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From the Department of Anesthesia, Division of Management Consulting, University of Iowa, Iowa City, Iowa. Financial disclosure: As director of the Division of Management Consulting, FD receives no funds personally other than his salary from the State of Iowa, including no travel expenses or honoria, and has tenure with no incentive program.
Address correspondence and reprint requests to Franklin Dexter, MD, PhD, Department of Anesthesia, Division of Management Consulting, University of Iowa, Iowa City, IA 52242. Address e-mail to franklin-dexter{at}uiowa.edu. Website www.FranklinDexter.net
Abstract
BACKGROUND: Deciding when patients should arrive for same-day-admission or ambulatory surgery is a problem at many hospitals and surgery centers. Although staff can often start cases earlier than scheduled, the potential start times are not known when each case is scheduled. Patient availability must therefore be balanced against patient waiting times and fasting times. Knowing the earliest time that a case might begin, given its scheduled start time, provides a rational basis for telling patients when to report for surgery and when to refrain from eating or drinking before their procedure.
METHODS: We describe and validate a simple method for determining the earliest possible start time for a case, with only a 5% probability that staff would be able to start the case even earlier. Calculations use nonparametric methods to determine the 0.05 lower prediction bound for the start time of a case, using historical values for the scheduled and actual start times of cases performed by the same surgical suite/surgical service/day of the week combination as the case of interest. Information is not needed regarding the preceding cases performed in the same operating room. No patient or surgeon identifiable information is used.
RESULTS: We use results from earlier studies to provide a derivation and theoretical justification for these methods. New data confirm the validity of the results obtained and show that the required calculations are easy to implement. Individualized patient instructions can be accessed via a public website without disclosing confidential information.
CONCLUSIONS: We have developed a simple method for determining when patients should be ready on the day of surgery based on the start times of historical cases performed by the same surgical suite/surgical service/day of the week combination as the case of interest.
Deciding when patients should arrive for same-day-admission or ambulatory surgery is a problem at many hospitals and surgery centers. If patients are told to arrive early in the morning for surgery scheduled in the afternoon, they often wait for several hours and remain NPO (nulla per os, nothing by mouth) most of the day. Such a policy results in diminished patient satisfaction. If patients do not arrive early enough, they may not be ready and available in the event an earlier case is cancelled or finishes ahead of schedule. Medical personnel will then have to wait for the patient, and the operating room (OR) will be idle.
Telling patients to report a fixed number of hours before their scheduled procedure has no rational foundation (1). Statistical methods are needed to determine when patients should report for surgery based on the earliest time that their scheduled case might actually begin:
Despite the existence of recommended guidelines, patients are often denied food and water for longer than the recommended periods. Many patients are still told to remain NPO after midnight (16–20). Patients are often fasted for long periods due to uncertainties over the actual times their procedures will begin (16,18,19). Patients may also choose to fast for periods longer than instructed (16,20,21).
Anesthesiologists in our department have chosen to adopt the newer NPO guidelines (10,11) in the interest of patient safety and satisfaction. Implementation has required the joint efforts of both nursing and anesthesia (22).
Knowledge of "the earliest possible start time" for each case, taking into consideration that preceding cases may finish early or be cancelled, provides a rational foundation for determining when patients should be ready for surgery and how long they should have to remain in a waiting area before their procedure. In conjunction with revised NPO guidelines, it allows some patients to eat or drink on the morning of a procedure with little chance their case would have to be postponed because of food consumption.
Studies have examined methods for deciding when patients should be ready on the day of surgery when the types of cases to be performed earlier in the same OR are known (1,4,6). These methods estimate the 0.05 percentile of the durations of cases performed previously that are of the same type as the preceding case(s) (4,23). The 0.05 percentile may be considered the shortest conceivable duration of the preceding case, since cases would be shorter than the 0.05 percentile only 5% of the time. When turnover time is added, the 0.05 percentile thus provides an estimate for the "earliest possible start time" of the next case.
Historical data from large numbers of cases are rarely available for the duration of the type of procedure associated with the preceding case. Many types of procedures are rare (6,24–26). If the number of historical cases is very small, Bayesian methods (6) can predict case durations by combining any historical data that might be available with the scheduled duration of the case.
Methods based on the expected durations of the preceding case(s) function well for updates and decision-making on the day of surgery (4,6), but not for decisions made days in advance. Changes in the schedule between the day the patient is given instructions and the time of surgery cause information to become outdated and incomplete. One or more of the preceding cases may be moved from one OR to another. The scheduled procedure(s) of one or more of the preceding case(s) may be changed, altering the probability that the preceding case(s) will finish as scheduled.
We report a new method to choose the earliest possible start time of a case. The method does not rely on knowledge of the preceding case(s), surgeon and/or procedure. Instead, the method uses the scheduled and actual start times of historical cases, classified by surgical suite, service, and day of the week. The method is easy to implement and does not require complex modeling.
