Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results

Vogel, Justus; Cordier, Johannes; Filipovic, Miodrag Working Paper Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results Working Paper Series in Health Economics, Management and Policy, No. 2025-01 Provided in Cooperation with: University of St.Gallen, School of Medicine, Chair of Health Economics, Policy and Management Suggested Citation: Vogel, Justus; Cordier, Johannes; Filipovic, Miodrag (2025) : Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results, Working Paper Series in Health Economics, Management and Policy, No. 2025-01, University of St.Gallen, School of Medicine, Chair of Health Economics, Policy and Management, St.Gallen This Version is available at: https://hdl.handle.net/10419/308439.2 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Working Paper Series in Health Economics, Policy, and Management 2025 – Nr. 01 Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results Justus Vogel, Johannes Cordier, Miodrag Filipovic I Working Paper Series in Health Economics, Policy, and Management Editor Prof. Dr. Alexander Geissler Professor and Chairholder Chair of Health Economics, Policy, and Management School of Medicine University of St.Gallen Editorial office Jonas Subelack Research Assistant Chair of Health Economics, Policy, and Management School of Medicine University of St.Gallen The entire series of publications is available on our website at: https://med.unisg.ch/en/research/health-caremanagement/publications/ © 2025. This publication is licensed by the CC license CC-BY-NC-ND 4.0 II Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results Keywords: Causal Machine Learning, Intensive Care Unit Management, Hospital Operations, Policy Learning JEL Classification: I10, C44 Authors: Justus Vogel Scientific Project Leader Chair of Health Economics, Policy, and Management, School of Medicine, University of St.Gallen [email protected] Johannes Cordier Research Assistant Chair of Health Economics, Policy, and Management, School of Medicine, University of St.Gallen Miodrag Filipovic Department Head Clinic for Operative Intensive Care Medicine, Cantonal Hospital St. Gallen Recommended citation: Vogel, Justus; Cordier, Johannes; Filipovic, Miodrag (2025): Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results. Working Paper Series in Health Economics, Management and Policy, No. 2025-01, University of St.Gallen, School of Medicine, Chair of Health Economics, Policy and Management, St.Gallen. Causal Effects and Optimal Policy Learning for Intensive Care Unit Discharge Decisions to Solve Hospital Process Bottlenecks: Approach, Methods, and First Results Dr. Justus Vogel*, C Johannes Cordier*, and Prof. Dr. med. Miodrag Filipovic+ * Chair of Health Economics, Policy and Management, School of Medicine, University of St. Gallen, St.- Jakob-Strasse 21, CH-9000 St. Gallen, Switzerland + Cantonal Hospital of St. Gallen, Rorschacher Strasse 95, CH-9000 St. Gallen C Corresponding Author Declarations Acknowledgements We thank Daria Bukanova-Berend for supporting us with the literature search which was necessary for this study. We thank Patrick Münger for extracting all data necessary for this study. We thank our student assistant David Klug for his support in data cleaning and descriptive analyses. Funding This research is supported by a Stepping Stone Grant of the University of St. Gallen (project number 2300181). Conflict of interest None. Author contributions JV: Conceptualization, Methodology, Formal analysis, Interpretation, Writing – Original Draft, Visualization, Supervision, Project administration, Final approval, Funding acquisition JC: Conceptualization, Methodology, Data curation, Formal analysis, Interpretation, Writing – Review & Editing, Final approval MF: Conceptualization, Data acquisition, Interpretation, Writing – Review & Editing, Final approval Ethics approval and consent to participate This study was approved by the Ethical Commission of Eastern Switzerland (project number 202400829). Consent for publication Not applicable. Data availability Anonymized personal health data with restricted access. Code availability Available from the corresponding author upon request. 1 Abstract Intensive care units (ICUs) operate with fixed capacities and face uncertainty such as demand variability, leading to demand-driven, early discharges to free up beds. These discharges can increase ICU readmission rates, negatively impacting patient outcomes and aggravating ICU bottleneck congestion. This study investigates how ICU discharge timing affects readmission risk, with the goal of developing policies that minimize ICU readmissions, managing demand variability and bed capacity. To define a binary treatment, we randomly assign hypothetical discharge days to patients, comparing these with actual discharge days to form intervention and control groups. We apply two causal machine learning techniques (generalized random forest, modified causal forest). Assuming unconfoundedness, we leverage observed patient data as sufficient covariates. For scenarios where unconfoundedness might fail, we discuss an IV approach with different instruments. We further develop decision policies based on individualized average treatment effects (IATEs) to minimize individual patients’ readmission risk. Our sample comprises 12,950 ICU stays (11,873 unique cases) from the Department of Surgical Intensive Medicine of the Cantonal Hospital of St. Gallen admitted between January 01, 2016, and December 31, 2023. We find that for 72% of our sample discharge at point in time 𝑡 as compared to 𝑡+1 increases patients’ readmission risk. Vice versa, 28% of cases profit from an earlier discharge in terms of readmission risk. The range of IATEs is quite large: For 91.4% of ICU stays, an earlier ICU discharge changes a patient’s readmission risk between -0.05 and 0.05 percentage points (-55% and 55% relative change as compared to the average readmission rate of 9.04%). To develop decision policies, we will exploit this treatment heterogeneity and rank patients according to their IATEs and compare IATEs of optimal and actual discharges across all decision points in our observation period. Finally, we outline how we will assess the potential reduction in readmissions and saved bed capacities under optimal policies in a simulation, offering actionable insights for ICU management. We aim to provide a novel approach and blueprint for similar operations research and management science applications in data-rich environments. 