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The Role of a Step-Down Unit in Improving Patient Outcomes Carri W. Chan Decision, Risk, and Operations, Columbia Business School, [email protected]

Linda V. Green Decision, Risk, and Operations, Columbia Business School, [email protected]

Lijian Lu Decision, Risk, and Operations, Columbia Business School, [email protected]

Gabriel Escobar Division of Research, Kaiser Permanente, [email protected]

This paper examines the role of a hospital Step-Down Unit (SDU) on patient flows and patient outcomes. An SDU provides an intermediate level of care for semi-critically ill patients who are not sick enough to require intensive care but not stable enough to be treated in the general medical/surgical ward (ward). Using data from 10 hospitals from a single hospital network, we use an instrumental variable approach to estimate the impact on patient outcomes of routing patients to the SDU following Intensive Care Unit (ICU) discharge. Our empirical findings suggest that SDU care is associated with reduced in-hospital mortality of 6%, shortened hospital length-of-stay of 1.08 days, a reduced ICU readmission rate of 4%, and a reduction in the hospital readmission rate of 8%. We use our empirical findings to calibrate a simulation model of critical care patient flows and examine how the size of an SDU may impact patient outcomes, even when there is ample capacity in the ICU. There is substantial debate in the medical community regarding whether SDU care is beneficial and, if so, how many SDU beds are needed. This work takes an important first step in addressing these issues. Key words : healthcare, empirical operations management, discharge controls, congestion, quality of service History :

1.

Introduction

Hospitals are responsible for the largest component of national health care expenditures and are therefore under pressure from government and private payers to become more cost efficient. Intensive care units (ICUs), which provide the highest level of care, are the most costly inpatient units 1

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to operate. The estimated annual cost of critical care in the U.S is between $121 and $263 billion, accounting for 17.4%-39% of total hospital costs (Coopersmith et al. 2012). Step-down units (SDUs), sometimes called transitional care or intermediate care units, have been used in many hospitals to mitigate critical care costs without jeopardizing the quality of care. SDUs provide an intermediate level of care for semi-critically ill patients who are not severe enough to require intensive care but not stable enough to be treated in the general medical/surgical ward (ward). SDUs are generally less expensive to operate than ICUs due primarily to lower nurse-to-patient ratios. While an ICU may have one nurse per one or two patients, an SDU would typically have one nurse per three to four patients. On the other hand, SDUs are more expensive than general wards where there are, generally, about 6 patients per nurse. There is ongoing debate about the role of the SDU in the medical community. Those who advocate the use of SDUs see them as an alternative to either maintaining larger ICUs or jeopardizing patient care due to premature, demand-driven, discharge of patients from ICUs to general care units. As the name suggests, the initial role of SDUs was to serve as a transition for patients after being discharged from the ICU and our study will focus on this particular use. In practice, SDUs are often used to treat other patients, for example, those who might have gone to an ICU but were blocked because the ICU was full. In general, the use of SDUs has evolved without substantial evidence as to their benefits and what their role should be. Some studies argue that these units provide a safe and cost-effective environment for semi-critical patients and can serve as a “bridge from hospital to home” thereby improving patient outcomes and efficiency (Byrick et al. 1986, Harding 2009, and Stacy 2011). Other studies argue that SDUs should not be used as there is not enough evidence of their cost-effectiveness (Keenan et al. 1998 and Hanson et al. 1999). Despite the lack of consensus in the medical community surrounding the use of SDUs, many hospitals have SDUs and others are considering introducing these units. Even within a single hospital, the use of SDUs is generally not standardized. Therefore, it is very important to understand their value and how they can best be used. This paper examines whether or not SDUs are associated with improved operational and/or clinical outcomes as well as how these outcomes may be affected by SDU size. As SDUs are much less expensive to operate than ICUs, improvements associated with SDU care may enable reductions in hospital operating costs without sacrificing patient outcomes. Given the increasing pressures for hospitals to reduce costs, such insights can be very valuable to hospital administrators. To the best of our knowledge, our work is the first to conduct a multi-hospital study to empirically examine the role of an SDU for patients who are discharged from the ICU. Our analyses are based on recent data from Kaiser Permanente Northern California, an integrated health care delivery system serving 3.6 million members that operates 21 hospitals, some of which do and some of

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which do not have SDUs. The cohort and type of data we employ have been described in previous studies (see Escobar et al. (2008), Kim et al. (2014) among others). Our data source is based on nearly 170,000 hospitalizations in a total of 10 hospitals over a course of one and half years. Each of the 10 hospitals in our study has an ICU and SDU, though the number of beds in each of the units varies across hospitals. We focus on patients who are admitted to the ICU and examine how being discharged from the ICU to the SDU impacts various clinical and operational outcomes. In conducting this research, we must deal with an important estimation challenge. The routing of a patient following ICU discharge may be affected by health factors which are known to the physician at the time of the decision on when and to where the patient is discharged, but are unobservable in the data. Ignoring this endogeneity could result in biased estimates. So we utilize an instrumental variable approach to identify the desired effects. Our empirical findings suggest that SDU care is associated with substantial improvements in various patient outcomes. In particular, on average, routing to the SDU following ICU discharge is associated with reductions in the likelihood of inhospital death by 6%, the remaining hospital length-of-stay by 1.08 days, the likelihood of ICU readmission by 4%, and the likelihood of hospital readmission by 8%. Based on our empirical findings, we develop and calibrate a simulation model of critical care patient flows to gain an understanding of how SDU size may impact patient flows and outcomes. In our simulation, we assume a critically ill patient receives care in the ICU until he enters one of two health conditions–semi-critical or stable. A stable patient is transferred to the ward or discharged out of the hospital, while a semi-critical patient is transferred to an SDU bed if one is available. Thus, depending on bed availability, a semi-critical patient could have three possible routings: (1) if an SDU bed is available, the patient is transferred to an SDU bed, (2) if the SDU is full, but the ICU has available beds, the patient remains in the ICU and is considered reverse access blocked (RAB), and (3) if both the SDU and the ICU are full, the patient is rerouted to the ward to make room for new ICU patients. A RAB patient may be subsequently rerouted to the ward if a critical patient arrives and there are no available ICU or SDU beds. Having a large number of RAB patients and/or having these patients blocked in the ICU for long periods of time can be very costly and may indicate a mismatch of capacity between the ICU and SDU. Semi-critical patients who are rerouted to the ward may have worse outcomes: a higher likelihood of in-hospital death and ICU readmission. While our simulation model does not capture every possible flow of patients through the ICU and SDU, it provides insight into the role of SDUs, especially in the case where an SDU is utilized as a true step-down unit (i.e. patients are only admitted to the SDU from the ICU). Our simulation reveals that SDU size is a very important factor in achieving better patient outcomes–a larger SDU results in improved patient outcomes even when there is ample ICU capacity. However, the marginal gains decrease for each additional SDU bed. Our findings suggest

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that a moderately sized SDU, e.g., 70% of ICU size, is sufficient to achieve patient outcomes as good as those associated with a much larger SDU. Our main contributions are summarized as follows: • Quantifying the impact of SDU care on patient outcomes: We develop an econometric

model to estimate the impact of discharging ICU patients to the SDU on patient outcomes. We find statistical evidence of reduced mortality, readmissions, and LOS when a patient is sent to the SDU following ICU discharge. • Understanding how SDU size affects patient outcomes: We develop a patient flow

model, with parameters calibrated using our empirical results, to quantify the impact of the SDU size on patient outcomes. Our results show that increasing the SDU size improves patient outcomes but the marginal improvement decreases as the SDU size increases. Our model provides insight for hospital administrators when determining 1) how many SDU beds to have and 2) how to allocate resources between the ICU and SDU. The rest of the paper is organized as follows. Section 2 reviews related papers in the literature. In Section 3, we present our econometric models and estimation results for the decision of where to transfer patients following ICU discharge. We present a general patient flow model in Section 4. Section 5 provides our simulation results and provides numerous robustness tests. Section 6 summarizes our results and provides directions for future research.

2.

Literature Review

Our work is related to existing literature in both the medical and operations management communities. First, our work contributes to the ongoing debate in the medical community about the role of SDUs. On one hand, some studies argue that SDUs are a cost-effective approach to treat patients by providing a safe and less expensive environment for patients who are not quite sick enough to require treatment in the ICU, but not quite stable enough to be treated in the ward. Without an SDU, most of these patients end up being cared for in the ICU. Byrick et al. (1986) suggests that the use of the SDU could alleviate the ICU congestion by reducing ICU length-of-stay (LOS) without increasing mortality rates. This reduction is possible because patients do not have to reach as high a level of stability to be discharged to an SDU rather than to a general medical-surgical ward. Other studies that have shown the cost-effectiveness of an SDU include Harding (2009), Stacy (2011), and Tosteson et al. (1996). On the other hand, a survey of studies on SDUs raises doubts about these benefits and argues that there is not enough evidence of cost-effectiveness (Keenan et al. 1998). The majority of these studies are conducted exclusively within a single hospital, whereas our study utilizes data from 10 different hospitals. Additionally, rather than conducting a

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before-and-after study, which may be limited by the inability to control for temporal changes such as staffing changes or closures of nearby hospitals, we utilize an instrumental variable approach to identify the impact of different care pathways (going to the SDU versus ward following ICU discharge). Our multi-center study provides compelling evidence that SDUs are associated with improved patient outcomes. Our work is also related to the body of literature in the operations management community regarding ICU care. There are a number of papers, including Suter et al. (1994), Azoulay et al. (2001), Shmueli et al. (2003), Escher et al. (2004), Simchen et al. (2004) and Kim et al. (2014), which examine the impact of ICU admission decisions on patient outcomes. In contrast, our work considers the bed transfer decision upon ICU discharge. Kc and Terwiesch (2012) also examines the ICU discharge process; however, the focus is on early discharges from the ICU due to demand pressures. This work examines the impact of the type of unit a patient is transferred to following ICU discharge. Our estimation model is most related to that considered in Kim et al. (2014). Like Kim et al. (2014) and Kc and Terwiesch (2012), among others, we utilize an instrumental variable which is based on an operational measure–congestion in an inpatient unit. While the general methodology is similar, the question we are considering is wholly different. Our focus is on the role of an SDU in the care of critical patients, whereas the aforementioned works focus on the ICU. Our work is also related to the voluminous literature on hospital capacity planning (see Green 2006 for an overview). These papers typically use a queueing approach. For example, Yankovic and Green (2011) develop a queuing model to identify nurse staffing levels that satisfy a particular service level. Yom-Tov and Mandelbaum (2014) introduce an Erlang-R model to guide staffing decisions in the Emergency Department. The work described in this paper complements some of the results from Armony et al. (2013), which uses fluid and diffusion analysis to study nurse allocation between the ICU and the SDU. The findings in that paper demonstrate that the nurse allocation decision depends upon the cost of two undesirable phenomena: The first occurs when critical patients requiring ICU care experience excessive delays in being admitted to the ICU. In many of these instances, they are sent to a different unit or hospital for care, or, in the worst case, die. We refer to this phenomenon of patients who need to be admitted to an ICU but don’t get admitted as ‘abandonment’ (e.g., in the queueing sense). The second undesirable phenomenon occurs when RAB semi-critical patients are demand-driven discharged (see, for example, Kc and Terwiesch (2012) and Chan et al. (2012)) from the ICU to the ward when new, more severe patients require ICU care. We refer to this phenomenon as ‘bumping’. Armony et al. (2013) find that the nurse allocation decision depends on the relative costs of bumping and abandonment. The work described in this paper is a first step towards estimating the impact of bumping on patient

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outcomes, which in turn can be used to estimate costs. We utilize these estimates to calibrate a simulation model to examine the impact of SDU size on patient outcomes. While there are a number of papers which utilize simulation approaches to examine capacity management (see e.g., Ballard and Kuhl 2006, Griffiths et al. 2010, and Zhu et al. 2012), to the best of our knowledge ours is the first to consider the role of an SDU.

3.

Empirical Estimation

In this section, we empirically estimate how SDU admission immediately following ICU discharge affects patient outcomes. Section 3.1 describes the data we use. Section 3.2 develops the estimation approach and Section 3.3 reports our empirical results. 3.1.