METHODS
Data were obtained on all surgical cases performed at a hospital's tertiary surgical suite (MAIN) and Ambulatory Surgery Center suite for the 3-yr period between June 1, 2002, and May 31, 2005. We studied elective cases representing 142 suite/service/day of the week combinations to determine the extent to which cases started earlier than scheduled.
First cases of the day were excluded from analysis. A case was considered a first-case-of-the-day start if it was scheduled to begin within 30 min of the start of the scheduled workday or was moved forward in the schedule so that it actually began within 30 min of the start of the scheduled workday (start of workday, 8:00 am Mondays and Tuesdays; 7:15 am Wednesdays, Thursdays, and Fridays). Cases scheduled to begin at the start of the workday were not included in the analysis because those cases could not have started earlier than scheduled. Cases moved forward to begin at the start of the workday were not included because they could not possibly have been moved on the day of surgery. The schedule must have been changed and the patient notified at least the evening before.
Inpatients were also excluded from analysis. Calculations to determine specific times when they should start fasting, report to the hospital, and be ready for surgery are irrelevant for these patients. Generally, they are receiving IV fluids and are available for surgery at any time.
Earliness Ratio
Historical data for the 3-yr period were analyzed in terms of an earliness ratio expressing the extent to which each case was performed earlier or later than scheduled.
For each case, an earliness ratio was determined:
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The theoretical basis for use of the earliness ratio is explained in Appendix 1.
The statistical distribution of surgical case durations is lognormal (6–8), meaning that the logarithms of case durations are normally distributed. The earliness ratio thus has a three-parameter lognormal distribution, where the three parameters are mean, standard deviation, and an offset or shift (equal to turnover time) that must be subtracted from the scheduled and actual start times before determining the logarithm of the ratio.
Appendix 1 also shows that this ratio can be interpreted as if each case is preceded by a single "virtual case" plus a turnover time of 30 min. The denominator depends on the scheduled duration of the virtual case, while the numerator depends on the actual duration of the virtual case. When the preceding virtual case finishes early and the case of interest is performed ahead of schedule, the earliness ratio is <1.0. We examined extreme examples of relative earliness by studying earliness ratios at the lower tail of the distribution, close to 0.05.
Quantiles were determined for different sets of earliness ratios, one set for each suite/service/day of the week combination. Quantiles are useful for examining the observed distribution of earliness ratios and the statistical properties of the tail of the distribution. The pth Cleveland quantile is a value between 0 and 1. For example, the 0.05 quantile is equivalent to the fifth percentile. Let I refer to the integer portion of the expression (N x p + 0.5) and let F refer to the fractional portion of the expression. Then, for a series of N ordered values x1, x2, ..., xN in ascending order, the pth quantile equals:
(1 - F) · xI + F · xI + 1.
In other words, if the expression N x p + 0.5 is not an integer, the quantile is determined by interpolation between two adjacent points. The Clopper–Pearson method was used to estimate 95% confidence intervals (CI) of the quantiles (27,28).
Lower Prediction Bound
The 0.05 lower prediction bound for the earliness ratio for each suite/service/day of the week combination was used to calculate a hypothetical "earliest possible start time" for each case. The "earliest possible start time" is the earliest time the case might actually begin, given its scheduled start time.
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The 0.05 prediction bound is appropriate for scheduling the next patient in a sequence because it provides a 5% probability that the earliness ratio for the next patient will be smaller than the prediction bound calculated from historical data. For each case, the probability is 5% that staff might want to begin the next case sooner than its earliest possible start time. Calculated values for earliest possible start times were rounded down to earlier start times in 15 min increments.
Values were determined independently for each suite and surgical service. Therefore, a surgeon who provides reasonably accurate estimates of case durations will invariably find the next patient ready at the end of each case. The surgeon will not be "penalized" by use of these methods, but modestly rewarded for his or her accuracy.
The 0.05 lower prediction bound for the earliness ratio was determined using a nonparametric method, described in Appendix 2. For a series of N earliness ratios, the value of 0.05 · (N + 1) is first determined. If it equals an integer, the 0.05 lower prediction bound is simply the [0.05 · (N + 1)]th smallest point. For example, for n = 199 historical cases, the 0.05 lower prediction bound is the 10th smallest value. For n = 59 cases, it is the third smallest value. For n = 39 cases, it is the second smallest value. For n = 19 cases, it is the smallest value. For other numbers of cases, it occurs between values (29, Appendix 2).
Differences Among Suite/Service/Day of Week
Earliest possible start times for scheduled cases were calculated from 0.05 prediction bounds for earliness ratios, considering each suite/service/day of the week combination separately. We reasoned that earliness ratios might vary substantially among surgical suites and services because of differences in the types of procedures performed and their average durations, the number of days in advance that cases are scheduled, and the scheduling paradigms used. For example, the service "otolaryngology" in a tertiary surgical suite is not the same as otolaryngology in an ambulatory surgery suite. The two locations have different procedures, cancellation rates, scheduling methods, and patient expectations concerning convenience. In addition, otolaryngology performed in an ambulatory surgery suite on Tuesdays may differ from otolaryngology on Thursdays. The different days are associated with different procedures, surgeons, cancellation rates, numbers of ORs run simultaneously, and numbers of cases moved from one OR to another.
