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Anesth Analg 2003;97:190-195 © 2003 International Anesthesia Research Society
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Abstract |
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 Top Abstract IntroductionMethodsResultsDiscussionReferences |
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Surgeons using the same amount of operating room (OR) time differ in their achieved hospital contribution margins (revenue minus variable costs) by >1000%. Thus, to improve the financial return from perioperative facilities, OR strategic decisions should selectively focus additional OR capacity and capital purchasing on a few surgeons or subspecialties. These decisions use estimates of each surgeon’s and/or subspecialty’s contribution margin per OR hour. The estimates are subject to uncertainty (e.g., from outliers). We account for the uncertainties by using mean-variance portfolio analysis (i.e., quadratic programming). This method characterizes the problem of selectively expanding OR capacity based on the expected financial return and risk of different portfolios of surgeons. The assessment reveals whether the choices, of which surgeons have their OR capacity expanded, are sensitive to the uncertainties in the surgeons’ contribution margins per OR hour. Thus, mean-variance analysis reduces the chance of making strategic decisions based on spurious information. We also assess the financial benefit of using mean-variance portfolio analysis when the planned expansion of OR capacity is well diversified over at least several surgeons or subspecialties. Our results show that, in such circumstances, there may be little benefit from further changing the portfolio to reduce its financial risk.
IMPLICATIONS: Surgeon and subspecialty specific hospital financial data are uncertain, a fact that should be taken into account when making decisions about expanding operating room capacity. We show that mean-variance portfolio analysis can incorporate this uncertainty, thereby guiding operating room management decision-making and reducing the chance of a strategic decision being made based on spurious information.
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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Financially strong hospitals can make capital improvements (e.g., buy anesthesia information management systems), grow vibrant surgical practices (e.g., provide sustained business to the anesthesiology group), and provide uncompensated community benefits (e.g., teaching and indigent care). Surgeons using the same amount of operating room (OR) time differ in their achieved hospital contribution margin (revenue minus variable costs) by >1000% (1–4). Thus, to improve financial return from perioperative facilities, OR strategic decisions can selectively focus additional OR capacity and capital purchasing on a few surgeons or subspecialties. Such strategic decisions are based on statistical summaries of data from OR information systems and hospital accounting databases (1–4). Financial summaries calculated at a surgeon and/or subspecialty level have uncertainty due to limited sample sizes, large variability, and outliers (3,4). Consequently, these strategic decisions (e.g., the anesthesiology group decides to hire a subspecialty trained anesthesiologist) can be based on spurious information. The purpose of this paper is to describe how to consider uncertainties in financial data in the strategic decision of which surgeons and/or subspecialties should have their OR capacity expanded.
The remainder of the Introduction reviews previous work in the field, to motivate our paper.
Lack of sufficient hospital accounting data is usually not a factor limiting these strategic analyses. Variable costs can be estimated sufficiently accurately for purposes of strategic decision-making using the patients’ OR times, hospital lengths of stay, intensive care unit (ICU) lengths of stay, and implant costs (3). Hospitals with a detailed hospital cost database would, of course, use their direct estimates of variable costs.
Patients undergoing urgent or emergent surgery are typically excluded from these analyses for two reasons. First, there is a commitment to provide timely care to such patients once admitted to the hospital (1–4). Decisions regarding OR capacity expansion do not change this. Second, cost accounting is easier when excluding such patients, because then the variable cost of each patient’s entire outpatient visit or hospitalization can be attributed to the surgeon or subspecialty who scheduled the patient for elective surgery. For example, the variable costs and revenues for a child undergoing tonsillectomy and adenoidectomy can reasonably be attributed to the child’s pediatric otolaryngologist. However, the costs of a 40-day hospitalization for cerebral trauma from a motor vehicle accident should not be attributed to the otolaryngologist who performed a tracheostomy on the 18th hospital day.
Hospital contribution margins per OR hour vary several-fold among surgeons (1,2), with 5th and 95th percentiles of $310 and $3370, respectively (Fig. 1). Thus, increasing OR capacity selectively is more advantageous financially for a hospital than increasing capacity equally. The conceptually easiest way to increase OR capacity selectively is to sort surgeons in descending order of their contribution margins per OR hour, increase capacity for the surgeon at the top of the list by the maximum allowable amount, increase capacity for the next highest surgeon by the maximum amount, and so forth, until no more additional capacity is available (1). For example, as each surgeon is considered for an increase in his or her OR time, the expansion of that surgeon’s OR time could be any value between 0% and 25% (2,4). This maximum expansion of 25% represents a feasible expansion of OR capacity without recruiting more surgeons. Alternatively, each surgeon could have his or her OR time increased by 100% (3). This represents hiring a new surgeon modeled after a surgeon currently practicing at the hospital (3). For example, we used a 15% total increase in OR capacity, a 100% maximum increase in each surgeon’s usage, and the data in Figure 1. Selective expansion of capacity increased expected contribution margin by 32% versus a 15% increase achieved by expanding capacity equally (i.e., proportional to current workload) (2,3).