Data

This project was approved by the Kaiser Permanente Northern California Institutional Review Board for the Protection of Human Subjects, which has jurisdiction over all study hospitals. We utilize patient data from 10 hospitals from Kaiser Permanente Northern California, containing 165,948 hospitalizations over a course of one and a half years. Our data contains operational level information, including every unit to which a patient is admitted during his hospital stay as well as the date and time of admission and discharge for each unit. For each inpatient unit in each hospital, we use these patient flow data to derive hourly occupancy levels and we define its capacity as the maximum occupancy level over the time horizon. Table 1 summarizes the capacity for each of the different levels of inpatient care in each hospital: the ICU, SDU, and ward. While each level of care may have further divisions based on specific services, e.g. medical versus surgical ICU, the boundaries are somewhat loose in these hospitals in the sense that if a critical patient requires ICU care and is under the medical service, but there are no medical ICU beds available, he will likely be cared for in the surgical ICU. We observe substantial heterogeneity across these hospitals; the SDU capacity varies from 11 to 32 and the ICU size ranges from one half to twice of the SDU size. Our data selection process is depicted in Figure 1. First, we restrict our cohort to the 12 months in the center of the 1.5 year time period to avoid censored estimation of capacity and occupancy. A patient’s admission category is defined as a combination of whether or not they were admitted through the ED, and whether they were a medical or surgical patient resulting in 4 categories: ED-medical, ED-surgical, non-ED-medical, or non-ED-surgical. Since our goal is to examine the impact of the ICU to SDU routing on patient outcomes, we only consider patients who are admitted to the ICU at least once during their hospital stay. Note that within the SDUs at this hospital network, there are patients who are not admitted via the ICU. Of these, we primarily focus on patients who are admitted via the ED to a medical service for two major reasons. First, this group

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Table 1

7

Capacity of various impatient units

Hosp ICU SDU Ward 1 11 24 61 2 11 25 76 3 16 14 77 4 16 19 76 5 16 24 78 6 23 19 124 7 24 20 145 8 26 27 110 9 31 11 188 10 32 32 100 is the largest, consisting of about 60% of the patients treated in these hospitals, and is similar to the cohort considered in Kim et al. (2014). Second, the care pathways of surgical patients tend to be fairly standardized, especially for non-ED-surgical patients, which is the larger surgical group. In contrast, the care pathways of ED-medical patients are more variable. For each patient, we focus on the initial ICU admission. We exclude patients who die in the ICU or are discharged directly home from the ICU, since there is no decision about whether to route these patients to the SDU or ward following ICU discharge. In Appendix C, we also consider the three other patient categories: ED-surgical, non-ED-medical, and non-ED-surgical. We also do some robustness checks by considering the last ICU admission and all ICU admissions within the hospital stay. Our dataset contains information about patient characteristics such as age, gender, admitting diagnosis and three different severity scores. One score is based on lab results taken within the first 72 hours of hospital admission and the second is based on comorbidities, such as diabetes, that may complicate patient recovery. These severity scores are assigned at hospital admission and are not updated during the hospital stay (more details on these scores can be found in Escobar et al. (2008)). The third severity score is the simplified acute physiology score 3 (SAPS3), which is a common severity score used for ICU patients. It is updated and assigned each time a patient is admitted to the ICU. Christensen et al. (2011), Mbongo et al. (2009), and Strand and Flaatte (2008) provide detailed descriptions and validation of the SAPS3 score. Table 2 provides summary statistics for these patient characteristics. 3.2.

Estimation Approach

We are interested in understanding whether SDU care following ICU discharge is associated with better patient outcomes and, if so, the magnitude of the improvement. Specifically, our objective is to determine if ICU patients who are transferred to the SDU have better outcomes than those transferred to the ward. To accomplish this, we must deal with an estimation challenge which arises due to the fact that the routing decision following ICU discharge is likely correlated with patient

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Figure 1

Data Selection

Patient cohort Total Hospitalizations:   165,948 *

Admitted during the 1‐year  period: 130,698

Admitted as ED‐Medical patients: 77,418

Admitted to ICU at least  once: 14,996

Admitted outside the study  period : 35,250 (21.24%) Admitted as a Surgical  or Non‐ED patient:  53,280(40.77%)

Never admitted to ICU:  62,422 (80.63%)

Out‐of‐hospital or non‐ Ward/SDU units :  3,938(26.26%)

Admitted to Ward/SDU after  1st ICU : 11,058 ** * to determine capacity and occupancy ** patient cohort used in our econometric model

Table 2

Summary Statistics

Variable mean std min p25 p75 max Age 68.13 15.91 18 56 78 105 Male 0.5513 0.4974 0 0 1 1 75.13 49.10 0 29 106 262 LAPS2 COPS2 46.63 44.61 0 10 66 267 SAPS3 45.41 11.79 15 37 52 100 ICU LOS (hrs) 61.87 84.75 0.02 22.02 68.07 2279.17 LOS before ICU (hrs) 32.99 108.88 0 0.9 18.52 4877.58 outcomes. That is, sicker patients are more likely to be transferred to the SDU, but also more likely to have bad outcomes. To overcome this potential endogeneity bias, we utilize an identification strategy using Instrumental Variables (IVs). 3.2.1.

Patient Outcomes We consider four patient outcomes: (1) in-hospital death

(M ortality), (2) ICU readmission (ICU Readm), (3) remaining hospital length-of-stay (HospRemLOS), and (4) hospital readmission (HospReadm). HospRemLOS measures the time between discharge from the first ICU admission and hospital discharge. In the model for HospRemLOS, we include patients with in-hospital death; the results are similar if we exclude these patients. HospReadm2w is defined as hospital readmission within two weeks after leaving

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hospital (e.g., see Doran et al. 2013 and Ouanes et al. 2012 which use these durations), and ICU Readm2d (ICU Readm5d ) indicates ICU readmission within two (five) days following ICU discharge (Brown et al. 2013 aims to define reasonable time periods for ICU readmission). In calculating hospital readmission rates, we exclude patients with in-hospital death. Similarly, we did not include patients who die within 2 days following ICU discharge in the analysis of ICU Readm2d . Table 3

Summary Statistics for Patient Outcomes

outcome Mortality ICUReadm - 2 days ICUReadm - 5 days HospReadm - 2 weeks HospRemLOS (days)

obs. 3,832 3,832 3,832 3,585 3,832

SDU mean std 0.06 0.04 0.08 0.14 7.32 15.68

obs. 7,226 7,226 7,226 6,685 7,226

Ward mean std 0.07 0.05 0.06 0.13 5.26 10.36

We also do robustness checks for different time windows in Appendix C. Summary statistics of these outcomes are provided in Table 3 and are grouped by the inpatient unit a patient goes to after completing ICU service. 3.2.2.

Econometric Model We now present our estimation model. For patient i, let yi denote

the patient outcome of interest (e.g., HospRemLOSi ) and Xi be a row-vector of covariates including age, gender, severity scores, admitting diagnosis, seasonality controls, care before the ICU admission, and ICU LOS. Table 7 in the Appendix provides detailed descriptions of these control variables. We also control for the daily average occupancy level, denoted as AvgOccV isitedi , patient i experiences for all impatient units s/he is admitted to after leaving the ICU and before leaving hospital. For ICU readmission, we modified AvgOccV isitedi to be the daily average occupancy level that patient i experiences in all impatient units s/he is admitted to between two consecutive ICU admissions. Appendix C provides robustness checks for different specifications of AvgOccV isitedi , as well as with this control excluded. We include such a measure as Kc and Terwiesch (2012) shows that patient outcomes are affected by congestion; Kim et al. (2014) provides additional discussion. Define ICU 2SDUi as an indicator that equals to 1 if patient i is transferred to the SDU after completing ICU service and 0 if patient i is routed to the ward, i.e.,  ICU 2SDUi =

1, if patient i is transferred to the SDU from the ICU . 0, if patient i is transferred to Ward from the ICU

(1)

The objective is to estimate the impact of the ICU discharge decision ICU 2SDUi on patient outcomes.

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Instrumental Variable: As discussed earlier, the decision to admit a patient to the SDU following ICU discharge is endogenous. A common approach to address endogeneity and obtain consistent estimates is by utilizing an instrumental variable (IV) approach. We use congestion in the SDU one hour before the ICU discharge as an IV. In particular, we define SDU Busyi as an indicator variable that equals one when the number of available beds in the SDU one hour prior to patient i’s discharge from the ICU is less than or equal to two, and zero otherwise1 . On average, about 11% patients are discharged from the ICU when the SDU is busy (SDU Busy = 1), though this varies quite a bit across hospitals (see Table EC.2). A valid instrument should be 1) correlated with the endogenous variable, which in our case is the routing at ICU discharge, ICU 2SDUi , and 2) unrelated to the unobservable factors captured in εi which affect patient outcomes. In Figure 2(a), we group patients into 20 groups based on their SAPS3 scores and compare the percentage of ICU patients who are routed to the SDU when the SDU is busy versus not busy. We find that with this coarse comparison, SDU congestion is associated with a reduction in the fraction of patients admitted to the SDU for all severity levels. When controlling for various patient characteristics in a Probit regression model (results in Table 8 of the Appendix), we also find at the 0.1% significance level that when the SDU is busy, patients are less likely to go to the SDU. In particular, we estimate that, on average, 21.14% percent of patients are routed to the SDU if SDU Busy = 1 and this percentage increases to 35.91% if SDU Busy = 0. Namely, a congested SDU is predicted to result in a 47% reduction in the likelihood of the SDU admission. For SDU Busyi to be a valid instrument, it also has to be uncorrelated with unobservable factors in patient outcomes captured in error εi . Since we cannot examine unobservable measures, we use patient severity, SAP S3, as a proxy for those unobservable factors. In particular, we perform a two-sample Kolmogorov-Smirnov test (see Gibbons and Chakraborti 2011 for details) to test the hypothesis that the distribution of SAPS3 for patients who are discharged from ICU when SDU Busy = 1 is not statistically different to that when SDU Busy = 0. The p-value for the combined Kolmogorov-Smirnov test is 0.136. Thus, we can not reject the null hypothesis and believe that patients who are discharged from the ICU when SDU Busy = 1 are statistically similar to patients who are discharged from the ICU when SDU Busy = 0. Figure 2(b) depicts the distributions for patients who are discharged from the ICU when SDU Busy = 1 and those when SDU Busy = 0. 1

We also do robustness checks in Appendix C by considering different specifications of SDU busy: (1) different cutoffs at one, two, three, four available beds; (2) dummy variables using occupancy level with cutoffs at 80th (or 85th , 90th , 95th ) percentile, i.e., SDU Busyi = 1 if occupancy level is larger than the cutoff percentile and zero otherwise; (3) congestion represented by a continuous piecewise linear spline variable with knots at the 80th (or 85th , 90th , 95th ) percentile; and (4) these measures 2 hours (instead of 1) prior to ICU discharge.

SDUBusy=0

SDUBusy=1 SDUBusy=0

0

0

20

.01

density .02

Observed ICU2SDU% 40 60

.03

80

SDUBusy=1

.04

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0

5

10 20 Quantiles of SAPS3

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(a) Routings of ICU Discharge Figure 2

20

20

40

60

80

100

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SAPS3

(b) Distributions of Patient Severity Scores

IV identification: SDU Busy = 1 vs SDU Busy = 0. (Left (a)) Observed percentage of ICU patients who are routed to SDU when SDU Busy = 1 vs SDU Busy = 0 with different severity levels characterized by SAPS3. (Right (b)) Distributions of severity scores for patients who are transferred to the SDU when SDU Busy = 1 vs SDU Busy = 0.

