Frontiers in Operations: Valuing Nursing Productivity in Emergency Departments

Abstract

Problem definition.

We quantify the increase in productivity in emergency departments (ED) from increasing nurse staff. We then estimate the associated revenue gains for the hospital and the associated welfare gains for society.

Academic/practical relevance.

The United States is over a decade into the worst nursing shortage crisis in history, fueled by chronic under-investment. To demonstrate to hospital managers and policymakers the benefits of investing in nursing, we clarify the positive downstream effects of doing so in the ED setting.

Methodology.

We use a high-resolution data set of patient visits to the ED of a major U.S. academic hospital. Time-dependent hazard estimation methods (nonparametric and parametric) are used to study how the realtime service speed of a patient varies with the state of the ED, including the time-varying workloads of the assigned nurse. A counterfactual simulation is used to estimate the gains from increasing nurse staff in the ED.

Results.

We find that lightening a nurse’s workload by one patient is associated with a 14% service speedup for every patient under the nurse’s care. Simulation studies suggest that adding one more nurse to the busiest 12-hour shift of each day can shorten stays and avert $160,000 in lost patient wages per 10,000 visits. The reduction in service times also frees up capacity for treating more patients and generate $470,000 in additional net revenues for the hospital per 10,000 visits. Extensive sensitivity analyses suggest that our key message—investing in nursing will more than pay for itself—is likely to hold across a wide range of EDs.

Managerial implications.

In determining whether to invest in more nursing resources, hospital managers need to look beyond whether payer reimbursements alone are sufficient to cover the upfront costs, to also account for the resulting downstream benefits.

1. Introduction

Nurses comprise the largest share of the healthcare workforce and shoulder most of the responsibility in provisioning direct patient care (Seervai, 2022). Nurses collect and interpret data upon which diagnoses and treatment plans are based, they administer and monitor treatments and recovery, as well as complete charting and clinical reporting. Overall, patients spend more than 80% of their time with nurses compared to around 10% with physicians, who generally spread their time amongst many patients at any given point in time (Butler et al., 2018). At the same time, the United States is over a decade into the worst nursing shortage crisis in history (Juraschek et al., 2012). In 2022, this shortage stands at 1.1 million nurses (American Hospital Association, 2022) and is expected to continue to grow through the 2030s (Mulisa et al., 2022). Berlin et al. (2022b) reports that nurses plan to quit at higher rates compared to the past decade, fueled in part by increased job dissatisfaction brought about by growing workloads (Aiken et al., 2002).

From the perspective of patient outcomes, there is substantial evidence in the medical literature that higher nurse workload (hereby abbreviated as nurseload) leads to more missed nursing care (Tubbs-Cooley et al., 2019), more preventable events such as post-operative infections, higher readmission rates, higher failure-to-rescue rates (Mark et al., 2004, 2013), and ultimately higher mortality rates (Mark et al., 2004; Aiken et al., 2014; Needleman et al., 2020). Each additional patient per nurse was associated with a 7% increase in the likelihood of patients dying within 30 days of admission, and a 7% increase in the odds of failure-to-rescue (Aiken et al., 2002).

While the negative impact of high nurseload on patient quality of care and on nursing job satisfaction is well documented, much less studied is the negative economic impact that high nurseload has on society and on the operational performance of healthcare systems. Take the emergency department (ED) as an example. Viewing patient flow through the ED as a queuing system, the natural measure of productivity is the service speed of a patient, which at any given point in time depends on the current workload of the assigned nurse and physician, among other things. While physician workload has previously been shown to impact service speed (Kc and Terwiesch, 2009; Batt and Terwiesch, 2012; KC, 2014; Kuntz et al., 2015; Berry Jaeker and Tucker, 2017), the nurseload effect is much less studied, despite the fact that nursing accounts for the lion’s share of the patient care process. Considering that nurses fulfill diverse roles as the main providers of direct patient care, the magnitude of the nurseload effects could be distinct from the known effects of physician workload. To determine whether hiring an additional nurse is economically viable, it is critical to understand the scale of nurseload’s influence on patient service speed.

This paper contributes to the discourse on the ongoing nursing shortage crisis by examining the following question: If nurseload is increased, how much would service slow down by, and what are the corresponding costs from the increased patient stay times? This cost has two components, a societal one in terms lost patient wages (Savva et al., 2019), and revenue loss for the ED itself due to reduced throughput (Becker’s Hospital Review, 2016). Viewing the same question from the other direction, what are the cost savings if additional nurses can be hired to reduce nurseload? The answer carries important implications for combating the nurse shortage.

First, Medicare currently reimburses nursing services as part of the fixed room rate or per-discharge payment (American Nurses Association, 2022). That is, for a given number of patients, facilities are paid the same amount regardless of the number of nurses employed to treat the patients. From the perspective of the facility’s manager, nursing is thus seen as a cost centre with no associated revenue. This exposes the nursing staff to downsizing and cost-cutting pressures, which contribute to burnout and churn (Gordon, 2012). Quantifying the economic contributions of nursing helps reframe it as a cost saving resource, and motivates the business case for both society and the facility to invest in nursing. This is a point that is echoed in Shalala et al. (2011) and Seervai (2022).

Second, in a recent poll of U.S. nurses (Berlin et al., 2022a), the two most commonly cited remedies for improving nurse retention are higher pay and lower patient-to-nurse ratios (lower nurseload). Indeed, these were the key points of the negotiations behind the recent nurses’ strike in New York City (Otterman et al., 2023). If cost savings can be achieved by hiring additional nurses to reduce nurseload, part of those savings can then be used to increase pay for nurses as well, thereby targeting both needs simultaneously.