To test whether each combination needed to be analyzed independently, a 0.05 prediction bound was determined for all data combined using Expression 6 in Appendix 2. If a single overall value were valid, then 5% of the earliness ratios for each suite/service/day of the week combination should be smaller than this overall 0.05 prediction bound. For each suite/service/day of the week combination, we determined the proportion of earliness ratios that were less than the overall 0.05 prediction bound. The proportion was considered to be different from 5%, if 5% did not fall within the upper and lower 95% CI of the proportion. The Clopper–Pearson method was used to estimate the 95% CI (27,28).
Determining the Number of Historical Cases to Use
If too few historical cases are used to determine the 0.05 prediction bounds of the earliness ratios, estimates may show considerable variability due to outlier values (30). On the other hand, use of too many earliness ratios that span a long period of time may reduce accuracy if surgical staff or scheduling procedures have changed. Lower prediction bounds were therefore calculated using different numbers of historical cases to determine an appropriate number.
Cases for each suite/service/day of the week combination were first sorted in chronological order. The oldest n = 19 cases were then sorted in ascending order according to earliness ratios. The 0.05 lower prediction bound equals the smallest of the 19 values, since 0.05 · (N + 1) = 0.05 · (19 + 1) is exactly equal to the integer 1. The lower prediction bound for the oldest 19 cases was compared to the earliness ratio for the 20th case. The comparison was considered "out of bounds" if the earliness ratio for the 20th case was less than the lower prediction bound based on the first 19 cases. This process was then repeated using the second through 20th cases to predict the value for the 21st case, etc., until the number of comparisons for each suite/service/day of the week combination equaled the number of cases for that combination minus 19. The total number of occurrences of out-of-bounds ratios was then summed across all suite/service/day of the week combinations for n = 19 historical cases.
Similar calculations were performed using moving windows of n = 39, 59, 79, 99, or 199 historical cases to determine the 0.05 lower prediction bound for the ratio of the next case. For each value of N, the total number of occurrences of out-of-bounds ratios was summed across all suite/service/day of the week combinations. These values of N were chosen because interpolation between adjacent earliness ratios, as described in Appendix 2, was not necessary. The 2nd, 3rd, 4th, 5th, or 10th smallest points could be used without further processing when the earliness ratios were sorted in ascending order. The number 199 was used because it is much larger than 99, yet still represents a realistic number of cases. We did not know that this approach of using only certain values of N, and excluding some data points from the analysis, would be valid until we performed the calculations and compared the results with those obtained using Expression 6.
In addition, the overall coefficient of variation of the 0.05 lower prediction bounds was determined for each number of historical cases (31). For each number of historical cases and each service/suite/day of the week combination, the individual values for the prediction bounds that were generated by the moving window were combined to yield a mean, standard deviation, and coefficient of variation. An overall average coefficient of variation for each number of historical cases was obtained by taking a weighted average of the individual coefficients of variation (31). Weights were based on the number of prediction bounds, or comparisons, generated for each suite/ service/day of the week.
Results are reported only for those suite/service/day of the week combinations with more than 99 cases, so that the number of out-of-bounds ratios and the coefficients of variation for the different values of N would always be based on the same cases.
Dependence of Earliness Ratio on Time of Day
The earliness ratio defined by Expression 1 is a valid measure of the extent to which cases are performed earlier than scheduled provided Expression 1 does not vary systematically by time of day. To verify that Expression 1 was not dependent on the time of day, each workday was divided into four time periods: before 10:30 am, 10:30 am to 12:29 pm, 12:30 pm to 2:29 pm, and 2:30 pm or later. The 0.05 quantiles and 95% CI of the 0.05 quantiles were determined for earliness ratios of cases scheduled to begin during each of these four time periods.
Choice of Lower Prediction Bound
A sensitivity analysis was performed to ascertain the impact of the choice of 5% as the fraction of cases for which staff would have to wait for the patient. Earliest possible start times for n = 59 historical cases were also calculated using lower prediction bounds of 0.02 and 0.10, assuming that staff would have to wait for the patient only 2% of the time or as often as 10% of the time.
Magnetic Resonance Imaging and Computerized Tomography
Anesthesia times for magnetic resonance imaging (MRI) and computerized tomography (CT) cases were obtained from anesthesia billing data for November 1, 2004, through January 6, 2006. Expert predictions for the duration of each case were provided by radiology technologists, as described previously (32). Anesthesia times and expert predictions for MRI and CT cases form the basis for the development of Expression 1, as explained in Appendix 1.
