Appendix. A. Simulation study on the effect of aggregate data

Transcription

1 36 Article submitted to Operations Research; manuscript no. Appendix A. Simulation study on the effect of aggregate data One of the main limitations of our data are that they are aggregate at the annual level, which introduces certain difficulties in our analysis and in interpreting our results. For example, we may be underestimating the importance of temporal undercapacity or overcapacity. In order to understand the impact of data aggregation, we conducted a simulation study. In this study we chose a sample of 30 random hospitals from our data set. For each of these hospitals, we had the arrival rate, the number of beds, and the diversion probability, which allowed us to find, the service rate that resulted in the reported diversion probability (using a simple line search). Based on this data, we simulated 12 months of data for this random cohort of 30 EDs. We then estimated an OLS model on the cross-sectional data that represented the monthly average of each hospital. Then, we also estimated a fixed effects model on the panel data (12 observations of 30 hospitals) that accounted for hospital and time fixed effects. The results of this analysis are shown in Table 9. Note that it is not possible to estimate the effect of ED size (REL ED SIZE) in the panel data because the number of ED beds does not vary over time in our simulation and hence is subsumed within the hospital fixed-effect. We find that the interaction effect is significant in panel estimation even though it is not so in the cross-sectional estimation. The coefficient of normalized ICU capacity is higher in absolute magnitude in the cross-sectional estimation but note that it is only at the average value of the relative ED size. In conclusion, based on this simulation study, we believe that our empirical findings will be stronger if a panel dataset is used instead of the cross-sectional dataset. B. Simulation In this section, we simulate the diffusion model described in section 4 to assess the accuracy of the simplifications and approximations made during our analysis in section 5.1. The performance measure of primary interest is the fraction of time on diversion but we also assess the accuracy of the delay probability estimation. We compute the fraction of time on diversion using the two approximations and a discrete-event simulation tool where we vary the size of the hospital and fix the other parameters. We choose our parameter values to roughly match with the median values in our data set. In the validated setting, we assume that K = 7, and N 1 = 15, the arrival rates are λ a = 6, λ w = 12, and λ n = 5, the service times are m 1 = 0.75, and m 2 = 2. We also assume that the likelihood that patients arriving to the emergency department continue to an inpatient department are p a = 0.6 and p w = Table 10 shows the estimates of fraction of time on diversion using both diffusion approximation and simulation, and the difference between along with the estimated traffic intensity level of the inpatient department. First, observe that the accuracy of the approximation is extremely high as long as the traffic intensity of the hospital is high enough. Second, as we fix the arrival rates and the size of the ED, the fraction of time on

2 Article submitted to Operations Research; manuscript no. 37 Table 9 Results of estimating fixed-effect and OLS models on panel and cross-sectional simulated data respectively Panel estimation Cross-sectional estimation INTERCEPT (0.09) (0.12) REL ED SIZE CTR 0.12 (0.04) NORM ICU CAP CTR (0.04) (0.09) REL ED SIZE CTR*NORM ICU CAP CTR 0.11 (0.03) (0.02) CS (0.1) CS (0.11) CS (0.22) CS (0.11) CS (0.11) CS6 0.1 (0.11) CS (0.11) CS (0.12) CS (0.13) CS (0.11) CS (0.11) CS (0.11) CS (0.12) CS (0.13) CS15 0 (0.16) CS (0.14) CS (0.1) CS (0.11) CS (0.14) CS (0.12) CS (0.18) CS (0.1) CS (0.28) CS (0.16) CS (0.16) CS (0.12) CS (0.23) CS (0.14) CS (0.12) TS (0.07) TS (0.07) TS (0.07) TS (0.07) TS (0.07) TS6-0.5 (0.07) TS7-0.4 (0.07) TS (0.07) TS (0.07) TS (0.07) TS (0.07) Notes: Standard Errors are shown in parentheses. Log Likelihood for the LIML estimation method is corresponding to the Probit model for the selection equation. p < 0.01; p < 0.05; p < 0.10 diversion decreases with increasing number of inpatient beds, as estimated by the simulation and computed by the approximation.