We address the question above by empirically assessing the nurseload effect on service speed using data from a major academic ED in the U.S. from April 2017 to March 2019. The dataset tracked patients, nurses, and physicians in the ED in realtime, allowing us to reconstruct changes in physician and nurse workloads over the course of a patient stay. Adopting a survival analysis framework for time-dependent covariates, we estimate the patient-level service rate μ(t, X_t) as a function of time-in-service t and time-dependent state variables X_t for the patient. The latter includes the workloads of the nurse and physician assigned to the patient, and other control variables. We find that:

• The model-free estimate of $\mu \left(t,X_t\right)$, obtained using the survival machine learning approach BoXHED (Lee et al., 2021; Wang et al., 2020; Pakbin et al., 2024), resembles the hazard of a log-normal distribution. This corroborates the literature that find log-normality to be a reasonable model for service durations (Brown et al., 2005; Armony et al., 2015).

• Increased nurseload is unambiguously associated with a decrease in service speed. Based on the log-normal hazard specification for $\mu \left(t,X_t\right)$, adding an extra patient to a nurse’s load is associated with a 17% increase in service duration for every patient under the nurse’s care. Equivalently, reducing nurseload by one patient is associated with a 14% reduction in service duration. While physician workload also exhibits the same negative association with service speed, the effect is an order of magnitude smaller (1.4%).

To perform a counterfactual analysis on the gains from adding more nurses to the ED, we develop a queuing simulation of the ED using the estimated service rate. From a social welfare perspective, we find that adding just one more nurse to the busiest shift of each day would shorten aggregate patient stays to the tune of averting $160,000 in lost opportunity wages for every 10,000 visits. Note that this number excludes the additional benefits from improved patient outcomes via reduced nurseloads. Taking the perspective of the hospital, the reduction in patient service times frees up capacity for treating more patients, estimated to be worth $470,000 in additional net revenues for every 10,000 visits. This is net of the additional staffing cost, so there is plenty left for retention pay raises. Sensitivity analyses on the key inputs to the counterfactual analysis indicate that our key message—investing in nursing will more than pay for itself—is likely to hold for a wide range of EDs beyond the study ED.

These estimates highlight the value proposition for hospital managers to invest in nursing, and suggest that significant Pareto improvement is possible for patient outcomes, provider earnings, and nursing job satisfaction and retention. This type of win-win-win scenario has also been demonstrated in other healthcare settings such as the dialysis industry (Lee and Zenios, 2012).

The remainder of this paper is organized as follows. In the next section, we review relevant literature. Section 3 describes our data and research setting. In section 4, we then introduce the econometric model specification and present the results. In section 5, we use the estimated service rate function from section 4 to perform a counterfactual analysis. Finally, section 6 discusses the implications of our results, limitations of our study, and potential avenues for future studies.

2. Literature review

Our work is related to three streams of literature—work that examines the effect of nurses, in particular the effect of nurseload, work that examines workers’ productivity and service speed, and work on staffing level-related care delivery in the emergency department.

2.1. The effect of nurseload in healthcare

Given nurses’ critical roles in the healthcare system, the medical literature has extensively studied the effect of nurseload under different contexts, with a focus on its effect on service quality. Substantial evidence shows that higher nurse workload negatively impacts patient care, leading to more missed nursing care (Tubbs-Cooley et al., 2019), more preventable events (e.g., post operative infections), higher readmission rates, higher rescue failure rates (Mark et al., 2004, 2013), and ultimately higher mortality rates (Aiken et al., 2002; Mark et al., 2004; Aiken et al., 2014; Needleman et al., 2020).

In addition, the literature has also shown that high workload has a direct and significant impact on nurses themselves. Lenehan (1999) finds that an overtime increase of 15% in clinical nursing hours (6–7 hours per week) results in higher incidence of nurse burnout. Similarly, Aiken et al. (2002) find that each additional patient per nurse was associated with a 23% increase in the odds of burnout and a 15% increase in the odds of job dissatisfaction. Interestingly, nurseload does not significantly change physician workload in the short-term, suggesting that nurses are used as supplements to general practitioners rather than substitutes (Laurant et al., 2004).

While much of the existing work in the medical literature has examined the effect of nurse workload with regard to patients and nurses themselves, in comparison, few studies have examined the effect of nurseload on service speed. We complement the literature by providing empirical evidence on the nurseload effect on service speed and the associated economic value.

2.2. Workload and productivity

Our work also builds on and contributes to the literature on productivity and service speed—two central topics in operations management. Recent empirical work has provided evidence that workload has a significant impact on service speeds, although there is less agreement over how. Indeed, a wide spectrum of relationships have been reported ranging from slow-down as workload increases (Lucas et al., 2009; Armony et al., 2015; Batt et al., 2019), to workload having no effect (Lucas et al., 2009; McCarthy et al., 2009), to speed-up (Kc and Terwiesch, 2009; Anderson et al., 2011), to slow-down and then speed-up (Batt and Terwiesch, 2012; Kuntz et al., 2015), and to still more complicated patterns (Berry Jaeker and Tucker, 2017).

Kc and Terwiesch (2009) study the impact of workload on service rates in the cardiac surgery and hospital transport. They find that an increase in system workload (bed occupancy and number of busy transporters, respectively) decreases the length of stay for patients and transport time. However, this speed-up is not sustainable. A consistent level of overwork reverses the relationship, leading to longer lengths of stay and transport times. Anderson et al. (2011) study a large medical center and find that the number of discharged patients increases when utilization is high and if more surgeries are scheduled for that day. However, they find that really high utilization also increases lengths of stay for patients. Kc and Terwiesch (2012) find that a higher ICU occupancy reduces ICU length of stay. Berry Jaeker and Tucker (2017) identify an N-shaped relationship between bed occupancy and patient length of stay. They find that service rates first decelerate, then accelerate, until they reach the second tipping point where they decelerate again. Batt and Terwiesch (2017) find an inverted U-shape relationship between the ED waiting room census and the patient treatment time.

Empirical research on the impact of workload on service rates builds on several possible mechanisms. Service speeds can decrease due to fatigue, mental strain, more frequent interruptions and physical capacity limitations as workload increases. At the same time, servers can speed up their work as workload increases or rely on multitasking to process a higher workload. The underlying mechanisms are all active on the individual level, since it is the servers who increase or decrease their respective service speed. However, the literature has mainly examined the system level workload (ward census) instead.