Calculations and Statistics
Calculations were programmed in Microsoft Excel 2003 using Visual Basic for Applications. Standard errors for the weighted coefficients of variation for the lower prediction bounds were based on the method of Tian using 1000 random samples (33). Lilliefors' test and the Kolmogorov–Smirnov test were performed using StatXact-7, Cytel Software Corporation, Cambridge, MA. LOWESS smoothing for Figure 7A was performed using Systat 11, Systat Software, Point Richmond, CA.
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RESULTS
Figure 1 illustrates use of the 0.05 lower prediction bound for the earliness ratio to calculate the earliest possible start time for a case based on its scheduled start time.
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The results of our analyses are summarized in Table 1. The results provide a foundation for the use of historical data and the calculation of lower prediction bounds based on an earliness ratio.
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Suite/Service/Day of the Week Combination
Data for each combination of surgical suite/surgical service/day of the week were analyzed separately. Results shown in Figures 1 and 2 support this approach. The 0.05 lower prediction bounds for the earliness ratios, and thus the earliest possible start times, differ greatly between Thursdays and Fridays for adult General Surgery patients in the MAIN surgical suite (Fig. 1). Figure 2A shows cumulative distribution plots of the earliness ratios for these two groups of cases. The distributions have very different shapes (P = 0.016 by Kolmogorov– Smirnov test). Figure 2B is a corresponding two-sample quantile plot showing that the 0.05 quantiles differ between the 2 days of the week.
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To verify whether a separate analysis was necessary for each suite/service/day of the week combination, we determined the effect of pooling all surgical data into a single group. An overall 0.05 lower prediction bound for the earliness ratio was calculated based on all OR data. The earliness ratio for each case was then compared to this overall lower prediction bound. The proportion of all earliness ratios that are less than the overall lower prediction bound equals 0.05. The proportion of earliness ratios that were <0.05 was determined separately for each suite/service/day of the week combination, as shown in Figure 3. The filled circles show combinations with proportions whose 95% CI did not encompass the expected proportion of 0.05, because the values were either too low or too high. The 95% CI should include 0.05 for 95% of the combinations. Only 5% of combinations should have proportions that differ from 0.05. Instead, 34% of combinations had proportions whose 95% CI did not include 0.05. This result shows that 0.05 lower prediction bounds should be computed separately for each suite/service/day of the week. A single overall lower prediction bound derived from all suite/service/ days of the week combined does not accurately reflect values at the lower tail of the distribution for 34% of combinations.
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Determining the Number of Historical Cases to Use
Table 1 and Figure 4 show that 94% of cases belong to combinations with at least 99 points. In addition, 80% of cases belong to suite/service/day of the week combinations with a minimum of 199 cases.
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Since small sample sizes are not a limitation, lower prediction bounds for the earliness ratio can conveniently be determined using any number of cases ranging from 19 to 199. Use of only 19 cases will result in relatively large imprecision (30), increasing patient waiting times on the day of surgery (Table 1) and increasing variability in estimates from one month to the next. An advantage in using n = 39 historical data points is that the 0.05 lower prediction bound is equal to the second smallest value, and thus 1 possible outlier is excluded (30). Increasing the number of historical data points even more will result in more precise estimates by trimming more outliers. However, use of too much data may cause biased estimates due to seasonal variations, turnover in surgeons, and changes in the types of procedures performed (30).
Table 1 shows the results of an "out of bounds" analysis using different numbers of historical data points ranging from 19 to 199. The analysis determined the frequency at which a subsequent case had an earliness ratio less than the 0.05 lower prediction bound of the historical cases. Frequencies were all very close to 5.0%, demonstrating that, for each sample size, the model tested was valid. Use of only 19 observations could easily have generated a different out of bounds rate if, for example, the data were highly correlated over the time frame studied. Outlier points could have also increased the rate. We did not know that 19 points would be sufficient until we performed this analysis. The out-of-bounds rate was not improved by using 199 instead of 19 historical cases.
Any number of cases greater than 19 can theoretically be used, because differences in the resulting prediction bounds will average out over time. Earliest possible start times posted on our website are based on the results of only a single determination made once a month, however. Thus, some variability will occur from month to month. To test the sensitivity of this variability to differences in the number of cases used, patient waiting times and coefficients of variation were estimated for the series of 0.05 lower prediction bounds that were calculated for each suite/service/day of the week combination.
Table 1 shows patient waiting times for different numbers of historical cases. When the number of historical cases was increased from 19 to 199, average patient waiting time was reduced by 14 min. Thus, as expected, an increase in the number of historical cases resulted in a small but statistically significant reduction in the patient waiting times. Long waiting times experienced by a few patients, which could be a cause of significant dissatisfaction for those patients, would be reduced.
As the number of historical cases was increased, the coefficient of variation of the lower prediction bounds decreased significantly (Table 1). Scatter among estimates for the prediction bounds was reduced monotonically as the number of cases increased. Thus, use of more historical cases will reduce variability between monthly estimates of the prediction bounds.