3 38 Article submitted to Operations Research; manuscript no. Table 10 Estimation of fraction of time on diversion: approximation vs simulation N ρ 2 Approximation Simulation Difference % % % % % % % Table 11 Estimation delay probability: approximation vs simulation N 1 ρ 1 Approximation Simulation Difference % % % % % % % Next, we validate the accuracy of the delay probability estimates via the diffusion approximation relative to a simulation-based estimation. We fix the parameter as described above. In addition, we fix the number of inpatient beds to N = 18. Table 11 reports, for different sizes of the ED, the delay probability as computed by the diffusion approximation and a discrete-event simulation, as well as the difference between the two estimates. The table also reports the traffic intensity of the emergency department. Again, observe that the accuracy of the delay probability estimate increases as the traffic intensity of the emergency department increases. These results suggest that the series of approximations introduced in Section 4 and the diffusion approximation provide a very good description of the actual dynamics of the original network. We also conducted a numerical study to validate the accuracy of the approximation when varying the threshold level at which the hospital begin diverting ambulances. We fix the parameters described above as well as the ICU size at 18 beds and the ED size at 15 beds. We vary the threshold and compute the delay probability and the diversion probability. Table 12 reports these probabilities as well as the accuracy of the diversion probability approximation. Consistent with our explanation above, as the hospital increases the threshold, the diversion probability decreases, yet the probability of being delayed increases. Lastly, we conducted a numerical study that aimed at testing the sensitivity of the approximation to violation of the condition λ a (1 δ)p a >> λ w p w. We fixed the parameters discussed above as well as the size of the hospital to 30 beds and computed the diversion probability for different values of p w. As we

4 Article submitted to Operations Research; manuscript no. 39 Table 12 Estimation diversion probability, approximation vs simulation, as a function of the boarding-beds-threshold K Delay probability Diversion Approximation Diversion Simulation Difference % % % % % % % Table 13 Estimation diversion probability, approximation vs simulation, as a function of the probability of admitting a walk-in patient p w ρ 2 Approximation Simulation Difference % % % % % % increase the probability above 0.2 the condition is violated. Table 13 reports the comparison between the approximation and the simulation and shows that even when the condition is violated the approximation is still fairly accurate. The main conclusion from this table is that while the assumption is required for the purpose of analytical tractability, even if the condition is not satisfied, the accuracy of the model is quite high. C. Background of heavy traffic theory To illustrate the approach, consider an M/M/s system with arrival rate λ and service rate µ for each server. To derive the heavy traffic limit, we consider a sequence of M/M/n systems, each with service rate µ and indexed by n = 1, 2,.., that satisfies the following conditions: (A1) For the n th system, the number of servers is n and the arrival rate is λ n, where the superscript n denotes the n th system (A2) λ s = λ, i.e., the s th system in the sequence is the original system, and (A3) n(1 ρ n ) β for some constant β as n, where ρ n = λ n /(nµ) is the traffic intensity of the n th system. Condition (A3) is critical and needs more explanation. It states that for large systems, the excess capacity (1 ρ) should be approximately inversely proportional to the square-root of the number of servers. It is not