1 1 Introduction The intensive care unit (ICU) of a hospital treats critically ill patients often suffering from lifethreatening diseases (Gopalan and Pershad, 2019; Milbrandt et al., 2008; Nates et al., 2016). An ICU is characterized by scarce, costly capacities such as high-end hospital beds and medical equipment, and specialized physicians and nurses. These capacities are fixed in the shortto mid-term. In addition, patient demand and the patients’ complexity mix are uncertain (Dobson et al., 2010; Thirumalai et al., 2024). Patients arrive due to scheduled surgeries, e.g., for post-surgery observation, or as external or internal emergencies. Prediction of the volume of incoming patients is difficult, and rarely done in practice. Additionally, patients’ length of stay in the ICU is also uncertain as it depends, among other factors, on patients’ main diagnosis, co-morbidities, performed procedures, and the progression of patients’ health status. Lastly, process times of prior process steps, i.e., emergency care or surgery, are much smaller (hours) than patients’ length of stay in the ICU (days). Fixed capacities, uncertain and unscheduled incoming demand, uncertain process times, and faster throughput time of upstream processes make the ICU into a classic example of a process bottleneck (Bai et al., 2021; Chan et al., 2012). Unlike classic production settings, inventory cannot be built between process steps and areas, however: Arriving patients are in great need for care and thus need intensive care immediately. If the ICU’s capacity is fully utilized, one option is to discharge a patient with a relatively low need for intensive care (Berk and Moinzadeh, 1998; Dobson et al., 2010). Discharging a patient in such a setting is referred to as demand-driven or early discharge, that is, if there were no newly arriving patient, the patient to be discharged would stay longer in the ICU, profiting from additional intensive care (Bai et al., 2021, 2018; Chan et al., 2012; Ouyang et al., 2020). Demand-driven discharges, in turn, are linked to higher ICU readmission rates (Kramer et al., 2013, 2012; Niven et al., 2014). Indeed, the health status of patients discharged in demand-driven settings are more likely to worsen downstream, requiring a readmission to and additional stay in the ICU. Such readmissions not only gravely negatively impact patients’ health and outcomes (Mcneill and Khairat, 2020; Rosa et al., 2020) and hospitals’ bottom line, but they also additionally clog the ICU process 2 bottleneck – and might trigger additional demand-driven discharges, potentially setting off a vicious cycle (KC and Terwiesch, 2011). We define an ICU discharge on a given day as the variable of interest (“treatment” in the econometric sense), causally linked to an outcome (i.e., ICU readmission). We formulate how a patient’s discharge on a given day compared to a discharge the following day causally effects a patient’s risk of readmission. Our goal is to discover a decision policy that minimizes ICU readmissions causally linked to (early) ICU discharge. To this end, we plan to show that in situations where there is more than one candidate that could be discharged to accommodate a newly arriving patient, readmission risk is minimized by discharging the patient for whom the effect of discharge on readmission risk is smallest. The gold standard for causal inference is a randomized experiment or a randomized controlled trial. Such a study design is very difficult or rather impossible to implement – both from an ethical as well as operational point of view – for our research endeavor. Thus, we present an approach that uses observational data and causal inference under the selection on observables assumption, also referred to as unconfoundedness or conditional independence assumption. Our research questions for developing an empirical approach and methodology are: I How can the causal effect of an ICU discharge on a given day on the ICU readmission risk of a patient be estimated with observational data? II How can a decision policy be learned that minimizes ICU readmissions? Research questions I and II structure our research objective in two concrete steps: First, estimating the causal effect and second, learning a decision policy based on the estimated causal relationship, exploiting effect heterogeneity. Note that both steps need to consider individual patient characteristics and individualized average treatment effects (IATEs) to enable decision makers to make a discharge decision in a setting of uncertainty. There is a rich operations management (OM) and medical literature on ICU management (Bai et al., 2018; Gopalan and Pershad, 2019; Niven et al., 2014), some of which we will review in the next chapter. As for many operations research settings (Ho et al., 2017), OM models address the ICU process 3 bottleneck problem with normative mathematical models, aiming to develop decision policies from a theoretical framework. Commonly, studies then test and/ or calibrate their models with (small) samples from one or several hospitals (e.g., Bai et al., 2021; Chan et al., 2012), or perform a simulation study (Ouyang et al., 2020). In this paper, we outline that due to the causal relationship of ICU discharge and ICU readmission and individual patient characteristics creating variation and uncertainty, a different approach might be more effective, generalizable, and better scalable than traditional OM methods. While we also develop our model from a theoretical start point, its main strength comes from its empirical application and practicality. We aim to show that machine learning methods from the family of doubly robust learners can answer our research questions yielding data-driven decisions that optimize medical quality and free up scarce capacity. We believe that our approach is not only valid for ICU management and other hospital operations settings but that our study can serve as a blueprint for a more general application of causal machine learning methods in operations research. This section continues with a review of related work. Section 2 presents a formal problem statement. Section 3 discusses our approach for estimating causal effects and developing optimal decision policies. We also present first descriptive results, and first estimations of causal effects. Lastly, we outline a simulation study to assess the practical utility of our approach. Section 4 summarizes and gives an outlook. Related work We apply machine learning methods for causal inference and policy learning to a hospital operations problem, more specifically for resolving one reason for bottleneck congestion of a key hospital process area, the ICU. Thus, related work of our study are studies in the OM literature focusing on (1) machine learning applications, (2) causal inference and causal machine learning, (3) policy learning, and (4) hospital operations. 