Earliness Ratio Does Not Vary by Time of Day
To verify the validity of the earliness ratio for these calculations, we confirmed that earliness ratios did not vary by time of day. Cases starting later in the day may be moved forward many hours, while those starting in the morning can only be moved forward a limited amount of time. Figure 1 shows that the earliness ratio corrects for these differences. Figure 5A shows earliness ratios <1.0 for orthopedics on Wednesdays. As expected from the theory presented in Appendix 1, the ratios show no evidence of trend throughout the day based on the scheduled time of the procedure. Figure 5B shows the same data separated into four intervals of scheduled times. For each time interval, 0.05 quantiles and corresponding 95% CI were calculated from the observed ratios. The CI for the quantiles overlap in each time interval, indicating no evidence of significant variation based on time of day. Similar results were obtained with General Surgery-Adult (Fig. 5C), and for all other services with sufficient numbers of cases.
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Choice of Lower Prediction Bound
We chose 5% for the percent of time that staff should wait for patients. Values of 2% and 10% were also tested to determine their impact on predictions for the earliest possible start times of cases. For waiting 2% of the time, a minimum of 49 historical cases instead of 19 were needed for each suite/ service/day of the week combination. Only 2 of 53 combinations from the MAIN surgical suite did not meet this criterion. For waiting 10% of the time, only nine historical cases were needed. An additional eight suite/service/day of the week combinations had sufficient numbers of cases.
Differences in earliest possible start times for 0.05 vs 0.02 or 0.05 vs 0.10 lower prediction bounds increased in a linear fashion throughout the day. Patients scheduled early in the day would barely be affected, while those scheduled for late afternoon would be affected the most. At 4:00 pm, when the time differentials were nearly their greatest, patients needed to be ready 1 h 32 min earlier when staff waited for patients 2% of the time instead of 5%. Patients could be ready 53 min later when staff waited for patients 10% of the time instead of 5%. The average patient waiting time of 1 h 53 min for 5% waiting and n = 59 historical cases was increased to 2 h 5 min for 2% waiting or decreased to 1 h 40 min for 10% waiting. The value of 5% is an reasonable balance between staff waiting and patient waiting, especially for those patients scheduled in the late afternoon who may already be asked to arrive at the hospital several hours before the scheduled start times of their cases.
DISCUSSION
Prior methods for determining when patients should be ready on the day of surgery have depended on knowledge of the types of procedures being performed earlier that day in the same OR (1,4,6). Such data are available on the day of surgery and are useful for providing updated information. Detailed data are generally not available days in advance, however, and thus these prior methods cannot usually be applied before the day of surgery.
We have developed a much simpler technique that can be used days in advance for determining when patients should be ready on the day of surgery based on the earliest possible start time of each case. The earliest possible start time is calculated from the scheduled and actual start times of historical cases performed by the same suite, service, and day of the week. This method is simple to use and requires only a limited amount of historical data. While 99 historical cases by the same suite, service, and day of the week are desirable, a minimum of 19 will suffice. The technique is thus useful and practical. We currently use this method not just for surgery, but also for procedures outside the OR that involve anesthesia, such as diagnostic radiology.
The results section demonstrates face validity (the approach seems reasonable), content validity (the method is appropriate for the ambulatory and same day admission patients for whom it is intended), construct validity (a theoretical basis for the method has been established), external validity (results are not dependent on the number of historical cases used, one sample of historical cases can be generalized to other cases), convergent validity (earliest possible start times compare favorably to the times patients are currently told to report to the hospital), and reliability over time. We have not, however, established predictive validity. We do not yet know how often staff will have to wait for patients once these methods are used universally for all surgical services at our hospital.
The number of historical cases to be used in the calculations is not crucial. Use of too few historical cases may yield inaccurate predictions if successive earliness ratios are correlated. If a morning case started much earlier than scheduled on one particular day, then subsequent cases performed that same day are more likely to start early. Results could thus be distorted by a single outlier day if too few historical cases are used. Large numbers of historical cases improves precision, reducing the coefficient of variation of lower prediction bounds and producing a slight reduction in the patient waiting times (Table 1). We consider 99 cases to be a reasonable balance, although fewer are adequate. If data on 99 cases are not available, we use the largest number possible that does not require interpolation, either 79, 39, or 19 cases. Initial calculations involving numbers of data points in between these near multiples of 20, with interpolation between points, showed that use of every available historical data point was not necessary. The method could thus be simplified to use N cases, where 0.05 (N + 1) is an integer.
Use of historical cases collected over too long a time period may yield misleading earliness ratios if data were collected before changes in scheduling policies or surgical personnel. The time frame from which historical data are derived can be adjusted to account for unique situations that may alter start times, such as the departure of a surgeon who consistently underestimated case durations. Since OR allocations and service-specific staffing should be adjusted at least once a year (35), and they affect case scheduling, we recommend that lower prediction bounds be calculated using data collected during a time period of no longer than 1 yr.
