Vogel, Jus us; Co die , Johannes; Filipo ic, Miod ag
Wo king Pape
Causal E ec s and Op imal Policy Lea ning o In ensi e Ca e Uni
Discha ge Decisions o Sol e Hospi al P ocess Bo lenecks: App oach,
Me hods, and Fi s Resul s
Wo king Pape Se ies in Heal h Economics, Managemen and Policy, No. 2025-01
P o ided in Coope a ion wi h:
Uni e si y o S .Gallen, School o Medicine, Chai o Heal h Economics, Policy and Managemen
Sugges ed Ci a ion: Vogel, Jus us; Co die , Johannes; Filipo ic, Miod ag (2025) : Causal E ec s and
Op imal Policy Lea ning o In ensi e Ca e Uni Discha ge Decisions o Sol e Hospi al P ocess
Bo lenecks: App oach, Me hods, and Fi s Resul s, Wo king Pape Se ies in Heal h Economics,
Managemen and Policy, No. 2025-01, Uni e si y o S .Gallen, School o Medicine, Chai o Heal h
Economics, Policy and Managemen , S .Gallen
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Wo king Pape Se ies in Heal h Economics, Policy, and Managemen
2025 – N . 01
Causal E ec s and Op imal Policy Lea ning o In ensi e Ca e Uni
Discha ge Decisions o Sol e Hospi al P ocess Bo lenecks: App oach,
Me hods, and Fi s Resul s
Jus us Vogel, Johannes Co die , Miod ag Filipo ic
I
Wo king Pape Se ies in Heal h Economics, Policy, and Managemen
Edi o
P o . D . Alexande Geissle
P o esso and Chai holde
Chai o Heal h Economics, Policy, and Managemen
School o Medicine
Uni e si y o S .Gallen
Edi o ial o ice
Jonas Subelack
Resea ch Assis an
Chai o Heal h Economics, Policy, and Managemen
School o Medicine
Uni e si y o S .Gallen
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© 2025. This publica ion is licensed by he CC license CC-BY-NC-ND 4.0
II
Causal E ec s and Op imal Policy Lea ning o In ensi e Ca e Uni Discha ge
Decisions o Sol e Hospi al P ocess Bo lenecks: App oach, Me hods, and
Fi s Resul s
Keywo ds: Causal Machine Lea ning, In ensi e Ca e Uni Managemen , Hospi al Ope a ions, Policy
Lea ning
JEL Classi ica ion: I10, C44
Au ho s:
Jus us Vogel
Scien i ic P ojec Leade
Chai o Heal h Economics, Policy, and Managemen , School o Medicine, Uni e si y o S .Gallen
[email p o ec ed]
Johannes Co die
Resea ch Assis an
Chai o Heal h Economics, Policy, and Managemen , School o Medicine, Uni e si y o S .Gallen
Miod ag Filipo ic
Depa men Head
Clinic o Ope a i e In ensi e Ca e Medicine, Can onal Hospi al S . Gallen
Recommended ci a ion:
Vogel, Jus us; Co die , Johannes; Filipo ic, Miod ag (2025): Causal E ec s and Op imal Policy Lea ning o In ensi e
Ca e Uni Discha ge Decisions o Sol e Hospi al P ocess Bo lenecks: App oach, Me hods, and Fi s Resul s.
Wo king Pape Se ies in Heal h Economics, Managemen and Policy, No. 2025-01, Uni e si y o S .Gallen, School
o Medicine, Chai o Heal h Economics, Policy and Managemen , S .Gallen.
Causal E ec s and Op imal Policy Lea ning o In ensi e Ca e Uni
Discha ge Decisions o Sol e Hospi al P ocess Bo lenecks: App oach,
Me hods, and Fi s Resul s
D . Jus us Vogel*, C Johannes Co die *, and P o . D . med. Miod ag Filipo ic+
* Chai o Heal h Economics, Policy and Managemen , School o Medicine, Uni e si y o S . Gallen, S .-
Jakob-S asse 21, CH-9000 S . Gallen, Swi ze land
+ Can onal Hospi al o S . Gallen, Ro schache S asse 95, CH-9000 S . Gallen
C Co esponding Au ho
Decla a ions
Acknowledgemen s
We hank Da ia Bukano a-Be end o suppo ing us wi h he li e a u e sea ch which was necessa y o
his s udy. We hank Pa ick Münge o ex ac ing all da a necessa y o his s udy. We hank ou
s uden assis an Da id Klug o his suppo in da a cleaning and desc ip i e analyses.
Funding
This esea ch is suppo ed by a S epping S one G an o he Uni e si y o S . Gallen (p ojec numbe
2300181).
Con lic o in e es
None.
Au ho con ibu ions
JV: Concep ualiza ion, Me hodology, Fo mal analysis, In e p e a ion, W i ing – O iginal D a ,
Visualiza ion, Supe ision, P ojec adminis a ion, Final app o al, Funding acquisi ion
JC: Concep ualiza ion, Me hodology, Da a cu a ion, Fo mal analysis, In e p e a ion, W i ing – Re iew
& Edi ing, Final app o al
MF: Concep ualiza ion, Da a acquisi ion, In e p e a ion, W i ing – Re iew & Edi ing, Final app o al
E hics app o al and consen o pa icipa e
This s udy was app o ed by he E hical Commission o Eas e n Swi ze land (p ojec numbe 2024-
00829).
Consen o publica ion
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Da a a ailabili y
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Code a ailabili y
A ailable om he co esponding au ho upon eques .
1
Abs ac
In ensi e ca e uni s (ICUs) ope a e wi h ixed capaci ies and ace unce ain y such as demand
a iabili y, leading o demand-d i en, ea ly discha ges o ee up beds. These discha ges can inc ease
ICU eadmission a es, nega i ely impac ing pa ien ou comes and agg a a ing ICU bo leneck
conges ion. This s udy in es iga es how ICU discha ge iming a ec s eadmission isk, wi h he goal o
de eloping policies ha minimize ICU eadmissions, managing demand a iabili y and bed capaci y.
To de ine a bina y ea men , we andomly assign hypo he ical discha ge days o pa ien s, compa ing
hese wi h ac ual discha ge days o o m in e en ion and con ol g oups. We apply wo causal machine
lea ning echniques (gene alized andom o es , modi ied causal o es ). Assuming uncon oundedness,
we le e age obse ed pa ien da a as su icien co a ia es. Fo scena ios whe e uncon oundedness
migh ail, we discuss an IV app oach wi h di e en ins umen s.
We u he de elop decision policies based on indi idualized a e age ea men e ec s (IATEs) o
minimize indi idual pa ien s’ eadmission isk. Ou sample comp ises 12,950 ICU s ays (11,873 unique
cases) om he Depa men o Su gical In ensi e Medicine o he Can onal Hospi al o S . Gallen
admi ed be ween Janua y 01, 2016, and Decembe 31, 2023. We ind ha o 72% o ou sample
discha ge a poin in ime 𝑡 as compa ed o 𝑡+1 inc eases pa ien s’ eadmission isk. Vice e sa, 28%
o cases p o i om an ea lie discha ge in e ms o eadmission isk. The ange o IATEs is qui e la ge:
Fo 91.4% o ICU s ays, an ea lie ICU discha ge changes a pa ien ’s eadmission isk be ween -0.05 and
0.05 pe cen age poin s (-55% and 55% ela i e change as compa ed o he a e age eadmission a e o
9.04%).
To de elop decision policies, we will exploi his ea men he e ogenei y and ank pa ien s acco ding
o hei IATEs and compa e IATEs o op imal and ac ual discha ges ac oss all decision poin s in ou
obse a ion pe iod. Finally, we ou line how we will assess he po en ial educ ion in eadmissions and
sa ed bed capaci ies unde op imal policies in a simula ion, o e ing ac ionable insigh s o ICU
managemen .