Kc and Terwiesch 2012 demonstrates that ICU congestion could result in early discharge, which could, in turn, affect the routing decision of ICU patients. While ICU congestion has been used as an IV in a number of hospital studies (e.g. Kc and Terwiesch 2012, Kim et al. 2014), we find that ICU congestion is not a valid IV. This is because the impact of ICU congestion does not exhibit a consistent effect on ICU routing, i.e., a congested ICU could result in both a higher and a lower percentage of patients being admitted to the SDU depending on a patient’s severity score. Joint Estimation of Two Equations. Since the ICU to SDU routing decision, ICU 2SDUi , is a binary variable, we model the ICU discharge decision via a latent variable model: ICU 2SDUi = 1 {ICU 2SDUi∗ > 0} and ICU 2SDUi∗ = Xi θ + αSDU Busyi + ωh(i) + ξi ,

(2)

where ICU 2SDUi∗ is a latent variable which represents the propensity towards SDU admission; Xi is a vector of control variables for patient information; ωh(i) is the hospital fixed effect (h(i) is patient i’s hospital); and, ξi represents unobservable factors that affect the routing at ICU discharge. For a binary outcome (M ortality, HospReadm, ICU Readm), we model the patient outcome as: yi = 1 {yi∗ > 0} and yi∗ = Xi β + γ · ICU 2SDUi + δ · AvgOccV isitedi + νh(i) + εi ,

(3)

where yi∗ is a latent variable which represents the propensity for the outcome; νh(i) is the hospital fixed effect; and εi captures unobservable factors that affect patient outcomes. The error terms (ξi , εi ) in the above two equations (2) - (3) may be correlated to model the endogeneity between the routing at ICU discharge and patient outcomes. We assume that (ξi , εi ) follows a Standard

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Bivariate Normal distribution with correlation coefficient ρ. This Bivariate Probit model can be jointly estimated via Full Maximum Likelihood Estimation “FLME” (see Cameron and Trivedi 1998, Greene 2012, and Kim et al. 2014). The presence of endogeneity can be tested through a likelihood ratio test of null ρ = 0. For HospRemLOS, which is a continuous variable, equation (3) is replaced by log(HospRemLOSi ) = Xi β + γ · ICU 2SDUi + δ · AvgOccV isitedi + νh(i) + εi ,

(4)

Similar to the binary outcomes, weassume  that (ξi , εi ) follows a Bivariate Normal distribution with σρ . This model can be jointly estimated using a treatment mean zero and covariance matrix ρ1 effect model via FLME ( Greene 2012). Similarly, a likelihood ratio test of null ρ = 0 can be used to test the presence of endogeneity. 3.3.

Estimation Results

This section summarizes our estimation results. We are primarily interested in estimating the effects of SDU admission on patient outcomes; thus, we report only the coefficient of SDU admission on the patient outcomes, i.e., γ in (3) and (4). The full estimation results are provided in Appendix B. Table 4

Effect of SDU Admission (γ) on Patient Outcomes

With IV Predicted Outcome Outcome γ (SE) ρ (SE) PˆSDU PˆW ard M ortality -0.60** (0.22) 3.96% 10.26% 0.26+ (0.14) log(HospRemLOS) -0.35*** (0.10) 0.91 1.27 0.44*** (0.05) -0.51** (0.20) 2.34% 6.62% 0.32* (0.12) ICU Readm2d ICU Readm5d -0.51**(0.18) 3.93% 10.10% 0.36** (0.11) -0.43* (0.21) 8.66% 17.08% 0.21+ (0.12) HospReadm2w

Without IV Test ρ=0 0.07 0.00 0.02 0.05 0.09

γ (SE) -0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.09* (0.04) 0.05 (0.04)

Note. Standard error in parentheses. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%). Predicted outcome: PˆSDU - Average predicted outcome if all patients could be routed to the SDU and PˆW ard if no SDU and everyone is rerouted to Ward.

Table 4 summarizes the relationship between SDU admission right after ICU discharge and patient outcomes. The sign of SDU admission is negative and statistically significant in all outcome measures, suggesting that routing an ICU discharge to the SDU is associated with improved patient outcomes. We also use our estimation results to predict patient outcomes under two extreme scenarios: (i) the SDU has ample capacity so that all patients will be routed to the SDU (referred to as PˆSDU ) versus (ii) the hospital does not have an SDU and all patients will be routed to the ward (referred to as PˆW ard ). We find that, on average, SDU care is associated with significant improvements in patient outcomes: the relative reduction is 72% in the likelihood of in-hospital death,

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34% in the hospital remaining length-of-stay, 67.7% (63.9%) in the likelihood of ICU readmission within 2 (5) days, and 52% in the likelihood of hospital readmission within 2 weeks. We note that the large effect on mortality rate could be partially due to “do not resuscitate (DNR)” orders, which are patients’ end-of life wishes not to undergo Cardiopulmonary resuscitation (CPR) or advanced cardiac life support if their heart were to stop or they were to stop breathing. In speaking with intensivists, we learned it is possible that patients with DNRs are more likely to be sent to the ward, but also may be more likely to die, resulting in an overestimate of the effect of SDU care. Unfortunately, we do not have access to patients’ DNR status, so cannot control for this. That said, DNR orders only represent 9% of ICU patients (Jayes et al. 1993), so this is likely to affect a small percentage of patients. Additionally, there is evidence that DNR orders do not change the quality of care (Baker et al. 2003). We do not expect DNR orders to impact our results for hospital readmission since we exclude patients who died in hospital in this model. For remaining LOS, we find that our results are robust to including and excluding patients who died. Our empirical findings also suggest strong evidence of an endogeneity bias between the routing following ICU discharge and patient outcomes. The p-value of the likelihood ratio test with null hypothesis ρ = 0 is small, as seen in Table 4, implying a strong correlation between the routing at ICU discharge and patient outcomes. Ignoring this endogeneity tends to result in underestimates of the benefit of SDU care and could result in a qualitatively different insight; see the column titled with “Without IV”. We examine a number of robustness checks including alternative specifications of patient characteristics, different specifications for SDU congestion, and different time windows for readmission. The results are summarized in Appendix C. We find that the main results hold with slight quantitative changes. For example, the effect of the ICU to SDU routing on hospital readmission is weaker when the elapsed time between two consecutive hospital stays is longer. This may be because the complication requiring readmission will be less likely to be related to the routing following ICU discharge and more likely related to post-discharge events. Given our findings indicating the benefits of SDU care following ICU discharge, we use these empirical results to develop an understanding of the role of SDU capacity on patient outcomes.

4.

Patient Flow Model

We present an ICU patient flow model that we use to explore the impact of SDU capacity on the percentage of patients routed to the ward following ICU discharge and the associated patient outcomes such as in-hospital mortality and ICU readmissions. The model is illustrated in Figure 3.

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14Article submitted to Manufacturing & Service Operations Management; manuscript no. Figure 3

Simulation Model Reverse Access Blocked

ICU ߣூ஼௎ New

Readmitted

BICU

p

1- pSC

pSC: semi-critical

R SDU

SDU

Survival pSDU R 1 − pSDU

BSDU R pWard

1− p

Survival SDU

Ward Survival pWard R Ward

1− p

… Survival 1 − pWard

ICU Admissions: New patients arrive at the ICU according to a Poisson process (see e.g. Shmueli et al. (2003), Kim et al. (2014)) with rate λICU and are assumed to be in a critical state. Their service times, defined as the duration of time for which they remain critical, are independent and 0 0 identically lognormally distributed with mean LOSICU and standard deviation σICU (see Armony

et al. (2011) and Shi et al. (2014) for evidence of lognormal LOS in hospital settings). As will be described in more detail later, patients may require readmission to the ICU after their initial R stay. The service time of readmitted patients is lognormally distributed with mean LOSICU and R standard deviation σICU . If, upon arrival of a new or readmitted ICU patient, there is an available

ICU bed, he will be admitted directly to the unit. If the ICU is full, the patient will join either the new or readmission queue, depending on how he arrived, to wait for an available bed. When an ICU bed becomes available, priority is given to readmitted patients. A first-come-first-serve (FCFS) rule is used within each queue. Inpatient Unit Routing following ICU Discharge: After service completion in the ICU, a patient becomes semi-critical following completion of time spent as critical with probability pSC . If a patient is semi-critical, medical needs would warrant placement in the SDU. Otherwise, with probability 1 − pSC , the patient does not require SDU service. For example, if the patient is sufficiently stable, he can be sent directly to the ward or home. Alternatively, the patient may die in the ICU. For newly semi-critical patients, there are three possibilities (1) an SDU bed is available and the patient is placed in the SDU; (2) the SDU is full and so the patient becomes a RAB patient and

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stays in the ICU, or (3) both the ICU and the SDU are full and so the patient is rerouted to the ward. Rerouting patients is associated with worse outcomes: a higher likelihood of in-hospital mortality and a higher readmission rate. In particular, if a semi-critically ill patient is admitted to the Survival SDU, the patient will die in the SDU with probability pM ). If the patient survives SDU (= 1 − pSDU

to SDU discharge, with probability pR SDU the patient’s health condition deteriorates and requires readmission to the ICU. The same process applies to patients who are rerouted to the ward with Survival pM ) and pR W ard (= 1 − pW ard W ard denoting the likelihood of in-hospital death and ICU readmission, R M M respectively. We assume that pR SDU < pW ard and pSDU < pW ard to reflect the worse outcomes asso-

ciated with rerouting semi-critically ill patients to the ward. We also assume that the service times of semi-critically ill patients are independent and identically lognormally distributed: with mean SC SC SC LOSSDU and standard deviation σSDU for the service time in the SDU, with mean LOSW ard and SC standard deviation σW ard for the service time at the ward. We assume that there is ample ward

capacity so that there is no possibility of patients discharged to the ward being blocked. Reverse Access Blocked (RAB) Patients in the ICU: An RAB patient leaves the ICU under the following scenarios: (1) An SDU bed becomes available. We refer this as a regular discharge. (2) A new or readmitted critical patient arrives to the ICU and there are no available ICU beds. The critical patient has priority for ICU care over the semi-critical patient, so the RAB will be discharged from the ICU. Note that because the SDU is full, this patient must go to the ward. We call this type of RAB discharge a demand-driven discharge. (3) It is also possible that a RAB patient leaves the ICU after stablizing in the ICU. For the regular and demand-driven discharge of RAB patients, we assume that the FCFS rule is used when there are multiple RAB patients in the ICU, namely, the patient with the longest blocked time in the semi-critical state leaves the ICU first. For each RAB patient, we assume that the time spent in the ICU while in a semi-critical state is perfectly substitutable for the service time in the SDU. This is because the time needed for a patient to become stable is mainly determined by the physiological state of the patient and this recovery time does not depend on where the patient is treated (e.g., see Zimmerman et al. 2006). In particular, at the epoch of entering the RAB state, we sample patient i’s service time LOSi,SDU as if he were transferred to the SDU and LOSi,W ard as if he were transferred to the ward according to the corresponding distributions. Let Blocki,ICU be the blocked time in the ICU for RAB patient i; then, patient i’s remaining service time in the SDU would be LOSi,SDU − Blocki,ICU if patient i is transferred to the SDU and LOSi,W ard − Blocki,ICU if patient i is transferred to a general ward. When Blocki,ICU = min {LOSi,SDU , LOSi,W ard }, a RAB patient i becomes stabilized in the ICU and either goes directly home or to the ward.

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16Article submitted to Manufacturing & Service Operations Management; manuscript no.

4.1.

Operating Regime

There are many operational factors that could impact patient outcomes. For example, patient outcomes can deteriorate as a result of actions taken by physicians when the ICU becomes congested (Chan et al. 2012, Kc and Terwiesch 2012, and Kim et al. 2014). Since our objective is to understand the impact of the SDU size on patient outcomes, we will assume in our analyses that the ICU is generally uncongested.2 In particular, we assume that the arrival rate of new patients to the ICU is such that at least 95% of patients have less than T hours blocked waiting for ICU admission, i.e., P(Waiting time ≤ T ) ≥ 95%.

(5)

Here T is a patient’s tolerance for waiting. For example, Chalfin et al. (2007) demonstrates that waits greater than 6 hours leads to worse outcomes, so we set T = 6 hours. To explicitly calibrate the arrival rate of new ICU patients, we approximate the waiting time distribution by an M/M/c queue (see Gross et al. 2008). In particular, we assume that all semi-critical patients are routed to the SDU, which will provide a lower bound of the true ICU readmission rate and subsequently, a lower bound on the total time a patient is critical, thereby the total required ICU LOS, T OT AL LOSICU . Note that this does not include time a patient is reverse access blocked since these patients are semi-critical and could be treated in an SDU if space were available. The average total ICU service time (T OT AL LOSICU ) for each patient, defined as the total time a patient spends in the critical state, including the first ICU admission and all sequential ICU admissions, is given by T OT AL LOSICU = Average Total LOS in the ICU ∞ X 0 k R = LOS + (pSC pR SDU ) LOSICU | {zICU} LOS for 1st ICU admission |k=1 {z }

0 = LOSICU +

pSC pR SDU R LOSICU . 1 − pSC pR SDU

LOS for readmitted ICU admissions

Let µICU =

1 T OT AL LOSICU

be the service rate of the ICU, r =

λICU µICU

and ρ =

λICU BICU µICU

be the traffic

intensity in the ICU, one has P(Waiting time > 6hours) =

rBICU BICU !(1 − ρ) ·



r BICU BICU !(1−ρ)

+

PBICU −1 rn  · e n=0

−6·(BICU ·µICU −λICU )

.

n!

