o e ansplan a ion (1 o mo e han 100 ×109pe L, o 0 o less). We apply he semipa ame ic p opo ional haza ds model o he cause-specific haza d unc ion o he da a se wi h he wo co a ia es: pla ele and age. We also conside cell (T-cell In e na ional Jou nal o Applied Ma hema ics and S a is ics 8 deple ed BMT; 1 o yes, 0 o no) o auxilia y a iable. In he da a se , he causes o ailu e a e all known. We can ob ain  βF. Fo illus a ion pu poses, we dele e some ailu e causes by he h ee ollowing missing mechanisms; missing comple ely a andom (MCAR), missing a andom (MAR), and no missing a andom (NMAR). Fo he MCAR, he causes o ailu e a e andomly selec ed o yield abou 23% missing causes. Fo he MAR, he logis ic model is chosen as logi { (W)}=0.1+0.5T−1.0age which yields abou 23% missing causes, whe e Tis he ailu e ime. Fo he NMAR, he logis ic model is chosen as logi { (W)}=0.1+1.0T−1.0age −0.5I(Δ = 1) which yields abou 22% missing causes, whe e Δ=1co esponds o he dea h in comple e emission, Δ=2 o elapse, and Δ=0does o being censo ed. Because co ec ly speci ying a s a is ical model is o en di ficul , i no impossible, in eal-wo ld da a analysis, we posi wo logis ic models o (W) wi h logi { (W, ψ)}=ψ1+ψ2Tb+ψ3pla ele +ψ4age +ψ5 cell o ob ain he es ima o s o β, whe e whe e Tb=Tλ−1 λwi h λ=0.06 (Model 1) and logi { (W, ψ)}=ψ1+ψ2T+ψ3pla ele + ψ4age +ψ5 cell (Model 2). The esul s o he es ima ion o βa e summa ized in Tables 3 and 4. Fo compa ison hey also includes he ull-case es ima o ,  βF, based on he o iginal da a wi hou a ificial missing. The esul s a e close unde all he missing mechanisms and almos all es ima o s a e close o he ull-case es ima o han he IPW es ima o om misspecified logis ic eg ession. The analyses a e consis en wi h he findings om he ea lie s udy ha pa ien s wi h high pla ele coun s ha e a lowe isk o ea men ela ed mo ali y han hose wi h low pla ele coun s and a highe isk a e is seen among he olde pa ien s. 4.3 Women’s In e agency HIV S udy (Lau e al., 2009) analyzed he Women’s In e agency HIV S udy (WIHS) da a. The WIHS was es ablished in Augus 1993 o in es iga e he impac o HIV in ec ion on US women. In 1994–1995, 2,054 HIV-posi i e and 569 HIV-nega i e women we e en olled. The s udy sample consis ed o 1,164 women en olled in WIHS, who we e in ec ed wi h HIV, ali e, and ee o clinical AIDS on Decembe 6, 1995. Women we e ollowed un il he fi s o he ollowing: ime o ini ia ion o he apy p io o AIDS o dea h (679 women, cause=1), ime o he occu ence o AIDS o dea h p io o ini ia ion he apy (359 women, cause=2), o adminis a i e censo ing on Sep embe 28, 2006 (126 women, cause=0). The da a se also includes his o y o injec ion d ug use a WIHS en ollmen (d ug, 1=yes, 0=no), whe he an indi idual was A ican Ame ican (black, 1=yes, 0=no), age on Decembe 6, 1995, s anda dized a mean 36.4 yea s, and CD4 nadi p io o Decembe 6, 1995 (CD4). We apply he p opo ional haza ds model o he cause-specific haza d unc ion o he da a se wi h he h ee co a ia es: he his o y o injec ion d ug use, age, A ican Ame ican, and CD4 as auxilia y a iable. Because he cause o ailu e a e all known, we dele e some causes o ailu e by missing a andom (MAR) mechanism by logi { (W)}=0.1+0.5T+0.5d ug −1.5age +0.5black which yields abou 20% missing causes. Howe e , we posi wo logis ic models o (W)wi h logi { (W, ψ)}=ψ1+ψ2Tb+ψ3d ug +ψ4age +ψ5CD4 o ob ain he es ima o s o β, whe e whe e Tb=Tλ−1 λwi h λ=0.06 (Model 1) and logi { (W, ψ)}=ψ1+ψ2T+ψ3d ug +ψ4age + ψ5CD4 (Model 2). The s udy ound ha women wi h an injec ion d ug use his o y we e less likely o ini ia e he apy In e na ional Jou nal o Applied Ma hema ics and S a is ics 9 Table 4: Es ima ion o he e ec s o pla ele and age in he bone ma ow ansplan da a, Model 2. Pla ele Age Missingness Es ima o Es . SE p- alue Es . SE p- alue None  βF−0.543 0.186 0.0035 0.377 0.087 <0.0001  βI−0.422 0.217 0.0522 0.393 0.107 0.0002 MAR  βIL −0.428 0.214 0.0405 0.372 0.103 0.0003  βIQ −0.558 0.287 0.0522 0.392 0.109 0.0003  βIN −0.636 0.219 0.0038 0.375 0.097 0.0001  βI−0.775 0.209 0.0002 0.362 0.087 <0.001 MCAR  βIL −0.774 0.209 0.0002 0.362 0.088 <0.001  βIQ −0.412 0.229 0.0725 0.443 0.144 0.0012  βIN −0.830 0.221 0.0002 0.366 0.086 <0.001  βI−0.568 0.228 0.0128 0.392 0.128 0.0021 NMAR  βIL −0.563 0.222 0.0113 0.330 0.109 0.0025  βIQ −0.735 0.230 0.0014 0.378 0.105 0.0003  βIN −0.683 0.235 0.0037 0.453 0.141 0.0013 Es ., he es ima e; SE, he s anda d e o es ima e; p- alue pe aining o es ing no co a ia e e ec ;  βF, he ull-case es ima o wi h no missing causes;  βI, he IPW es ima o unde logis ic eg ession;  βIL and  βIQ, he IPW es ima o s unde linea and quad a ic disc iminan analysis, espec i ely;  βIN, he IPW es ima o unde neu al ne wo k. In e na ional Jou nal o Applied Ma hema ics and S a is ics 16 Table 5: Es ima ion o he e ec s o d ug use, age, and A ican Ame ican in he WIHIV s udy. D ug Use Age A ican Ame ican Missingness Es ima o Es . SE p- alue Es . SE p- alue Es . SE p- alue None  βF−0.383 0.187 <0.0001 0.132 0.038 0.0005 −0.394 0.077 <0.0001 (Model 1)  βI−0.494 0.117 <0.0001 0.126 0.069 0.0659 −0.321 0.090 0.0004 MAR  βIL −0.513 0.120 <0.0001 0.129 0.072 0.0720 −0.317 0.091 0.0005  βIQ −0.943 0.244 0.0001 0.246 0.096 0.0103 −0.181 0.123 0.1418  βIN −0.472 0.108 <0.0001 0.127 0.056 0.0225 −0.320 0.089 0.0003 (Model 2)  βI−0.511 0.117 <0.0001 0.165 0.069 0.0173 −0.308 0.090 0.0006 MAR  βIL −0.511 0.121 <0.0001 0.126 0.075 0.0801 −0.309 0.090 0.0006  βIQ −0.960 0.201 <0.0001 0.342 0.070 <0.0001 −0.102 0.115 0.3745  βIN −0.490 0.107 <0.0001 0.149 0.058 0.0102 −0.320 0.088 0.0003 Es ., he es ima e; SE, he s anda d e o es ima e; p- alue pe aining o es ing no co a ia e e ec ;  βF, he ull-case es ima o wi h no missing causes;  βI, he IPW es ima o unde logis ic eg ession;  βIL and  βIQ, he IPW es ima o s unde linea and quad a ic disc iminan analysis, espec i ely;  βIN, he IPW es ima o unde neu al ne wo k. In e na ional Jou nal o Applied Ma hema ics and S a is ics 17