10 3) Bias and uncertainty of transition probabilities: Markov decision processes require identifying probabilities for transitioning from one state to another. In dynamic and complex settings, such probabilities have to be identified for many parameters (Bertsekas, 2012). A single biased transition probability will bias the whole model. Additionally, the uncertainty of a point estimate of transition probabilities is usually not considered in Markov decision process models (Zhang et al., 2019). In other words, Markov decision process models implicitly assume point estimates from empiric data or expert estimates to be true probabilities. Indeed, uncertain transition probabilities have received attention in the literature for decades (e.g., Satia and Lave, 1973), and there are several approaches to address this issue (Delgado et al., 2011; Mastin and Jaillet, 2012; Zhang et al., 2019). Practical ICU decision making: A novel problem statement We propose a novel approach for addressing the limitations presented above. Firstly, we define a model based on practical ICU decision making which needs to resolve the bed capacity constraint 𝐵𝑎≥0 over the course of each day. Secondly, to help decision makers solve this constraint, we design policies that are generalizable, learnable with causal machine learning methods and re-learnable with the same algorithms and similar data from other hospitals, and thus scalable. Note also that state and action space models were shown to improve ICU decision making on a tactical level, e.g., when deciding whether and how much ICU capacity should be reserved to avoid more costly rejection and/ or early discharge (Bai et al., 2021). With our model, we will provide decision support for operational decision making in the ICU, i.e., for decisions that must be made routinely and daily. We use the flow of a patient through a hospital as the starting point of our problem statement (cf. Figure 1, loosely based on Bai et al. (2021, 2018) and Litvak et al. (2008)). We are interested in reducing readmission flows (1), (2), and (3). Note, however, that at our partner hospital, we investigate a mixed ICU also accommodating intermediate care patients. Thus, we are specifically investigating readmission flows (1) and (3), i.e., the readmission of a patient to a higher level of care unit. 11 Kramer et al. (2013), for instance, report that the median readmission rate at the more than 100 ICUs they investigated was at 5.9% (interquartile range between 5.1% and 7.0%) and Hosein et al. (2014) found in a meta-analysis that readmission rates typically are between 4% and 6%. While a certain percentage of ICU readmissions appears to be nonpreventable (Al-Jaghbeer et al., 2016), preventable readmissions and especially those causally linked to discharges could be termed as rework and correction in Lean Management terms, adding to bottleneck congestion. Figure 1: Hospital patient flow Annotations: Patients can enter a hospital as scheduled, plannable cases (full arrows) through the outpatient clinic or as unscheduled/ emergency cases through the emergency department (dashed arrows). From the emergency department, patients are pushed onto the process area with free capacity and/ or where they need to receive care. Unscheduled transfers from other hospitals disrupt hospital system management further. Patients are pushed through the different process areas in a scheduled or oftentimes unscheduled manner. Demand planning and corresponding supply planning only regularly occur in the central operating room area, other process areas are commonly staffed corresponding to their full capacity (e.g., to serve all beds on a ward). We are particularly interested in how to minimize readmission flows (1) and (2), often causally linked to early, demanddriven discharges of patients in the ICU. In such situations, patients with the smallest risk of being readmitted due to the early discharge should be discharged, and not necessarily those with the lowest predicted readmission risk (cf. Athey (2017), Feuerriegel et al. (2024), and Prosperi et al. (2020) for a discussion of this topic). The basis for our formal problem statement is the capacity constraint 𝐵𝑎≥0, denoting that the available ICU bed capacity 𝐵𝑎 must always be equal to or greater than zero operatable ICU beds throughout a given day. In a naïve state, 𝐵𝑎 is solely defined by the overall ICU bed capacity on a given day, and the sum of patients who will not be discharged and thus occupy the ICU ∑𝑥𝑖 𝐼𝑖=1 with 𝑖= {1,2,…,𝐼} and 𝑥=1 if 𝑥𝑖 resides in the ICU at least for one more day, 𝑥=0 otherwise: 𝐵𝑎=𝐵−∑𝑥𝑖 𝐼 𝑖=1 (1) Consulting Figure 1, the first patient flow decision-makers in a surgical ICU, as at our partner hospital, must consider and incorporate into (1) is the sum of elective surgery patients with a planned postOutpatient Clinic (surgery indication, and pre-medication) Emergency Department Ambulance Helicopter Central Operating Room Area Intensive Care Unit Intermediate Care Unit Normal Care Unit Transfers from other hospitals Admissions/ discharges Scheduled / elective Unscheduled/ emergency Scheduled appointments Walk-ins 12 3 xReadmission flows 12 surgery ICU stay ∑𝑎𝑒 𝐸 𝑒=1 on a given day, with 𝑒= {1,2,…,𝐸}, and 𝑎=1 if 𝑎𝑒 has a planned postsurgery ICU stay, 𝑎=0 otherwise. In addition, planned discharges occur because a patient has reached ICU service completion (natural discharge), denoted by 𝑑𝑛 with 𝑛= {1,2,…,𝑁}, and 𝑑=1 if 𝑑𝑛 is naturally discharged during the day, 𝑑=0 otherwise. This expands the capacity constraint given in (1) into: 𝐵𝑎=𝐵−(∑𝑥𝑖 𝐼 𝑖=1 +∑𝑎𝑒 𝐸 𝑒=1 )+∑𝑑𝑛 𝑁 𝑛=1 (2) If overall ICU capacity and natural discharges were sufficient to accommodate ∑𝑥𝑖 𝐼𝑖=1 and new arrivals ∑𝑎𝑒 𝐸 𝑒=1 , and if we stayed in this simplified decision framework, we could be satisfied with the definition of available ICU bed capacity as outlined in (2). However, even