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)
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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
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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
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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.
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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.
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