Research that has focused on the impact of individual workload has often found a deterioration of work quality. Powell et al. (2012) find that an increase in individual physician workload reduces the quality of their note-keeping and, in turn, reduces the reimbursement rates of the hospital. Xu et al. (2021) find that workload has a U-shaped impact on the rate of operational errors. Tan and Netessine (2014) find an inverse-U relationship between servers and their selling efforts. Servers in a restaurant setting first increase their sales efforts (at the expense of table turnover) while occupancy increases but then reduce them at higher levels.

The different findings on the impact of system workload on service rates suggest the need for additional research on the topic. Furthermore, the findings on the impact of individual workload on service quality suggest a separate and simultaneous effect of individual workload compared to the system workload. In this paper, we reconcile the literature by simultaneously considering the impact of different workloads on service rates. Our paper also answers the call for more time-dependent measures of workload as service speeds may vary dynamically with the state of the system (Armony et al., 2015).

2.3. Emergency department staffing

An emerging body of literature has examined how staffing levels impact workplace dynamics and the overall performance of the ED. For example, multitasking has been identified as a common response to the increasing demand of time and attention of the physician (Chisholm et al., 2000). KC (2014) finds an optimal level of multitasking – between 4 and 5 on average that maximizes both physician productivity (e.g. throughput) and quality of care as measured by ED revisit rate. Physician workload in the ED has also been found to impact downstream healthcare resource use, including inpatient admission (Gorski et al., 2017) and the use of post-discharge care resources. For example, Soltani et al. (2022) find that post-ED discharge care intensity, as measured by the number of episodes of care, increased if the ED physician that was assigned to the patient happened to have been busy.

The organization of work has also been found to affect ED performance. For example, batching is a tactic employed when dispositioning groups of ED patients for inpatient admission. Feizi et al. (2022) find that batching introduces additional variability into the patient flow process, increasing the time taken to secure an inpatient bed, even as it increased patient throughput. Batt and Terwiesch (2017) find that given delays in turnaround times for various tasks in the patient care process, early initiation of tasks such as test ordering by the physician had the effect of improving patient flow.

The discontinuity associated with the end of shift has also been found to impact ED performance. Chan (2018) finds that physicians are less likely to pick up patients towards the end of their shift, and that such end-of-shift patients incurred more tests. Relatedly, Batt et al. (2019) finds a hand-off effect arising from the potential lapse in continuity of care between physician shifts, leading to increased ED revisit incidence. The makeup of the care team (Kim et al., 2022), process ownership-related queuing effects (Song et al., 2015), as well as team familiarity also influences ED performance. For example, Niewoehner III et al. (2022) report a familiarity effect, finding that physicians who worked together frequently in the past were more likely to pick up new patients from the waiting area, resulting in reduced patient wait time.

The extant work has overwhelmingly focused on the physician’s role in the ED care process. Although nurses are integral members of the patient care process, responsible for critical care delivery and process flow tasks, the role of nurses has been largely ignored in the literature. One possible reason is that the nurse identifier is often missing from administrative data sets that have been used in much of the literature.

3. Data

We acquire a unique high-frequency dataset from the ED of a large academic hospital in the United States. The data captures patient visits to the ED between April 2017 and March 2019. Each patient is tracked in realtime over the course of their stay. For example, the data includes the time when the patient arrives at the ED, when they move from the waiting room to the treatment ward (i.e., the start of the service), the nurse and physician they are assigned to (or subsequently reassigned), and when they depart from the ED (i.e., the end of the service). The granularity of our data allows us to construct realtime workload measures for each nurse and each physician, as well as for the aggregate ward census.

Upon arrival to the ED, a patient’s information and chief complaint (which categorizes their symptoms at arrival) are recorded, and they are triaged into one of the five Emergency Severity Index (ESI) levels. ESI is a measure of patient illness severity, with level 1 being the most severe and level 5 the least. We exclude ESI level 1 visits from this study because they are treated in a separate dedicated area. For all other visits, the patient is assigned to a nurse and separately to a physician, each via rotational patient assignment (Traub et al., 2016). This is a common practice in EDs that assigns patients to staff in a round-robin fashion, resulting in randomized patient assignment. If the assigned nurse is currently at capacity, then the patient remains in the waiting room until the nurse regains capacity (a stylized description of the process is presented in Figure 3 in Section 5). While there may be trading of patients to prioritize treatment for those with more severe conditions, this is rare in practice (Hodgson and Traub, 2020).

Figure 3:

An illustration of patient transition in the ED. The service rate experienced by each patient is a function of their time in service, time-varying nurseload and other time-varying covariates.

After the patient commences treatment, Figure 1 illustrates how the time-varying workload measures are constructed. In essence, the nurseload for nurse j at time t is the number of patients assigned to j at that moment. The workload for physician k is defined analogously. The nurseload and physician workload for patient i at time t are simply the workloads of the nurse and physician assigned to i at that point in time. Note that our definition of nurseload is more granular than the aggregate patient-to-nurse ratio frequently used in the medical literature, which is the total number of patients in the facility divided by the total number of nurses. Table 1 reports summary statistics for nurseload, physician workload, and ward census.

Figure 1:

An illustration of how physician and nurse workload measures evolve over the course of Patient1’s stay. This example focuses on two of the physicians and one of the nurses on duty.

Table 1:

Workload summary statistics

	Apr 2017-Mar 2018		Apr 2018-Mar 2019		Apr 2017-Mar 2019
	mean	s.d.	mean	s.d.	mean	s.d.
Nurseload	3.05	1.83	2.56	1.84	2.77	1.85
Physician workload	13.10	7.04	15.68	7.26	14.56	7.28
Ward census	37.98	7.53	42.71	8.90	40.66	8.66

Summary statistics for workload variables. Summary by each year, and for both years.