Implementation Using Website
In addition to developing the mathematics necessary to test the validity of this method, we implemented these calculations at our hospital for surgical cases and for diagnostic and interventional radiology procedures involving general anesthesia. Clerks currently use our website www.CaseDuration.com (accessed November 11, 2006). They choose the patient's age group, anesthetizing location (suite), surgical service or diagnostic procedure, date, and scheduled procedure time. The server uses this information and accesses a lookup table to determine the time that the patient should be ready and the time the patient should report to the hospital. The patient is also told when to begin fasting and when last to drink clear liquids. The lookup table is updated once a month based on the past 1 yr of data. If 99 cases are available for a given suite/service/day of the week combination, then the fifth smallest earliness ratio is used as the lower prediction bound. For smaller numbers of cases, the largest number possible that does not require interpolation is used, 79, 39, or 19 cases.
We are able to use a public website that is not password protected because each case does not have to be identified uniquely. Thus, patient-identifiable information that might be confidential in nature need not be entered. In addition, the web site does not indirectly reveal information about the number of cases performed over time by a specific surgeon or the number of rooms staffed simultaneously by a single surgeon.
Similar calculations could be performed for other hospitals or facilities using data on scheduled and actual start times from those hospitals. Each hospital could have its own customized website to display times that patients should be ready for surgery and times that patients should report to the hospital. These times would be unique for each hospital or surgery center, depending on how often cases are started earlier than scheduled. Alternatively, data for other facilities could be incorporated into www.CaseDuration.com, with users selecting the facility of interest from a drop-down box or series of radio buttons.
Use of 0.05 Lower Prediction Bound
Before implementation of this method, staff waited for patients who were not available 5.0% of the time. Specifically, patients entered the holding area after the nurses in the OR were ready for the patient 5.0% of the time (95% confidence interval 4.7%–5.3%, n = 13,277 non-inpatient cases). Without making a deliberate effort to do so, the hospital had already achieved the goal of having patients ready 95% of the time when a case could be started earlier than scheduled. The finding that staff waited for patients 5% of time represents the results of an unintentional revealed preference study. This percentage, found to be appropriate from a societal perspective (1), matches the value the hospital chose without making a deliberate and conscious decision. These results suggest that our choice of 5% waiting is reasonable.
However, the observed waiting percentage of 5% represents waiting due to all causes, not just the patient being unavailable or not being NPO. OR staff may have waited for the patient because paperwork had not been completed or because lab tests had not been run. Thus, determining patient report times and NPO times to produce a 5% probability of waiting may result in a slight increase in the incidence of waiting from all causes combined.
Limitations
Data cannot be pooled for the various suite/service/ day of the week combinations. Earliness ratios, lower prediction bounds, and uncertainties in case durations may vary considerably based on the types of cases performed. Thus, calculations of patient report times and the organization of the website are slightly more complex than if all surgical services could be pooled together.
Surgical services form the basis for OR allocations on a short-term basis and thus the scheduling of cases into ORs (3–5). At the studied hospital, the services are large groups of surgeons (e.g., orthopedics). We do not know if our method would be valid for a facility at which services are individual surgeons. Surely the concept of a public website would not apply.
The calculations might be less accurate for determining the earliest possible start time of a case if that case were scheduled into allocated, but otherwise unused, time assigned to another service. For example, the earliest possible start time for an oral surgery case scheduled at 2 pm to fill an open slot at the end of a series of general surgery cases should be based on the historical start times of general surgery cases, not oral surgery cases. Updates provided on the day of surgery would be useful for providing more accurate estimates of case start times (1,4,6).
Combining data for a single suite/service/day of the week may result in systematic bias if surgeons within that combination are heterogeneous with regard to the accuracy of case duration predictions. At the study hospital, the default duration for each case was the mean duration of historical cases (5) classified by scheduled procedure(s), surgeon, and type of anesthetic (23,24). The implication is that no surgeon always takes more time or less time than scheduled. At a hospital that does not use historical data for scheduling each case, this method may not apply.
Since calculations were based on actual start times of historical cases, earliness ratios may have been skewed by the censoring of cases that were not done as early as desired because the patient was not ready. In other words, sample bias could have occurred due to the elimination of cases that would have had very small earliness ratios, except that the staff had to wait for the patient. In the MAIN tertiary suite, the patient entered the holding area after the nurses in the OR were ready for the patient 5.0% of the time (see above). Thus, some censoring did occur, although the effect was minimal. Results were slightly skewed toward elimination of very low earliness ratios. The number of long patient waiting times, in which patients would have been told to be ready for their surgery several hours earlier than scheduled, was thus reduced. This bias is not necessarily bad. It does illustrate, however, that historical data are not appropriate for determining when patients should be ready on the day of surgery if historical start times do not accurately reflect desired start times for the future.
Situations in which 0.05 lower prediction bounds for earliness ratios are inflated due to data censoring are unlikely to occur in practice. Censoring, with elimination of the lower tail of the distribution, would be expected only if staff frequently waited for patients who were not available. Hospitals tend to lean toward the other extreme, with policies that direct all patients to be NPO after midnight and report very early in the morning, regardless of the scheduled start time of their procedure (16–20). In many countries, patients are still asked to check into hospitals the evening before their surgery. This assures patient availability in the event cases can be started earlier than scheduled.