5 40 Article submitted to Operations Research; manuscript no. only consistent with the common wisdom that the appropriate level of server utilization in service systems should increase with the size of the system but also further specifies that this increases at a rate proportional to the square root of the system size. This condition is also the theoretical underpinning for the famous square-root staffing rule that is used in call center management. Halfin and Whitt (1981) provide theoretical justification for (A3) by showing that the probability of delay is approximately equal for each of the systems in the sequence if and only if this condition is satisfied. If the fraction of excess capacity goes down at a rate faster than 1/ n, then an arriving customer has to wait almost surely. On the other hand, if the fraction of excess capacity goes down at a rate slower than 1/ n, an arriving customer almost never waits. Halfin and Whitt (1981) further show that under condition (A3), in steady state, the normalized queue length Q n ( ) = ( Q n ( ) n ) / n converges to a diffusion process Q( ) in distribution. Therefore, we can approximate Q n ( ) by n Q( ) + n, a property that will be used extensively to characterize the performance measures for the system of our interest. Similar approach can be adopted for an M/M/s/K system (Whitt 2004) by considering the limit of a sequence of systems, each with service rate µ and indexed by n such that: (B1) The n th system is an M/M/n/K n system with arrival rate λ n (B2) λ s = λ and K s = K, i.e., the s th system in the sequence is the original system, and (B3) n(1 ρ n ) β and K n / n κ for some constants β and κ as n. D. Estimation on Delay Probability P d In this section, we derive the function P d (N 1, N 2, β 2, κ) that we use to approximate the delay probability P d. Let ρ 1 denote the traffic intensity of station 1 (the ED). It is clear that ρ 1 = (λ w + (1 δ)λ a )m 1 /(N 1 B). (5) [ Using Proposition ] 1 of Halfin and Whitt (1981), we can approximate the delay probability in the ED by β 1Φ(β 1 ) φ(β 1, where ) β 1 = N 1 B(1 ρ 1 ). (6) Therefore, to obtain an estimate of P d, we need estimates of B, the average number of boarding patients, and ρ 1, the traffic intensity of the ED. We first approximate B using the steady state distribution of Q 2 given in Whitt (2004), B(N 2, β 2, κ) = N 2 1 e κβ 2(1 + κ) β 2 (1 e κβ 2 + β φ(β 2 ) Φ(β 2 ) ) (7)

6 Article submitted to Operations Research; manuscript no. 41 Next, from (5), we can approximate ρ 1 as ρ 1 (N 1, N 2, β 2, κ) = λ a(1 δ(n 2, β 2, κ)) + λ w N 1 B(N m 1, (8) 2, β 2, κ) where δ and B are given in (1) and (7) respectively. [ We can then substitute ] (8) and (7) in (6) to obtain 1 β 1 (N 1, N 2, β 2, κ). And P d (N 1, N 2, β 2, κ) = 1 + β 1 (N 1,N 2,β 2,κ)Φ( β 1 (N 1,N 2,β 2,κ)) φ( β. 1 (N 1,N 2,β 2,κ)) E. Proofs Proof of Proposition 1 To prove result (i), let g 1 (β 2 ) = ( N 2 β 2 )e κβ 2/2 and g 2 (β 2 ) = eκβ 2 /2 e κβ 2 /2. We know that β 2 +e κβ 2 /2 Φ(β 2 ) φ(β 2 ) δ(n 2, β 2, κ) = [g 1 (β 2 )g 2 (β 2 )] 1. Hence, it is sufficient to show that g 1 is nondecreasing and g 2 is increasing in β 2. To show g 1 is nondecreasing, we look at the first order derivative: g 1(β 2 ) =e κβ 2/2 [κ( N 2 β 2 )/2 1] (9) =e κβ 2/2 (Kρ 2 /2 1) 0. (10) For g 2, it is easy to see that e κβ 2/2 Φ(β 2 )/φ(β 2 ) is increasing in β 2. Therefore, it remains to show that g 3 (β 2 ) = (e κβ 2/2 e κβ 2/2 )/β 2 is nondecreasing in β 2. Again, we take the first order derivative: g 3(β 2 ) = [(κβ 2 /2)(e κβ 2/2 + e κβ 2/2 ) e κβ 2/2 + e κβ 2/2 ]/β 2 2 = f(κβ 2 /2)/β 2 2, where f(x) = x(e x + e x ) e x + e x. Obviously f(0) = 0. Moreover f(x) is nondecreasing in x because f (x) = x(e x e x ) 0. Therefore, f(x) 0 for x 0, proving g 3(β 2 ) 0. To prove result (ii), we can rewrite (1) as δ(n 2, β 2, κ) = The result then follows because the function Proof of Proposition 2 We can rewrite equation (7) as B(N 2, β 2, κ) = N 2 ( 1 β 2 κe κβ 2 β 2 ( ( ). N 2 β 2 ) e κβ e κβ 2β2 Φ(β 2 ) φ(β 2 ) ( ) e κβ e κβ Φ(β 2β 2 ) 2 φ(β 2 is increasing in κ. ) (1 e κβ 2 ) ) ( ) β 2 Φ(β 2 ) / 1 +. φ(β 2 )(1 e κβ 2 ) It is easy to see that B is increasing in κ because κe κβ 2 = 1 e κβ 2 κ/(eκβ 2 1) is decreasing in κ. Since δ is decreasing in κ, β 1 is decreasing in κ. Therefore, P d is increasing in κ. Proof of Proposition 3 It is equivalent to show that κ is increasing in N 1. It is easy to see that P d is decreasing in N 1 because, clearly, β 1 is increasing in N 1. Now for any N 1,1 < N 1,2, let κ 1 = κ (N 1,1, N 2, β 2, P d ) and κ 2 = κ (N 1,2, N 2, β 2, P d ). It suffices to show κ 1 < κ 2. Suppose κ 1 κ 2. Because N 1,1 < N 1,2, we have P d (N 1,1, N 2, β 2, P d ) > P d (N 1,2, N 2, β 2, P d ), which contradicts with the fact that P d (N 1,1, N 2, β 2, P d ) = P d (N 1,2, N 2, β 2, P d ) = P d.