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Increasing OR capacity for a subspecialty using many ICU beds would be counter-productive if ICU occupancy is nearly 100%. Thus, hospitals also need to consider constraints from limited ICU and hospital ward beds (2). Then, strategic decisions are made not simply by ranking surgeons or subspecialties by the contribution margin per OR hour (2). Instead, the more sophisticated method of linear programming should be used, because it considers constraints (2–5). On the far right of Figure 1, we label those surgeons who would be provided more OR capacity when the constraints on ICU and hospital ward beds are considered along with the expected contribution margin per OR hour. Several surgeons with the largest contribution margins per OR hour do not have increases in their OR capacity. These surgeons’ patients need ICU beds, even though few are available (2). Several surgeons with relatively small contribution margins per OR hour receive the maximum increase in OR capacity. These surgeons use few of the scarce hospital ward and ICU beds relative to surgeons with larger contribution margins per OR hour (2). In the Methods and Results sections, the percentage increases in contribution margins are less than that described in the preceding paragraph, because of the important effects of limited hospital ward and ICU beds.
Decisions to change or maintain resources are uncertain for many surgeons or subspecialties when the decisions are based on the estimated financial impact on the hospital (4). Typically, one fiscal year of historical data is used for estimation, thereby achieving as large a sample size as possible, while excluding annual re-negotiations of managed care contracts, changes in national reimbursement, trends in practice patterns, and alternations in hospital accounting for fixed costs (1–4). However, the estimates are not always reliable. The relative standard error in estimating a surgeon’s mean contribution margin per OR hour is 23% on average (n = 98; SD = 46%) (Fig. 1) (4). This uncertainty in each surgeon’s estimated financial performance is sufficiently large to alter decision-making for many surgeons (4). For example, in one study, 45% of the surgeons not having their OR capacity increased should have received more OR resources, the discrepancy being due to random error (4).
Confidence intervals for expected contribution margins are not routinely included in hospital financial reports. However, for this application, uncertainty is sufficiently important that it should be incorporated into the analysis. Reports listing each surgeon’s or subspecialty’s contribution margin per OR hour in rank order are misleading otherwise (4). The purpose of this new paper is to improve upon just measuring the uncertainty, by determining how to use the measure of uncertainty to improve the quality of strategic decisions.
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Methods |
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The data have been described previously (2–4). The population consists of all patients undergoing outpatient or same day admit surgery during the 2000 fiscal year at a large, academic, multiple specialty hospital in the United States (US). The data were extracted from the hospital’s activity based costing system. Variable costs, revenue, hours of OR time, hours of regular ward time, and hours of ICU time were calculated for each physician using year 2000 US dollars. We limited the analysis to the 98 physicians who did at least 15 cases at the hospital during the study year (2–4).
Linear programming (5) (Excel 2000, Microsoft, Redmond, WA) with estimates of the surgeons’ mean contribution margins per OR hour was used to find the portfolio of surgeons’ OR time allocations that maximizes expected contribution margin. As a typical example, we included the following constraints on the availability of resources:
- Each surgeon’s OR capacity can be expanded by as much as 25% or 100% (see above) of the number of OR hours that he or she used during the past fiscal year.
- OR time for a surgeon is not reduced (i.e., we studied expansion of OR capacity).
- Total OR time used can be increased by up to 15% by extending the workday. The additional OR time will cost $50 more per hour more than existing OR time.
- Hospital ward use can be 10% more without hiring additional staff.
- ICU use can be 5% more without hiring additional staff.
Linear programming determines the single best portfolio of surgeons to maximize expected contribution margin subject to the above constraints. The number of surgeons in the optimal portfolio, which surgeons are in the optimal portfolio, and the increase in OR time provided to each surgeon are chosen automatically. Generally, each increase in the maximum expansion of OR time for each surgeon results in a decrease in the number of surgeons in the optimal portfolio. Likewise, each increase in the total available OR time results in an increase in the number of surgeons in the optimal portfolio.
We used Fieller’s result for the statistical distribution of the ratio of two random variables to obtain the standard error of each surgeon’s mean contribution margin per OR hour (4,6). Variables used were the contribution margins and OR times of all cases performed by each surgeon (4). The corresponding 95% confidence intervals are shown in Figure 1.
The standard deviation of the expected return quantifies the risk of a portfolio, whether of stocks and bonds or of surgeons (5). For a specified level of expected return, a rational investor chooses the portfolio having the least risk. A portfolio is "efficient" if there is no portfolio having the same return with a lesser standard deviation, and vice versa. The "efficient frontier" is the collection of all efficient portfolios.