The method we studied is not the best choice to provide updates on the day of surgery or fill holes in the surgery schedule (1,4,6,36). When the patient arrives on the day of surgery, a preceding case may be underway. Since the type of procedure being performed is known, Bayesian methods should be used to estimate the waiting time for that patient (6). Bayesian methods will consider procedure-specific information for the case in progress, and will also consider the surgeon's expert estimate for the time remaining. If the preceding case is expected to last longer than 2 h, the patient can continue to drink clear liquids if desired. If one or more additional cases are to be performed before this patient's case, Monte-Carlo simulation will provide the most accurate updated information (4,36). Because the information used is procedure and surgeon specific, waiting times will always be shorter and more accurate than those obtained using the methods described in this article.
Inevitably, regardless of the accuracy of case start time predictions, some cases will have to be postponed because patients did not follow NPO instructions. Updates on the day of surgery, however, will help reduce the impact of last-minute changes in the schedule due to postponement of cases.
CONCLUSION
We have developed and validated a simple method for determining when patients should be ready on the day of surgery based solely on the scheduled start time of the procedure and historical data from other cases of the same suite/service/day of the week. Patient instructions can be accessed via a publicly available website without compromising confidential or sensitive information about individual patients or specific surgeons. The methods are designed to be used before, but not on, the day of surgery.
Several statistical models have been applied to the distribution of surgical procedure times, including log normal (8,37–40) and three-parameter log normal (7,41). Surgical times are not normally distributed, but are skewed to the right. When classified by surgeon and scheduled procedure(s), they are reasonably well fit by a two-parameter log normal distribution, where the logarithms of the surgical times are normally distributed with a mean µ and standard deviation 
(1,6). Prediction bounds estimated from two-parameter log normal distributions are highly accurate (1,4,6,36,40). A three-parameter model provides a better fit when cases are classified by the procedure(s) that is actually performed (7,41), rather than the scheduled procedure(s). The three-parameter model adds a location parameter that describes shifting of the data in the event that surgical times must always exceed some minimum value, such as 15 min. A constant value is subtracted from each surgical time before taking its logarithm. The two-parameter model without the shift includes procedures that are terminated prematurely (e.g., opening with immediate closure because of metastases).
A recent study involving the durations of anesthetics for MRI and CT (32) illustrates the theoretical justification for development of the earliness ratio of Expression 1.
MRI and CT data are used for several reasons. (a) Sample sizes for a single type of procedure are very large. Few surgical procedures have such large sample sizes (26). (b) Variability in case durations based on surgeon and type of anesthetic is irrelevant (23). (c) Scheduled procedure times are known. (d) A detailed analysis of the shape of the distribution of anesthesia times has already been published (32). (e) We are aware of no dataset from the OR that satisfies these four criteria.
Anesthesia times for MRI and CT cases follow two-parameter log normal distributions centered at the expert estimate for the duration of the case (32). This means that the 0.05 lower prediction bound for the anesthesia time of the ith case is (32):
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where Zi is the expert estimate for the duration of the case, s is the standard deviation of all N observed cases, and T–1[N – 1,
] is the percentile (0.05) of the cumulative Student's t distribution with N – 1 degrees of freedom. Because the anesthesia times are log normally distributed, the term
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has a mean of zero, as shown previously (32).
We now extend these results to determine the earliest possible start time for the next case. Suppose that the ith case were a first case of the day. Then
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would follow the same normal distribution with a mean of zero.
Adding a turnover time of 30 min,
actual finish time of the ith case + 30 min = actual start time of the next case.
As a result,
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also has a normal distribution with a mean of zero. The expert estimate for the start time of the next case is the same as its scheduled start time (6,32).
Expression 5 is simply the logarithm of the earliness ratio of Expression 1. Since anesthesia times for case durations follow two-parameter log normal distributions in practice (1,6,36,40), the actual start times of subsequent cases should follow a three-parameter log normal distribution (7,41), where the turnover time is the third parameter.
The above derivation provides a rationale for using the earliness ratio and Expression 2 to provide an estimate for the earliest possible start time of a case. The derivation assumes that each case in question is preceded by a single case whose duration follows a two-parameter log normal distribution. Expression 5 shows how the earliness ratio can be interpreted as if there is one long virtual case preceding the case of interest. Using the 0.05 prediction bound for the earliness ratio is then analogous to using the 0.05 prediction bound for the duration of the one preceding case.
Although the concept of a virtual case provides an intuitive explanation for the earliness ratio, this interpretation is not necessary for the earliness ratio to provide a valid model for determining the earliest possible start time of a case. The usefulness of the ratio is not predicated on any assumptions about the statistical distribution of the duration of the preceding case(s) performed in the same OR.