We aim o p o ide a no el app oach and bluep in o simila ope a ions esea ch and managemen
science applica ions in da a- ich en i onmen s.
1
1 In oduc ion
The in ensi e ca e uni (ICU) o a hospi al ea s c i ically ill pa ien s o en su e ing om li e-
h ea ening diseases (Gopalan and Pe shad, 2019; Milb and e al., 2008; Na es e al., 2016). An ICU is
cha ac e ized by sca ce, cos ly capaci ies such as high-end hospi al beds and medical equipmen , and
specialized physicians and nu ses. These capaci ies a e ixed in he sho - o mid- e m. In addi ion,
pa ien demand and he pa ien s’ complexi y mix a e unce ain (Dobson e al., 2010; Thi umalai e al.,
2024). Pa ien s a i e due o scheduled su ge ies, e.g., o pos -su ge y obse a ion, o as ex e nal o
in e nal eme gencies. P edic ion o he olume o incoming pa ien s is di icul , and a ely done in
p ac ice. Addi ionally, pa ien s’ leng h o s ay in he ICU is also unce ain as i depends, among o he
ac o s, on pa ien s’ main diagnosis, co-mo bidi ies, pe o med p ocedu es, and he p og ession o
pa ien s’ heal h s a us. Las ly, p ocess imes o p io p ocess s eps, i.e., eme gency ca e o su ge y, a e
much smalle (hou s) han pa ien s’ leng h o s ay in he ICU (days). Fixed capaci ies, unce ain and
unscheduled incoming demand, unce ain p ocess imes, and as e h oughpu ime o ups eam
p ocesses make he ICU in o a classic example o a p ocess bo leneck (Bai e al., 2021; Chan e al., 2012).
Unlike classic p oduc ion se ings, in en o y canno be buil be ween p ocess s eps and a eas, howe e :
A i ing pa ien s a e in g ea need o ca e and hus need in ensi e ca e immedia ely. I he ICU’s
capaci y is ully u ilized, one op ion is o discha ge a pa ien wi h a ela i ely low need o in ensi e
ca e (Be k and Moinzadeh, 1998; Dobson e al., 2010). Discha ging a pa ien in such a se ing is e e ed
o as demand-d i en o ea ly discha ge, ha is, i he e we e no newly a i ing pa ien , he pa ien o
be discha ged would s ay longe in he ICU, p o i ing om addi ional in ensi e ca e (Bai e al., 2021,
2018; Chan e al., 2012; Ouyang e al., 2020).
Demand-d i en discha ges, in u n, a e linked o highe ICU eadmission a es (K ame e al., 2013,
2012; Ni en e al., 2014). Indeed, he heal h s a us o pa ien s discha ged in demand-d i en se ings a e
mo e likely o wo sen downs eam, equi ing a eadmission o and addi ional s ay in he ICU. Such
eadmissions no only g a ely nega i ely impac pa ien s’ heal h and ou comes (Mcneill and Khai a ,
2020; Rosa e al., 2020) and hospi als’ bo om line, bu hey also addi ionally clog he ICU p ocess
2
bo leneck – and migh igge addi ional demand-d i en discha ges, po en ially se ing o a icious
cycle (KC and Te wiesch, 2011).
We de ine an ICU discha ge on a gi en day as he a iable o in e es (“ ea men ” in he econome ic
sense), causally linked o an ou come (i.e., ICU eadmission). We o mula e how a pa ien ’s discha ge
on a gi en day compa ed o a discha ge he ollowing day causally e ec s a pa ien ’s isk o eadmission.
Ou goal is o disco e a decision policy ha minimizes ICU eadmissions causally linked o (ea ly) ICU
discha ge. To his end, we plan o show ha in si ua ions whe e he e is mo e han one candida e ha
could be discha ged o accommoda e a newly a i ing pa ien , eadmission isk is minimized by
discha ging he pa ien o whom he e ec o discha ge on eadmission isk is smalles .
The gold s anda d o causal in e ence is a andomized expe imen o a andomized con olled ial.
Such a s udy design is e y di icul o a he impossible o implemen – bo h om an e hical as well as
ope a ional poin o iew – o ou esea ch endea o . Thus, we p esen an app oach ha uses
obse a ional da a and causal in e ence unde he selec ion on obse ables assump ion, also e e ed o
as uncon oundedness o condi ional independence assump ion. Ou esea ch ques ions o de eloping
an empi ical app oach and me hodology a e:
I How can he causal e ec o an ICU discha ge on a gi en day on he ICU eadmission isk o a
pa ien be es ima ed wi h obse a ional da a?
II How can a decision policy be lea ned ha minimizes ICU eadmissions?
Resea ch ques ions I and II s uc u e ou esea ch objec i e in wo conc e e s eps: Fi s , es ima ing he
causal e ec and second, lea ning a decision policy based on he es ima ed causal ela ionship,
exploi ing e ec he e ogenei y. No e ha bo h s eps need o conside indi idual pa ien cha ac e is ics
and indi idualized a e age ea men e ec s (IATEs) o enable decision make s o make a discha ge
decision in a se ing o unce ain y.
The e is a ich ope a ions managemen (OM) and medical li e a u e on ICU managemen (Bai e al.,
2018; Gopalan and Pe shad, 2019; Ni en e al., 2014), some o which we will e iew in he nex chap e .
As o many ope a ions esea ch se ings (Ho e al., 2017), OM models add ess he ICU p ocess
3
bo leneck p oblem wi h no ma i e ma hema ical models, aiming o de elop decision policies om a
heo e ical amewo k. Commonly, s udies hen es and/ o calib a e hei models wi h (small) samples
om one o se e al hospi als (e.g., Bai e al., 2021; Chan e al., 2012), o pe o m a simula ion s udy
(Ouyang e al., 2020).
In his pape , we ou line ha due o he causal ela ionship o ICU discha ge and ICU eadmission and
indi idual pa ien cha ac e is ics c ea ing a ia ion and unce ain y, a di e en app oach migh be mo e
e ec i e, gene alizable, and be e scalable han adi ional OM me hods. While we also de elop ou
model om a heo e ical s a poin , i s main s eng h comes om i s empi ical applica ion and
p ac icali y. We aim o show ha machine lea ning me hods om he amily o doubly obus lea ne s
can answe ou esea ch ques ions yielding da a-d i en decisions ha op imize medical quali y and
ee up sca ce capaci y. We belie e ha ou app oach is no only alid o ICU managemen and o he
hospi al ope a ions se ings bu ha ou s udy can se e as a bluep in o a mo e gene al applica ion
o causal machine lea ning me hods in ope a ions esea ch.
This sec ion con inues wi h a e iew o ela ed wo k. Sec ion 2 p esen s a o mal p oblem s a emen .
Sec ion 3 discusses ou app oach o es ima ing causal e ec s and de eloping op imal decision policies.
We also p esen i s desc ip i e esul s, and i s es ima ions o causal e ec s. Las ly, we ou line a
simula ion s udy o assess he p ac ical u ili y o ou app oach. Sec ion 4 summa izes and gi es an
ou look.
Rela ed wo k
We apply machine lea ning me hods o causal in e ence and policy lea ning o a hospi al ope a ions
p oblem, mo e speci ically o esol ing one eason o bo leneck conges ion o a key hospi al p ocess
a ea, he ICU. Thus, ela ed wo k o ou s udy a e s udies in he OM li e a u e ocusing on (1) machine
lea ning applica ions, (2) causal in e ence and causal machine lea ning, (3) policy lea ning, and (4)
hospi al ope a ions.