We choose the arrival rate λICU such that the above P(Waiting time > 6hours) equals to 5%. In order to focus on the role of the SDU, we assume that there is ample capacity in the general ward. We vary the capacity of the SDU in order to understand the impact of these capacity limits (or lack thereof) on the patient outcomes. 2

To be more precise, the ICU has limited capacity in our settings, but we assume that the ICU is not congested by selecting a sufficiently low arrival rate such that the utilization in the ICU is small.

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While this model does not capture every possible patient flow into and out of the ICU and SDU, it describes the most common patient care pathways. In particular, the SDU is used as a true StepDown Unit in the sense that patients treated in the SDU come from the ICU as an intermediate step down before being sent to the ward.

5.

Simulation

We now examine how the SDU size affects patient outcomes and utilize these insights to explore capacity management of the SDU via simulation for our general patient flow model described in Section 4. 5.1.

Parameter Calibration and Selection

We start by using our patient data to determine the parameters for patient LOS. Recall that the LOS in the ICU, SDU, and ward are assumed to follow log-normal distributions. We estimate the mean and standard deviation of LOS in an impatient unit using all patients who are discharged from that unit when it is not congested. The reason that we use data when the hospital unit is not congested is due to adaptive mechanisms physicians use when congestion is high, which may censor our estimates (e.g. Kc and Terwiesch 2012 and Chan et al. 2012). For instance, we find ICU LOS tends to be longer for patients who are discharged when the SDU is congested. Therefore, in estimating the average ICU LOS, we consider patients who are discharged from the ICU when both the ICU and the SDU are not-busy; this should mitigate the effects of demand-driven discharges and reverse access blocks. We use results of our econometric model from Section 3.2 to calibrate the patient outcomes for different ICU to SDU routing decisions. In particular, we set the probability of in-hospital death as 3.96% for patients who are routed to the SDU following their ICU discharge and 10.26% for patients who are rerouted to the ward (see Table 4). As 73% of readmitted patients return to the ICU within 5 days of ICU discharge, we consider the ICU readmission rate within five days to estimate the ICU readmission probabilities of 3.93% and 10.10% for patients who are routed from the ICU to the SDU and wards, respectively (see Table 4). We also estimated the likelihood of a patient entering the semi-critical state following ICU discharge by examining the percentage of patients who are discharged from the ICU and routed to the SDU when both the ICU and the SDU are not-busy. In the baseline model, we set this percentage to be 60%, i.e., pSC = 0.6, which is the average in our data. However, this percentage varies across the 10 hospitals in our dataset, so we test the sensitivity of our results by considering pSC ∈ {0.5, 0.6, 0.7, 0.8}. Finally, to address the question of what SDU size is appropriate, we varied the SDU size from 10% to 100% of the ICU size, i.e.,

BSDU BICU

∈ {0.1, 0.2, 0.3, . . . , 1}. Our baseline model considers 20 ICU

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18Article submitted to Manufacturing & Service Operations Management; manuscript no. Table 5

The Value of Parameters

Parameter Number of beds LOS for 1st ICU admission (hours) LOS for Readmitted ICU admission (hours) LOS for SDU (hours) LOS for ward (hours) Percentage of SDU requests ICU Readmission rate Mortality rate Sensitivity test

Value BICU = 20, BSDU = {0.1, 0.2, 0.3, . . . , 1} × BICU 0 0 LOSICU = 56, σICU = 60 R R LOSICU = 66, σICU = 74 SC SC LOSSDU = 67, σSDU = 77 SC SC LOSW ard = 74, σW ard = 108 pSC = 0.6 R pR SDU = 3.93%, pW ard = 10.10% M pSDU = 3.96%, pM W ard = 10.26% BICU ∈ {10, 20, 30, 40}, pSC ∈ {0.5, 0.6, 0.7, 0.8}

beds, which is the average ICU size in the 10 hospitals. However, Table 1 shows substantial variation in ICU size across different hospitals. We test the sensitivity of simulation results by considering four different ICU sizes, BICU ∈ {10, 20, 30, 40}. The value of these parameters is summarized in Table 5. We run the simulation for 100,000 iterations over 14 months, where the first 2 months are used as a warm-up period. Validating the M/M/c approximation. Recall that the length-of-stay in the ICU is a log-normal distributed random variable in our simulation model, while the arrival rate is chosen to meet a desired performance benchmark (the probability a critical patient needs to wait more than 6 hours for ICU admission is less than 5%) by using an approximation based on the M/M/c queueing model. We verify in our simulation that choosing the arrival rate in this manner is reasonably accurate. Indeed, we find that on average 94.5% patients spend less than 6 hours waiting for an ICU bed. 5.2.

Simulation Results

The baseline model considers 20 ICU beds and 60% of patients enter the semi-critical state after ICU discharge. The simulated ICU utilization (excluding RAB patients) is close to 70% with slight fluctuations for different SDU sizes. (The ICU utilization calculated according to M/M/c approximation is 69.99%.) Table 6 reports the simulation results for ICU readmission rate, mortality rate, average queue length and waiting time for new and readmitted ICU patients, and average number of RAB patients who are blocked in the ICU after completing ICU service as well as their blocked time. • ICU readmission and mortality: It is intuitive that a large SDU size reduces the likelihood

of ICU readmission and in-hospital death since adding SDU capacity reduces the need for rerouting patients to the ward. The marginal improvement of adding one SDU bed is significant when the SDU is small, for example a 21.5% reduction in the ICU readmission rate is achieved when the SDU size is increased from 2 beds to 4 beds and another 20% reduction from 4 SDU beds to 6 SDU beds. This marginal improvement decreases with the SDU size. In particular, 14 SDU beds results in practically the same outcomes as a large SDU (20 beds).

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Table 6

BSDU 2 4 6 8 10 12 14 16 18 20

Simulation Results (BICU = 20, pSC = 0.6)

Avg Queue length New Readmitted RAB 0.22 0.0036 4.01 0.19 0.0024 2.81 0.18 0.0018 1.42 0.18 0.0016 0.63 0.18 0.0016 0.29 0.18 0.0016 0.12 0.18 0.0017 0.04 0.18 0.0016 0.01 0.18 0.0017 0.00 0.18 0.0017 0.00

Avg Waiting time New Readmitted 0.91 0.43 0.79 0.39 0.75 0.35 0.74 0.31 0.72 0.30 0.75 0.28 0.74 0.29 0.75 0.27 0.75 0.29 0.74 0.28

(hrs) RAB ICUReadm Mortality 36.46 8.23% 8.25% 32.09 6.46% 6.41% 21.85 5.17% 5.18% 14.10 4.57% 4.57% 10.38 4.29% 4.26% 7.91 4.09% 4.15% 6.16 4.03% 4.03% 4.73 3.98% 4.01% 3.45 3.98% 3.98% 1.04 4.00% 3.98%

• Queue length and waiting time: For readmitted patients, the queue length is negligible

due to the operating regime we’ve selected and the fact that priority is given to readmitted patients. We also observe that the queue length of new arrivals is insensitive to the SDU size. At first glance, one may expect that a large SDU will reduce the ICU readmission rate due to fewer reroutings, which, in turn, will result in a smaller queue length. However, the change in readmission rate is very small and so does little in affecting the overall arrival rate of new and readmitted critical patients. As such, the system load, and subsequently queue length, does not change significantly. By Little’s Law, the same insights carry over to explain the negligible impact of SDU size on the waiting time for new and readmitted patients. • RAB Patients: Adding SDU capacity reduces the number of RAB patients and their time

in the ICU. With more SDU beds, fewer semi-critical patients will find the SDU full, so will not be blocked. Additionally, if they are blocked, the time until an SDU bed becomes available is decreased, thereby reducing the total time spent blocked in the ICU. Our key observation from the baseline simulation is that the SDU size is a major factor associated with patient outcomes. In particular, we find that adding SDU capacity improves patient outcomes, but the marginal improvement decreases with the SDU size. An SDU which is 70% of the ICU size (14 SDU beds in this case) results in patient outcomes that are very similar to those associated with a much larger SDU. 5.2.1.

Sensitivity Analysis We now test the sensitivity of our baseline simulation results by

changing the number of ICU beds and the percentage of patients who become semi-critical and need SDU care. In the first scenario, we test the sensitivity to ICU beds by varying the ICU size. In the second scenario, we test the sensitivity to pSC . Similar to the baseline model, we find that the SDU size has very little impact on the average queue length and waiting time for new patients and readmitted patients in the ICU. As such, we report only mortality, ICU readmission, the number of reverse access blocked patients and their blocked time.

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20Article submitted to Manufacturing & Service Operations Management; manuscript no.

Sensitivity to ICU size. We vary the ICU size from {10, 20, 30, 40} and fix the percentage of SDU requests to be pSC = 60%. Figure 4 reports the mean and 95% confidence intervals of our Figure 4

Sensitivity Analysis of ICU Size (p = 0.6) Readmission

Mortality

9%

9%

8%

8%

7%

7%

6%

6%

5%

5%

4%

4% 3%

3% 0.1

0.2

0.3

10 ICU Beds

0.4

0.5

0.6

0.7

0.8

The SDU size / the ICU size 20 ICU Beds 30 ICU Beds

0.9

0.1

1

0.2

0.3

10 ICU Beds

40 ICU Beds

(a) ICU Readmission Rate

0.4

0.5

0.6

0.7

0.8

The SDU size / the ICU size 20 ICU Beds 30 ICU Beds

0.9

1

40 ICU Beds

(b) Mortality Rate Avg Blocked Time for RAB Patients

Avg Number of RAB Patients 50

7

45

6

40

5

35

4

30 25

3

20

2

15 10

1

5

0

0 0.1

0.2

0.3

10 ICU Beds

0.4

0.5

0.6

0.7

The SDU size / the ICU size 20 ICU Beds 30 ICU Beds

0.8

0.9 40 ICU Beds

(c) Number of RAB patients

1

0.1

0.2

0.3

10 ICU Beds

0.4

0.5

0.6

0.7

The SDU size / the ICU size 20 ICU Beds 30 ICU Beds

0.8

0.9

1

40 ICU Beds

(d) RAB blocked time (hours)

simulation. Similar to what we saw with the baseline simulation, we find that adding SDU beds improves patient outcomes, but the marginal benefits are diminishing. An artifact of our simulation model is that increasing ICU capacity results in worse outcomes. If demand were held constant, an increase in ICU capacity would certainly result in improved outcomes. However, in the simulation, ICU capacity and demand are tightly intertwined, so that an increase in capacity also results in an increase in demand. Recall that the ICU arrival rate is endogenously chosen so that 95% of patients wait less than 6 hours. By the economies of scale in queueing systems, a bigger ICU allows for a higher ICU utilization by Critical patients. In particular, the ICU utilization is ∼57% for 10 ICU beds, ∼70% for 20 ICU beds, ∼76% for 30 ICU beds, and ∼80% for 40 ICU beds. Because of the higher utilization, more semi-critical patients are rerouted, which, in turn, increases readmission and mortality rates. Thus, we see that an increased ICU size results in higher ICU workload and, subsequently, higher mortality and ICU readmission rates. We also find that a large ICU increases the number of RAB patients but reduces their average blocked time in the ICU. The former is intuitive as more ICU beds implies more capacity for RAB

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patients. However, it is somewhat surprising that the time spent blocked decreases with ICU size. The reason is because, as discussed before, with an endogenously selected arrival rate, a large ICU means higher ICU utilization. As such, there are more demand-driven discharges of RAB patients to accommodate new ICU arrivals, thereby reducing the average blocked time of RAB patient in the ICU. These demand-driven discharged patients are rerouted to the ward, which results in a higher likelihood of death and/or ICU readmission. Sensitivity to the probability of semi-critical state. We now perform sensitivity analysis to another important parameter–the percentage of ICU patients that become semi-critical and request SDU care (pSC )–by changing pSC in {0.5, 0.6, 0.7, 0.8}. The simulation results are reported in Figure 5, which also displays the 95% confidence intervals. Figure 5

