^{1}

^{1,2}A lower prediction bound for the OR time of a case is the value that will be exceeded by the next randomly selected case of the same type at the specified rate. There is a 5% chance that the OR time of a case will be briefer than its 5% lower prediction bound.

^{1,2}

*e.g.*, their 90% upper prediction bounds).

^{1,3}

*i.e.*, the one with the microscope) should be

^{1}the add-on case performed in OR 2. That decision is good provided the case in OR 1 will finish before the first case of the day in OR 2. That probability depends not only on the expected OR time of the cases, but also on the uncertainty of the estimates.

^{1,4}

^{1,5}A brief mediastinoscopy should be performed before esophagectomy. The unpredictability of the preceding case in an OR can be quantified by its probability of finishing more than 1 h late.

^{6}

^{1}For example, if the 90% upper prediction bound for the OR time of a newly scheduled case is 3.0 h, the actual probability of that case having an OR time longer than 3.0 h is likely somewhere between 90% and 91%. The calculated probability of one case lasting longer than another case matches the actual probability within 1%.

^{1}For example, if the calculated probability that one case will take longer than another case is 67%, the actual probability is likely somewhere between 66% and 68%.

^{6}This limitation is commonplace. Among outpatient cases in the United States with an anesthesia provider, 20% were of a procedure(s) performed four times or less per workday

*nationwide*, and 36% of cases were of a procedure(s) likely performed an average of less often than once per facility per year.

^{7}At private hospitals, there can be more than 5,600 surgeon preference cards (

*i.e.*, at least 5,600 combinations of surgeon and procedure(s)).

^{8}At academic hospitals, there can be more than 13,000 surgeon preference cards.

^{9}More than 75% of procedure(s) can be performed just once or twice annually.

^{10}More than half of cases can be of a procedure(s) scheduled by the surgeon less than three times per year.

^{11}

^{2,3,6,4}methods. The more cases performed per OR per day, the less useful the existing science is.

^{2,12,13}Despite some encouraging results,

^{2,12,13}these methods have problems in implementation.

^{10}provides only modest gains, because the problem is generally not that a surgeon schedules a procedure(s) that usually is performed by another surgeon at the facility, but rather that many procedure(s) are rare.

^{10,14}There are thousands of Current Procedural Terminology or International Classification of Diseases’ procedure codes, and surgeons persistently produce new technologies.

*versus*a mean of 2.0 h and 90% upper prediction bound of 4.0 h for 2 cases of the surgeon scheduling the new case. Qualitative instructions cannot reconcile such results.

^{15}Some surgeons are consistently slower or faster than others.

^{15}Therefore, excluding classification by surgeon appropriately reduces face validity of recommendations.

#### Methods

##### Data Set Used

^{2,3,6,4,10}can be compared. Scheduled (

*vs.*actual) procedure code(s) were used, because for a future case for which a prediction bound is being calculated, only the scheduled procedure(s) would be known.

^{16}

##### Lower and Upper Prediction Bounds

*Xk*refer to the natural logarithms of OR times classified by

^{15}the

*k*th combination of surgeon, scheduled procedure(s), and anesthetic,

*k*= 1, 2, …,

*p*. For brevity, we henceforth refer to each of the

*p*combinations as “surgeon and procedure(s).” The

*nk*previously observed (historic) OR times in hours for the

*k*th combination of surgeon and procedure(s) are exp(

*x*

_{k1}, exp(

*x*

_{k2}, … exp(

*x*

_{knk}. The sample mean of the

*nk*historic data

*x*

_{k1},

*x*

_{k2}, …

*x*

_{knk}. equals x̄

_{k}, and the sample variance equals μ̂

^{2}

_{k}. The logarithm of the scheduled OR time in hours for the next case is

*xs**

_{k}. The objective is to predict the OR time of the next case

*x*k*. We use * to represent the next case throughout the article.

Equation 8 Image Tools |
Equation 17 Image Tools |

*t*

_{0.05, 2a*k}percentile of the

*t*distribution with α*

_{k}degrees of freedom (

*e.g.*,

*t*

_{0.05, 2α*k}= −1.65 for large α*

_{k}). The 90% upper bound is calculated with.

Equation 1 Image Tools |
Equation 3 Image Tools |
Equation 4 Image Tools |

*n*

_{k}OR times become available for the

*k*th surgeon and procedure(s). Equations 3 and 4 show how the sample variability revises the prior estimates of α and β, which affect the variance component in the predictive distribution (

*i.e.*, the square root term in equation 1). In addition, the parameter τ expresses how the prior (logarithm of the scheduled OR time)

*xs**

_{k}and the historic mean OR time x̄

_{k}should be combined. Equation 1 shows that has a substantial influence for

*n*

_{k}< τ but that its influence diminishes with larger sample sizes. For large sample sizes, the Bayes approach of equation 1 converges to the result previously reported and given in equation 8.