7 42 Article submitted to Operations Research; manuscript no. F. Details of the fluid approximation F.1. Diversion Probability Let Q 2 (t) denote the number of patients that have completed service in the ED but are yet to complete service in the inpatient department including the boarding patients (Figure 4(c)). K 0 (< N 2 ) denotes the number of patients in the inpatient department at time 0 and let τ represent the first time that diversion starts, i.e., τ = min{t : Q 2 (t) = N 2 + K}. From t = 0 to t = τ, i.e., when not on diversion, the nominal arrival rate to the inpatient department can be approximated by λ h ap a + λ h wp w since ambulance arrivals receive priority over walk-in arrivals and we assume that λ h ap a >> λ h wp w and N 1 /m 1 > λ h ap a + λ h wp w > N 2 /m 2. Thus, from t = 0 to t = τ, the number of patients in the inpatient department including the boarding patients increase at a rate of r 1 = λ h ap a + λ h wp w N 2 /m 2. Thus, N 2 + K = K 0 + r 1 τ. (11) Next, consider the diversion period, i.e., from t = τ to t = τ. Since the fluid model does not capture individual patient level dynamics, we model diversion as a fraction of ambulances that are not accepted by the ED, using a parameter 0 < δ < 1. Thus, the nominal arrival rate to the inpatient department during diversion is λ h ap a (1 δ ) + λ h wp w. Since Q 2 (t) is equal to N 2 + K at all times during the diversion, the flows in and out of the ED must be balanced. Hence, λ h ap a (1 δ ) + λ h wp w = N 2 /m 2. (12) At time τ, when the peak period ends and arrival rates jump from high to low, the number of patients Q 2 (t) depletes at a rate of r 2 = N 2 /m 2 λ l ap a λ l wp w and drops below N 2 + K thereby ending diversion. Since time T is also time 0 of next decision cycle, Q 2 (T ) = Q 2 (0) = K 0. Note that balancing the flow between times t = τ and t = T, we obtain N 2 + K K 0 = r 2 (T τ). (13) Now, we can approximate the diversion probability as δ = δ (τ τ )/T since the ED diverts a fraction δ of the ambulances during the time between τ and τ. Thus, combining (12), (11) and (13) we have δ = λh ap a + λ h wp w N 2 /m ( 2 λ h ap a 1 (N 2 + K K 0 )(r 1 + r 2 ) r 1 r 2 T While the above is a closed-form expression for the diversion probability δ, it contains several parameters that are not observable in the data. Hence, we use the fact that the utilization of the inpatient department is given by ρ 2 = 1 ( ) (N 2 K 0 ) 2 2r + (N 2 K 0 )2 1 2r 2 T N 2. In addition, we restrict ourselves to analyzing the diversion probability δ for a sequence of hospitals where the arrival rate scales linearly with the size of the inpatient department, i.e., λ j i = β j i N 2, i {a, w}, j {h, l}. ).