Sensitivity Analysis of Percentage of SDU Request (BICU = 20) Mortality

Readmission 9%

9%

8%

8%

7%

7%

6%

6%

5%

5%

4%

4%

3%

3% 2

4

6

8

10

12

14

16

The SDU size P(ICU2TCU) = 0.6 P(ICU2TCU) = 0.7

P(ICU2TCU) = 0.5

18

20

2

4

6

8

10

12

14

16

The SDU size P(ICU2TCU) = 0.6 P(ICU2TCU) = 0.7

P(ICU2TCU) = 0.5

P(ICU2TCU) = 0.8

(a) ICU Readmission Rate

18

20

P(ICU2TCU) = 0.8

(b) Mortality Rate

Avg Number of RAB Patients

Avg Blocked Time for RAB Patients

6

45 40

5

35

4

30 25

3

20

2

15 10

1

5

0

0 2

4

6

P(ICU2TCU) = 0.5

8

10

12

14

16

The SDU size P(ICU2TCU) = 0.6 P(ICU2TCU) = 0.7

(c) Number of RAB patients

18

20

P(ICU2TCU) = 0.8

2

4

P(ICU2TCU) = 0.5

6

8

10

12

14

The SDU size P(ICU2TCU) = 0.6 P(ICU2TCU) = 0.7

16

18

20

P(ICU2TCU) = 0.8

(d) RAB blocked time

We note that changing pSC does not substantially alter the qualitative insight of how SDU size impacts patient outcomes, but does introduce slight changes in the exact magnitude of the effects. An increase in pSC increases the demand for SDU beds and so, as a result, more patients are rerouted to the ward. This, in turn, increases the likelihood of ICU readmission, in-hospital death, and the number of RAB patients. As for the blocked time of RAB patients, a high pSC has two opposing effects for a given SDU size. First, it increases the ICU readmission rate since a higher pSC increases the rerouting percentage and rerouted patients are more likely to be readmitted to the

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ICU. The increased ICU readmission rate increases the frequency of the demand-driven discharge of RAB patients to accommodate readmitted patients and thus reduces the average RAB blocked time. On the other hand, the arrival rate of new patients decreases with a higher pSC , as the total pSC pR

0 R + 1−pSCSDU ICU LOS (T OT AL LOSICU = LOSICU LOSICU ) increases. Recall that the arrival rate pR SDU

of new patients is endogenously chosen to satisfy the waiting time constraint (e.g. the arrival rate of new patients is 0.2442 when pSC = 0.5 compared to 0.2407 when pSC = 0.8). The reduced arrival rate reduces the frequency of transferring out RAB patients and increases the blocked time. The dominance of the above two forces of pSC on blocked time depends on various model parameters including the SDU size. 5.3.

Managerial Insights

The simulation results provide important insights about how to allocate resources between ICUs and SDUs. We summarize these insights in the follows: • Adding SDU capacity helps: Our simulation shows that adding SDU capacity could

improve patient outcomes, even when the ICU has ample capacity. • Marginal improvement diminishes: The marginal benefit of adding one SDU bed decreases

with the SDU size. In all scenarios we tested, we observed that an SDU with 70% of ICU size could achieve outcomes that are as good as those associated with a much larger SDU. Applying basic economic principles, hospitals could leverage this insight to determine the ideal SDU capacity via a cost-benefit analysis. Of course, this is for the case where the SDU is used only for ICU patients; further analysis is necessary for SDUs which include non-ICU demand. • Blocked time indicates capacity mismatch: The measure of reverse access blocked

patients provides useful insights on how resources might be reallocated across the ICU and SDU. The regular presence of many RAB patients with long blocked times indicates ample ICU capacity but scarce SDU capacity. Thus, hospitals could potentially improve efficiency by allocating more resources to the SDU and reducing ICU capacity. On the other hand, when there are many RAB patients with negligible blocked time, there are two possible scenarios: (1) when the rerouting percentage is large, this implies a shortage of capacity in both the SDU and the ICU; (2) when the rerouting percentage is small, this suggests a scarcity of ICU capacity and an abundance of SDU capacity. Thus, one can consider investments in new beds or reallocation of current beds depending on which scenario exists. Our simulation and econometric models provide a way for physicians to estimate the marginal benefit, in terms of better patient outcomes, of adding SDU capacity. Our findings provide hospital decision makers very useful insights in determining the size of the SDU and resource reallocation between the ICU and the SDU, to achieve good patient outcomes while considering the economic costs of adding beds. Most importantly, our study lends weight to support the use of SDUs in critical care–an issue which continues to be debated by the medical community.

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6.

Conclusions and Future Research

This paper studies the role of a hospital step-down unit (SDU) in the care of critical patients. To that end, we consider two fundamental questions regarding the SDU: (1) Does admitting a patient to the SDU following ICU discharge improve patient outcomes and, if so, what is the magnitude of these improvements? (2) How does the size of the SDU affect patient outcomes, and, more importantly, how many SDU beds should hospitals operate and what is the best allocation of resources between the SDU and the ICU? Our work represents an important first step towards answering these questions. Using patient data from 10 hospitals, we use econometric models to estimate the benefit of SDU care. Our empirical results suggest that routing an ICU discharge to the SDU is associated with substantial improvements in patient outcomes: reduced in-hospital mortality rate of 70%, ICU readmission rate of 67.7%, hospital readmission rate of 52%, and decreased hospital length-of-stay of 34%. As such, we find compelling evidence that there are measurable benefits to having an SDU. Using our empirical results to calibrate a simulation model, we find that increasing SDU capacity is associated with better patient outcomes, but the marginal improvement is decreasing. We also find that the number of reverse access block patients who are in the ICU, but do not medically need to be there, provides a useful indicator about potential capacity mismatches between the ICU and SDU. It is important to note that our findings provide guidelines and demonstrate the potential for using SDUs in a way that not only results in better clinical outcomes, but also can result in significant cost savings for hospitals. Because SDUs are significantly less expensive to operate than ICUs, and have the potential to significantly decrease ICU and hospital readmissions as well as remaining hospital LOS, properly used they can result in decreased hospital bed utilization and staffing costs. In the current healthcare environment in which hospitals are under increasing pressure to be more cost-effective, findings like the ones in this paper can be very valuable in achieving that goal. Our work complements the results of Armony et al. (2013). When ICU and SDU capacity is limited, Armony et al. (2013) provides insights into how to allocate resources between the two units depending on the relative costs of abandonment of critical patients and bumping of semicritical patients. This work is a first step towards estimating the impact of bumping semi-critical patient on patient outcomes, thereby providing insights into the bumping cost component which drives the capacity allocation decision. In contrast to Armony et al. (2013), this work examines how much SDU capacity is required when a hospital has the ability to invest in more capacity and the ICU is relatively uncongested. These two studies can be used by hospital administrators when

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determining the SDU size and resource reallocation between the ICU and the SDU. Depending on the availability of critical care resources, Armony et al. (2013) or this work will be more pertinent. There are a lot of opportunities for future work. Our empirical analysis relies on the variation in patient routings following ICU discharge due to capacity constraints. As such, it is not possible to make any statements about the impact of SDU care for patients whose care pathway is invariant to SDU bed availability. Because SDUs go in and out of favor at individual hospitals, there may be opportunities for natural experiments to make such inferences without requiring an instrumental variable analysis. Alternatively, at a hospital system such as Kaiser Permanente, it might be possible to conduct a controlled randomized trial by randomizing which hospitals have SDUs. Of course, such a study would require substantial buy-in from hospital administrators and staff. The purpose of our work is to measure the relationship between SDU care and patient outcomes rather than to build a predictive model to determine the role of SDU care for each individual patient. In such a setting, a split-validation approach would be useful to verify the out of sample predictive power of such a model. From an analysis point of view, it would be interesting to study the optimal control policies regarding where to transfer patients following ICU discharge in the presence of an SDU. This would provide a system-level view of the SDU admission decision, rather than focusing on the medical needs of an individual patient at each decision epoch as we do in our simulation model by assuming a semi-critical patient will be admitted to the SDU as long as there is an available bed. Also, in this paper, we have considered an SDU as serving only as a step-down unit; patients are only admitted to the SDU following ICU discharge. In a number of hospitals, patients may be directly admitted to the SDU or may be transferred from the ward or operating rooms. It would be interesting to extend our work to include non-ICU demand for the SDU and examine the tradeoff between ICU demand and non-ICU demand. Unfortunately, due to data limitations, we are unable to estimate the benefits of SDU care for non-ICU demand. However, as more data becomes available, e.g. that which includes requests and denials for SDU care for non-ICU patients, such estimates will become possible. Our work demonstrates that SDUs play an important role in patient outcomes in critical care; however, more work needs to be done to fully grasp the potential benefits these units may provide.

Acknowledgments References Armony, M., C. W. Chan, B. Zhu. 2013. Critical care in hospitals: When to introduce a step down unit? Working Paper, Columbia Business School.

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Armony, M., S. Israelit, A. Mandelbaum, Y. Marmor, Y. Tseytlin, G. Yom-Tov. 2011. Patient flow in hospitals: A data-based queueing-science perspective. Working Paper, New York University. Azoulay, E., F. Pochard, S. Chevret, C. Vinsonneau, M. Garrouste, Y. Cohen, M. Thuong, C. Paugam, C. Apperre, B. De Cagny. 2001. Compliance with triage to intensive care recommendations. Critical care medicine 29 2132 – 2136. Baker, D., D. Einstadter, S. Husak, R. Cebul. 2003. Changes in the use of do-not-resuscitate orders after implementation of the patient self-determination act. Journal of General Internal Medicine 18 343 – 349. Ballard, S., M. Kuhl. 2006. The use of simulation to determine maximum capacity in the surgical suite operating room. Proceedings of the 2006 Winter Simulation Conference . Brown, S., S. Ratcliffe, S. Halpern. 2013. An empirical derivation of the optimal time interval for defining ICU readmissions. Medical Care 51 706 – 714. Byrick, R. J., J. D. Power, J. O. Ycas, K. A. Brown. 1986. Impact of an intermediate care area on ICU utilization after cardiac surgery. Critical care medicine 14 869–872. Cameron, A. C., P. K. Trivedi. 1998. Regression analysis of count data. Cambridge University Press. Chalfin, D. B., S. Trzeciak S, A. Likourezos A, B. M. Baumann, R. P. Dellinger. 2007. Impact of delayed transfer of critically ill patients from the emergency department to the intensive care unit. Critical care medicine 35 1477 – 1483. Chan, C. W., V. F. Farias, N. Bambos, G. Escobar. 2012. Optimizing ICU discharge decisions with patient readmissions. Operations Research 60 1323–1341. Christensen, S., M. Johansen, C. Christiansen, R. Jensen, S. Lemeshow. 2011. Comparison of charlson comorbidity index with saps and apache scores for prediction of mortality following intensive care. Journal of Clinical Epidemiology 3 203–211. Coopersmith, C., H. Wunsch, M. Fink, W. Linde-Zwirble, K. Olsen, M. Sommers, K. Anand, K. Tchorz, D. Angus, C. Deutschman. 2012. A comparison of critical care research funding and the financial burden of critical illness in the United States. Critical Care Medicine 40 1072 – 1079. Doran, K., K. Ragins, A. Iacomacci, A. Cunningham, K. Jubanyik, G. Jenq. 2013. The revolving hospital door: hospital readmissions among patients who are homeless. Medical Care 51 767 – 773. Escher, M., T. Perneger, J. Chevrolet. 2004. National questionnaire survey on what influences doctors’ decisions about admission to intensive care. BMJ 329 425 – 429. Escobar, G. J., J. D. Greene, P. Scheirer, M. N. Gardner, D. Draper, P. Kipnis. 2008. Risk-adjusting hospital inpatient mortality using automated impatient, outpatient, and laboratory databases. Medical Care 46 232–239. Gibbons, J., S. Chakraborti. 2011. Nonparametric Statistical Inference. 5th ed. Boca Raton, FL: Chapman & Hall/CRC.