##### Testing for the First New Case Taking Longer Than the Second New Case

Equation 18 Image Tools |
Equation 19 Image Tools |
Equation 20 Image Tools |

##### Probability That a New Case Will Finish More Than 1 h Late

*L*hours late equals

*t*(2α*

_{k})refers to the

*t*distribution with 2α*

_{k}degrees of freedom. For testing, we use

*L*= 1 h.

##### Mean Operating Room Time

##### Testing the Accuracy and Usefulness of the Estimates and Predictions

*xs**

_{k}) was calculated.

*i.e.*,

*k*) was determined.

*k*th combination minus the one case selected at random.

_{k}, α*

_{k}, and β*

_{k}were determined from equations 2, 3, and 4, respectively.

^{4}

#### Results

##### Lower and Upper Prediction Bounds

*i.e.*, were conservative by 0.3%). The 5% lower bounds exceeded 4.9% of the actual OR times. Excluding the 33% of cases used to estimate the parameters (

*i.e.*, only studying surgeon and procedure(s) combinations with fewer than 29 historic cases), the 90% upper bound accuracy was 10.2%, and the 5% lower bound accuracy was 5.5%.

Fig. 1 Image Tools |
Fig. 2 Image Tools |

*versus*trying to use the scheduled OR time itself plus or minus some value (

*e.g.*, 1.5 h).

^{2}Nevertheless, it would be a poor choice to skip use of the historic data and instead simply take a percentage of the scheduled OR time. Figure 1 shows that the variance differs among combinations of surgeon and procedure(s). Using the same percentages for all combinations results in an overall accurate coverage rate, but a rate that is too high or low for many combinations. Figure 2 shows this result in a histogram of the ratio of the 90% upper Bayesian prediction bound to the scheduled OR time for all cases.

##### Testing for the First New Case Taking Longer Than the Second New Case

##### Probability That a New Case Will Finish at Least 1 h Late

Equation 5 Image Tools |
Fig. 3 Image Tools |
Fig. 4 Image Tools |

^{1,5}Theoretical justification for use of this heuristic for ORs has not previously been described. When historic data are absent or ignored, in equations 2–4. Equation 5 shows that, then, each increase in the scheduled OR time results in an increase in the probability that the case will finish more than 1 h late. Figure 3 shows the result graphically. The equations apply for any other interval desired (

*e.g.*, case finishing 15 min late).

*i.e.*, set equal to 0, while our estimated values for α and β were used). The predicted percentage of cases finishing more than 1 h late was 13%. The actual percentage was 15% (fig. 4).

##### Mean Operating Room Time

^{1,17,18}

*P*< 0.0001) more accurate on both criteria, the differences were too small to be of practical advantage when focusing on the mean OR time.

#### Discussion

*e.g.*, the surgeon or an expensive piece of equipment), and (4) sequencing a surgeon’s list of cases.

##### Benefits

^{1,17,18}Provided OR staffing and allocations have been made based on finding the optimal balance of underutilized and overutilized OR time, such decisions can then reasonably be made using the expected mean OR time of each case.

^{1,17,18}When there are at least two historic cases of the same combination of surgeon and procedure(s), the incremental reduction in overutilized OR time has been shown to be negligible from increasing the accuracy of the prediction of the OR time of a new case

*versus*using the mean of the OR times of the historic cases.

^{1}The reason is that decisions to reduce overutilized OR time are affected much more by the day-to-day variability in the total hours of cases than by variability in OR time prediction.

^{1}People generally work late because of extra cases, not underestimation in the time to complete each case, because rarely does the underestimation change whether and when the case is performed.

^{19}In the current article, we show that the scheduled OR time alone is nearly as good a predictor of the expected mean OR time of a new case as is the Bayesian method. Therefore, the incremental reduction in overutilized OR time from further improvements in predicting OR times is likely negligible.