We construct a set of control variables using the extensive information on patient-level demographics available in our dataset. In particular, we account for individual patient characteristics, including gender, race, age, and clinical information such as the chief complaint and ESI level. Beyond patient-specific attributes, we also incorporate controls for the temporal aspects of patient arrivals, including arrival day of the week (e.g., Friday), arrival month (e.g., December), and arrival hour of the day. Summary statistics for ED visits are presented in Table 2.

Table 2:

Patient summary statistics

	Apr 2017-Mar 2018		Apr 2018-Mar 2019		Apr 2017-Mar 2019
	N	%	N	%	N	%
Gender
F	23,647	0.61	25,466	0.60	49,113	0.60
M	15,383	0.39	16,787	0.40	32,170	0.40
Race
African American or Black	17,247	0.44	18,332	0.43	35,579	0.44
Caucasian or White	15,139	0.39	16,370	0.39	31,509	0.39
Other	6,644	0.17	7,551	0.18	14,195	0.17
ESI
2	12,040	0.31	11,081	0.26	23,121	0.28
3	20,815	0.53	24,637	0.58	45,452	0.56
4	5,951	0.15	6,346	0.15	12,297	0.15
5	224	0.01	189	0.00	413	0.01
Chief complaint
Cardiorespiratory	7,652	0.20	8,827	0.21	16,479	0.20
Gastrointenstinal	6,057	0.16	6,954	0.16	13,011	0.16
General medicine	9,774	0.25	9,157	0.22	18,931	0.23
Neurology	2,480	0.06	2,480	0.06	4,960	0.06
Non-traumatic	2,960	0.08	3,456	0.08	6,416	0.08
Trauma	3,324	0.09	3,678	0.09	7,002	0.09
Other	6,783	0.17	7,701	0.18	14,484	0.18
	mean	sd	mean	sd	mean	sd
Age	50.09	20.12	50.29	20.03	50.19	20.07
Service duration (hours)	5.76	5.38	5.81	5.11	5.79	5.24
Total visits	39,030	-	42,253	-	81,283	-

Summary statistics of patient characteristics on the visit level. Summary by each year, and for both years.

Chief complaint categories are grouped as shown in Table 2. For patient race, the categories are White (Caucasian), Black (African-American), and Other. Informed by the tree splits from the nonparametric machine learning model that we use in the model-free exploration in Section 4, we group some of the ordinal control variables as follows. For ESI levels, levels 4 and 5 are grouped together, likely because they both represent relatively low acuities. Recall that level 1 visits are omitted as discussed earlier. Patient arrival hours are discretized into dummy variables for {midnight to 11am}, {11am to 2pm}, {2pm to 6pm}, and {6pm to midnight}.

Lastly, motivated by Batt et al. (2019), we also introduce shift control variables. These track how far along each staff is in their shift and how long each staff has attended to a particular patient. For example, if nurse j’s shift started at noon, at 2pm they would be 2 hours into their shift. If nurse j has attended a patient for 4 hours, then the variable for time with patient will show 4 hours. If nurse j finishes their shift and hands over the patient to another nurse, this variable resets for that particular patient. We monitor these durations for both the assigned physician and nurse, yielding four control variables in total.

4. Empirical analysis

The most natural way to understand the relationship between nurseload and productivity in the ED is through the lens of time-dependent survival analysis. Specifically, the instantaneous service rate experienced by a patient who is t hours into treatment and endowed with time-dependent state variables X_t is

$$\mu \left(t,X_t\right).$$

The state variables X_t include the time-varying nurseload, physician workload, and ward census for the patient, as well as the control variables discussed in Section 3. We emphasize that the service rate is at the patient level rather than at the nurse or physician level.

The service rate function $\mu \left(t,X_t\right)$ is also known as the hazard function in survival analysis. While the traditional event of interest in survival analysis is time to death, here we are interested in time to service completion. The relationship between nurseload and ED productivity is then captured by the dependence of the service rate $\mu \left(t,X_t\right)$ on nurseload, the estimate of which depends on the empirical specification chosen for μ(·, ·). In our analysis we first estimate μ(·, ·) nonparametrically. We then use the resulting fit to inform an appropriate parametric model to use for estimating the nurseload effect.

4.1. Model-free exploration

We use a survival machine learning method called BoXHED (Wang et al., 2020; Pakbin et al., 2024) to nonparametrically fit $\mu \left(t,X_t\right)$ to the first half of the data from April 2017 to Mar 2018. BoXHED is a gradient-boosted tree estimator that inherits the consistency properties of the more general boosted hazard estimator in Lee et al. (2021).

The first three panels of Figure 2 display bivariate profiles of the estimated service rate μ(t, X_t) as functions of time-in-service t as well as: i) nurseload; ii) physician workload; and iii) ward census. All other variables are fixed at their median values. We highlight three findings. First, by fixing the value of workload, the service rate as a function of t resembles the hazard of a log-normal distribution, which typically exhibits a unimodal shape. As an example, panel iv) of Figure 2 shows a slice of panel i) along with an approximating log-normal hazard function. Our model-free estimates corroborate the literature that finds log-normality to be a reasonable model for service duration (Brown et al., 2005; Armony et al., 2015).

Figure 2:

Service rates as functions of time-in-service and workload variables.

Second, by holding time-in-service t fixed in panels i) and ii) of Figure 2, we see that both nurseload and physician workload exhibit an unambiguous negative association with service speed. However, nurseload has a far greater impact on slowing down service than physician workload does, and it also has a far greater impact than the ward census effect. To rank the impact of each variable on service speed, Table 3 reports the BoXHED variable importance scores. Each score quantifies a variable’s contribution to increasing the nonparametric likelihood of the BoXHED estimator (Pakbin et al., 2024), with larger values indicating higher importance. We scale the scores to be between 0 and 1. Nurseload is the second most important variable (0.96), with physician workload (0.23) and ward census (0.03) far behind.

Table 3:

BoXHED variable importance scores

Variable	Relative importance
Time-in-service t	1
Nurseload	0.96
ESI	0.95
Physician workload	0.23
Age	0.12
Hour of arrival	0.07
Ward census	0.03
Month of arrival	0.02
Day of week	0.01
Gender	0.00

Larger values indicate higher importance. Workload variables are highlighted in bold. Shift controls omitted from table.