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Green, L. V. 2006. Queueing Analysis in Healthcare, chap. Patient Flow: Reducing Delay in Healthcare Delivery. Springer, New York, N.Y. Greene, W. H. 2012. Econometric Analysis. 7th ed. Upper Saddle River, NJ: Prentice Hall. Griffiths, J., M. Jones, M. Read, J. Williams. 2010. A simulation model of bed-occupancy in a critical care unit. Journal of Simulation 4 52 – 59. Gross, D., J. Shortle, J. Thompson, C. Harris. 2008. Fundamentals of Queueing Theory. 4th Ed., Wiley, New York. Hanson, C. W., C. S. Deutschman, H. L. Anderson, P. M. Reilly, E. C. Behringer, C. W. Schwab, J. Price. 1999. Effects of an organized critical care service on outcomes and resource utilization: A cohort study. Critical Care Medicine 27 270 – 274. Harding, A. D. 2009. What can an intermediate care unit do for you? Journal of Nursing Administration 39 4 – 7. Jayes, R., J. Zimmerman, D. Wagner, E. Draper, W. Knaus. 1993. Do-not-resuscitate orders in intensive care units. current practices and recent changes. JAMA 270 2213 – 2217. Kc, D. S., C. Terwiesch. 2012. An econometric analysis of patient flows in the cardiac intensive care unit. Manufacturing and Service Operations Management 14 50 – 65. Keenan, S. P., W. J. Sibbald, K. J. Inman, D. Massel. 1998. A systematic review of the cost-effectiveness of noncardiac transitional care units. Chest 113 172 – 177. Kim, S. H., C. W. Chan, M. Olivares, G. Escobar. 2014. ICU Admission Control: An Empirical Study of Capacity Allocation and its Implication on Patient Outcomes. Management Science, to appear . Mbongo, C., P. Monedero, F. Guillen-Grima, M. Yepes, M. Vives, G. Echarri. 2009. Performance of saps3, compared with apache ii and sofa, to predict hospital mortality in a general ICU in southern europe. European Journal of Anaesthesiology 26 940–945. Ouanes, I., C. Schwebel, A. Francais, C. Bruel, F. Philippart, A. Vesin, L. Soufir L, C. Adrie, M. GarrousteOrgeas, J. Timsit, B. Misset. 2012. A model to predict short-term death or readmission after intensive care unit discharge. Journal of Critical Care 27 422 e1 – e9. Shi, P., M. Chou, J. Dai, D. Ding, J. Sim. 2014. Models and insights for hospital inpatient operations: Time-dependent ED boarding time. Management Science, to appear . Shmueli, A., C. Sprung, E. Kaplan. 2003. Optimizing admissions to an intensive care unit. Health Care Management Science 6 131–136. Simchen, E., C. L. Sprung, N. Galai, Y. Zitser-Gurevich, Y. Bar-Lavi, G. Gurman, M. Klein, A. Lev, L. Levi, F. Zveibil, et al. 2004. Survival of critically ill patients hospitalized in and out of intensive care units under paucity of intensive care unit beds. Critical care medicine 32 1654 – 1661. Stacy, K. M. 2011. Progressive care units: Different but the same. Critical Care Nurse 31 77 – 83.

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Appendix A:

Supporting Table Table 7

Variable Gender Age

Control variables for patient characteristics and hospital care

Description Dummy variable: Males were coded 1 and females 0 Continuous variable: Coded as piecewise linear spline variables with knots at its 50th and 80th percentiles (65 and 81) LAPS2 Laboratory-based Acute Physiology Score; measures physiologic derangement at admission and is mapped from 14 laboratory test results, such as arterial pH and white blood cell count, obtained 72 hours preceding hospitalization to an integer value that ranges from 0 to 262 in our data set(higher scores indicate poorer condition); coded as piecewise linear spline variables with knot at its 50th and 80th percentiles (94 and 134) COPS2 Comorbidity Point Score; measures the chronic illness burden and is based on 41 comorbidities, such as diabetes, to which patients are categorized using outpatient and inpatient data from the 12 months preceding hospitalization. It ranges from 0 to 267 in our data set, a higher score indicates a higher comorbid illness burden, it was coded as piecewise linear spline variables with knot at its 50th and 80th percentiles (33 and 87) SAPS3 Simplified Acute Physiology Score; measures the severity of illness and predict vital status at hospital discharge based on ICU admission data. SAPS3 score is associated with each ICU admission and is calculated based on data obtained within on hour of ICU admission. SAPS3 ranges from 14 to 100 in our data set; coded as piecewise linear spline variables with knot at its 50th and 80th percentiles (52 and 61) Admitting diagnosis A way of classifying ICD9 codes. This clinical classification system was developed by HCUP and buckets ICD9’s into about 200 groups. A further grouping of the variable HCUP developed by Gabriel Escobar to condense the HCUP grouping into 38 groups so it could be used in a similar fashion as PRIMCOND3. Seasonality Month/day-of-week/time-of-day; Category variable for each month and dayof-week. For time-of-day, we use category variables for nurse shifts happening three times a day at 7am, 15pm, and 23pm. Previous unit Category variable to track impatient unit a patient is admitted to immediately before ICU admission. LOS before ICU Continuous variable that is the total lengh-of-stay (hrs) prior to the ICU admission. It measures how long a patient has been in hospital before being admitted to the ICU, coded as piecewise linear spline variables with knot at its 50th and 80th percentiles (2 and 31). ICU LOS Continuous variable that is the lengh-of-stay (hrs) at the first ICU. It measures how long a patient has been taking care of at ICU, coded as piecewise linear spline variables with knot at its 50th and 80th percentiles (38 and 83).

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Appendix B:

Detailed Estimation Results for the ICU to SDU Routing Decision

This section briefly summarizes the estimation results for the ICU to SDU routing decision given in Equation (2). Recall that the ICU to SDU decision and the patient outcomes are jointly estimated using full MLE. As such the coefficients for the ICU to SDU decision may vary slightly for different outcomes. That said, we observe that differences are very minor. For the sake of brevity, we present estimation results associated with M ortality. We report estimates for a subset of patient and operational controls in Table 8; the complete set of estimation results are provided in Table EC.7 in the online Appendix.

Table 8

Estimation Results for the ICU to SDU Routing Decision

Covariate SDUBusy SAPS3 [0, 52] SAPS3 [52,61] SAPS3 [61, ] LOSB4ICU [0, 2] (hrs) LOSB4ICU [2,31] LOSB4ICU [31, ] ICULOS [0, 38] ICULOS [38,83] ICULOS [83, ] NightShift[11pm,7am] MorningShift[7am,3pm] AfternoonShift[3pm,11pm] Pseudo R2 Number of obs

Estimation b/se -0.5110*** (0.0503) 0.0016 (0.0030) 0.0025 (0.0060) -0.0182** (0.0055) -0.0119 (0.0340) 0.0114*** (0.0021) 0.0005** (0.0002) -0.0108*** (0.0016) 0.0046*** (0.0010) 0.0004+ (0.0002) base -0.6186*** (0.0506) -0.5018*** (0.0490) 0.1396 11057

Note. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%)

The empirical results for the ICU to SDU routing provide some interesting managerial insights. We summarize these insights in the follows: • SDU congestion: The coefficient for SDU Busyi is negative and highly statistically significant, suggesting that the SDU congestion reduces the likelihood of routing an ICU patient to the SDU. We also predict the percentage of patients routed to SDU under two scenario: (a) the percentage is 35.91% when the SDU is not congested and (b) the percentage is 21.14% if SDU is congested. Thus, a congested SDU results in a 46% relative reduction in the likelihood of transferring an ICU patient to the SDU. • Severity: The impact of severity scores (LAPS2, COPS2, and SAPS3) is very marginal on the ICU discharge decision. The coefficients are very small and most of them are not significant. This is likely because these severity scores are not updated at time of ICU discharge. LAPS2 and COPS2 are assigned at hospital admission and SAPS3 is assigned at ICU admission. The time the patient spends in the ICU (ICU LOS) seems to more accurately measure patient severity and has a significant effect on where a patient is transferred after ICU discharge.

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• LOS before ICU: We find that an increase in hospital length-of-stay (hours) before a patient is admitted to the ICU, denoted by LOSB4ICUi , increases the likelihood of going to the SDU. One possible explanation is that a longer LOSB4ICUi indicates a longer ‘delay’ for the ICU bed. As a result, the patient’s health condition may be worse and is therefore more likely to be routed to the SDU. • Nurse shift: Our results also show that the ICU to SDU routing decision is highly related to the time of ICU discharge. In particular, a patient is less likely to be transferred to the SDU when the patient is discharged in the morning shift [7am, 3pm] or afternoon shift [3pm, 11pm], compared to the night shift [11pm, 7am]. This time-of-day effect is likely due to the fact that more patients are discharged from ICU in the morning and afternoon, i.e., the demand for SDU beds is high in the morning and afternoon; therefore, a high percentage of patients will need to be rerouted. Overall, we find that operational factors, especially congestion in the SDU, have a significant impact on the ICU to SDU routing decision.

Appendix C:

Robustness Checks

We now describe a number of robustness checks for our main results described in Section 3.3. First, we tried different specifications of control variables. Recall that, some of our control variables – severity scores (LAPS2, COPS2, SAPS3), length-of-stay at ICU, and length-of-stay before ICU admission – are modeled as spline variables to account their possible non-linear effects on the ICU to SDU routings and patient outcomes. We repeated the analysis with different specifications, including changing the number of cutoffs and values of these cutoffs. Our results are qualitatively similar to these changes. The second robustness check we did is with respect to the length of time window for readmission. For ICU readmission, we varied the time window of the ICU readmission from time of ICU discharge from 2 to 7 days and also during any time frame during the same hospital stay. For hospital readmission, we consider hospital readmission within 1 week, 2 weeks, 3 weeks, and 30 days after a patient is discharged from the hospital. We found that our main results are similar to those in Table 4 with slight quantitative changes. For example, the effect of the ICU to SDU routing is weaker when the elapsed time between two consecutive hospital stays is longer. We provide the detailed estimation results in Table EC.3 in the online Appendix. For ICU readmission, we also checked alternative specifications of AvgOccV isited. In the paper, we define AvgOccV isited for ICU readmission as the average occupancy for all units a patients is admitted to between two consecutive ICU admissions. We considered three alternatives: 1) the average occupancy for all units a patient is admitted to in the remaining hospital stay before leaving hospital ; 2) the average occupancy for all units a patient is admitted to within 3 days after the ICU discharge and before the next ICU admission; and 3) excluding a control for AvgOccV isited. While these three specifications yield similar results, there are slight differences in the significance level – the first is the strongest and the last is the weakest, see Table EC.1. We also vary the definition of a busy SDU, considering different cutoffs for the number of available beds, ranging from one bed to four. On average, the percentage of patients, who are discharged from ICU when SDU is congested, varies from 34% to 3% when the cutoff is decreased from four beds to one (Table EC.2). While the quantitative effect of a busy SDU varies for these different specifications, the main results do not

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change–being admitted to the SDU is associated with better patient outcomes. See Table EC.3. We also considered alternative measures of SDU congestion based on percentiles of the SDU occupancy level. First, we use a dummy variable that equals one when the occupancy level in the SDU exceeds its 95th percentile. Next, we use is a piece-wise linear spline variable for the SDU occupancy level with a knot at the 95th percentile. The estimation results are similar to our results obtained in the other specifications. We also repeated the analysis with different cutoff values replacing the 95th percentile by the 80th , 85th , and 90th percentile, and the results were qualitatively similar. See Table EC.4 and Table EC.5. Finally, we also ran our econometric model for other admission categories: ED-Surgical, Non-ED Medical, Non-ED Surgical. These three categories contain much fewer observations than ED-Medical; as such, the joint estimation does not converge for most of patient outcome models due to the small sample size. Thus, we could not examine the impact of the ICU to SDU transfer decision on patient outcomes. We were able to estimate the first-stage of these models, which captures the decision to admit (or not admit) patients to the SDU following ICU discharge. We present these results in Table EC.7. The results for all admitting categories are similar to those for ED-Medical. Namely, the operational factors (particularly SDU congestion and nurse-shift) have a significant impact on the ICU to SDU routing decision. Potentially, with a larger patient cohort, one may be able to use a similar estimation strategy used in this work to examine the impact of SDU care for the remaining admission categories.