10
3) Bias and unce ain y o ansi ion p obabili ies: Ma ko decision p ocesses equi e iden i ying
p obabili ies o ansi ioning om one s a e o ano he . In dynamic and complex se ings, such
p obabili ies ha e o be iden i ied o many pa ame e s (Be sekas, 2012). A single biased
ansi ion p obabili y will bias he whole model. Addi ionally, he unce ain y o a poin es ima e
o ansi ion p obabili ies is usually no conside ed in Ma ko decision p ocess models (Zhang e
al., 2019). In o he wo ds, Ma ko decision p ocess models implici ly assume poin es ima es
om empi ic da a o expe es ima es o be ue p obabili ies. Indeed, unce ain ansi ion
p obabili ies ha e ecei ed a en ion in he li e a u e o decades (e.g., Sa ia and La e, 1973), and
he e a e se e al app oaches o add ess his issue (Delgado e al., 2011; Mas in and Jaille , 2012;
Zhang e al., 2019).
P ac ical ICU decision making: A no el p oblem s a emen
We p opose a no el app oach o add essing he limi a ions p esen ed abo e. Fi s ly, we de ine a model
based on p ac ical ICU decision making which needs o esol e he bed capaci y cons ain 𝐵𝑎≥0 o e
he cou se o each day. Secondly, o help decision make s sol e his cons ain , we design policies ha
a e gene alizable, lea nable wi h causal machine lea ning me hods and e-lea nable wi h he same
algo i hms and simila da a om o he hospi als, and hus scalable. No e also ha s a e and ac ion space
models we e shown o imp o e ICU decision making on a ac ical le el, e.g., when deciding whe he
and how much ICU capaci y should be ese ed o a oid mo e cos ly ejec ion and/ o ea ly discha ge
(Bai e al., 2021). Wi h ou model, we will p o ide decision suppo o ope a ional decision making in
he ICU, i.e., o decisions ha mus be made ou inely and daily.
We use he low o a pa ien h ough a hospi al as he s a ing poin o ou p oblem s a emen (c . Figu e
1, loosely based on Bai e al. (2021, 2018) and Li ak e al. (2008)). We a e in e es ed in educing
eadmission lows (1), (2), and (3). No e, howe e , ha a ou pa ne hospi al, we in es iga e a mixed
ICU also accommoda ing in e media e ca e pa ien s. Thus, we a e speci ically in es iga ing eadmission
lows (1) and (3), i.e., he eadmission o a pa ien o a highe le el o ca e uni .
11
K ame e al. (2013), o ins ance, epo ha he median eadmission a e a he mo e han 100 ICUs
hey in es iga ed was a 5.9% (in e qua ile ange be ween 5.1% and 7.0%) and Hosein e al. (2014)
ound in a me a-analysis ha eadmission a es ypically a e be ween 4% and 6%. While a ce ain
pe cen age o ICU eadmissions appea s o be nonp e en able (Al-Jaghbee e al., 2016), p e en able
eadmissions and especially hose causally linked o discha ges could be e med as ewo k and
co ec ion in Lean Managemen e ms, adding o bo leneck conges ion.
Figu e 1: Hospi al pa ien low
Anno a ions: Pa ien s can en e a hospi al as scheduled, plannable cases ( ull a ows) h ough he ou pa ien clinic o as
unscheduled/ eme gency cases h ough he eme gency depa men (dashed a ows). F om he eme gency depa men , pa ien s
a e pushed on o he p ocess a ea wi h ee capaci y and/ o whe e hey need o ecei e ca e. Unscheduled ans e s om o he
hospi als dis up hospi al sys em managemen u he . Pa ien s a e pushed h ough he di e en p ocess a eas in a scheduled o
o en imes unscheduled manne . Demand planning and co esponding supply planning only egula ly occu in he cen al
ope a ing oom a ea, o he p ocess a eas a e commonly s a ed co esponding o hei ull capaci y (e.g., o se e all beds on a
wa d). We a e pa icula ly in e es ed in how o minimize eadmission lows (1) and (2), o en causally linked o ea ly, demand-
d i en discha ges o pa ien s in he ICU. In such si ua ions, pa ien s wi h he smalles isk o being eadmi ed due o he ea ly
discha ge should be discha ged, and no necessa ily hose wi h he lowes p edic ed eadmission isk (c . A hey (2017),
Feue iegel e al. (2024), and P ospe i e al. (2020) o a discussion o his opic).
The basis o ou o mal p oblem s a emen is he capaci y cons ain 𝐵𝑎≥0, deno ing ha he a ailable
ICU bed capaci y 𝐵𝑎 mus always be equal o o g ea e han ze o ope a able ICU beds h oughou a
gi en day. In a naï e s a e, 𝐵𝑎 is solely de ined by he o e all ICU bed capaci y on a gi en day, and he
sum o pa ien s who will no be discha ged and hus occupy he ICU ∑𝑥𝑖
𝐼𝑖=1 wi h 𝑖= {1,2,…,𝐼} and
𝑥=1 i 𝑥𝑖 esides in he ICU a leas o one mo e day, 𝑥=0 o he wise:
𝐵𝑎=𝐵−∑𝑥𝑖
𝐼
𝑖=1
(1)
Consul ing Figu e 1, he i s pa ien low decision-make s in a su gical ICU, as a ou pa ne hospi al,
mus conside and inco po a e in o (1) is he sum o elec i e su ge y pa ien s wi h a planned pos -
Ou pa ien Clinic
(su ge y
indica ion, and
p e-medica ion)
Eme gency
Depa men
Ambulance
Helicop e
Cen al
Ope a ing Room
A ea
In ensi e Ca e
Uni
In e media e
Ca e Uni
No mal Ca e
Uni
T ans e s om
o he hospi als
Admissions/
discha ges
Scheduled /
elec i e
Unscheduled/
eme gency
Scheduled appoin men s
Walk-ins
12 3
xReadmission
lows
12
su ge y ICU s ay ∑𝑎𝑒
𝐸
𝑒=1 on a gi en day, wi h 𝑒= {1,2,…,𝐸}, and 𝑎=1 i 𝑎𝑒 has a planned pos -
su ge y ICU s ay, 𝑎=0 o he wise. In addi ion, planned discha ges occu because a pa ien has eached
ICU se ice comple ion (na u al discha ge), deno ed by 𝑑𝑛 wi h 𝑛= {1,2,…,𝑁}, and 𝑑=1 i 𝑑𝑛 is
na u ally discha ged du ing he day, 𝑑=0 o he wise. This expands he capaci y cons ain gi en in (1)
in o:
𝐵𝑎=𝐵−(∑𝑥𝑖
𝐼
𝑖=1 +∑𝑎𝑒
𝐸
𝑒=1 )+∑𝑑𝑛
𝑁
𝑛=1
(2)
I o e all ICU capaci y and na u al discha ges we e su icien o accommoda e ∑𝑥𝑖
𝐼𝑖=1 and new a i als
∑𝑎𝑒
𝐸
𝑒=1 , and i we s ayed in his simpli ied decision amewo k, we could be sa is ied wi h he de ini ion
o a ailable ICU bed capaci y as ou lined in (2).