We calculated efficient frontiers by minimizing the variance of resulting hospital contribution margins, subject to the constraint that the expected return was at least equal to a specified value (5). This mean-variance portfolio analysis was accomplished using Excel’s Solver function’s quadratic programming (5). The analysis automatically chooses the number of surgeons in the portfolio, which surgeons are included in the portfolio, and the increase in OR time provided to each surgeon in the portfolio.
We also examined the sensitivity of our results to the number of surgeons in the portfolio of surgeons with expanded OR capacity. We varied the numbers of surgeons by progressively increasing the maximum allowable increase in each surgeon’s capacity in increments of 0.1%. We evaluated sensitivity of our results to constraints from limited ICU and hospital ward capacity by repeating the sensitivity analysis while excluding these two constraints.
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Results |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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Each point along an efficient frontier represents a portfolio of surgeons. Each portfolio differs in the number of surgeons provided additional OR capacity and/or in the increase in OR capacity provided to each such surgeon.
The portfolio that corresponds to the peak of the efficiency frontier guarantees maximum expected contribution margin, but usually at the expense of sizable risk. The peak is at the point at the far right of Figure 2. That portfolio is the same as that obtained by linear programming (see Introduction).
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Moving to the left on the efficiency frontier, the portfolios have smaller risk, but also smaller expected financial return (i.e., contribution margin). The objective of the mean-variance portfolio analysis is to draw the efficient frontier and then choose a portfolio along the frontier that has an acceptable expected return, but appreciably less risk. The desired portfolio is identified by choosing a point along the curve where, from right to left, the initial steepness of the curve has leveled off. There is certainly no good reason for a portfolio to be selected that corresponds to points that are to the left of this value.
For example, with a 100% maximum increase in each surgeon’s OR time, the desired point where the steepness levels off has a risk of 1.7% and expected return of 13.7%. The smallest attainable risk is 0.2% less, but at a relatively large expense in expected return. In contrast, the risk at the peak of the efficiency frontier is 0.2% more, but with little (only 0.4% increase) effect on the expected return.
The portfolios, of surgeons provided expanded OR capacity, differ between the peak of the efficient frontier and the point at which the slope in the efficient frontier has started to level off. Differences in the portfolios reveal which surgeons’ uncertainties in contribution margins per OR hour influence strategic decisions. For example, with a 100% maximum increase in each surgeon’s OR time, the portfolio at the peak of the efficient frontier increases the OR capacity of 16 surgeons. The portfolio at the point where the slope of the curve starts to level off (moving from right to left) increases the allocation of 14 surgeons.
The two excluded surgeons had the largest uncertainties in the mean contribution margin per OR hour among all surgeons. The large uncertainties were not the result of overly small sample sizes. The two surgeons had samples of 72 and 74 cases per year, respectively. Instead, the primary reason for their large standard errors was the presence of outliers with large positive contribution margins relative to the used OR time. Specifically, both surgeons had two patients with diagnosis related groups 483 (tracheostomy except for face, mouth, and neck diagnosis). This diagnosis related group has the second largest Diagnosis Related Groups’ Medicare reimbursement. Yet, the four patients had lengths of stays of only 3 to 10 days.
Our exploration of Figure 2 illustrates how the mean-variance portfolio analysis is used in practice. The remaining results describe sensitivity analyses that we performed to test hypotheses about possible reasons for our findings.
The peaks of the efficient frontiers in Figure 2 were obtained by increasing OR capacity of 34 and 16 of the 98 surgeons, when the maximum increases were 25% and 100% of baseline OR capacity, respectively. There was some reduction in the risk by accepting a smaller expected contribution margin. However, the risk reduction in Figure 2 was not particularly impressive. We know that if OR capacity was expanded based solely on the data from the one surgeon with the largest mean contribution margin per OR hour, then the risk would be very high. Thus, we hypothesized that the number of surgeons in the peak portfolio of Figure 2 was sufficiently large to have provided a portfolio that was diversified enough to account for the limited incremental reductions in risk achieved by the quadratic programming.
To test this hypothesis, the maximum possible increase in each surgeon’s OR capacity was varied to achieve a wide range in the numbers of surgeons with expanded OR capacity (Fig. 3). With the decision based on data from progressively fewer surgeons, the risk increased by <1% (lower pane, lower line), even though the expected return increased from 5% to 17% (upper pane, lower line). Each reduction in the number of surgeons’ data used for the strategic decision caused the ratio of the risk (lower pane) to the expected increase in contribution margin (upper pane) to be reduced until the strategic decision was based on data from fewer than five surgeons.