Third, the nonparametric estimates also provide guidance on how to specify a parametric log-normal model for the service rate. Let us hold time-in-service t fixed again in panel i) of Figure 2. We see that the service rate decreases at a diminishing rate with nurseload. Holding t fixed in panel ii) shows a linearly decreasing physician workload effect, and holding t fixed in panel iii) shows a linearly increasing ward census effect. This suggests a log-linear workload effect for t fixed:

$$\exp \left(-{\beta }_{\mathit{nurse}}\cdot W{L}_{\mathit{nurse}}-{\beta }_{\mathit{phys}}\cdot W{L}_{\mathit{phys}}-{\beta }_{\mathit{ward}}\cdot W{L}_{\mathit{ward}}+{\gamma }^{\prime }\mathit{Controls}\right)$$ (1)

where WL denotes the different types of workload. A positive coefficient ${\beta }_{nurse}$ would qualitatively yield the observed nonlinear nurseload effect. A positive but much smaller coefficient ${\beta }_{phys}$ would recover the observed linear physician workload effect, because $e^{-x}\approx 1-x$ for $\left|x\right|\ll 1$. Likewise, a negative and small ${\beta }_{ward}$ would recover the observed linear ward census effect. We will explain next how (1) justifies a linear workload specification for the log-normal hazard model.

4.2. Parametric analysis

Given the alignment of the previous section with the literature on the log-normality of service durations (Brown et al., 2005; Armony et al., 2015), we use a log-normal hazard specification for time-dependent covariates to estimate μ(t, X_t) from the second half of the data (April 2018 to March 2019). Our results independently confirm that the workload effects are statistically significant and in the same directions as the model-free results.

The basic log-normal regression model for time-static covariates follows log T = βx + ϵ where ϵ ~ N (0, σ²). Hence a unit increase in x increases the service duration T by factor of $e^{\beta }$, thus a positive coefficient implies service slowdown. Equivalently, a unit increase in x changes the service speed by the reciprocal factor of e^−β. That is, the service rate function that results from a log-normal linear specification behaves log-linearly like (1). Thus the model-free evidence suggests that it is reasonable to specify the log-normal model with linear workload terms.

We note from the literature review that a number of existing studies have identified non-monotonic relationships between length of stay and workload. For example, if an inverted U-shaped relationship exists, a quadratic workload term would be necessary to capture that. By contrast, in our setting the nonparametric workload effects are all monotonic. While adding local polynomial spline terms with monotonicity constraints can make the log-normal model more accurate (albeit in an ad-hoc way), this comes at the cost of interpretability. If accuracy is the goal, then the theoretically justified nonparametric service rate estimates should be used instead.

Table 4 presents the regression estimates for the log-normal hazard specification for the service rate. Column 1 reports results for a simplified specification that only includes the workload variables, while column 2 also includes the control variables. The results are in line with the exploratory analysis performed on the first half of the data (April 2017 to Mar 2018): Increasing a nurse’s load by one patient is associated with a e^0.155 = 17% increase in service duration for every patient under the nurse’s care. Equivalently, lightening the nurse’s load by one patient leads to a 1 − e^−0.155 = 14% speedup for every patient under their care. All told, increasing nurseload is associated with significant slowdowns in service speed. Whereas increasing the workload of a physician by one patient is associated with a much smaller e^0.014 = 1.4% increase in service duration for patients under the physician’s care. The directions and magnitudes of the workload coefficients are in agreement with our predictions based on (1).

Table 4:

Regression estimates from the log-normal hazard specification

Variables	(1)	(2)
Nurseload	0.181*** (0.003)	0.155*** (0.003)
Physician workload	0.023*** (0.001)	0.014*** (0.001)
Ward census	−0.024*** (0.001)	−0.010*** (0.001)
ESI 3		−0.27*** (0.001)
ESI 4-5		−1.01*** (0.016)
Age		0.006*** (0.000)
Female		0.039*** (0.008)
Race control		X
Chief complaint control		X
Temporal controls		X
Shift controls		X
Log σ	−0.167*** (0.005)	−0.446*** (0.006)
Constant	5.87** (0.032)	6.21*** (0.035)

^*p < 0.1;

^**p < 0.05;

^***p < 0.01.

Robust standard errors are reported in parentheses.

4.3. Robustness checks

Correlation among workload variables.

There are three different workload measures in our specification. Since these measures might be correlated, one might wonder whether our observed effects are driven by multicollinearity. To rule this out, we repeat our log-normal hazard estimation with different subsets of workload measures.

Table 5 presents the results. Column 1 are our main results with all workload variables included. In column 2, we drop the nurseload variable from the specification. In columns 3 and 4, we drop the physician workload and ward census from the specification, respectively. In column 5, we drop both physician workload and ward census. The results remain consistent throughout all specifications. Whenever a specification includes nurseload, its estimated impact remains remarkably stable between 0.15 to 0.16, whether we include or exclude physician workload and/or census.

Table 5:

Regression estimates from the log-normal hazard specification

Variables	(1) All	(2) No Nurse	(3) No Phys	(4) No Ward	(5) Nurse Only
Nurseload	0.155*** (0.003)		0.158*** (0.003)	0.151*** (0.003)	0.155*** (0.003)
Physician workload	0.014*** (0.001)	0.017*** 0.001		0.013*** (0.001)
Ward census	−0.010*** (0.001)	−0.001* (0.001)	−0.005*** (0.001)
ESI 3	−0.27*** (0.010)	−0.222*** (0.010)	−0.267*** (0.010)	−0.265*** (0.010)	−0.266*** (0.010)
ESI 4-5	−1.01*** (0.016)	−0.896*** (0.016)	−1.006*** (0.00163)	−1.006*** (0.00164)	−1.009*** (0.00164)
Age	0.006*** (0.000)	0.005*** (0.000)	0.006*** (0.000)	0.006*** (0.000)	0.006*** (0.000)
Female	0.039*** (0.008)	0.039*** (0.008)	0.038*** (0.008)	0.038*** (0.008)	0.038*** (0.008)
Race control	X	X	X	X	X
Chief complaint control	X	X	X	X	X
Temporal controls	X	X	X	X	X
Shift controls	X	X	X	X	X
Log σ	−0.446*** (0.006)	−0.423*** (0.007)	−0.448*** (0.007)	−0.444*** (0.007)	−0.443*** (0.007)
Constant	6.21*** (0.035)	6.83*** (0.072)	6.84*** (0.073)	6.46*** (0.072)	6.72*** (0.071)

^*p < 0.1;

^**p < 0.05;

^***p < 0.01.