Howe e , e en i he e we e no unscheduled, ex e nal eme gency pa ien a i als, he e a e h ee
addi ional conside a ions o make (c . Figu e 1). Gi en he capaci y cons ain 𝐵𝑎≥0, he e will be days
whe e ∑𝑎𝑒
𝐸
𝑒=1 will equi e discha ges in addi ion o ∑𝑑𝑛
𝑁
𝑛=1 , which we de ine as planned ea ly, demand-
d i en discha ges deno ed by 𝑑𝑝
𝑒𝑎𝑟𝑙𝑦 wi h 𝑝= {1,2,…,𝑃} and 𝑑=1 i 𝑑𝑝
𝑒𝑎𝑟𝑙𝑦 is planned o be ea ly
discha ged, 𝑑=0 o he wise. Secondly, ano he le e o sa is y 𝐵𝑎≥0 is o ejec pa ien s, i.e. o cancel
o pos pone elec i e su ge ies wi h a planned pos -su ge y ICU s ay, deno ed by 𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 wi h 𝑐=
{1,2,…,𝐶}, and 𝑎=1 i 𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 is cancelled, 𝑎=0 o he wise. Thi dly, he capaci y cons ain mus also
hold when conside ing eadmissions om downs eam uni s, deno ed by 𝑎𝑟 wi h 𝑟= {1,2,…,𝑅}, and
𝑎=1 i 𝑎𝑟 is eadmi ed, 𝑎=0 o he wise. These h ee conside a ions expand 𝐵𝑎 o:
𝐵𝑎=𝐵−(∑𝑥𝑖
𝐼
𝑖=1 +∑𝑎𝑒
𝐸
𝑒=1 −∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒
𝐶
𝑐=1 +∑𝑎𝑟
𝑅
𝑟=1 )+(∑𝑑𝑛
𝑁
𝑛=1 +∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 )
(3)
In a se ing wi hou unscheduled a i als, e.g., in an o hopedic hospi al exclusi ely ea ing elec i e
su ge y pa ien s, (3) would su ice o desc ibe 𝐵𝑎 and he discha ge decision p ocesses o sa is y 𝐵𝑎≥0.
No e ha he e s ill is unce ain y in se ice ime, i.e., he numbe o ICU inpa ien days o each 𝑥𝑖, and
hus p edic ion o ∑𝑥𝑖
𝐼𝑖=1 and ∑𝑑𝑛
𝑁
𝑛=1 a e p one o unce ain y. Fu he no e ha a he s a o a day
(e.g., be ween 07:00 a.m. o 08:00 a.m. a ou pa ne hospi al), ICU decision-make s migh ini ia e hei
13
decision p ocess a (2), and i hey o esee ha he capaci y cons ain will no be sa is ied a any poin
in ime h oughou he day, hey will y o sa is y he capaci y cons ain by balancing ∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒
𝐶
𝑐=1 ,
and ∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 . While we app ecia e ha pas s udies mus make his assump ion o hei models,
cancella ions and planned demand-d i en discha ges do no happen simul aneously o pa ien a i al.
ICU decision-make s a he an icipa e he numbe o planned ICU admissions acco ding o he
ope a ing oom schedule a he s a o a day and compa e hese wi h he a ailable ICU bed capaci y
a e na u al discha ges (c . (2)).
Conside ing unscheduled o eme gency/ u gen pa ien s coming om ups eam uni s, i.e., di ec ly om
he eme gency depa men o , by a oundabou ou e, om he cen al ope a ing oom a ea, o as
ans e -ins om o he hospi als, he decision p ocess o sa is y 𝐵𝑎≥0 becomes mo e complex. No e
ha in p ac ice, expe ienced ICU decision-make s migh an icipa e unscheduled ICU admissions and
hus ese e some ICU capaci y by discha ging mo e pa ien s han absolu ely needed o sa is y 𝐵𝑎≥0
as de ined in (3). Indeed, pas s udies such as Bai e al. (2021) quan i a i ely de i e exac ly how much
capaci y should be ese ed balancing he cos s o pa ien ejec ion, and demand-d i en discha ge. In
ac , one could conside ha he capaci y cons ain ac ually is 𝐵𝑎≥∑𝑎𝑢
𝑎𝑛𝑡
𝑈
𝑢=1 , ha a ailable ICU
capaci ies mus be a leas he o al numbe o unscheduled a i als/ admissions he expe ienced ICU
decision-make an icipa es on a gi en day (𝑢= {1,2,…,𝑈}, and 𝑎=1 i 𝑎𝑢
𝑎𝑛𝑡 is an icipa ed as
unscheduled a i al, 𝑎=0 o he wise). As i is unce ain how many unscheduled pa ien s will a i e
exac ly, we addi ionally de ine 𝑎𝑣𝑎𝑑𝑑 as he un o eseen unscheduled a i als in addi ion o ∑𝑎𝑢
𝑎𝑛𝑡
𝑈
𝑢=1 ,
wi h 𝑣= {1,2,…,𝑉} and 𝑎=1 i he a i al o 𝑎𝑣𝑎𝑑𝑑 is unscheduled and is no an icipa ed, 𝑎=0
o he wise. No e ha decision make s migh inc ease he sum o planned ea ly discha ges ∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1
and/ o he sum o cancella ions o elec i e su ge ies ∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒
𝐶
𝑐=1 o o se ∑𝑎𝑢
𝑎𝑛𝑡
𝑈
𝑢=1 , ye o manage any
14
𝑎𝑣𝑎𝑑𝑑, he only le e is o demand-d i en discha ge a pa ien in an unplanned manne , deno ed by 𝑑𝑞𝑒𝑎𝑟𝑙𝑦
wi h 𝑞= {1,2,…,𝑄} and 𝑑=1 i 𝑑𝑞𝑒𝑎𝑟𝑙𝑦 is ea ly discha ged in an unplanned way, 𝑑=0 o he wise.
1
To sa is y 𝐵𝑎≥0, decision make s hen conside ha
𝐵𝑎=𝐵−(∑𝑥𝑖
𝐼
𝑖=1 +∑𝑎𝑒
𝐸
𝑒=1 −∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒
𝐶
𝑐=1 +∑𝑎𝑟
𝑅
𝑟=1 +∑𝑎𝑢
𝑎𝑛𝑡
𝑈
𝑢=1 )+(∑𝑑𝑛
𝑁
𝑛=1 +∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 )
+(∑𝑑𝑞𝑒𝑎𝑟𝑙𝑦
𝑄
𝑞=1 −∑𝑎𝑣𝑎𝑑𝑑
𝑉
𝑣=1 )
(4)
No e ha planned discha ges, bo h na u al and demand-d i en, happen a a la e poin in ime han he
discha ge decision. A ou pa ne hospi al, ∑𝑑𝑛
𝑁
𝑛=1 and ∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 a e decided in he ea ly mo ning o
a (week-) day, while discha ges occu in he la e mo ning o ea ly a e noon, in close alignmen wi h he
downs eam ca e uni s admi ing he pa ien (s). Should a pa ien ha e wo sened be ween ini ial
discha ge decision and ac ual discha ge, decision-make s migh e- hink and change hei ini ial
decision.
Empi ical e idence o o mally de ined decision cons ain
In ou da a, he e is empi ical e idence o he dynamics o he decision cons ain as o mula ed in
equa ion (4). Figu e 2 shows he ypical a i al pa e n a ou pa ne hospi al, depic ing he sha e o
pa ien s a i ing pe hou o he day.
1
Addi ionally, one could a gue ha i 𝑎𝑣𝑎𝑑𝑑 a i ed ea ly in he day be o e su ge y has s a ed o he las 𝑎𝑒, decision-make s
migh also ha e he op ion o cancel one addi ional 𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒. Fo simplici y, we do no conside his op ion.
15
Figu e 2: A i al pa e n
Anno a ions: The analysis includes all ICU s ays o pa ien s wi h a posi i e o wai ed gene al consen admi ed a ou pa ne
hospi al be ween Janua y 01, 2016 and Decembe 31, 2023, no exclusion c i e ia applied (14,121 s ays and 12,932 unique cases; c .
Figu e 4 below).
