Ann Thorac Surg 1998;66:1969-1971
© 1998 The Society of Thoracic Surgeons
Original Articles
What is the marginal cost for marginal risk in cardiac surgery?
Thomas E. Williams, Jr, MDa,
William J. Fanning, MDa,
W.C. Benton, PhDa,
Gerard S. Kakos, MDa,
Randall L. Miller, MDa,
William J. Esterline, MDa,
Thomas D. Hankins, CPa
a Grant Hospital, Columbus, Ohio, USA
Accepted for publication June 1, 1998.
Address reprint requests to Dr Williams, 300 East Town Str, 12th Floor, Columbus, OH 43215
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Abstract
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Background. It has been shown that postoperative length of stay (LOS) correlates highly with mortality risk for cardiac surgical procedures. Similar correlations have been found for charges with LOS and costs with risk.
Methods. Postoperative LOS and risk scores were obtained, tabulated, and compiled into the five original Parsonnet risk groups for 2,589 patients who underwent cardiac operations from 1992 through 1996 at one hospital. The correlation of the group mean LOS with the group mean risk was tested.
Results. The correlation coefficient was 0.9827; 96.58% of the variance was removed using risk to predict LOS. A calculation of the difference in cost for difference in risk for cohorts of patients is developed.
Conclusions. The high correlation of mean LOS with mean risk permits calculation of marginal cost for marginal risk based on clinical data. The marginal cost is equal to the difference in variable costs for cohorts.
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Introduction
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It seems intuitively obvious to both laymen and cardiac surgeons that sicker patients will be hospitalized longer, consume more resources, and cost the hospital more for their care. As physicians and hospitals deal with diagnosis-related group reimbursement, the arrival of fixed price discounted fee-for-service contracts, and capitation, we also have seen the evolution of a pool of higher risk surgical candidates [1]. As these risks increase and the mechanisms of reimbursement change, it is useful to evaluate the increase in cost for an increase of risk in cardiac surgical patients. Further development of the analysis relating mean length of stay (LOS) to mean risk for our open heart program permits us to derive a relationship between marginal costs and marginal risks for that program.
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Material and methods
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Patient population
The records of all 2,589 patients who underwent cardiac operations performed at the Grant Hospital, Columbus, OH, in the five calendar years 1992 through 1996 are included in this study. No cases were censored for any reason. The types of cases, their respective numbers, and the various mortality rates are summarized in Table 1. A total of 214 patients (8.3%) underwent reoperative procedures; 257 patients (9.9%) were classified as having emergency operations. There were 107 patients (4.1%) who were 80 years or older in age at the time of their operations.
Database
The Grant Medical Center cardiac surgical database includes surgical records for all patients operated on since its inception in 1983. A total of 5,431 patients were enrolled through the end of December 1996. The Parsonnet additive risk algorithm was incorporated into the database in 1991 [2]. This risk stratification method uses data elements concerning sex, obesity, diabetes, hypertension, reoperation, emergency status, and certain other clinical information to determine a risk level and category for each patient. The methods of data collection and management, described previously, have permitted us to acquire risk stratification and LOS data in 3,756 patients from January 1, 1989, through December 31, 1996 [3]. The records of the 2,589 patients in the calendar years 1992 through 1996 were used for this analysis. The records for the initial 1,167 risk-stratified patients were used as the basis for the development of the risk–reward curve in our 1994 report. The distribution of cases by risk category is essentially unchanged from the distribution described in our previous study.
For this study, the individual data for each patient’s stay (LOS) and risk were compiled into the five original Parsonnet risk categories. These data were then tabulated and mean risk and mean postoperative LOS were obtained for each risk category. Mean postoperative LOS is compared with mean risk for each of the Parsonnet risk groups for 1992 through 1996 in Table 2; the data are plotted in Fig 1.
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Fig 1. Mean postoperative length of stay (LOS) versus mean risk for original Parsonnet risk groups for patients at Grant Hospital, 1992 through 1996.
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Results
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Correlation of risk and length of stay
The correlation coefficient for mean postoperative LOS with mean risk was 0.9827; mean risk accounted for 96.58% of the variance in predicting mean postoperative LOS. A linear regression was then performed for mean LOS against mean risk using a statistical and graphics program [4]; the regression gave the following equation:
Calculation of marginal cost for marginal risk
Several reports have shown a nearly linear relationship between postoperative LOS and risk; the relationship appears to hold for both the Parsonnet and the Bayesian probability models of risk estimation. In addition, Lahey and coworkers [5] and Dickinson [6] have shown that hospital charges are a nearly linear function of LOS. If charges have a similar relationship to costs, then
where K represents the cost per day (per diem) for the hospital open heart program, LOS represents the length of stay for a given patient, or the mean length of stay for a cohort of patients, and Fxdcost represents the total of both hospital and program fixed costs. Hospital fixed costs include debt service, costs of regulatory compliance, and other costs common to all patients. Program fixed costs are those peculiar to the open heart program—amortization of pumps and instruments, payment for tubing packs and drapes, and so forth.