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Online Appendix: Full Estimation Results and Robustness Analysis Table EC.1

Robustness Test on ICU Readmissions: Alternative Specifications for AvgOccV isited

With IV Estimate (SE) AME ARC ρ(SE)

Outcome

Test ρ = 0

Without IV Estimate (SE)

AvgOccV isited : for all units before next ICU admission (Baseline)

ICU Readm2d ICU Readm

-0.51** (0.20) -0.24 (0.20)

-0.04 -68% -0.04 -36%

0.32* (0.12) 0.25* (0.12)

ICU Readm2d ICU Readm

-0.62*** (0.19) -0.05 -74% 0.38** (0.11) -0.40* (0.18) -0.06 -53% 0.34** (0.11)

ICU Readm2d ICU Readm

-0.50** (0.20) -0.23 (0.20)

-0.04 -67% 0.31* (0.12) -0.04 -35% 0.24+ (0.12)

ICU Readm2d ICU Readm

-0.40* (0.20) -0.15 (0.20)

-0.03 -58% -0.02 -24%

0.02 0.05

0.01 (0.05) 0.17*** (0.04)

AvgOccV isited : for all units before hospital discharge

0.00 0.00

-0.00 (0.05) 0.16*** (0.04)

AvgOccV isited : for all units within 3 days before next ICU admission

0.02 0.06

-0.00 (0.05) 0.16*** (0.04)

AvgOccV isited : not included

0.25* (0.12) 0.19 (0.12)

0.05 0.11

0.00 (0.05) 0.17*** (0.04)

Note. Standard error in parentheses. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%). AME - Average Marginal Effect; ARC - Average Relative Change.

Table EC.2

Percentage of patients who are discharged from ICU when SDU is busy

Hosp SDU Size 1 24 25 2 3 14 4 19 24 5 6 19 20 7 8 27 9 11 10 32 All hosp

% when number of available SDU beds ≤1 ≤2 ≤3 ≤4 0.93 3.57 7.80 12.17 0.66 2.95 7.54 12.46 0.56 7.94 24.29 45.63 3.17 12.68 27.07 41.59 0.28 1.54 3.93 7.87 0.82 3.34 6.76 15.37 0.00 2.84 16.74 36.77 2.81 9.34 18.80 31.74 9.76 37.72 63.94 80.34 0.34 2.66 6.19 12.71 2.52 10.64 21.70 34.00

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Table EC.3

Robustness Test on Patient Outcomes: Binary SDU Busy Based on Number of Available Beds

Outcome

With IV Estimate (SE) AME ARC

ρ(SE)

Test ρ = 0

Without IV Estimate (SE)

IV: SDU Busy = 1 {SDU F reeBeds <= 4}, P (SDU F reeBeds <= 4) = 34.00%

M ortality -0.56* (0.27) -0.06 -70% 0.23 (0.17) log(HospRemLOS) -0.42*** (0.03) -0.42 -37% 0.47*** (0.05) ICU Readm2d -0.70*** (0.20) -0.06 -78% 0.43*** (0.12) ICU Readm3d -0.30 (0.31) -0.03 -47% 0.21 (0.19) -0.41 (0.27) -0.05 -56% 0.30+ (0.16) ICU Readm5d ICU Readm -0.28 (0.25) -0.04 -41% 0.27+ (0.15) HospReadm1w -0.49* (0.25) -0.07 -62% 0.24 (0.15) HospReadm2w -0.26 (0.22) -0.05 -36% 0.10 (0.13) HospReadm3w -0.22 (0.20) -0.05 -29% 0.11 (0.12) -0.18 (0.20) -0.05 -23% 0.08 (0.12) HospReadm30d

0.17 0.00 0.00 0.28 0.08 0.08 0.12 0.46 0.35 0.49

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDU Busy = 1 {SDU F reeBeds <= 3}, P (SDU F reeBeds <= 3) = 21.70%

M ortality -0.47* (0.23) -0.05 -64% 0.17 (0.14) log(HospRemLOS) -0.38*** (0.10) -0.38 -36% 0.46*** (0.05) ICU Readm2d -0.76*** (0.17) -0.07 -81% 0.47*** (0.10) ICU Readm3d -0.52* (0.22) -0.05 -67% 0.35* (0.13) -0.55** (0.20) -0.07 -67% 0.39*** (0.12) ICU Readm5d ICU Readm -0.35+ (0.20) -0.05 -48% 0.32** (0.12) HospReadm1w -0.27 (0.25) -0.04 -41% 0.10 (0.15) HospReadm2w -0.32 (0.22) -0.06 -42% 0.14 (0.13) HospReadm3w -0.18 (0.20) -0.04 -24% 0.09 (0.12) -0.20 (0.21) -0.05 -24% 0.09 (0.12) HospReadm30d

0.22 0.00 0.00 0.02 0.00 0.01 0.49 0.30 0.45 0.45

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDU Busy = 1 {SDU F reeBeds <= 2}, P (SDU F reeBeds <= 2) = 10.64%

M ortality -0.60** (0.22) log(HospRemLOS) -0.35*** (0.10) ICU Readm2d -0.51** (0.20) -0.45* (0.20) ICU Readm3d -0.51** (0.18) ICU Readm5d ICU Readm -0.24 (0.20) HospReadm1w -0.22 (0.23) HospReadm2w -0.43* (0.21) -0.39* (0.19) HospReadm3w HospReadm30d -0.44* (0.20)

-0.06 -0.35 -0.04 -0.05 -0.06 -0.04 -0.03 -0.08 -0.09 -0.11

-72% 0.26+ (0.14) -34% 0.44*** (0.05) -68% 0.32* (0.12) -61% 0.31* (0.12) -64% 0.37*** (0.10) -36% 0.25* (0.12) -34% 0.07 (0.13) -52% 0.21+ (0.12) -45% 0.22+ (0.11) -47% 0.24+ (0.12)

0.07 0.00 0.02 0.02 0.00 0.05 0.60 0.09 0.06 0.05

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDU Busy = 1 {SDU F reeBeds <= 1}, P (SDU F reeBeds <= 1) = 2.52%

M ortality log(HospRemLOS) ICU Readm2d ICU Readm3d ICU Readm5d ICU Readm HospReadm1w HospReadm2w HospReadm3w HospReadm30d

-0.69*** (0.25) -0.36*** (0.12) -0.39 (0.27) -0.18 (0.35) -0.15 (0.29) -0.02 (0.27) -0.34 (0.31) -0.79*** (0.21) -0.70*** (0.20) -0.71*** (0.23)

-0.07 -0.36 -0.03 -0.02 -0.02 -0.002 -0.05 -0.15 -0.16 -0.18

-77% 0.31+ (0.16) -34% 0.44*** (0.06) -58% 0.24 (0.17) -32% 0.14 (0.21) -26% 0.14 (0.18) -3% 0.11 (0.16) -49% 0.15 (0.18) -74% 0.43*** (0.13) -66% 0.41*** (0.12) -64% 0.41** (0.14)

0.06 0.00 0.16 0.53 0.42 0.49 0.43 0.00 0.00 0.01

Note. Standard error in parentheses. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%). AME - Average Marginal Effect; ARC - Average Relative Change.

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

ec4

e-companion to Author: The Role of a Step-Down Unit in Improving Patient Outcomes

Table EC.4

Outcome

Robustness Test on Patient Outcomes: Binary SDU Busy Based on Occupancy

With IV Estimate (SE) AME ARC ρ(SE)

Test ρ = 0

Without IV Estimate (SE)

IV: SDU Busy = 1 {SDU Occ >= 80th percentile}

M ortality log(HospRemLOS) ICU Readm2d ICU Readm3d ICU Readm5d ICU Readm HospReadm1w HospReadm2w HospReadm3w HospReadm30d

-0.55* (0.27) -0.50*** (0.08) -0.63*** (0.20) -0.44 (0.27) -0.54*** (0.22) -0.54*** (0.18) -0.37 (0.23) -0.23 (0.23) -0.14 (0.22) -0.06 (0.23)

-0.06 -0.50 -0.05 -0.04 -0.07 -0.08 -0.05 -0.05 -0.03 -0.02

-70% -43% -75% -61% -66% -64% -51% -32% -19% -8%

0.23 (0.17) 0.51*** (0.04) 0.39*** (0.11) 0.30+ (0.17) 0.38*** (0.13) 0.43*** (0.11) 0.27+ (0.14) 0.17 (0.14) 0.12 (0.13) 0.07 (0.14)

0.19 0.00 0.00 0.09 0.01 0.00 0.06 0.24 0.37 0.63

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDU Busy = 1 {SDU Occ >= 85th percentile}

M ortality -0.72** (0.29) log(HospRemLOS) -0.47*** (0.09) ICU Readm2d -0.55* (0.23) ICU Readm3d -0.19 (0.32) -0.40 (0.26) ICU Readm5d ICU Readm -0.49** (0.19) HospReadm1w -0.55** (0.22) HospReadm2w -0.37+ (0.23) HospReadm3w -0.22 (0.22) -0.08 (0.22) HospReadm30d

-0.08 -0.47 -0.05 -0.02 -0.05 -0.07 -0.08 -0.07 -0.05 -0.02

-78% 0.33+ (0.18) -42% 0.50*** (0.04) -71% 0.34* (0.14) -34% 0.14 (0.20) -55% 0.30+ (0.16) -60% 0.40*** (0.12) -65% 0.38** (0.13) -47% 0.26+ (0.14) -29% 0.17 (0.13) -11% 0.08 (0.13)

0.09 0.00 0.02 0.47 0.08 0.00 0.01 0.00 0.20 0.53

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDU Busy = 1 {SDU Occ >= 90th percentile}

M ortality -0.72** (0.27) log(HospRemLOS) -0.42*** (0.09) ICU Readm2d -0.38 (0.30) -0.05 (0.38) ICU Readm3d -0.19 (0.32) ICU Readm5d ICU Readm -0.12 (0.26) HospReadm1w -0.06 (0.25) HospReadm2w -0.18 (0.22) -0.20 (0.22) HospReadm3w HospReadm30d -0.13 (0.24)

-0.08 -0.43 -0.03 -0.01 -0.02 -0.02 -0.01 -0.04 -0.05 -0.03

-78% 0.33+ (0.17) -39% 0.48*** (0.05) -58% 0.24 (0.18) -11% 0.06 (0.23) -31% 0.16 (0.19) -21% 0.18 (0.16) -11% 0.08 (0.15) -26% 0.14 (0.14) -26% 0.16 (0.13) -17% 0.11 (0.14)

0.07 0.00 0.20 0.80 0.41 0.28 0.58 0.33 0.25 0.45

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDU Busy = 1 {SDU Occ >= 95th percentile}

M ortality log(HospRemLOS) ICU Readm2d ICU Readm3d ICU Readm5d ICU Readm HospReadm1w HospReadm2w HospReadm3w HospReadm30d

-0.81*** (0.27) -0.42*** (0.10) -0.54* (0.24) -0.24 (0.33) -0.36 (0.26) -0.29 (0.23) -0.54* (0.23) -0.71*** (0.21) -0.67*** (0.20) -0.69*** (0.23)

-0.09 -0.42 -0.05 -0.02 -0.04 -0.04 -0.08 -0.14 -0.16 -0.18

-82% 0.39* (0.17) -37% 0.47*** (0.05) -70% 0.33* (0.14) -39% 0.17 (0.20) -52% 0.27 (0.15) -42% 0.27+ (0.14) -65% 0.38** (0.14) -70% 0.47*** (0.13) -65% 0.45*** (0.12) -63% 0.45** (0.14)

0.04 0.00 0.03 0.40 0.10 0.06 0.01 0.00 0.00 0.01

Note. Standard error in parentheses. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%). AME - Average Marginal Effect; ARC - Average Relative Change.