*whether*a case is scheduled into an OR allocation on a specific day,

^{1,18}some surgeons resist reduction in their autonomy in choosing the value. Such organizational resistance to change is irrelevant to the Bayesian method, because the scheduled OR time alone can still be used for that purpose. Second, the premise of relying on expert judgment when there are few data

*versus*predominantly historic data when there are many data has face validity to hospital audiences without statistical training. Third, the information system using the Bayesian method can always provide an estimate or decision recommendation, eliminating the need to educate scores of clerks, physicians, and nurses as to the interpretation of absence. Fourth, equations 2–4 show that the information system need not use the raw data for calculation, only the sample size, mean, and variance (

*i.e.*, one record for each combination of surgeon and procedure(s)). Finally, we speculate about the advantage of combining the Bayesian method with new real-time methods of collecting OR time data (

*e.g.*, from vital signs).

^{20}An information system can now function autonomously, providing recommendations and progressively increasing its usefulness as its historic data increases.

##### Limitations

^{21}Without adjustment, the Bayesian method may be inaccurate, and the argument of the preceding paragraph will be incorrect. Bias can be monitored and incorporated into the equations in the Methods simply by changing each listed to

*xs**

_{k}to (

*xs**

_{k}+ Δ), where Δ is the overall or specialty-specific proportional bias. Feedback can alternatively be provided immediately at the time when the case is scheduled if the Bayesian estimate for the expected mean OR time differs substantially from the scheduled OR time.

Equation 7 Image Tools |
Equation 9 Image Tools |
Equation 10 Image Tools |

Equation 11 Image Tools |
Equation 12 Image Tools |
Equation 13 Image Tools |

Equation 14 Image Tools |
Equation 15 Image Tools |
Equation 16 Image Tools |

##### Appendix

##### Previous Studies Based on Assumption of Log Normal Distribution

*x*

_{k}, the natural logarithm of OR time, follows a normal distribution:

Fig. 5 Image Tools |
Fig. 6 Image Tools |

^{13,22}For example, figure 5 shows the natural logarithms of OR times for a surgeon’s 105 strabismus surgery cases with one muscle (chi-square test of normality,

*P*= 0.71; Lilliefors test of normality,

*P*= 0.69). Figure 6 shows the corresponding probability plot. Figures 5 and 6 of Dexter and Traub

^{2}show data for laparoscopic cholecystectomy, and dependence of the skewness of these distributions on.

*X**

_{k}, can be obtained accurately

^{1,2}by taking

_{k}and μ̂

^{2}

_{k}are the sample estimates from the

*n*

_{k}historic data. The 90% upper bound is calculated by

*t*

_{0.90.nk-1}substituting into equation 8 (

*e.g.*,

*t*

_{0.90.nk-1}= 1.28 for large

*n*

_{k}).

^{1,3,6}If

*n*

_{k}< 2, the prediction bounds μ̂

^{2}

_{k}cannot be calculated, because cannot be estimated from 0 or 1 cases. Furthermore, although the prediction bounds are valid for small

*n*

_{k},

^{2,3,6}they are often not useful being so wide.

*i.e.*, Student

*t*test with unequal variances).

^{1,4,23}We refer to the first new case with subscript 1 and the second new case with subscript 2, whichever combinations of surgeons and procedures(s) they are from. To test for the first new case being longer than the second new case, that is for exp(

*X**

_{1}) ≥ exp(

*X**

_{2}) let

^{23}

^{23}

*df*are not integers, we calculate the probabilities of the

*t*distribution from its relation with the Incomplete Beta Function.

^{24}Press

*et al.*

^{24}provide the computer code. The primary limitation to equations 9–12 is that to calculate μ̂

^{2}

_{1}and μ̂

^{2}

_{2}and

*n*

_{1}≥ 2 and

*n*

_{2}≥ 2.

##### Two Additional Assumptions for Inference and Prediction in the Absence of Historic Data

^{2}

_{k}and the other about

*M*

_{k.}

^{2}

_{k}follows an inverse gamma distribution:

*et al.*

^{13,22}in their investigations of the statistical distributions of OR times, we considered those combinations of surgeon and procedure(s) that were performed a moderate to large (

*n*

_{k}≥ 30) number of times. There were 302 such combinations of surgeon and procedure(s) corresponding to 21,541 cases. The probability plot in figure 1 shows that the assumption of the inverse gamma distribution is reasonable (chi-square test,

*P*= 0.51; Kolmogorov-Smirnov,

*P*= 0.88).

*M*

_{k}given ς

_{k}is normal with prior mean μ

_{k}and variance μ̂

^{2}

_{k}/τ:

*e.g.*, τ = 1 historic case), our prior information about

*M*

_{k}would be vague, making the incremental value of any historic data relatively influential.