if there were no unscheduled, external emergency patient arrivals, there are three additional considerations to make (cf. Figure 1). Given the capacity constraint 𝐵𝑎≥0, there will be days where ∑𝑎𝑒 𝐸 𝑒=1 will require discharges in addition to ∑𝑑𝑛 𝑁 𝑛=1 , which we define as planned early, demanddriven discharges denoted by 𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 with 𝑝= {1,2,…,𝑃} and 𝑑=1 if 𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 is planned to be early discharged, 𝑑=0 otherwise. Secondly, another lever to satisfy 𝐵𝑎≥0 is to reject patients, i.e. to cancel or postpone elective surgeries with a planned post-surgery ICU stay, denoted by 𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 with 𝑐= {1,2,…,𝐶}, and 𝑎=1 if 𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 is cancelled, 𝑎=0 otherwise. Thirdly, the capacity constraint must also hold when considering readmissions from downstream units, denoted by 𝑎𝑟 with 𝑟= {1,2,…,𝑅}, and 𝑎=1 if 𝑎𝑟 is readmitted, 𝑎=0 otherwise. These three considerations expand 𝐵𝑎 to: 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 setting without unscheduled arrivals, e.g., in an orthopedic hospital exclusively treating elective surgery patients, (3) would suffice to describe 𝐵𝑎 and the discharge decision processes to satisfy 𝐵𝑎≥0. Note that there still is uncertainty in service time, i.e., the number of ICU inpatient days for each 𝑥𝑖, and thus prediction of ∑𝑥𝑖 𝐼𝑖=1 and ∑𝑑𝑛 𝑁 𝑛=1 are prone to uncertainty. Further note that at the start of a day (e.g., between 07:00 a.m. to 08:00 a.m. at our partner hospital), ICU decision-makers might initiate their 13 decision process at (2), and if they foresee that the capacity constraint will not be satisfied at any point in time throughout the day, they will try to satisfy the capacity constraint by balancing ∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 𝐶 𝑐=1 , and ∑𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 𝑃 𝑝=1 . While we appreciate that past studies must make this assumption for their models, cancellations and planned demand-driven discharges do not happen simultaneously to patient arrival. ICU decision-makers rather anticipate the number of planned ICU admissions according to the operating room schedule at the start of a day and compare these with the available ICU bed capacity after natural discharges (cf. (2)). Considering unscheduled or emergency/ urgent patients coming from upstream units, i.e., directly from the emergency department or, by a roundabout route, from the central operating room area, or as transfer-ins from other hospitals, the decision process to satisfy 𝐵𝑎≥0 becomes more complex. Note that in practice, experienced ICU decision-makers might anticipate unscheduled ICU admissions and thus reserve some ICU capacity by discharging more patients than absolutely needed to satisfy 𝐵𝑎≥0 as defined in (3). Indeed, past studies such as Bai et al. (2021) quantitatively derive exactly how much capacity should be reserved balancing the costs of patient rejection, and demand-driven discharge. In fact, one could consider that the capacity constraint actually is 𝐵𝑎≥∑𝑎𝑢 𝑎𝑛𝑡 𝑈 𝑢=1 , that available ICU capacities must be at least the total number of unscheduled arrivals/ admissions the experienced ICU decision-maker anticipates on a given day (𝑢= {1,2,…,𝑈}, and 𝑎=1 if 𝑎𝑢 𝑎𝑛𝑡 is anticipated as unscheduled arrival, 𝑎=0 otherwise). As it is uncertain how many unscheduled patients will arrive exactly, we additionally define 𝑎𝑣𝑎𝑑𝑑 as the unforeseen unscheduled arrivals in addition to ∑𝑎𝑢 𝑎𝑛𝑡 𝑈 𝑢=1 , with 𝑣= {1,2,…,𝑉} and 𝑎=1 if the arrival of 𝑎𝑣𝑎𝑑𝑑 is unscheduled and is not anticipated, 𝑎=0 otherwise. Note that decision makers might increase the sum of planned early discharges ∑𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 𝑃 𝑝=1 and/ or the sum of cancellations of elective surgeries ∑𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 𝐶 𝑐=1 to offset ∑𝑎𝑢 𝑎𝑛𝑡 𝑈 𝑢=1 , yet to manage any 14 𝑎𝑣𝑎𝑑𝑑, the only lever is to demand-driven discharge a patient in an unplanned manner, denoted by 𝑑𝑞𝑒𝑎𝑟𝑙𝑦 with 𝑞= {1,2,…,𝑄} and 𝑑=1 if 𝑑𝑞𝑒𝑎𝑟𝑙𝑦 is early discharged in an unplanned way, 𝑑=0 otherwise. 1 To satisfy 𝐵𝑎≥0, decision makers then consider that 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) Note that planned discharges, both natural and demand-driven, happen at a later point in time than the discharge decision. At our partner hospital, ∑𝑑𝑛 𝑁 𝑛=1 and ∑𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 𝑃 𝑝=1 are decided in the early morning of a (week-) day, while discharges occur in the late morning to early afternoon, in close alignment with the downstream care units admitting the patient(s). Should a patient have worsened between initial discharge decision and actual discharge, decision-makers might re-think and change their initial decision. Empirical evidence for formally defined decision constraint In our data, there is empirical evidence for the dynamics of the decision constraint as formulated in equation (4). Figure 2 shows the typical arrival pattern at our partner hospital, depicting the share of patients arriving per hour of the day. 1 Additionally, one could argue that if 𝑎𝑣𝑎𝑑𝑑 arrived early in the day before surgery has started for the last 𝑎𝑒, decision-makers might also have the option to cancel one additional 𝑎𝑐𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒. For simplicity, we do not consider this option. 15 Figure 2: Arrival pattern Annotations: The analysis includes all ICU stays of patients with a positive or waived general consent admitted at our partner hospital between January 01, 2016 and December 31, 2023, no exclusion criteria applied (14,121 stays and 12,932 unique cases; cf. Figure 4 below). In sum, roughly 72% of all patients arrive between 10:00 a.m. and 09:00 p.m. Still, the volume of patient admissions (28% of all admissions) between 09:00 p.m. and 10:00 a.m. is not negligible. The share of discharges in the same time window is quite small (see Figure 3). This is evidence that ICU decision makers incorporate planned admissions after surgery ∑𝑎𝑒 𝐸 𝑒=1 into their decision making process and that they do anticipate unscheduled admissions ∑𝑎𝑢 𝑎𝑛𝑡 𝑈 𝑢=1 and possibly also readmissions ∑𝑎𝑟 𝑅 𝑟=1 . Discharges typically occur between 09:00 a.m. and 04:00 p.m. (roughly 95% of all discharges for Panel A, 97% for Panel B), and most discharges happen in an even closer time window between 10:00 a.m. and 02:00 p.m. (roughly 86% for Panel A and 89% for Panel B). This is evidence for the planned discharge decisions ∑𝑑𝑛 𝑁 𝑛=1 and ∑𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 𝑃 𝑝=1 happening early in the morning with corresponding discharges a few hours later. 