In sum, oughly 72% o all pa ien s a i e be ween 10:00 a.m. and 09:00 p.m. S ill, he olume o pa ien
admissions (28% o all admissions) be ween 09:00 p.m. and 10:00 a.m. is no negligible. The sha e o
discha ges in he same ime window is qui e small (see Figu e 3). This is e idence ha ICU decision
make s inco po a e planned admissions a e su ge y ∑𝑎𝑒
𝐸
𝑒=1 in o hei decision making p ocess and
ha hey do an icipa e unscheduled admissions ∑𝑎𝑢
𝑎𝑛𝑡
𝑈
𝑢=1 and possibly also eadmissions ∑𝑎𝑟
𝑅
𝑟=1 .
Discha ges ypically occu be ween 09:00 a.m. and 04:00 p.m. ( oughly 95% o all discha ges o Panel
A, 97% o Panel B), and mos discha ges happen in an e en close ime window be ween 10:00 a.m.
and 02:00 p.m. ( oughly 86% o Panel A and 89% o Panel B). This is e idence o he planned discha ge
decisions ∑𝑑𝑛
𝑁
𝑛=1 and ∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 happening ea ly in he mo ning wi h co esponding discha ges a ew
hou s la e .
16
Figu e 3: Discha ge pa e n
Anno a ions: Panel A includes all ICU s ays o pa ien s wi h a posi i e o wai ed gene al consen admi ed a ou pa ne hospi al
be ween Janua y 01, 2016 and Decembe 31, 2023, no exclusion c i e ia applied (14,121 s ays and 12,932 unique cases; c . Figu e 4
below). In Panel B, we exclude pa ien s who died du ing hei ICU s ay (589 s ays, 518 unique cases), as he iming o hese
pa ien s’ “discha ge” does no occu acco ding o he decision p ocess o malized in equa ion (4).
17
We also see ha discha ges in “o -hou s” be ween 02:00 p.m. and 10:00 a.m. (14% (11%) o all discha ges
o Panel A (B)) and especially be ween 04:00 p.m. and 09:00 a.m. a e a he a e (5.4% (2.9%) o all
discha ges o Panel A (B)). Especially o Panel B, whe e we exclude pa ien s who died du ing hei
ICU s ay, we may assume ha a conside able sha e o hese o -hou discha ges a e unplanned demand-
d i en discha ges ∑𝑑𝑞𝑒𝑎𝑟𝑙𝑦
𝑄
𝑞=1 igge ed by un o eseen unscheduled admissions ∑𝑎𝑣𝑎𝑑𝑑
𝑉
𝑣=1 . Figu e 3
clea ly shows ha hese si ua ions occu compa a i ely a ely. A he same ime, a conside able sha e
o admissions does occu in hese o -hou s, namely oughly 57% be ween 04:00 p.m. and 09:00 a.m.
This is e idence ha – a leas in ou empi ical se ing – demand-d i en discha ges a ely occu as
de ined in con en ional ope a ions esea ch s udies.
Summa y
In summa y, we ha e asce ained h ee di e en ypes o poin s in ime whe e ICU discha ge decisions
occu o a e changed:
1) Planned discha ges o any kind a e made in he mo ning in he i s hou s o a physician’s shi ,
ypically be ween 07:00 a.m. and 08:00 a.m. a ou pa ne hospi al
2) A discha ge decision is changed should a pa ien ’s heal h s a us change conside ably be ween he
ime o he discha ge decision and he planned discha ge ime
3) Unplanned discha ge decisions a e made a any poin du ing he day (and nigh ) i 𝑎𝑣𝑎𝑑𝑑 occu s
In equa ion (4), we o mula e he decision p oblem o sa is ying he capaci y cons ain 𝐵𝑎≥0 ha
decision make s ace a each o hese h ee ypes o poin s in ime. Ou model supplies decision-make s
a each o hese ypes o poin s in ime wi h he e ec o a discha ge on a pa ien ’s eadmission isk.
No e ha while ou model can in o m any kind o demand-d i en discha ge, i also p o ides decision
suppo o na u al discha ges. Should he esul o ou model e eal ha he change in eadmission
isk is oo la ge due o he discha ge, he “na u al” discha ge could be pos poned.
3 Causal E ec s and Op imal Policy Lea ning
We s a wi h a sho desc ip ion o ou da ase and desc ip i e analyses. Then, we ou line a h ee-s ep
app oach o enable op imal ICU discha ge decision making: (1) We p esen how o es ima e he ATE,
18
CATEs, and IATEs o an ICU discha ge a a poin in ime 𝑡 on he ICU eadmission isk in an
obse a ional s udy se ing wi h double obus lea ne s, (2) we ou line how we plan o de elop decision
policies based on pa ien s’ IATEs, and (3) we p esen how we plan o apply hese policies o ou empi ic
da a o gauge how many ICU bed capaci ies could ha e been sa ed wi h op imal decisions. In he i s
sec ion, we also discuss iden i ying assump ions o causal in e ence in he con ex o ou s udy.
O e iew o Da ase and Desc ip i e Analyses
Da ase
Ou da ase con ains clinical and basic da a o all ICU s ays wi hou a documen ed nega i e gene al
consen admi ed o he Depa men o Su gical In ensi e Ca e Medicine be ween Janua y 01, 2016, and
Decembe 31, 2023. Figu e 4 shows he inclusion and exclusion c i e ia and co esponding samples o
he di e en analyses we conduc ed.
Figu e 4: Inclusion and exclusion c i e ia and co esponding samples
Anno a ions: Reasons o excluding cases ha a e no candida es o eadmission a e discha ge o ano he ICU, o home, o a
ehabili a ion clinic o nu sing home, o o a di e en hospi al (c . K ame e al. (2013)).
Ou da a is eco ded a ICU s ay le el. Each ICU s ay has a unique iden i ie . One case, gi en by a
unique case numbe , has a leas one ICU s ay. Theo e ically, one pa ien could be admi ed o a hospi al
19
se e al imes in a yea . In such cases, a new unique case numbe is de ined o each hospi al admission.
A case numbe o which wo o mo e ICU s ays a e eco ded hus indica es ha a pa ien was
eadmi ed o he ICU wi hin he same hospi al s ay.
Sample A was used o analyses ega ding a i al and discha ge pa e ns (e.g., Figu e 2 abo e). Sample
B includes only hose cases and s ays ha a e useable o causal in e ence. To his end, we excluded
pa ien s who died in he ICU as hese a e eco ded as discha ges ye a e no connec ed o a delibe a e
discha ge decision, excluded cases ha could no be eadmi ed o he ICU as hey we e discha ged o
a di e en ICU, o home, o a ehabili a ion clinic o nu sing home, o o a di e en hospi al, and we
excluded cases o whom he discha ge eason was missing.
O e all, we can ex ac and enginee mo e han 4,600 ea u es om ou da ase (see Table 1). Mos o
hese ea u es a e ela ed o medica ion and d ugs (mo e han 4,000 ea u es). We include one ea u e
pe subs ance, dosage and olume uni and alues ep esen he gi en olume (ei he 0 o a con inuous
numbe ) wi hin he las 24 hou s be o e discha ge (in e en ion g oup) o “simula ed” discha ge
(con ol g oup, see Figu e 6 below). O he majo ea u e ca ego ies a e labo a o y diagnos ic alues
(mo e han 200 ea u es), i al signs (20 ea u es), clinical ( isk) sco es (close o 40 ea u es) and basic
da a (age, gende , weigh , heigh , BMI).
Table 1: O e iew o used ea u es
Fea u e
ca ego y
Numbe o
ea u es
Nume ic ype
Conside ed
measu emen s
Desc ip ion and examples
Medica ion
4,119
Con inuous
Las 24 hou s
Hund eds o di e en subs ances wi h a
leas one, o en imes se e al dosages and
olume uni s, e.g., No ad enalin
pe iphe al in mic og am, No ad enalin in
mic og am
Labo a o y
es s
208
Con inuous
Las
measu emen
208 di e en labo a o y alues, e.g.,
Kalium, C ea inine, Choles e ol, HbA1c
Clinical
sco es
40
Con inuous
Las 2
measu emen s
10 di e en sco es, e.g., SAPS II, SAPS 3,
NEMS, GCS, e c.