Robust standard errors are reported in parentheses.

Let us draw attention to columns 1 and 2. When all three workload measures are included in the specification (column 1), we find that the nurseload effect is an order of magnitude larger than that for physician workload and ward census. If this result is largely driven by multicollinearity, then dropping nurseload from the model should result in much larger estimated effects for physician workload and ward census. Column 2 shows that this is not the case at all, with the physician workload and ward census effects remaining at the same levels.

One might wonder why the results are so stable, given that it seems intuitive for the workloads to be highly correlated. This is because of the timescale we use in this study: Since we track changes second-by-second, the different types of workloads are not very correlated when sampled at such high frequency. As shown in Table 6, no two workloads are more than 25% correlated with one another.

Table 6:

Correlations among workload measures

	Nurse	Physician	Census
Nurse	1
Physician	0.061	1
Census	0.18	0.25	1

Heterogeneity by disposition status.

To investigate the potential impact of patient discharge/admit status on workload effect, we perform a heterogeneous effects analysis. The impact of nurseload can be more pronounced for patients that are eventually admitted to inpatient care compared to patients that are discharged from the ED. This is because admitted patients might suffer from increased levels of severity or comorbidity, and as a result require more nurse-intensive care.

We construct a binary variable Discharge with a value of 1 if the patient is discharged, and 0 if the patient is admitted. We also construct interaction terms of Discharge with nurseload, physician workload, and ward census. We find statistically significant heterogeneous workload effects based on patient discharge status. For example, increasing nurseload by one patient increases the LOS for a discharged patient by e^{0.197–0.054} − 1 = 15%. By contrast, the increase in LOS for an admitted patient is higher at e^0.197 − 1 = 21%, with both being similar in direction and magnitude to the aggregate nurseload effect of e^0.155 − 1 = 17% from the main model in Table 4. The full results of the heterogeneity analysis are presented in the Online Supplement.

Potential impact of unobserved heterogeneity.

We now turn to the potential issue of workload variables being confounded by unobserved patient illness severity. Empirical studies have consistently found that patient arrivals to the ED are well modelled by non-homogeneous Poisson processes (Brown et al., 2005; Kim and Whitt, 2014; Chen et al., 2023). This is in line with our expectation that the occurrences of medical emergencies are random events. Kc and Terwiesch (2017) and Soltani et al. (2022) thus argue that the pool of patients for staff assignment is random and independent of workload. Furthermore, as mentioned in Section 3, the study hospital uses rotational assignment to assign patients to staff, which restricts selection. As shown in Soltani et al. (2022), the impact of physician workload on post-ED care use is virtually the same with or without instrumental variable adjustments for workload. Consistent with their work, we also find a very weak correlation between ESI severity and workload: Soltani et al. (2022) reports a correlation of −0.13 for physician workload, whereas the correlations from our data range from 0.02 for physician workload to 0.11 for nurseload. If the correlations were in fact high, it might still be possible for ESI and workload to be conditionally independent given the day of week and hour of day. Appendix B in Ding et al. (2019) provides a clever way for testing for conditional independence.

To further investigate the impact of potential unobserved heterogeneity, we employ frailty models for the service rate. In survival analysis, this is a random effects model for the hazard function (service rate). Batt et al. (2019) also employ frailty models to control for unobserved heterogeneity when studying the effect of physician shift handoffs on patient service rates in a discrete time setting. The description of the model and the results are presented in the Online Supplement. We find that the results are largely unchanged.

Dependence of service rate on workload history.

In the previous subsections we model the instantaneous service rate as a function of current workloads. In principle, the service rate can depend on the histories of the workloads as well. To investigate this, we extend our model to include a number of measures that describe history of nurseload, physician workload, and ward census up to time t: The running average $t^{-1}{\int }_0^{1}W{L}_{s}ds$, the running maximum max_s≤t WL_s, and the running minimum min_s≤t WL_s.

The results are presented in the Online Supplement. The chief takeaway from this exercise is that including workload history in the service rate model does not change our main findings on the economic benefits of increasing nurse staffing (see the counterfactual analysis in Section 5).

5. Counterfactual analysis

The results from Section 4 point to nurseload as a leading factor in determining the productivity of the ED. We use the estimated service rate function from Section 4.2 to develop a digital twin of the ED. This is a virtual reconstruction of the ED that allows us to run counterfactual simulations to examine how productivity improves with more nursing staff. The digital twin simulated 42,253 patient visits, which corresponds to the actual number of visits to the study ED during the one-year study period from April 2018 to March 2019.

We run three different scenarios. The first is the baseline that employs the existing staffing levels at our study ED during the study period. We also run two counterfactual scenarios: (1) Adding one more nurse to the first shift between 7am and 7pm every day, which is the busiest shift of the day; and (2) adding another nurse to the second shift from 7pm to 7am as well, so that there is always one more nurse on duty relative to the baseline. These counterfactuals represent perturbations to the existing staffing levels given that the study ED employs about 17 nurses per shift.