16 Figure 3: Discharge pattern Annotations: Panel A includes all ICU stays of patients with a positive or waived general consent admitted at our partner hospital between January 01, 2016 and December 31, 2023, no exclusion criteria applied (14,121 stays and 12,932 unique cases; cf. Figure 4 below). In Panel B, we exclude patients who died during their ICU stay (589 stays, 518 unique cases), as the timing of these patients’ “discharge” does not occur according to the decision process formalized in equation (4). 17 We also see that discharges in “off-hours” between 02:00 p.m. and 10:00 a.m. (14% (11%) of all discharges for Panel A (B)) and especially between 04:00 p.m. and 09:00 a.m. are rather rare (5.4% (2.9%) of all discharges for Panel A (B)). Especially for Panel B, where we exclude patients who died during their ICU stay, we may assume that a considerable share of these off-hour discharges are unplanned demanddriven discharges ∑𝑑𝑞𝑒𝑎𝑟𝑙𝑦 𝑄 𝑞=1 triggered by unforeseen unscheduled admissions ∑𝑎𝑣𝑎𝑑𝑑 𝑉 𝑣=1 . Figure 3 clearly shows that these situations occur comparatively rarely. At the same time, a considerable share of admissions does occur in these off-hours, namely roughly 57% between 04:00 p.m. and 09:00 a.m. This is evidence that – at least in our empirical setting – demand-driven discharges rarely occur as defined in conventional operations research studies. Summary In summary, we have ascertained three different types of points in time where ICU discharge decisions occur or are changed: 1) Planned discharges of any kind are made in the morning in the first hours of a physician’s shift, typically between 07:00 a.m. and 08:00 a.m. at our partner hospital 2) A discharge decision is changed should a patient’s health status change considerably between the time of the discharge decision and the planned discharge time 3) Unplanned discharge decisions are made at any point during the day (and night) if 𝑎𝑣𝑎𝑑𝑑 occurs In equation (4), we formulate the decision problem for satisfying the capacity constraint 𝐵𝑎≥0 that decision makers face at each of these three types of points in time. Our model supplies decision-makers at each of these types of points in time with the effect of a discharge on a patient’s readmission risk. Note that while our model can inform any kind of demand-driven discharge, it also provides decision support for natural discharges. Should the result of our model reveal that the change in readmission risk is too large due to the discharge, the “natural” discharge could be postponed. 3 Causal Effects and Optimal Policy Learning We start with a short description of our dataset and descriptive analyses. Then, we outline a three-step approach to enable optimal ICU discharge decision making: (1) We present how to estimate the ATE, 18 CATEs, and IATEs of an ICU discharge at a point in time 𝑡 on the ICU readmission risk in an observational study setting with double robust learners, (2) we outline how we plan to develop decision policies based on patients’ IATEs, and (3) we present how we plan to apply these policies to our empiric data to gauge how many ICU bed capacities could have been saved with optimal decisions. In the first section, we also discuss identifying assumptions for causal inference in the context of our study. Overview of Dataset and Descriptive Analyses Dataset Our dataset contains clinical and basic data of all ICU stays without a documented negative general consent admitted to the Department of Surgical Intensive Care Medicine between January 01, 2016, and December 31, 2023. Figure 4 shows the inclusion and exclusion criteria and corresponding samples for the different analyses we conducted. Figure 4: Inclusion and exclusion criteria and corresponding samples Annotations: Reasons for excluding cases that are not candidates for readmission are discharge to another ICU, to home, to a rehabilitation clinic or nursing home, or to a different hospital (cf. Kramer et al. (2013)). Our data is recorded at ICU stay level. Each ICU stay has a unique identifier. One case, given by a unique case number, has at least one ICU stay. Theoretically, one patient could be admitted to a hospital 19 several times in a year. In such cases, a new unique case number is defined for each hospital admission. A case number for which two or more ICU stays are recorded thus indicates that a patient was readmitted to the ICU within the same hospital stay. Sample A was used for analyses regarding arrival and discharge patterns (e.g., Figure 2 above). Sample B includes only those cases and stays that are useable for causal inference. To this end, we excluded patients who died in the ICU as these are recorded as discharges yet are not connected to a deliberate discharge decision, excluded cases that could not be readmitted to the ICU as they were discharged to a different ICU, to home, to a rehabilitation clinic or nursing home, or to a different hospital, and we excluded cases for whom the discharge reason was missing. Overall, we can extract and engineer more than 4,600 features from our dataset (see Table 1). Most of these features are related to medication and drugs (more than 4,000 features). We include one feature per substance, dosage and volume unit and values represent the given volume (either 0 or a continuous number) within the last 24 hours before discharge (intervention group) or “simulated” discharge (control group, see Figure 6 below). Other major feature categories are laboratory diagnostic values (more than 200 features), vital signs (20 features), clinical (risk) scores (close to 40 features) and basic data (age, gender, weight, height, BMI). Table 1: Overview of used features Feature category Number of features Numeric type Considered measurements Description and examples Medication 4,119 Continuous Last 24 hours Hundreds of different substances with at least one, oftentimes several dosages and volume units, e.g., Noradrenalin peripheral in microgram, Noradrenalin in microgram Laboratory tests 208 Continuous Last measurement 208 different laboratory values, e.g., Kalium, Creatinine, Cholesterol, HbA1c Clinical scores 40 Continuous Last 2 measurements 10 different