To describe the impact of choosing 0.05 for the lower prediction bound, we used MRI data illustrating Expression 5 above (taken from Fig. 2 of Ref. 32, µ = –0.01 and = 0.36). We assumed the expert estimate for the duration of the MRI procedure was 1 h. If the cost of having staff wait for a patient is equal to 19 times the value of the patient's time, then staff should wait for the patient for only 5% of cases. Patients should wait for staff for 95% of cases. From Expression 4, patients should be ready 37 min before the end of the preceding MRI. We generated a random series of 64,000 MRI case durations that were log normally distributed. Total patient and staff waiting times and the costs involved were totaled across all simulated case durations. The cost of patient waiting was only 86% of the total costs. However, patient waiting time was 99% of the total waiting time. Thus, the 5% probability of staff waiting results in a relatively greater burden for staff in terms of cost, but a smaller burden in terms of waiting time.
For MRI alone, Expression 5 follows the perfect bell shape of a normal distribution (P = 0.69) (Fig. 2 in Ref. 32). However, Figure 6 shows that the earliness ratios are not log normally distributed for CT and MRI combined (P < 10–6 by Lilliefors' test). This result is expected. The combination of two log normal distributions is not log normal. The bell-shaped curve in Figure 6A is not symmetric, and the probability plot of Figure 6B is not linear. The finding that the earliness ratios of Expression 1 do not follow log normal distributions, meaning that the logarithms from Expression 5 do not follow normal distributions, does not mean that earliness ratios are inappropriate for calculating earliest possible start times. It simply means that the statistical methods used to calculate 0.05 prediction bounds should be distribution-free. The methods should not be contingent on the shape of the distribution. Expression 3 is not appropriate for determining a lower prediction bound for the earliness ratio when the distribution is not log normal. The nonparametric method of Expression 6 in Appendix 2 should be used instead.
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For the earliness ratio to be a useful and valid metric for determining the earliest possible start time of a case, the ratio should not vary systematically throughout the day. Figure 7A shows a plot of the ratio as a function of the scheduled start time for each MRI and CT case. We are primarily interested in ratios <1.0, which represent cases that started earlier than scheduled. Figure 7B shows those ratios <1.0 on an expanded scale. No trend throughout the day is apparent.
The 0.05 lower prediction bound for the earliness ratio of pooled data was determined using a nonparametric method. This method is not dependent on the shape of the statistical distribution of earliness ratios. Given a series of earliness ratios xi from the N most recent historical cases of the same combination of suite/service/day of the week, the ratios are first sorted in ascending order, x1, x2, ..., xN, where xi < xi + 1. The expression 0.05 · (N + 1) is then evaluated to yield an integer portion I and a fractional portion F.
If 0.05 · (N + 1) is exactly equal to an integer I, then the 0.05 lower prediction bound equals xI. For example, for n = 199 historical cases, the 0.05 lower prediction bound is the 10th smallest value. For n = 59 cases, it is the third smallest value. For n = 39 cases, it is the second smallest value, and for n = 19 cases it is the smallest value. For other numbers of cases with N > 39, the lower prediction bound equals (29):
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For 19 < N < 39, the lower prediction bound is:
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Expression 6 contains three terms, each considered an order statistic. Interpolation among three adjacent order statistics (29) reduces the coverage error relative to other methods of calculating prediction bounds, such as those used previously for OR information system data (1,40). Expression 6 is nonparametric in that it does not depend on the values of ratios above or below xI–1, xI, and xI+1. Data on either side of these points are important only for determining which ratio is xI. The precise magnitude of the other points does not matter as long as they are less than xI–1 or greater than xI+1.
Pooling of earliness ratios for different types of procedures and different surgeons based on suite, service, and day of the week overcomes a major limitation in the availability of sufficient numbers of data points. Although earliest possible start times should ideally be determined from the expected duration of preceding cases involving the same type of procedure(s) and surgeon, data on even 19 historical cases are rarely available (1,26). For decision-making on the day of surgery, this limitation is addressed by using a Bayesian method, which supplements the historical data with other available information (6). For decisions before the day of surgery, we show in this article that pooling of ratios is an acceptable method.
Footnotes
Accepted for publication March 12, 2007.
Dr. Dexter, Section Editor for Economics, Education, and Policy, was recused from all editorial decisions related to this manuscript.
A calculator to perform the analyses described in this paper and generate patient ready times for the study hospital is currently available (November 11, 2006) on-line at www.CaseDuration.com, a site of the Division of Management Consulting of the Department of Anesthesia, University of Iowa.
An abstract of this work was presented at the annual meeting of the Institute for Operations Research and the Management Sciences (INFORMS) in Pittsburg, PA, November 5, 2006.
REFERENCES
This article has been cited by other articles:
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  F. Dexter, Y. Xiao, A. J. Dow, M. M. Strader, D. Ho, and R. E. Wachtel Coordination of Appointments for Anesthesia Care Outside of Operating Rooms Using an Enterprise-Wide Scheduling System Anesth. Analg., December 1, 2007; 105(6): 1701 - 1710. [Abstract] [Full Text] [PDF]
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