Therefore, the difference in costs for two cohorts of patients whose risks differ can be written as
or,
But, the regression equation for Fig 1 is
Hence, the increase in cost for an increase in risk is
or
in our hospital, where K is the cost per day of hospitalization excluding fixed costs, and (Risk2 - Risk1) represents the difference in mean risks between two cohorts of patients.
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Comment
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The patient population, type of surgical procedure, and surgical results have been reviewed previously and are shown in Table 1. The fraction of reoperative cases and the fraction of patients age 80 years or older are similar to those given in the most recent report from the Society of Thoracic Surgeons (STS) National Database. The fraction of emergency cases is smaller. The surgical mortality rates (Table 1) are comparable to those reported by the STS for the same time period (1992 through 1996). The ranges for mortality rates given by the STS were 2.9% to 3.4% for coronary artery bypass grafting, 3.7% to 4.2% for aortic valve replacement, 5.3% to 6.6% for mitral valve replacement, 6.7% to 8.2% for coronary artery bypass grafting and aortic valve replacement, and 12.3% to 16.6% for coronary artery bypass grafting and mitral valve replacement.
These data further confirm that postoperative LOS correlates highly with the risk score obtained using the Parsonnet additive risk algorithm. This observation of a nearly linear relationship between mean postoperative LOS and mean risk was demonstrated by Parsonnet and associates in their 1989 article [2]. Dickinson [6] confirmed this for the Parsonnet model in 1992. The report of the STS National Database, which uses a different risk estimation algorithm, also indicates a nearly linear relationship between postoperative LOS and risk [7]. Even though substantive changes were made in our open heart program in 1992 and a fast track program was added in 1995, the nearly linear relationship between postoperative LOS and mean risk for the Parsonnet groups remained valid. In fact, both the correlation coefficient and the fraction of the variance removed were slightly greater than those reported earlier (correlation coefficient, 0.9761; 95.41% variance removed in the 1989 through 1991 data set). The regression equation for the 1992 through 1996 data differs from that for the 1989 through 1991 data ( 
) only slightly. The change in the constant term reflects a reduced mean LOS of 1 day. The change in the slope of the regression line indicates some reduction in LOS for an increment of risk.
The choice to use LOS as a surrogate for cost has precedence in other studies. Lahey and coworkers [5] developed a model for predicting LOS from preoperative risk factors in patients who underwent coronary artery bypass grafting. They found that congestive heart failure, the use of intraaortic balloon assist devices, obesity, age, and impaired renal function were the most significant factors in the prediction of hospital LOS. A further significant finding of their data was that their average hospital charges were nearly linear with LOS through about 35 days of hospitalization. Thus, they felt that LOS was a good predictor of hospital costs. Dickinson [6] further confirmed this by showing that hospital charges were a nearly linear function of LOS for all cases in the open heart program. Loop [8] has recently shown cost accounting data that demonstrate a nearly linear relationship between cost and predicted risk at the Cleveland Clinic. This substantially validates the assumptions we have used to derive both the risk–reward curve and the marginal cost for marginal risk.
Examination of the data permits the calculation of marginal cost based on marginal risk. A change of only 6% in risk adds approximately 1 hospital day to a postoperative LOS in our experience. This then permits the forecasting of differences in hospital LOS for groups of patients when the basic demographics of age and sex and certain clinical risk factors are known.
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References
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Williams T.E., Fanning W.J., Link L., et al. Can we afford to do cardiac operations in 1996? A risk-reward curve for cardiac surgery. Ann Thorac Surg 1994;58:815-821.[Abstract]
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Parsonnet V., Dean D., Bernstein A. A method of uniform stratification of risk for evaluating the results of surgery in acquired adult heart disease. Circulation 1989;79(Suppl 1):103-112.
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Williams T.E., Benton W.C., Fanning W.J., Hankins T.D., Kakos G.S. Quantitative quality descriptors for an open-heart program. Qual Prog 1992;25:29-32.
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Statgraphics, version 6.0. Rockville, MD: Manugistics, Inc.
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Lahey S.J., Borlase B.C., Lavin P.T., Levitsky S. Preoperative risk factors that predict hospital length of stay in coronary artery bypass patients >60 years old. Circulation 1992;86(Suppl 2):181-185.
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Dickinson T.A. How to do it: utilizing risk stratification to evaluate outcomes in adult open-heart operations. J Extra Corporeal Technol 1993;24:135-140.
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The Society of Thoracic Surgeons. Data Analyses of The Society of Thoracic Surgeons National Cardiac Surgery Database, the Seventh Year—January, 1998.
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Loop F.D. You are in charge of cost. Ann Thorac Surg 1995;60:1509-1512.[Abstract/Free Full Text]
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Abstract
Introduction