scores, e.g., SAPS II, SAPS 3, NEMS, GCS, etc. Diagnoses 114 Dummy Time invariant 57 indication areas for first and follow-up diagnosis, e.g., "diseases of the liver and biliary tract" Infusions 35 Continuous Last 24 hours 22 different infusions, partially with more than one dosage, e.g., Glucose 5%, 10%, 20%, 40%, 50%; all in ml 26 For our current results, we only run the last two steps once per ICU stay. For our final results, we plan to run these two steps multiple times for separate (I)ATE estimations. Results from each run will be averaged in the end to receive the final results. Outcome definition There is an ongoing discussion in the medical literature regarding a meaningful measurement and use of ICU readmission rates (Hosein et al., 2014; Kramer et al., 2013; Woldhek et al., 2017). In our main model, we use readmission regardless of the time between discharge and readmission as outcome. We plan to perform one to two sensitivity analyses, using more narrowly defined readmission rates such as readmission within 48 hours and 96 hours (two to four days). In our dataset, we observe a raw readmission rate of roughly 9.0% while readmission rates with defined times between discharge and readmission are between 2.8% and 6.1% (see Figure 7). Figure 7: Readmission rates according to time between discharge and readmission Annotations: The plot is based on Sample B (12,950 ICU stays and 11,873 unique cases). 27 Identifying assumptions Identifying assumptions for causal inference are (Angrist et al., 1996; Imbens, 2000; Lechner, 2001): (1) Unconfoundedness, (2) Common Support (CS) or overlap, (3) Stable-Unit-Treatment-Value Assumption (SUTVA), and (4) exogeneity. With unconfoundedness, we assume that we observe all variables that might influence both the treatment selection (i.e., to be or not to be discharged at point 𝑡) and the potential outcome of a discharge. Discharge decisions are made by senior physicians (Nates et al., 2016). We observe all data and decision variables that are available to these physicians when they make discharge decisions. This includes most variables that were judged by most Swiss ICUs as relevant for ICU discharge decisions (Heidegger et al., 2005). Thus, we argue for unconfoundedness in our study setting. There are three lines of counterargumentation, however. The first argument is that situations may occur in which a discharge decision (i.e., our treatment assignment) and thus discharge are determinate. Unconfoundedness would be violated if such a treatment determination was dependent on patient characteristics (𝑋𝑖). (i) The decision must be positive every time when there is only one patient candidate for discharge at 𝑡. We may still assume unconfoundedness, as the number of potential discharge candidates is exogenous, i.e., we can view treatment assignment in these situations still as random. (ii) A discharge decision must be negative if there is a contraindication, e.g., a patient is intubated or a patient receives a certain drug or substance (e.g., catecholamines) (Heidegger et al., 2005). This does not violate unconfoundedness in our context, however, as we always observe the discharge at a point in time when it is not contraindicated (cf. treatment definition). The second argument is that in practice, physicians actually consider more variables than “only” the thousands of clinical parameters available to them (and to us): Physicians might additionally collect “soft” data, e.g., during daily patient visiting rounds by visual control of the patient and discussions with nurses (Nates et al., 2016; Ofoma et al., 2018). Data obtained this way might include degree of paleness, sweating, communicated pain, mental confusion and disorientation, or therapy compliance. 28 Soft data is not available to us (or readable by any machine). Unconfoundedness would be violated if these soft data were not correlated with the thousands of clinical parameters available to us. This is rather unlikely. The third argument is that a discharge, and possibly also a later readmission, are influenced by the available downstream skill mix and/ or capacity. Physicians might decide to discharge a certain patient on a given day if they know that there are experienced, well-qualified physicians and nurses available in downstream units who can manage the patient, even if this patient is sicker given clinical parameters than another patient who was not discharged another day when downstream skill mix and capacity were (allegedly) inadequate. In such situations, physicians might decide against a discharge as they expect a worsening of the patient’s health status downstream, rendering an ICU readmission more likely. To address such doubts that might still remain after controlling for several thousand patient characteristics, we could implement an instrumental variable (IV) approach. An IV approach is available for binary instruments in GRF (Athey et al., 2019) and was developed by Wang et al. (2021) for continuous IVs in causal forests. As IV, we have several options: (1) the daily number of admitted patients as a degree of ICU busyness both as continuous variable (only useable if we can incorporate a continuous IV in GRF) and as dummy (1=busy, 0=not busy; threshold to be determined, e.g., one third of overall ICU capacity), as this has shown to increase the number of ICU discharges (Nates et al., 2016), (2) the number of potential discharge candidates throughout ICU day shifts (e.g., 06:00 a.m. to 08:00 p.m.), or (3) proxies for downstream unit capacity (and skill-mix), e.g., weekday vs. weekend, or days until weekend. Regarding (3), actual utilization data would be preferable, yet such data is not collected on a daily level in a digital manner or highly unreliable if collected manually. Note that all three IV options are exogenous of treatment assignment. For future studies, collecting data on daily downstream skill mix and capacity might be perceivable. This comes with considerable effort, however, as such data is not available in a structured manner for all IMCUs and NCUs of a hospital. While staff schedules could be available (often only available in 29 analog form or in e-mails, manually administered Microsoft Excel tables, or similar), these alone will not explain the skill mix perceived by the ICU decision maker. To accurately account for this perceived skill mix, we would have to label all physician and nurse staff members of all downstream units