Diagnoses
114
Dummy
Time in a ian
57 indica ion a eas o i s and ollow-up
diagnosis, e.g., "diseases o he li e and
bilia y ac "
In usions
35
Con inuous
Las 24 hou s
22 di e en in usions, pa ially wi h mo e
han one dosage, e.g., Glucose 5%, 10%,
20%, 40%, 50%; all in ml
26
Fo ou cu en esul s, we only un he las wo s eps once pe ICU s ay. Fo ou inal esul s, we plan
o un hese wo s eps mul iple imes o sepa a e (I)ATE es ima ions. Resul s om each un will be
a e aged in he end o ecei e he inal esul s.
Ou come de ini ion
The e is an ongoing discussion in he medical li e a u e ega ding a meaning ul measu emen and use
o ICU eadmission a es (Hosein e al., 2014; K ame e al., 2013; Woldhek e al., 2017). In ou main
model, we use eadmission ega dless o he ime be ween discha ge and eadmission as ou come. We
plan o pe o m one o wo sensi i i y analyses, using mo e na owly de ined eadmission a es such as
eadmission wi hin 48 hou s and 96 hou s ( wo o ou days).
In ou da ase , we obse e a aw eadmission a e o oughly 9.0% while eadmission a es wi h de ined
imes be ween discha ge and eadmission a e be ween 2.8% and 6.1% (see Figu e 7).
Figu e 7: Readmission a es acco ding o ime be ween discha ge and eadmission
Anno a ions: The plo is based on Sample B (12,950 ICU s ays and 11,873 unique cases).
27
Iden i ying assump ions
Iden i ying assump ions o causal in e ence a e (Ang is e al., 1996; Imbens, 2000; Lechne , 2001): (1)
Uncon oundedness, (2) Common Suppo (CS) o o e lap, (3) S able-Uni -T ea men -Value
Assump ion (SUTVA), and (4) exogenei y.
Wi h uncon oundedness, we assume ha we obse e all a iables ha migh in luence bo h he
ea men selec ion (i.e., o be o no o be discha ged a poin 𝑡) and he po en ial ou come o a discha ge.
Discha ge decisions a e made by senio physicians (Na es e al., 2016). We obse e all da a and decision
a iables ha a e a ailable o hese physicians when hey make discha ge decisions. This includes mos
a iables ha we e judged by mos Swiss ICUs as ele an o ICU discha ge decisions (Heidegge e
al., 2005). Thus, we a gue o uncon oundedness in ou s udy se ing. The e a e h ee lines o coun e -
a gumen a ion, howe e .
The i s a gumen is ha si ua ions may occu in which a discha ge decision (i.e., ou ea men
assignmen ) and hus discha ge a e de e mina e. Uncon oundedness would be iola ed i such a
ea men de e mina ion was dependen on pa ien cha ac e is ics (𝑋𝑖). (i) The decision mus be posi i e
e e y ime when he e is only one pa ien candida e o discha ge a 𝑡. We may s ill assume
uncon oundedness, as he numbe o po en ial discha ge candida es is exogenous, i.e., we can iew
ea men assignmen in hese si ua ions s ill as andom. (ii) A discha ge decision mus be nega i e i
he e is a con aindica ion, e.g., a pa ien is in uba ed o a pa ien ecei es a ce ain d ug o subs ance
(e.g., ca echolamines) (Heidegge e al., 2005). This does no iola e uncon oundedness in ou con ex ,
howe e , as we always obse e he discha ge a a poin in ime when i is no con aindica ed (c .
ea men de ini ion).
The second a gumen is ha in p ac ice, physicians ac ually conside mo e a iables han “only” he
housands o clinical pa ame e s a ailable o hem (and o us): Physicians migh addi ionally collec
“so ” da a, e.g., du ing daily pa ien isi ing ounds by isual con ol o he pa ien and discussions
wi h nu ses (Na es e al., 2016; O oma e al., 2018). Da a ob ained his way migh include deg ee o
paleness, swea ing, communica ed pain, men al con usion and diso ien a ion, o he apy compliance.
28
So da a is no a ailable o us (o eadable by any machine). Uncon oundedness would be iola ed i
hese so da a we e no co ela ed wi h he housands o clinical pa ame e s a ailable o us. This is
a he unlikely.
The hi d a gumen is ha a discha ge, and possibly also a la e eadmission, a e in luenced by he
a ailable downs eam skill mix and/ o capaci y. Physicians migh decide o discha ge a ce ain pa ien
on a gi en day i hey know ha he e a e expe ienced, well-quali ied physicians and nu ses a ailable
in downs eam uni s who can manage he pa ien , e en i his pa ien is sicke gi en clinical pa ame e s
han ano he pa ien who was no discha ged ano he day when downs eam skill mix and capaci y
we e (allegedly) inadequa e. In such si ua ions, physicians migh decide agains a discha ge as hey
expec a wo sening o he pa ien ’s heal h s a us downs eam, ende ing an ICU eadmission mo e
likely.
To add ess such doub s ha migh s ill emain a e con olling o se e al housand pa ien
cha ac e is ics, we could implemen an ins umen al a iable (IV) app oach. An IV app oach is a ailable
o bina y ins umen s in GRF (A hey e al., 2019) and was de eloped by Wang e al. (2021) o
con inuous IVs in causal o es s. As IV, we ha e se e al op ions: (1) he daily numbe o admi ed
pa ien s as a deg ee o ICU busyness bo h as con inuous a iable (only useable i we can inco po a e a
con inuous IV in GRF) and as dummy (1=busy, 0=no busy; h eshold o be de e mined, e.g., one hi d
o o e all ICU capaci y), as his has shown o inc ease he numbe o ICU discha ges (Na es e al., 2016),
(2) he numbe o po en ial discha ge candida es h oughou ICU day shi s (e.g., 06:00 a.m. o 08:00
p.m.), o (3) p oxies o downs eam uni capaci y (and skill-mix), e.g., weekday s. weekend, o days
un il weekend. Rega ding (3), ac ual u iliza ion da a would be p e e able, ye such da a is no collec ed
on a daily le el in a digi al manne o highly un eliable i collec ed manually. No e ha all h ee IV
op ions a e exogenous o ea men assignmen .
Fo u u e s udies, collec ing da a on daily downs eam skill mix and capaci y migh be pe cei able.
This comes wi h conside able e o , howe e , as such da a is no a ailable in a s uc u ed manne o
all IMCUs and NCUs o a hospi al. While s a schedules could be a ailable (o en only a ailable in
29
analog o m o in e-mails, manually adminis e ed Mic oso Excel ables, o simila ), hese alone will
no explain he skill mix pe cei ed by he ICU decision make . To accu a ely accoun o his pe cei ed
skill mix, we would ha e o label all physician and nu se s a membe s o all downs eam uni s
acco ding o he expe ience and quali ica ion o all ICU decision make s. This migh esul in di e en
labels o indi idual downs eam physicians and/ o nu ses depending on wha ICU decision make is
asked o label. Indeed, i expe ience and skill le el a e pe cei ed di e en ly by physicians, his migh
in u n supply an a gumen ha uncon oundedness does hold, as downs eam skill mix would hen no
sys ema ically in luence ea men assignmen and ou come in he same (pe cei ed) way. Las ly,
ano he a gumen ha uncon oundedness s ill holds is ha i is unlikely ha a physician will be able o
judge he expe ience and skill le el o all ea ing nu ses and physicians ac i e in downs eam uni s, o
e en ha an ICU physician always has ull anspa ency o downs eam uni s a schedules. G an ed,
i a discha ge decision o a pa icula ly complex pa ien we e made, a physician could ake he ime o
ge anspa ency o e s a schedules and also expe ience and skill le el. A sys ema ic in luence s ill is
unlikely, howe e .