The patient arrival times and patient characteristics (age, gender, ESI level, etc.) are the same as in the data from the study period. In each scenario, each patient is assigned to a physician and a nurse upon arrival according to the rotational assignment process described in Section 3. If the assigned nurse is already at the maximum nurseload (caring for six patients), the patient is placed in the waiting room until the nurse regains capacity, even if another nurse is available at the time. This is illustrated in the left panel of Figure 3, where the blue patient i is assigned to nurse A, who is already caring for patients in the pod of six beds assigned to A. For physicians, we do not impose a cap on their capacity. The resulting maximum physician load observed in the simulation is 17, which is consistent with the maximum of 15 observed in the data. The service duration for each patient is simulated using the estimated service rate function. This is done dynamically as the workloads of the assigned staff and the state of the virtual ED evolve over time, altering the instantaneous service rate experienced by the patient with each change. For example in the middle panel of Figure 3, patient i has just been admitted to the treatment unit from the waiting room. The calendar time at this point is t₁, while the elapsed service time for patient i is 0. Using the latter to index time for the patient, the service rate they experience is μ(0, 6, X_i(0)) where the second argument is the nurseload, and X_i(0) represents the values of all other state variables at that time. At calendar time t₂ (right panel of Figure 3), nurse A’s load drops to five, and patient i’s service rate is μ(t₂ – t₁, 5, X_i(t₂ – t₁)).

The final outputs from each scenario are the reduction in patient wait times in the waiting room and the reduction in the service times in the treatment unit, relative to the baseline. We quantify the net economic gain from the additional nurse staffing along two dimensions. First, we look at societal welfare, where each hour of reduction in patient length of stay (wait time plus service time) translates to an extra hour of opportunity wages for patients. Second, we examine the potential for increasing hospital revenue due to the increased ED throughput. Both gains are weighed against the additional staffing costs.

5.1. Social welfare gains from reduced length of stays

Table 7 reports the reduction in length of stay when nurse staffing is increased in the virtual ED. This is further broken down into its two components, the time spent in the waiting room and the service duration in the treatment unit. We observe that adding one more nurse to the busier shift from 7am to 7pm shortened the length of stay by 45 minutes on average per visit, of which 23 minutes were due to the reduction in service time. These numbers are averages across all visits, not just those during the 7am-7pm shift. Adding one more nurse to the second shift as well reduces the average length of stay by another 61 – 45 = 16 minutes, although the average service time is only reduced by 26 – 23 = 3 minutes.

Table 7:

Average LOS reduction from incrementing nurse staff

Additional nurse?		Service time reduction (minutes)	Length of stay reduction (minutes)
7am - 7pm	7pm - 7am
✓		23	45
✓	✓	26	61

Length of stay is the sum of the time spent in waiting room and the service duration in the treatment ward.

The social cost implications are reported in Table 8. The second column of Table 8 reports the aggregate reduction in length of stay across all visits to the virtual ED, and we also normalize this figure to be in terms of every 10,000 visits. The reduced length of stay can be viewed as opportunity wages for patients (Savva et al., 2019), so we convert this time saving into a dollar amount using the May 2021 U.S. wage rate from the U.S. Bureau of Labor Statistics. We also report in terms of the wage rate in Mississippi (the state with the lowest median wage) and in terms of the one in Massachusetts (the state with the highest median wage).

Table 8:

Patient lost wages averted from incrementing nurse staff

Additional nurse?		LOS reduction (hours)	@Mississippi wages ($17/hr)	@U.S. wages ($22/hr)	@Massachusetts wages ($28/hr)
7am - 7pm	7pm - 7am
Savings per year at study ED
✓		32,000	$540K	$700K	$900K
✓	✓	43,000	$730K	$940K	$1.2MM
Savings for every 10,000 visits
✓		7,400	$130K	$160K	$210K
✓	✓	10,000	$170K	$220K	$280K

Numbers reported to two significant figures. We multiply time savings by median wage levels in May 2021, sources from the U.S. Labor Bureau. We compare savings in the state with the lowest median wage (Mississippi) to the median U.S. wage, and to the state with highest median wage (Massachusetts).

At the median U.S. wage rate, our counterfactual analysis suggests that the additional nurse can save society $700,000 in patient opportunity wages per year for the study ED, or $160,000 for every 10,000 visits (dollar amounts reported to 2 significant figures). Since the cost of the extra staffing is borne by the hospital (discussed below), this entire amount is the net surplus to society.

5.2. Financial impact on ED from reduced service times

We now explore the financial implications for the ED if it hires more nurses to reduce patient service times in the unit. Quantifying this economically is difficult because, unlike for in-patient care, reimbursements for ED services are not adjusted for patient length of stay. Furthermore the relationship between staffing levels and hospital revenue is relatively unexplored in the literature. However, prior industry analysis (Becker’s Hospital Review, 2016) shows that, for an ED that sees 30,000 visits a year, a 15-minute reduction in average service time translates into $1.4 million in additional revenues for the ED due to increased throughput. This is consistent with results from other published work, which suggests an estimated benefit between $1.8 and $2.5 million in 2021 dollars, adjusted for patient volume at our study ED (Pines et al., 2011). In 2021 dollars, that translates into $530,000 for every 10,000 visits.

Our counterfactual analysis suggests that service time can be reduced by 23 minutes on average per visit if one more nurse is staffed during the 7am-7pm shift. Since this achieves the 15-minute reduction target, our study ED can potentially increase its revenues by $2.3MM per year based on its size. This is significantly more than the $310,000 it costs per year for the extra staffing. The cost is calculated based on the $83,000 average nursing salary in the U.S. (U.S. Bureau of Labor Statistics, 2022), plus 50% fringe rate. To staff an additional nurse from 7am to 7pm every day requires 2.5 FTEs, where 1 nurse FTE is based on a 36-hour work week with 4 weeks of vacations. The net hospital revenue gain is thus $2MM for our study ED per year, or $470,000 per 10,000 visits.

Since a 15-minute service time reduction per visit translates into $2.3MM in additional revenue for the study ED, a 2-minute reduction per visit is sufficient to cover the cost of adding an extra nurse to a shift. This simple break-even analysis shows that staffing an additional nurse to the 7pm-7am shift is also economically viable (recall from Section 5.1 that this reduces the service time by another 3 minutes per visit).