8 Article submitted to Operations Research; manuscript no. 43 Making these two substitutions, we obtain: δ = βh a p a + βwp h w 1/m ( 2 2(1 ρ2 ) 1 βa h p a T K ) N 2 T It is clear from the above expression that δ is decreasing in θ 2 = 1 ρ 2 and decreasing in κ = K N 2 where we use κ to distinguish it from κ used in the previous section. Note that if the arrival rate increases less than linearly with the size of the hospital, the diversion probability will tend to zero for very large hospitals. On the other hand, if the arrival rate increases more than linearly with the size of the hospital, the diversion probability will converge to one for very large hospitals. Since we do not observe both of these scenarios in our data, we believe that choosing a linear scaling provides the best explanatory power to the model based on fluid approximation. F.2. Delay Probability Figure 4(d) depicts the build-up and draw-down of the number of patients that have not completed their service in the ED, Q 1 (t). We assume that the day begins with an empty ED, i.e., Q 1 (0) = 0. Before the diversion begins, i.e., from t = 0 to t = τ, the number of patients in the ED builds up since λ h a + λ h w > N 1 /m 1. During diversion, i.e., from t = τ to t = τ, rate of build up (or draw down) changes to λ h w (N 1 K)/m 1 due to simultaneous changes in arrival rate (due to ambulance diversion) and service rate (due to blocking). After diversion, i.e., from t = τ to t = T, the number of patients draws down at the rate of N 1 /m 1 λ a l λ w l until all patients are discharged. Note that in this model, the ED operates under two extreme regimes: arrivals to the ED either surely wait for a bed when Q 1 (t) < N 1 or surely do not wait for the bed when Q 1 (t) > N 1. We define the fraction of time spent in the latter regime as the delay probability for this model. In order to show that an optimal threshold K is increasing in N 1, we make two observations. First, for fixed N 1, an increase in K leads to an increase in the slope of Q 1 (t) during periods of diversion, i.e., from t = τ t = τ thereby increasing Q 1 (t) t > τ and hence increasing the probability of delay. Second, for fixed K, an increase in N 1 leads to a reduction in slope of Q 1 (t) from t = 0 to t = τ and an increase in the slope for t > τ thereby reducing Q 1 (t) t and hence decreasing the probability of delay. Combining the above two arguments, we can see that for a fixed delay probability, the threshold K is increasing in N 1. G. Estimates for alternative specifications This section contains the estimation results for alternate specifications B and C for all three families of models - diffusion, fluid, and basic. These are shown in Tables H. FIML estimates for all models and specifications This section contains the estimation results of various specifications using the Full Information Maximum Likelihood (FIML) method. These are shown in Tables 19 to 21. Here, the likelihood function corresponding to the bivariate normal distribution is constructed and maximized using standard methods to obtain ML estimates. (14)

9 44 Article submitted to Operations Research; manuscript no. Table 14 Estimates for alternate specifications based on diffusion approximation. Standard Errors are shown in parentheses. p < 0.01; p < 0.05; p < Specification B Specification C LEVEL EQUATION INTERCEPT 6.34 (0.83) 6.69 (0.3) REL ED SIZE HT CTR (0.09) (0.06) NORM ICU CAP CTR (0.11) (0.08) (REL ED SIZE HT CTR)*(NORM ICU CAP CTR) (0.04) (0.04) RURAL (1.55) INVESTOR 0.17 (0.76) GOVERNMENT (0.53) INVERSE MILLS RATIO (1.63) (0.54) R F-value CHOICE EQUATION INTERCEPT 0.46 (0.11) 0.47 (0.11) REL ED SIZE HT CTR 0.13 (0.04) NORM ICU CAP CTR (0.06) (REL ED SIZE HT CTR)*(NORM ICU CAP CTR) 0.02 (0.03) RURAL (0.27) (0.26) INVESTOR 0.68 (0.22) 0.5 (0.21) GOVERNMENT (0.3) (0.29) TRAUMA 0.38 (0.23) 0.36 (0.22) Log Likelihood Table 15 Simple slopes for Relative ED size for different values of normalized ICU spare capacity. σ N is the standard deviation of the normalized ICU spare capacity in the sample. Standard Errors are shown in parentheses. p < 0.01; p < 0.05; p < 0.10 Specification B Specification C NORM ICU CAP CTR = -2*σ N 0.13 (0.13) 0.15 (0.12) NORM ICU CAP CTR = (0.11) (0.06) NORM ICU CAP CTR = 2*σ N (0.19) (0.14) I. Results of nonlinear specifications of the basic model This section contains the results for two nonlinear specifications of the basic model shown in Table 22.