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The small risk reductions for portfolios corresponding to points on the left of the efficient frontier in Figure 2 may also have been caused, in part, by ICU and hospital ward constraints leading to already diversified portfolios with a sufficient number of surgeons’ data. To test this hypothesis, the sensitivity analysis was repeated without the constraints on ICU and hospital ward use. The risk did not increase by more than 1% until fewer than eight surgeons’ data were used for the strategic decision (lower pane, upper line). When the risk was expressed as a percentage of the expected increase, the risk was progressively reduced until the strategic decision was based on data from fewer than eight surgeons.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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The vibrancy of an anesthesiology group is inextricably linked to the financial strength of the hospital(s) where its anesthesia providers practice. In this paper, we continued previous work in improving methods of linking hospital financial data and OR information systems data for making good strategic decisions (1–4).
The premise of mean-variance portfolio management is that the economically rational investor accepts a smaller reduction in expected return to achieve a large reduction in risk (5). However, with respect to expanding OR capacity selectively, basing decision-making on trying to reduce that uncertainty is a low-yield proposition when at least several surgeons’ or subspecialties’ data are in the optimal portfolio. This is evident from the relatively unimportant absolute reduction in risk along the vertical axis of Figure 2.
Previously, linear programming was used to identify the portfolio of surgeons providing the maximum expected return (that is, the peaks in Fig. 2) (2,3). Such linear programming provides the mean part of mean-variance portfolio analysis. Next, we learned how to create graphs such as Figure 1, displaying uncertainty in each surgeon’s contribution margin per OR hour (4). That measured the variance part of mean-variance portfolio analysis. In this study, we showed that mean-variance portfolio analysis (i.e., quadratic programming) can be used to combine both the mean and the variance into decision-making. Specifically, the analysis can be used to assess the sensitivity of the portfolio with the maximum expected return to uncertainty in each surgeon’s contribution margin per OR hour. The assessment is made by comparing the portfolios of surgeons at the peak and at the point where the slope of the curve, which initially is steep, starts to level off.
Mean-variance portfolio analysis can also include the correlations in uncertainty among surgeons (5). Suppose that among 10 surgeons planned for expansion in OR capacity, 4 were cardiac surgeons in the same surgical practice. Then, a decision for one surgeon could affect the decision for others (i.e., risk could be correlated). A limitation of our work is that currently we do not know how to estimate such correlations using data from OR information systems and hospital accounting databases.
To simplify the presentation, we considered the amount by which each surgeon could expand his or her use of OR time to be the same for all surgeons. In practice, these values should be adjusted based on marketing data and knowledge about each of the surgeons.
In conclusion, there is marked uncertainty in each surgeon’s mean contribution margin per OR hour. We showed that this is, to a large extent, due to the large inherent variability in the contribution margins of a surgeon’s procedures and also due to the influence of outliers. Outliers are probably an inevitable part of health care economics. Consequently, the decision to maintain or expand capacity will always be uncertain for each particular surgeon or subspecialty. The primary advance of this paper is its use of mean-variance portfolio analysis as a method to reduce the likelihood of making strategic decisions on the basis of spurious information. However, we also learned that if selective expansion of OR capacity is well-diversified over at least several surgeons or subspecialties, any additional efforts to change decisions to reduce risk are unlikely to yield financially important benefits.
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Footnotes |
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Presented, in part, at the Institute for Operations Research and Management Science meeting, November 19, 2002, San Jose, CA, and the Association of Anesthesia Clinical Directors meeting, March 23, 2003, Carlsbad, CA.
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References |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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- Macario A, Dexter F, Traub RD. Hospital profitability per hour of operating room time can vary among surgeons. Anesth Analg 2001; 93: 669–75.
[Abstract/Free Full Text] - Dexter F, Blake JT, Penning DH, Lubarsky DA. Calculating a potential increase in hospital margin for elective surgery by changing operating room time allocations or increasing nursing staffing to permit completion of more cases: a case study. Anesth Analg 2002; 94: 138–42.
[Abstract/Free Full Text] - Dexter F, Blake JT, Penning DH, et al. Use of linear programming to estimate impact of changes in a hospital’s operating room time allocation on perioperative variable costs. Anesthesiology 2002; 96: 718–24.[Web of Science][Medline]
- Dexter F, Lubarsky DA, Blake JT. Sampling error can significantly affect measured hospital financial performance of surgeons and resulting operating room time allocations. Anesth Analg 2002; 95: 184–8.
[Abstract/Free Full Text] - Ragsdale CT. Spreadsheet modeling and decision analysis, a practical introduction to management science. 2nd ed. Cincinnati, OH: South-Western College Publishing, 1998: 45–64, 128–41, 334–41.
- Briggs AH, Mooney CZ, Wonderling DE. Constructing confidence intervals for cost-effectiveness ratios: an evaluation of parametric and non-parametric techniques using Monte-Carlo simulation. Stat Med 1999; 18: 3245–62.[Web of Science][Medline]
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