according to the experience and qualification of all ICU decision makers. This might result in different labels for individual downstream physicians and/ or nurses depending on what ICU decision maker is asked to label. Indeed, if experience and skill level are perceived differently by physicians, this might in turn supply an argument that unconfoundedness does hold, as downstream skill mix would then not systematically influence treatment assignment and outcome in the same (perceived) way. Lastly, another argument that unconfoundedness still holds is that it is unlikely that a physician will be able to judge the experience and skill level of all treating nurses and physicians active in downstream units, or even that an ICU physician always has full transparency of downstream unit staff schedules. Granted, if a discharge decision for a particularly complex patient were made, a physician could take the time to get transparency over staff schedules and also experience and skill level. A systematic influence still is unlikely, however. Another possibility could be to measure experience by the number of years a medical professional has been actively working, and qualification by academic degrees, further education certificates, and scientific publications. This approach would also pose a major challenge in terms of required effort, and at least some of these data will not be available in a structured form. Lastly, measuring experience and skill level in this way also has limitations. For instance, one might argue that experience is in fact built by being exposed to adverse events and difficult situations. This should correlate with the number of years a professional has been working but this must not necessarily be true. In summary, we believe we can viably argue that unconfoundedness holds (1) as we control for all confounders also available to senior physicians making discharge decisions, (2) patient characteristics do not systematically influence treatment assignment and outcome and if they do, serve as exclusion criteria (e.g., mechanical ventilation), and (3) while downstream capacity and skill-mix might influence ICU decision makers, it is not perceivable that they have transparency over both for all downstream 30 units for every discharge decision, or even the majority of decisions, thus impeding systematic influence. Still, we to acknowledge any concerns possibly left and the observational setting of our study, we will implement an IV approach as sensitivity analysis in our final manuscript. We may assume CS, if we can show that propensity scores “overlap”, i.e., that each patient could be observed with or without a discharge at point 𝑡: 0<𝑝(𝑊𝑖=1|𝑋𝑖=𝑥)<1 ∀ 𝑥∈𝑋 (16) In Figure 8, we plot the distribution of propensity scores for our sample (cf. Wager and Athey, 2018). Figure 8: Common support and overlap analysis of propensity scores Annotations: Propensities were estimated with GRF, and for Sample B (12,950 ICU stays and 11,873 unique cases). The plot shows that for our sample and treatment definition, there is a medium to strong selectivity into and out of treatment, respectively. Lechner and Mareckova (2024) show that their Modified Causal Forest (MCF), delivers more robust estimates than GRF in cases of (medium and) strong selectivity, especially when estimating IATEs. Thus, we estimate ATEs with both algorithms, one example of CATEs for exemplification with GRF, and IATEs with MCF. For our final results, we will estimate all results with both MCF and GRF. 31 The SUTVA requires that spillover effects between discharged patients are absent. More concretely, the discharge of one patient can only affect the outcome of the same patient and not the outcome of another patient. This is given as discharging one patient does not directly influence the readmission risk of another patient. One might argue that once the capacity constraint 𝐵𝑎≥0 holds, no additional patient(s) is discharged any more, even if this were medically possible, and thus the treatment state of the discharged patient(s) to satisfy 𝐵𝑎≥0 might influence the treatment state of some of the patients remaining in the ICU. Still, the treatment state of the discharged patient(s) will not directly influence the outcome of the patient(s) remaining in the ICU. Exogeneity stipulates that patient characteristics used as confounders (𝑋𝑖) are not influenced by the discharge at point in time 𝑡 (the treatment). This assumption holds as we observe 𝑋𝑖 before the discharge occurs. In summary, we should be able to fulfill all four identifying assumptions or at least be able to implement an empirical strategy and methods to handle potential violations, ensuring robust ATE, CATE, and IATE estimations. First results In Table 3, we present the ATE estimation, employing both MCF and GRF. Table 3: Estimated Average Treatment Effects Algorithm Estimate Standard Error Absolute change of readmission risk (in %-pts.) Relative change of readmission risk (in %) MCF 0.00767 0.0083903 0.77 8% GRF -0.00179 0.0047603 -0.18 -2% Annotations: Estimates were made with Sample B (12,950 ICU stays and 11,873 unique cases), >4,600 included features (cf. Table 1). The relative change of readmission risk is calculated in comparison to the raw readmission rate of 9.04%. The ATE of discharging an ICU patient to a downstream unit at 𝑡 as compared to one decision cycle later increases the readmission risk by 8% according to MCF but decreases the readmission risk by -2% according to GRF. Evidently, both point estimates are insignificant, while the standard error for GRF is much larger as compared to the estimate than for MCF. Besides, the magnitude of the effect, regardless of significance and sign of the coefficient, is more than four times higher for MCF than for GRF. Still, simply put, according to both MCF and GRF, on average, there is no effect. It is important to note, 32 however, that we are not interested in the ATE as it does not enable discriminative discharge decisions between two or more individual ICU discharge candidates (cf. below). In fact, it is sensible that on average, discharging a dischargeable, comparatively stable patient one day (or rather decision cycle) later will not affect the readmission risk significantly. Table 4 shows the estimated CATEs for four groups, stratified by the number of times a patient passes the regular discharge decision time after becoming a potential candidate for discharge, denoted by 𝑛. The CATEs show that there is ample treatment heterogeneity. While there is no effect for the group 𝑛= 2, the magnitude of the estimate is quite large for all other 𝑛 and standard errors are smaller for 𝑛=0 and 𝑛=1, but effects are still insignificant. Table 4: Estimated Conditional Average Treatment Effects according to 𝒏 𝒏 Estimate Standard Error Absolute change of readmission risk (in %-pts.) Relative change of readmission risk (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% Annotations: Estimates were made with GRF and with Sample B (12,950 ICU stays and 11,873 unique cases), >4,600 included features (cf. Table 1). The relative change of readmission risk is calculated in comparison to the raw readmission rate of 9.04%. 