Ano he possibili y could be o measu e expe ience by he numbe o yea s a medical p o essional has
been ac i ely wo king, and quali ica ion by academic deg ees, u he educa ion ce i ica es, and
scien i ic publica ions. This app oach would also pose a majo challenge in e ms o equi ed e o , and
a leas some o hese da a will no be a ailable in a s uc u ed o m. Las ly, measu ing expe ience and
skill le el in his way also has limi a ions. Fo ins ance, one migh a gue ha expe ience is in ac buil
by being exposed o ad e se e en s and di icul si ua ions. This should co ela e wi h he numbe o
yea s a p o essional has been wo king bu his mus no necessa ily be ue.
In summa y, we belie e we can iably a gue ha uncon oundedness holds (1) as we con ol o all
con ounde s also a ailable o senio physicians making discha ge decisions, (2) pa ien cha ac e is ics
do no sys ema ically in luence ea men assignmen and ou come and i hey do, se e as exclusion
c i e ia (e.g., mechanical en ila ion), and (3) while downs eam capaci y and skill-mix migh in luence
ICU decision make s, i is no pe cei able ha hey ha e anspa ency o e bo h o all downs eam
30
uni s o e e y discha ge decision, o e en he majo i y o decisions, hus impeding sys ema ic
in luence. S ill, we o acknowledge any conce ns possibly le and he obse a ional se ing o ou s udy,
we will implemen an IV app oach as sensi i i y analysis in ou inal manusc ip .
We may assume CS, i we can show ha p opensi y sco es “o e lap”, i.e., ha each pa ien could be
obse ed wi h o wi hou a discha ge a poin 𝑡:
0<𝑝(𝑊𝑖=1|𝑋𝑖=𝑥)<1 ∀ 𝑥∈𝑋
(16)
In Figu e 8, we plo he dis ibu ion o p opensi y sco es o ou sample (c . Wage and A hey, 2018).
Figu e 8: Common suppo and o e lap analysis o p opensi y sco es
Anno a ions: P opensi ies we e es ima ed wi h GRF, and o Sample B (12,950 ICU s ays and 11,873 unique cases).
The plo shows ha o ou sample and ea men de ini ion, he e is a medium o s ong selec i i y in o
and ou o ea men , espec i ely. Lechne and Ma ecko a (2024) show ha hei Modi ied Causal
Fo es (MCF), deli e s mo e obus es ima es han GRF in cases o (medium and) s ong selec i i y,
especially when es ima ing IATEs. Thus, we es ima e ATEs wi h bo h algo i hms, one example o
CATEs o exempli ica ion wi h GRF, and IATEs wi h MCF. Fo ou inal esul s, we will es ima e all
esul s wi h bo h MCF and GRF.
31
The SUTVA equi es ha spillo e e ec s be ween discha ged pa ien s a e absen . Mo e conc e ely, he
discha ge o one pa ien can only a ec he ou come o he same pa ien and no he ou come o ano he
pa ien . This is gi en as discha ging one pa ien does no di ec ly in luence he eadmission isk o
ano he pa ien . One migh a gue ha once he capaci y cons ain 𝐵𝑎≥0 holds, no addi ional pa ien (s)
is discha ged any mo e, e en i his we e medically possible, and hus he ea men s a e o he
discha ged pa ien (s) o sa is y 𝐵𝑎≥0 migh in luence he ea men s a e o some o he pa ien s
emaining in he ICU. S ill, he ea men s a e o he discha ged pa ien (s) will no di ec ly in luence
he ou come o he pa ien (s) emaining in he ICU.
Exogenei y s ipula es ha pa ien cha ac e is ics used as con ounde s (𝑋𝑖) a e no in luenced by he
discha ge a poin in ime 𝑡 ( he ea men ). This assump ion holds as we obse e 𝑋𝑖 be o e he discha ge
occu s.
In summa y, we should be able o ul ill all ou iden i ying assump ions o a leas be able o implemen
an empi ical s a egy and me hods o handle po en ial iola ions, ensu ing obus ATE, CATE, and
IATE es ima ions.
Fi s esul s
In Table 3, we p esen he ATE es ima ion, employing bo h MCF and GRF.
Table 3: Es ima ed A e age T ea men E ec s
Algo i hm
Es ima e
S anda d
E o
Absolu e change o eadmission
isk (in %-p s.)
Rela i e change o eadmission
isk (in %)
MCF
0.00767
0.0083903
0.77
8%
GRF
-0.00179
0.0047603
-0.18
-2%
Anno a ions: Es ima es we e made wi h Sample B (12,950 ICU s ays and 11,873 unique cases), >4,600 included ea u es (c . Table
1). The ela i e change o eadmission isk is calcula ed in compa ison o he aw eadmission a e o 9.04%.
The ATE o discha ging an ICU pa ien o a downs eam uni a 𝑡 as compa ed o one decision cycle
la e inc eases he eadmission isk by 8% acco ding o MCF bu dec eases he eadmission isk by -2%
acco ding o GRF. E iden ly, bo h poin es ima es a e insigni ican , while he s anda d e o o GRF is
much la ge as compa ed o he es ima e han o MCF. Besides, he magni ude o he e ec , ega dless
o signi icance and sign o he coe icien , is mo e han ou imes highe o MCF han o GRF. S ill,
simply pu , acco ding o bo h MCF and GRF, on a e age, he e is no e ec . I is impo an o no e,
32
howe e , ha we a e no in e es ed in he ATE as i does no enable disc imina i e discha ge decisions
be ween wo o mo e indi idual ICU discha ge candida es (c . below). In ac , i is sensible ha on
a e age, discha ging a discha geable, compa a i ely s able pa ien one day (o a he decision cycle)
la e will no a ec he eadmission isk signi ican ly.
Table 4 shows he es ima ed CATEs o ou g oups, s a i ied by he numbe o imes a pa ien passes
he egula discha ge decision ime a e becoming a po en ial candida e o discha ge, deno ed by 𝑛.
The CATEs show ha he e is ample ea men he e ogenei y. While he e is no e ec o he g oup 𝑛=
2, he magni ude o he es ima e is qui e la ge o all o he 𝑛 and s anda d e o s a e smalle o 𝑛=0
and 𝑛=1, bu e ec s a e s ill insigni ican .
Table 4: Es ima ed Condi ional A e age T ea men E ec s acco ding o 𝒏
𝒏
Es ima e
S anda d
E o
Absolu e change o eadmission
isk (in %-p s.)
Rela i e change o eadmission
isk (in %)
0
-0.01504
0.006106
-1.50
-17%
1
0.01168
0.007846
1.17
13%
2
-0.00097
0.018317
-0.10
-1%
3
0.03302
0.036234
3.30
37%
Anno a ions: Es ima es we e made wi h GRF and wi h Sample B (12,950 ICU s ays and 11,873 unique cases), >4,600 included
ea u es (c . Table 1). The ela i e change o eadmission isk is calcula ed in compa ison o he aw eadmission a e o 9.04%. 𝑛
deno es he numbe o imes a pa ien passes he egula discha ge decision ime a e becoming a po en ial candida e o
discha ge.