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

ec5

e-companion to Author: The Role of a Step-Down Unit in Improving Patient Outcomes

Table EC.5

Outcome

Robustness Test on Patient Outcomes: Continuous SDU Busy Modeled as a Spline Variable

With IV Estimate (SE) AME ARC

ρ(SE)

Test ρ = 0

Without IV Estimate (SE)

IV: SDUOcc. modeled as a piece-wise linear spline with knot at 80th pct

M ortality -0.79*** (0.25) -0.08 -81% 0.38* (0.16) log(HospRemLOS) -0.44*** (0.09) -0.44 -44% 0.49*** (0.04) ICU Readm2d -0.43+ (0.24) -0.04 -61% 0.26+ (0.15) ICU Readm3d -0.06 (0.31) -0.01 -12% 0.06 (0.19) -0.23 (0.27) -0.03 -37% 0.19 (0.16) ICU Readm5d ICU Readm -0.24 (0.21) -0.04 -36% 0.25+ (0.13) HospReadm1w -0.37+ (0.22) -0.05 -51% 0.27+ (0.14) HospReadm2w -0.38+ (0.21) -0.07 -47% 0.26+ (0.13) HospReadm3w -0.30 (0.20) -0.07 -37% 0.22+ (0.12) -0.24 (0.22) -0.06 -29% 0.17 (0.13) HospReadm30d

0.03 0.00 0.09 0.74 0.25 0.07 0.05 0.06 0.09 0.19

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDUOcc. modeled as a piece-wise linear spline with knot at 85th pct

M ortality -0.79*** (0.25) -0.08 -81% 0.38* (0.16) log(HospRemLOS) -0.42*** (0.09) -0.42 -42% 0.48*** (0.05) ICU Readm2d -0.40 (0.26) -0.03 -59% 0.25 (0.16) ICU Readm3d -0.02 (0.33) -0.002 -3% 0.04 (0.20) -0.18 (0.28) -0.02 -31% 0.16 (0.17) ICU Readm5d ICU Readm -0.15 (0.23) -0.02 -24% 0.19 (0.14) HospReadm1w -0.33 (0.23) -0.05 -47% 0.25+ (0.14) HospReadm2w -0.39+ (0.21) -0.08 -49% 0.27+ (0.13) HospReadm3w -0.34+ (0.21) -0.08 -41% 0.25+ (0.12) -0.29 (0.22) -0.08 -34% 0.21 (0.13) HospReadm30d

0.03 0.00 0.14 0.85 0.35 0.18 0.09 0.05 0.06 0.13

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDUOcc. modeled as a piece-wise linear spline with knot at 90th pct

M ortality -0.77*** (0.25) -0.08 -81% 0.37* (0.16) log(HospRemLOS) -0.40*** (0.10) -0.40 -38% 0.46*** (0.05) ICU Readm2d -0.42 (0.26) -0.04 -61% 0.26 (0.16) -0.05 (0.34) -0.005 -10% 0.06 (0.20) ICU Readm3d -0.20 (0.28) -0.02 -32% 0.17 (0.17) ICU Readm5d ICU Readm -0.07 (0.25) -0.01 -12% 0.14 (0.15) HospReadm1w -0.29 (0.25) -0.04 -43% 0.23 (0.15) HospReadm2w -0.42+ (0.23) -0.08 -51% 0.29+ (0.14) -0.40+ (0.21) -0.09 -46% 0.28* (0.13) HospReadm3w HospReadm30d -0.39+ (0.23) -0.10 -43% 0.27+ (0.14)

0.04 0.00 0.11 0.78 0.33 0.34 0.14 0.05 0.04 0.07

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

IV: SDUOcc. modeled as a piece-wise linear spline with knot at 95th pct

M ortality -0.83*** (0.26) -0.09 -83% 0.40* (0.16) log(HospRemLOS) -0.35*** (0.11) -0.35 -34% 0.43*** (0.06) ICU Readm2d -0.50* (0.24) -0.04 -67% 0.31* (0.15) ICU Readm3d -0.12 (0.34) -0.01 -23% 0.10 (0.20) ICU Readm5d -0.31 (0.26) -0.04 -46% 0.24 (0.15) -0.12 (0.25) -0.02 -20% 0.17 (0.15) ICU Readm HospReadm1w -0.45+ (0.23) -0.06 -58% 0.33* (0.14) HospReadm2w -0.63*** (0.21) -0.12 -65% 0.42*** (0.13) HospReadm3w -0.55** (0.20) -0.13 -57% 0.37** (0.12) HospReadm30d -0.60** (0.22) -0.16 -58% 0.40** (0.14)

0.03 0.00 0.05 0.62 0.14 0.26 0.03 0.00 0.01 0.01

Note. Standard error in parentheses. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%). AME - Average Marginal Effect; ARC - Average Relative Change.

-0.18*** (0.05) 0.38*** (0.02) 0.01 (0.05) 0.04 (0.04) 0.09+ (0.04) 0.17*** (0.04) 0.08+ (0.04) 0.05 (0.04) 0.06+ (0.03) 0.05+ (0.03)

ec6 Table EC.6 Covariate

e-companion to Author: The Role of a Step-Down Unit in Improving Patient Outcomes

Full Estimation Results for the ICU to SDU Routing Decision – Supplementing Table 8 Estimation Covariate Estimation b (se) b (se) SDUBusy -0.51*** (0.05) HEMAT -0.18 (0.18) Male 0.06* (0.03) ILLDEF -0.07 (0.07) LAPS2 [0, 94] -0.00 (0.00) 0.03 (0.15) LIVPAN LAPS2 [94, 134] 0.00+ (0.00) MISC -0.44+ (0.26) LAPS2 [134, ] -0.00 (0.00) MLG -0.33 (0.29) -0.00+ (0.00) COPS2 [0, 33] NEURO 0.01 (0.08) COPS2 [33, 87] 0.00* (0.00) NUTRT -0.07 (0.28) COPS2 [87, ] 0.00 (0.00) OBSDIV -0.31* (0.15) SAPS3 [0, 52] 0.00 (0.00) -0.13 (0.16) OTHGI SAPS3 [52,61] 0.00 (0.01) OTHGU -0.31 (0.19) SAPS3 [61, ] -0.02*** (0.01) OTHINF -0.10 (0.08) 0.01*** (0.00) Age [,65] OTHRSP 0.25*** (0.06) Age [65,81] 0.00 (0.00) PSYCH -0.33** (0.12) Age [81, ] -0.01 (0.01) 0.10 (0.12) SURGCX LOSB4ICU [0, 2] -0.02 (0.03) TRAUMA 0.09 (0.10) LOSB4ICU [2,31] 0.01*** (0.00) 0.00** (0.00) LOSB4ICU [31, ] [Impatient unit before ICU admission] ICULOS [0, 38] -0.01*** (0.00) Out-of-hospital (base case) ICULOS [38,83] 0.00*** (0.00) -0.12 (0.10) ED ICULOS [83, ] 0.00+ (0.00) Ward -0.39** (0.12) -0.21+ (0.13) OR [Hospital] OTH -0.38 (0.27) 8 (base case) PAR -0.18 (0.13) 9 -0.74*** (0.06) SDU 0.03 (0.13) 0.85*** (0.06) 1 6 -0.31*** (0.05) [Nurse Shift] 2 0.28*** (0.08) night [23pm,7am] (base case) 3 -0.21*** (0.05) morning [7am,15pm] -0.62*** (0.05) 7 0.46*** (0.06) -0.50*** (0.05) afternoon [15pm,23pm] 4 -0.37*** (0.06) 0.49*** (0.06) 5 [Day of week] 10 -0.01 (0.05) Sun (base case) Mon -0.07 (0.05) [Admitting Diagnosis: HCUP] Tue -0.03 (0.05) Sepsis (base case) Wed -0.04 (0.05) -0.04 (0.12) Fluid and electrolyte disorders Thu -0.15** (0.05) Coma stupor and brain damage 0.02 (0.09) Fri -0.10* (0.05) AMI 0.42*** (0.09) Sat 0.01 (0.05) Atherosclerosis 0.61*** (0.11) Chest pain 0.44*** (0.07) [Month] Dysrhythmia 0.30** (0.10) Jan (base case) Cardiac arrest 0.43* (0.17) -0.08 (0.08) Feb CHF 0.30** (0.09) Mar -0.04 (0.07) Acute CVD 0.11 (0.08) Apr 0.02 (0.07) CAP 0.22** (0.08) May -0.01 (0.07) Asp pneumonia 0.18 (0.23) Jun -0.01 (0.07) GI bleed -0.13+ (0.07) Jul 0.04 (0.07) UTI 0.14 (0.15) Aug -0.00 (0.07) Hip fracture -0.05 (0.19) Sep -0.03 (0.07) -0.17 (0.12) Residual codes Oct 0.09 (0.07) ALLRF -0.14 (0.13) Nov -0.19* (0.08) CANCER -0.02 (0.27) Dec -0.22** (0.07) CATAS 0.16+ (0.08) Constant 0.10 (0.19) COPD 0.11 (0.10) Observations 11057 ENDOC -0.23** (0.08) 0.1396 Pseudo R2 GIMISC -0.38** (0.14) +p < 10%, ∗p < 5%, ∗ ∗ p < 1%, ∗ ∗ ∗p < 0.1% HART 0.10 (0.08)

ec7

e-companion to Author: The Role of a Step-Down Unit in Improving Patient Outcomes

Table EC.7

Estimation for the ICU to SDU Routing Decision for All Admitting ED,Medical ED,Surgical Non-ED, Surgical b (se) b (se) b (se) SDUBusy -0.5110*** (0.0503) -0.7018*** (0.1831) -0.4234*** (0.1013) LAPS2 [0, 94] -0.0013 (0.0008) 0.0010 (0.0022) -0.0019 (0.0018) LAPS2 [94, 134] 0.0022+ (0.0012) 0.0024 (0.0042) 0.0041 (0.0118) LAPS2 [134, ] -0.0001 (0.0012) -0.0009 (0.0046) -0.0070 (0.0264) COPS2 [0, 33] -0.0033+ (0.0018) -0.0045 (0.0055) -0.0034 (0.0034) 0.0022* (0.0010) 0.0019 (0.0033) -0.0011 (0.0022) COPS2 [33, 87] COPS2 [87, ] 0.0006 (0.0007) 0.0032 (0.0026) -0.0050+ (0.0027) SAPS3 [0, 52] 0.0016 (0.0030) 0.0053 (0.0077) 0.0307*** (0.0041) SAPS3 [52,61] 0.0025 (0.0060) 0.0104 (0.0198) 0.0264 (0.0168) SAPS3 [61, ] -0.0182** (0.0055) -0.0105 (0.0171) -0.0087 (0.0176) LOSB4ICU [0, 2] -0.0119 (0.0340) -0.3239* (0.1424) 0.8090+ (0.4730) 0.0114*** (0.0021) 0.0106* (0.0043) -0.0026 (0.0038) LOSB4ICU [2,31] LOSB4ICU [31, ] 0.0005** (0.0002) 0.0004 (0.0003) 0.0011* (0.0004) ICULOS [0, 38] -0.0108*** (0.0016) -0.0092+ (0.0054) -0.0039 (0.0043) ICULOS [38,83] 0.0046*** (0.0010) 0.0029 (0.0030) 0.0020 (0.0023) ICULOS [83, ] 0.0004+ (0.0002) 0.0006 (0.0004) 0.0003 (0.0004) NightShift[11pm,7am] (base case) MorningShift[7am,15pm] -0.6186*** (0.0506) -0.6301*** (0.1651) -0.9586*** (0.1329) AfternoonShift[15pm,23pm] -0.5018*** (0.0490) -0.4462** (0.1487) -0.7886*** (0.1242) √ √ √ Hospital-fixed effect √ √ √ Admitting Diagnosis √ √ √ Impatient unit before ICU √ √ √ Day of week √ √ √ Month Pseudo R2 0.1396 0.2298 0.4505 Number of obs 11057 1348 3836 Note. +(p < 10%), ∗(p < 5%), ∗ ∗ (p < 1%), ∗ ∗ ∗(p < 0.1%) Covariate

Categories Non-ED, Medical b (se) -0.5364*** (0.1591) -0.0002 (0.0018) -0.0014 (0.0083) -0.0104 (0.0117) -0.0027 (0.0053) 0.0016 (0.0029) -0.0018 (0.0023) -0.0043 (0.0077) 0.0188 (0.0162) -0.0131 (0.0151) -0.1903 (0.2407) 0.0173** (0.0056) 0.0005* (0.0002) -0.0099* (0.0048) 0.0106*** (0.0030) -0.0003 (0.0007) -0.6202*** (0.1521) -0.6323*** (0.1437) √ √ √ √ √ 0.2301 1545