𝑛 denotes the number of times a patient passes the regular discharge decision time after becoming a potential candidate for discharge. Note that neither the ATE, nor any kind of CATEs will be of use for decision support. For the ATE, this is obvious as all cases have the same ATE and it thus cannot suggest what patient should rather be discharged. CATEs encompass still too many patients, so that on a given day, two or more discharge candidates might fall into the same group with the same CATE. Then, the same issue as with using an ATE applies. Therefore, IATEs need to be estimated, forming the basis for developing decision policies that can truly support daily, operational ICU decision making. Given the results in Table 3 and our common support analysis in Figure 8, we chose MCF as causal machine learning method to estimate IATEs. Figure 9 shows the density of estimated IATEs. Almost the entire sample (97.8%, 12,669 ICU stays) shows IATE estimates between -0.05 and 0.10. Note that this variation is actually quite large: In percentage points, readmission risk is influenced between 33 negative 5 and 10 points, amounting to a relative change of readmission risk between -55% and 111% when compared to the average readmission rate of 9.04%. Considering a narrower IATE range between -0.05 and 0.05 (91.4% of sample, 11,833 ICU stays), the relative change of readmission risk still amounts to -55% to 55%. This range shows the existing treatment heterogeneity and showcases that these results could prove effective when developing discharge policies based on IATEs. Figure 9: Density of Individualized Average Treatment Effects Annotations: IATEs were estimated with MCF, and for Sample B (12,950 ICU stays and 11,873 unique cases). Still, the figure also shows that the IATE estimate for large share of ICU stays is between 0.00 and 0.03 (58.5%, 11,833 ICU stays). This means that for the majority of patients, a discharge one decision cycle later increases their readmission risk relatively mildly, between 0% and 33% compared to the readmission rate of 9.04%. Lastly, CATEs in Table 4 already indicate that there are certain patients that might not benefit from additional time on the ICU. Our results for the IATE estimation show that IATEs are indeed negative for roughly 28% of all patients. 34 Still, when interpreting IATEs, we must be cautious insofar that we do not show estimated standard errors of patients’ IATEs which would be needed to generalize statements about the effectiveness of additional intensive care for individual ICU patients. Accordingly, the above statements have a descriptive and not necessarily causal intention. For our research objective, i.e., supporting clinical decision making, standard errors are of a lesser importance, which we discuss in the next section. Development of Decision Policies and Simple Simulation Study We aim to design policies offering decision support for all ICU discharge types satisfying the capacity constraint 𝐵𝑎≥0, with 𝐵𝑎 as defined in equation (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 ) Specifically, policies should offer support when planning discharges ∑𝑑𝑛 𝑁 𝑛=1 +∑𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 𝑃 𝑝=1 , and when having to decide in unplanned demand-driven situations described by ∑𝑑𝑞𝑒𝑎𝑟𝑙𝑦 𝑄 𝑞=1 −∑𝑎𝑣𝑎𝑑𝑑 𝑉 𝑣=1 . Note that in practice, both 𝑑𝑛 and 𝑑𝑝 𝑒𝑎𝑟𝑙𝑦 are planned discharges and distinguishing clearly between what discharge is natural and what discharge is rather demand-driven due to unplanned but anticipated admissions 𝑎𝑢 𝑎𝑛𝑡, anticipated readmissions 𝑎𝑟, or planned admissions due to elective surgeries 𝑎𝑒, might be difficult. To support decision making, in a first step, we will rank all patients according to their IATE estimate. In a second step, we determine that patients should be selected for discharge based on their spot in this ranking as compared to other patients who are discharge candidates at the same regular decision cycle. More concretely, if the ICU decision maker were to decide between two patients, the patient with the lower change of readmission risk due to the discharge at point in time 𝑡, i.e., the smaller IATE, would be selected. The same applies if multiple patients were selected for discharge. For instance, if there are 35 seven discharge candidates and to comply with the capacity constraint 𝐵𝑎≥0, five patients would need to be discharged, the five patients with the lowest IATEs would be selected for discharge. Statistical significance, e.g., at the 5%-level of estimated IATEs would indicate the robustness of the magnitude of individual effects, i.e., how certain we can be about individual patients’ point estimate. We argue that statistical significance is of lesser importance for selecting discharge candidates as even insignificant IATEs can optimize decisions, at least as long as confidence intervals do not overlap. For the simulation and final version of our manuscript, we will address this potential challenge in more detail. To show the practical utility of our decision policy approach in terms of avoided readmissions and saved ICU capacity, we will start a simulation at the first day of our observation period. We then apply our decision policy and change the discharge decision for all patients where the policy would discharge a different patient than the patient that was actually discharged. We continue to apply this decision policy across all days of our observation period. To receive the total number of avoidable readmissions, we sum the difference between the minimal (i.e., optimal) IATE, 𝐼𝐴𝑇𝐸𝑑,𝑝, and empirical (i.e., actual) IATE, 𝐼𝐴𝑇𝐸𝑑,𝑎, across all discharge decisions 𝐷. 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