No e ha nei he he ATE, no any kind o CATEs will be o use o decision suppo . Fo he ATE, his
is ob ious as all cases ha e he same ATE and i hus canno sugges wha pa ien should a he be
discha ged. CATEs encompass s ill oo many pa ien s, so ha on a gi en day, wo o mo e discha ge
candida es migh all in o he same g oup wi h he same CATE. Then, he same issue as wi h using an
ATE applies.
The e o e, IATEs need o be es ima ed, o ming he basis o de eloping decision policies ha can uly
suppo daily, ope a ional ICU decision making. Gi en he esul s in Table 3 and ou common suppo
analysis in Figu e 8, we chose MCF as causal machine lea ning me hod o es ima e IATEs. Figu e 9
shows he densi y o es ima ed IATEs.
Almos he en i e sample (97.8%, 12,669 ICU s ays) shows IATE es ima es be ween -0.05 and 0.10. No e
ha his a ia ion is ac ually qui e la ge: In pe cen age poin s, eadmission isk is in luenced be ween
33
nega i e 5 and 10 poin s, amoun ing o a ela i e change o eadmission isk be ween -55% and 111%
when compa ed o he a e age eadmission a e o 9.04%. Conside ing a na owe IATE ange be ween
-0.05 and 0.05 (91.4% o sample, 11,833 ICU s ays), he ela i e change o eadmission isk s ill amoun s
o -55% o 55%. This ange shows he exis ing ea men he e ogenei y and showcases ha hese esul s
could p o e e ec i e when de eloping discha ge policies based on IATEs.
Figu e 9: Densi y o Indi idualized A e age T ea men E ec s
Anno a ions: IATEs we e es ima ed wi h MCF, and o Sample B (12,950 ICU s ays and 11,873 unique cases).
S ill, he igu e also shows ha he IATE es ima e o la ge sha e o ICU s ays is be ween 0.00 and 0.03
(58.5%, 11,833 ICU s ays). This means ha o he majo i y o pa ien s, a discha ge one decision cycle
la e inc eases hei eadmission isk ela i ely mildly, be ween 0% and 33% compa ed o he
eadmission a e o 9.04%. Las ly, CATEs in Table 4 al eady indica e ha he e a e ce ain pa ien s ha
migh no bene i om addi ional ime on he ICU. Ou esul s o he IATE es ima ion show ha IATEs
a e indeed nega i e o oughly 28% o all pa ien s.
34
S ill, when in e p e ing IATEs, we mus be cau ious inso a ha we do no show es ima ed s anda d
e o s o pa ien s’ IATEs which would be needed o gene alize s a emen s abou he e ec i eness o
addi ional in ensi e ca e o indi idual ICU pa ien s. Acco dingly, he abo e s a emen s ha e a
desc ip i e and no necessa ily causal in en ion.
Fo ou esea ch objec i e, i.e., suppo ing clinical decision making, s anda d e o s a e o a lesse
impo ance, which we discuss in he nex sec ion.
De elopmen o Decision Policies and Simple Simula ion S udy
We aim o design policies o e ing decision suppo o all ICU discha ge ypes sa is ying he capaci y
cons ain 𝐵𝑎≥0, wi h 𝐵𝑎 as de ined in equa ion (4):
𝐵𝑎=𝐵−(∑𝑥𝑖
𝐼
𝑖=1 +∑𝑎𝑒
𝐸
𝑒=1 −∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒
𝐶
𝑐=1 +∑𝑎𝑟
𝑅
𝑟=1 +∑𝑎𝑢
𝑎𝑛𝑡
𝑈
𝑢=1 )+(∑𝑑𝑛
𝑁
𝑛=1 +∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 )
+(∑𝑑𝑞𝑒𝑎𝑟𝑙𝑦
𝑄
𝑞=1 −∑𝑎𝑣𝑎𝑑𝑑
𝑉
𝑣=1 )
Speci ically, policies should o e suppo when planning discha ges ∑𝑑𝑛
𝑁
𝑛=1 +∑𝑑𝑝
𝑒𝑎𝑟𝑙𝑦
𝑃
𝑝=1 , and when
ha ing o decide in unplanned demand-d i en si ua ions desc ibed by ∑𝑑𝑞𝑒𝑎𝑟𝑙𝑦
𝑄
𝑞=1 −∑𝑎𝑣𝑎𝑑𝑑
𝑉
𝑣=1 . No e
ha in p ac ice, bo h 𝑑𝑛 and 𝑑𝑝
𝑒𝑎𝑟𝑙𝑦 a e planned discha ges and dis inguishing clea ly be ween wha
discha ge is na u al and wha discha ge is a he demand-d i en due o unplanned bu an icipa ed
admissions 𝑎𝑢
𝑎𝑛𝑡, an icipa ed eadmissions 𝑎𝑟, o planned admissions due o elec i e su ge ies 𝑎𝑒, migh
be di icul .
To suppo decision making, in a i s s ep, we will ank all pa ien s acco ding o hei IATE es ima e.
In a second s ep, we de e mine ha pa ien s should be selec ed o discha ge based on hei spo in his
anking as compa ed o o he pa ien s who a e discha ge candida es a he same egula decision cycle.
Mo e conc e ely, i he ICU decision make we e o decide be ween wo pa ien s, he pa ien wi h he
lowe change o eadmission isk due o he discha ge a poin in ime 𝑡, i.e., he smalle IATE, would
be selec ed. The same applies i mul iple pa ien s we e selec ed o discha ge. Fo ins ance, i he e a e
35
se en discha ge candida es and o comply wi h he capaci y cons ain 𝐵𝑎≥0, i e pa ien s would need
o be discha ged, he i e pa ien s wi h he lowes IATEs would be selec ed o discha ge.
S a is ical signi icance, e.g., a he 5%-le el o es ima ed IATEs would indica e he obus ness o he
magni ude o indi idual e ec s, i.e., how ce ain we can be abou indi idual pa ien s’ poin es ima e.
We a gue ha s a is ical signi icance is o lesse impo ance o selec ing discha ge candida es as e en
insigni ican IATEs can op imize decisions, a leas as long as con idence in e als do no o e lap. Fo
he simula ion and inal e sion o ou manusc ip , we will add ess his po en ial challenge in mo e
de ail.
To show he p ac ical u ili y o ou decision policy app oach in e ms o a oided eadmissions and
sa ed ICU capaci y, we will s a a simula ion a he i s day o ou obse a ion pe iod. We hen apply
ou decision policy and change he discha ge decision o all pa ien s whe e he policy would discha ge
a di e en pa ien han he pa ien ha was ac ually discha ged. We con inue o apply his decision
policy ac oss all days o ou obse a ion pe iod. To ecei e he o al numbe o a oidable eadmissions,
we sum he di e ence be ween he minimal (i.e., op imal) IATE, 𝐼𝐴𝑇𝐸𝑑,𝑝, and empi ical (i.e., ac ual)
IATE, 𝐼𝐴𝑇𝐸𝑑,𝑎, ac oss all discha ge decisions 𝐷. No e ha his di e ence is ze o whe e he op imal and
ac ual discha ge decisions a e he same.
𝑅=∑𝐼𝐴𝑇𝐸𝑑,𝑝
𝐷
𝑑=1 −𝐼𝐴𝑇𝐸𝑑,𝑎
(17)
Addi ionally, we can compu e he numbe o sa ed ICU bed capaci ies 𝐵 acco ding o ou decision
policy wi h
𝐵= 𝑅×𝐿𝑂𝑆
365 × 𝑏𝑒𝑑 𝑜𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦 𝑟𝑎𝑡𝑒
(18)
whe e 𝐿𝑂𝑆
ep esen s he a e age leng h o s ay o eadmi ed cases. We will assume di e en bed
occupancy a es ypical o ICUs, e.g., 90% and 95%, o ecei e a ange o ou a oidable ICU bed
capaci y es ima e.
42
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