5.3. Generalizability to other EDs in the U.S.

While this study is based on one hospital, hospital managers can apply the estimation strategy and counterfactual simulation developed here to estimate the impact of the extra staffing for their own ED. The tailored estimates can be used by the manager to decide if they should invest in more nurse staff.

From our perspective, without having access to detailed data from other hospitals, what can we infer about the generalizability of our results to other EDs? For this, we perform extensive sensitivity analyses on the key inputs to the counterfactual simulation. Due to space constraint, the full set of results are presented in the Online Supplement. We consider sensitivity along the following dimensions: Patient visit volume (±50% from study ED), nurseload effect coefficient (±50% from current estimate), patient-to-staff and physician-to-nurse ratios (±50% from study ED), and additional bottlenecks. In each analysis, our key message remains valid over the wide range we vary the input over: Investing in nursing will more than pay for itself. This provides some comfort to the idea that a wide range of EDs would experience a net revenue increase from investing in more nursing resources.

The robustness of our message is perhaps unsurprising, given the break-even analysis from Section 5.2. For the study ED, adding an extra nurse to the busiest shift reduces the average service time by 23 minutes per visit, which is 11 times more than what is required for the hospital to break even.

6. Discussion

Nurses play a central role in hospital care delivery and they provide direct, hands-on care to patients. Currently, the U.S. is experiencing the worst nursing shortage in history. High patient-to-nurse ratios lead to nurse burnout and contribute to rising churn. To alleviate this, hospital managers and policy makers need to address two major determinants of nursing retention – high nurseload and noncompetitive pay. Although there is substantial evidence that lower nurseload is associated with better patient outcomes, it is not straightforward to impute a financial value to these benefits. Without an economic estimate of the value that nurses bring, it is challenging to weigh the benefits against the costs of increasing nurse pay and headcount.

In this paper, we estimate the economic value of nursing productivity. We find that reducing a nurse’s load by one patient is associated with a 14% service speedup for every patient under the nurse’s care. From a social welfare perspective, we estimate that adding one more nurse to the busiest 12-hour shift of the day can save society $160,000 in lost patient time per 10,000 visits. For hospital units that are similar in size to the study ED, this scales to $700,000 per year. From the perspective of the hospital, the staffing addition can potentially generate $470,000 in net additional revenue for every 10,000 visits, or $2MM per year for the study ED. This is net of the costs associated with the extra staff, and a portion of this surplus can be used to increase nursing pay. Lower nurseload (more nursing staff for the same number of patients) and higher pay are the two most commonly cited remedies for retaining nurses in the profession, so this suggests that both needs can be addressed simultaneously.

These findings suggest that healthcare providers should reassess how they evaluate nursing. Currently, hospitals view nursing services, which are reimbursed at fixed rates, as cost centres that do not generate revenue. However, our results show that a substantial net benefit can be derived from investing in nursing capacity. This makes a clear case for providers to view nursing as a valuable resource rather than as a cost. All told, a significant Pareto improvement is possible for patient outcomes, provider earnings, and nursing job satisfaction and retention.

One might argue that it is infeasible to add nurses during a nurse shortage. However, it is important to bear in mind that the nurse supply problem is only a problem at current wage levels. The number of nurses leaving the profession suggests that there is a large pool of potential employees with the necessary skills in the labour force. To quote from a recent New York Times article on the nurses’ strike in New York City (Otterman et al., 2023):

“In New York State, there are a large amount of people that hold registered nursing licenses but aren’t working at the bedside as nurses… So there’s people out there, and they fled the bedside for various reasons.”

Indeed, hospitals already use travel nurses, who are compensated at much higher rates than salaried nurse employees, to temporarily fill the current shortage. This is evidence that higher wages can solve the supply problem, and even more so if coupled with hospitable working conditions involving lower nurseload.

In addition to implications for providers, our findings also carry significant weight for policymakers. While our analysis highlights the economic benefits for providers to invest in nurse staffing, it might still be a challenge to convince some hospital administrations to recognize the benefits that may materialize in the future. A more straightforward approach to strengthening nursing investments is to introduce policy incentives on the payer side. Currently, the Centers for Medicare & Medicaid Services (CMS) reimburses nursing services as part of a fixed room rate or per-discharge fee. This means hospitals receive a set amount, regardless of the nurse staffing level, effectively casting nursing as a cost centre without corresponding revenues. Modifying the payment structure to include nursing services as a separate line charge item and linking hospital revenue to the quality of nursing (e.g., higher nurse-to-patient ratio) would immediately clarify the economic advantages of a higher nurse staffing level for hospitals. By doing so, policymakers can help promote greater efficiency and higher revenues for hospitals, provide increased capacity and better outcomes for patients, and capture additional social welfare gains associated with reduced wait times.

There are several limitations to our analysis. First, our counterfactual analysis considered only a simple staffing heuristic where we add one nurse to the 7am-7pm shift of each day. Performing stochastic optimization on the simulation model could identify further efficiency gains from selecting specific shifts of the month or quarter to add variable numbers of nurse staff, but this is expected to be challenging and computationally taxing. Second, our data does not include information on nurse tenure, which can impact productivity. If such data become available, it would be interesting and useful to impute an economic value to nurse experience. Third, the calculation of social cost savings could be refined using patient-specific financial information. Although this information is not available in our dataset, we are aware that some hospitals collect data on patient incomes (grouped in annual salary ranges).

Finally, future studies can also leverage detailed task-level data to measure nurseload on an even more granular level. For example, a patient who requires many visits to radiology will have different demands on a nurse than someone with a puncture wound. It will be interesting to develop a case-mix adjusted nurseload measure for this. Another application is to examine how the level of workload influence the discretionary selection of tasks by physicians and nurses. These can be exciting avenues of research, which would allow hospitals to mitigate the effects of workload by optimizing and redesigning tasks, in addition to adjusting patient-to-nurse ratios.