10 Article submitted to Operations Research; manuscript no. 45 Table 16 Simple slope for Normalized ICU spare capacity for different values of relative ED size. σ R is the standard deviation of the relative ED size in the sample. Standard Errors are shown in parentheses. p < 0.01; p < 0.05; p < 0.10 Specification B Specification C REL ED SIZE HT CTR = -2*σ R 0.21 (0.21) 0.16 (0.19) REL ED SIZE HT CTR = (0.09) (0.08) REL ED SIZE HT CTR = 2*σ R (0.20) (0.2) Table 17 Estimates for alternate specifications based on fluid approximation. Standard Errors are shown in parentheses. p < 0.01; p < 0.05; p < 0.10 Specification B Specification C LEVEL EQUATION INTERCEPT 6.80 (0.37) 7.32 (0.4) SQRT ICU CAP 0.07 (1.24) (0.41) REL ED SIZE FL (5.05) (2.42) RURAL (1.64) INVESTOR (0.52) GOVERNMENT 0.38 (0.98) INVERSE MILLS RATIO -0.7 (2.23) (0.55) R F-value CHOICE EQUATION INTERCEPT 1.45 (0.24) 0.48 (0.11) SQRT ICU CAP (0.29) REL ED SIZE FL (1.37) RURAL (0.28) (0.26) INVESTOR 0.51 (0.21) 0.45 (0.2) GOVERNMENT -0.6 (0.3) (0.29) TRAUMA 0.20 (0.22) 0.26 (0.21) Log Likelihood

11 46 Article submitted to Operations Research; manuscript no. Table 18 Estimates for alternate specifications based on basic model. Standard Errors are shown in parentheses. p < 0.01; p < 0.05; p < 0.10 Specification B Specification C LEVEL EQUATION INTERCEPT 2.82 (2.00) 5.82 (0.45) ICU CAP 0.79 (0.47) (0.31) ICU SIZE 0.03 (0.01) 0.01 (0.01) ED SIZE 0.04 (0.03) 0.01 (0.01) RURAL (1.5) INVESTOR 0.93 (0.71) GOVERNMENT (0.79) INVERSE MILLS RATIO 2.93 (2.05) (0.53) R F-value CHOICE EQUATION INTERCEPT (0.28) (0.27) ICU CAP 0.49 (0.22) ICU SIZE 0.01 (0.01) ED SIZE 0.03 (0.01) RURAL (0.29) (0.26) INVESTOR 0.75 (0.22) 0.81 (0.22) GOVERNMENT (0.29) (0.30) TRAUMA (0.24) 0.36 (0.22) Log Likelihood

12 Article submitted to Operations Research; manuscript no. 47 Table 19 LEVEL EQUATION INTERCEPT REL ED SIZE HT CTR Estimation results for selection models based on diffusion approximation. Spec A Spec B Spec C 7.33 (0.16) (0.07) NORM ICU CAP CTR 0.01 (0.10) (REL ED SIZE HT CTR)*(NORM ICU CAP CTR) (0.04) 7.40 (0.19) (0.07) (0.10) (0.05) RURAL 1.11 (0.56) INVESTOR (0.35) GOVERNMENT 0.69 (0.59) CHOICE EQUATION INTERCEPT 0.55 (0.08) REL ED SIZE HT CTR 0.08 (0.13) NORM ICU CAP CTR (0.04) (REL ED SIZE HT CTR)*(NORM ICU CAP CTR) 0.03 (0.02) RURAL (0.04) INVESTOR GOVERNMENT (0.04) (0.03) TRAUMA 0.11 (0.08) 0.51 (0.10) 0.08 (0.04) (0.05) 0.03 (0.03) (0.24) 0.23 (0.16) (0.26) 0.08 (0.15) 7.35 (0.15) (0.05) (0.03) (0.02) 0.47 (0.09) (0.04) (0.03) (0.02) 0.15 (0.18) Log Likelihood Notes: Standard Errors are shown in parentheses. Log Likelihood for the LIML estimation method is corresponding to the Probit model for the selection equation. p < 0.01; p < 0.05; p < 0.10