*X*

_{k}given ς

_{k}is normal with prior mean μ

_{k}and variance ς

_{k}

^{2}+ (ς

_{k}

^{2}/τ). Combining terms, the variance equals ς

_{k}

^{2}·((τ +1)/τ). Bringing in the inverse gamma distribution for ς

_{k}

^{2}from equation 13, the predictive prior distribution of the logarithm of the OR time of the next case,

*X*

_{k}*, is a scaled Student

*t*distribution

^{25}:

##### Prior Values for Use in Inference and Prediction in the Absence of Historic Data

*xs**

_{k}, to be the prior value for μ

_{k}, and the prior prediction of

*xs**

_{k}. Figure 7 shows a histogram of the prediction errors, (

*xs**

_{k}–

*xs**

_{k}), for the 18,381 cases for which

*n*

_{k}≥ 2. The symmetric distribution around zero confirms that

*xs**

_{k}is a good prior value, because it provides an unbiased prediction of the logarithm of the actual OR time in hours. This implies that exp(

*xs**

_{k}), provides an unbiased prediction of the 50th percentile of (

*xs**

_{k}), which is slightly less than the expected (mean) value of (

*xs**

_{k}). We would therefore expect that the scheduled OR time would slightly, but significantly, underestimate the actual OR time. This was the finding reported in the final paragraph of the Results. In those analyses, the expected (mean) value averaged the 52nd percentile of (

*xs**

_{k}), with lower and upper quartiles of 51% and 52%, respectively.

*Var*(

*xs**

_{k}–

*xs**

_{k}) = 0.120. The variance of a

*t*distribution with 2α degrees of freedom equals

^{26}/(2α – 2). From the result of equation 15,

_{k}–

*xs**

_{k}), with x̄

_{k}and

*xs**

_{k}taking the places of

*M*

_{k}and μ

_{k}, respectively. When the variances from figure 1 were rounded to the nearest 0.05, the interval 0.05 ≤ μ̂

^{2}

_{k}< 0.10 included the most cases, 10,562, as well as many (156) different values of

*k*. Figure 8 shows a normal probability plot for The functional (normal) form of equation 14 is reasonable. From Strum

*et al.*,

^{22}the undulation around the straight line in figure 8 was the expected consequence of our data’s

*xs**

_{k}being in 15-min intervals, resulting in (x̄ –

*xs**

_{k}) taking a small set of values.

##### Updating the Estimates and Predictions Using Historic Data

^{2}

_{k}and

*M*

_{k}belong to the same family.

^{25}The Bayesian literature refers to such prior distributions as conjugate priors. Specifically, the posterior distribution of μ̂

^{2}

_{k}is again an inverse gamma distribution. The posterior of

*M*

_{k}and the posterior predictive distribution of

*x**

_{k}are scaled

*t*distributions, with revised parameters that update the prior values with the sample information:

##### Assessment of Sensitivity of Estimates to Prior Values of α, β, and τ

*n*

_{k}≥ 30

*versus*302 combinations with all the data. The estimated α = 1.70 and β = 0.089,

*versus*α = 2.32 and β = 0.142 with all the data. From these 7,405 cases, τ = 14.9 cases,

*versus*from all 21,541 cases, τ = 8.68 cases. Applying the α, β, and τ from 1996 and 1997 to all of the data, the 90% upper bounds were exceeded by 10.0% of cases. The 5% lower bounds exceeded 5.1% of cases. Therefore, the Bayesian method was insensitive to the choice of α, β, and τ. Furthermore, repeating with τ = 1.49 cases (an order of magnitude less), the results were the same, within 0.2%.

*i.e.*, 1996–1997 prior values were applied to 1998 data, and 1996–1998 prior values were applied to 1999 data). We assumed that the lookup table with the running totals of

*n*

_{k}, ∑

*x*

_{ki}, and ∑

*x*

_{ki}

^{2}for each combination of surgeon and procedure(s), for use in equations 2–4, is updated nightly. For example, to estimate prediction bounds for cases performed on Tuesday August 4, 1998, the running totals used were for January 1, 1996, through August 3, 1998. Then, the calculated 90% upper bounds were exceeded by 8.0% ± 0.1% of cases (n = 32,930). The 5% lower bounds exceeded 6.9% ± 0.1% of cases. Repeating using τ = 1.49 cases, results were the same, within 0.2%. Therefore, improvement in accuracy would rely on modifying the assumption of equation 13 of a common α and β for those combinations of surgeon and procedure(s) with little to no data, not those combinations with small to large numbers of historic cases. Those cases with little to no data are precisely those for which α and β cannot be estimated directly. Cited Here...