13 48 Article submitted to Operations Research; manuscript no. Table 20 LEVEL EQUATION Estimation results for selection models based on fluid approximation. Specification A Specification B Specification C INTERCEPT 6.3 (0.37) 6.5 (0.38) 7.73 (0.31) SQRT ICU CAP 1.55 (0.47) 1.37 (0.47) 0.04 (0.13) REL ED SIZE FL 4.04 (2.61) 2.85 (2.54) (0.06) RURAL 1.08 (0.57) INVESTOR (0.33) GOVERNMENT 0.69 (0.58) CHOICE EQUATION INTERCEPT 1.08 (0.17) 1.07 (0.18) 0.44 (0.09) SQRT ICU CAP (0.2) (0.22) REL ED SIZE FL (1.3) (1.15) RURAL -0.2 (0.17) (0.25) (0.06) INVESTOR (0.05) 0.21 (0.15) (0.06) GOVERNMENT (0.06) (0.26) 0.31 (0.16) TRAUMA 0.23 (0.12) 0.2 (0.07) (0.03) Log Likelihood Notes: Standard Errors are shown in parentheses. Log Likelihood for the LIML estimation method is corresponding to the Probit model for the selection equation. p < 0.01; p < 0.05; p < 0.10

14 Article submitted to Operations Research; manuscript no. 49 Table 21 LEVEL EQUATION Estimation results for selection models based on the basic model. Specification A Specification B Specification C INTERCEPT 7.44 (0.5) 5.21 (0.64) 7.19 (0.17) ICU CAP -0.1 (0.37) 0.46 (0.31) (0.08) ICU SIZE 0.01 (0.01) 0.02 (0.01) 0.00 (0.01) ED SIZE (0.02) 0.01 (0.02) 0.00 (0.01) RURAL (0.76) (0.24) INVESTOR 0.13 (0.34) GOVERNMENT 0.2 (0.61) CHOICE EQUATION INTERCEPT (0.24) (0.28) (0.24) ICU CAP 0.32 (0.19) (0.29) ICU SIZE 0 (0.01) 0.05 (0.25) ED SIZE 0.02 (0.01) 0.79 (0.22) RURAL -0.6 (0.23) (0.29) (0.04) INVESTOR 0.17 (0.19) 0.45 (0.22) 0.18 (0.19) GOVERNMENT (0.19) 0.01 (0.01) (0.19) TRAUMA (0.17) 0.03 (0.01) 0.01 (0.06) Log Likelihood Notes: Standard Errors are shown in parentheses. Log Likelihood for the LIML estimation method is corresponding to the Probit model for the selection equation. p < 0.01; p < 0.05; p < 0.10

15 50 Article submitted to Operations Research; manuscript no. Table 22 Two non-linear specifications of the basic model LEVEL EQUATION INTERCEPT 5.48 (0.72) 6.02 (0.64) ICU CAP 0.42 (0.35) 0.15 (0.47) ED SIZE (0.01) (0.01) ICU SIZE (0.02) ICU SIZE*ICU CAP 0.02 (0.03) SQRT ICU SIZE 0.16 (0.1) LAMBDA (0.57) (0.54) CHOICE EQUATION INTERCEPT (0.36) (0.3) ICU CAP 0.61 (0.23) 0.08 (0.3) ED SIZE 0.03 (0.01) 0.03 (0.01) ICU SIZE (0.01) ICU SIZE*ICU CAP 0.06 (0.02) SQRT ICU SIZE 0.11 (0.08) TRAUMA 0.06 (0.25) (0.26) INVESTOR 0.82 (0.22) 0.87 (0.23) GOVERNMENT (0.3) (0.3) RURAL (0.29) (0.29)