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Statistical Modeling of Average and Variability of Time to Extubation for Meta-Analysis Comparing Desflurane to Sevoflurane

Dexter, Franklin, MD, PhD*†; Bayman, Emine O., PhD*; Epstein, Richard H., MD

Section Editor(s): Dexter, Franklin

doi: 10.1213/ANE.0b013e3181b5dcb7
Economics, Education, and Policy: Research Reports
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BACKGROUND: The recovery profile of an ideal anesthetic or technique would be fast (e.g., mean of 5 min from end of surgery to extubation) with little variability (e.g., always 4–7 min). We used anesthesia information management system (AIMS) data to learn how to model the time from end of surgery to extubation. We applied that knowledge for meta-analyses of trials comparing extubation times after use of desflurane and sevoflurane.

METHODS: AIMS data studied were 32,792 cases performed by 95 surgeons that included tracheal intubation and extubation in the operating room (OR) and use of volatile anesthetic(s). Meta-analysis included the 29 randomized controlled trials through 2008 comparing extubation times with desflurane and sevoflurane. Percentage differences in means and standard deviations were studied using random effects meta-analysis and a Bayesian method.

RESULTS: Times to extubation were better fit by (skewed) Weibull distributions than by (symmetric) normal distributions. Drug choice had nearly equally proportional effects on the means and standard deviations of extubation times, as shown by unchanged coefficients of variation (P > 0.10 for 26 of 29 studies) and nonsignificant pooled difference in the coefficient of variation (sevoflurane − desflurane = −1%, 95% confidence interval [CI] −3% to 1%, P = 0.22). Applying these findings, desflurane reduced the mean extubation time by 25% (95% CI 17%–32%, P < 0.0001) and standard deviation by 21% (95% CI 16%–26%). To value the intangible costs (e.g., frustrated waiting surgeons) of prolonged extubation times, we considered the 15% of AIMS cases with times >15 min. These cases averaged 4.9 min longer times from out of the OR to the start of surgery of the surgeon's next case (95% CI 2.7–7.1 min, P < 0.0001). Reduction in the means and standard deviations by 20%–25% would likely reduce incidences of these prolonged extubation times by 71%–82% (95% CI 68%–84%).

CONCLUSIONS: Desflurane reduces the average extubation time and the variability of extubation time by 20%–25% relative to sevoflurane. The principal economic value of these end points is their reductions of direct (labor) costs of OR time. However, reductions in intangible costs of prolonged extubation are real, being associated with subsequent delays. Reductions in the average and variance of times to extubation can be interpreted and monitored in terms of corresponding expected 75% reductions in the incidences of prolonged extubation times by using desflurane relative to sevoflurane.

Published ahead of print October 9, 2009

From the *Departments of Anesthesia, and †Health Management and Policy, University of Iowa, Iowa City, Iowa; and ‡Department of Anesthesiology, Jefferson Medical College, Philadelphia, Pennsylvania.

Accepted for publication July 2, 2009.

Published ahead of print October 9, 2009

Franklin Dexter is the Section Editor of Economics, Education, and Policy for the Journal. The manuscript was handled by Steve Shafer, Editor-in-Chief, and Dr. Dexter was not involved in any way with the editorial process or decision.

Supported by the University of Iowa and Baxter Healthcare Corporation. Baxter physicians, scientists, and statisticians made recommendations about the study design before providing funding and reviewed the manuscript once written. They were not involved in the conduct of the study; collection, analysis, or interpretation of the data; or preparation of the manuscript. Drs. Dexter and Epstein previously performed research funded by Abbott Laboratories. Dr. Dexter receives no funds personally other than his salary from the State of Iowa, including no travel expenses or honoraria, and has tenure with no incentive program.

Address correspondence and reprint requests to Franklin Dexter, MD, PhD, Department of Anesthesia, University of Iowa, Iowa City, IA 52242. Address e-mail to Franklin-Dexter@UIowa.edu or web site www.FranklinDexter.net.

Randomized clinical trials are performed to compare recovery times of anesthetics (e.g., sevoflurane or desflurane), methods of IV drug titration (e.g., manual or based on effect site concentrations predicted from pharmacokinetic/dynamic models), and/or monitoring techniques (e.g., clinical assessment of anesthetic depth or processed electroencephalogram). The recovery profile of an ideal drug, method, or technique would be fast (e.g., mean of 5 min from end of surgery to extubation) and have low variability (e.g., always between 4 and 7 min). However, we are unaware of any study in which a statistical approach to quantify the variability of recovery times has been validated. Typically, there are many comparative studies with relatively few patients per group (Table 1).1–31 Thus, evaluation of differences in the variability of recovery would combine studies using meta-analysis techniques. The primary objective of our study was to perform a meta-analysis of trials comparing time to extubation after use of desflurane and sevoflurane (Table 1). To do so, we first used anesthesia information management system (AIMS) data to learn how to statistically model the times from end of surgery to extubation obtained in randomized clinical trials. We show that even though economic interpretation depends on the absolute reduction in times, quantitative pooling (meta-analysis) of the clinical trials' results needs to be performed on a percentage basis.

Table 1

Table 1

Differences among volatile anesthetics in operating room (OR) recovery times tend to be around 2 min (e.g., time to following commands after discontinuation of desflurane versus sevoflurane).32–34 The principal economic value of reductions in OR recovery times are achieved through these reductions in time per case.35 If an OR has more than 8 h of cases, then a 2-min time savings per case has a larger economic value than 2 min (e.g., saves the equivalent of 3 min of regularly scheduled time).35–39 However, many ORs routinely have fewer than 8 h of cases. Although generally no direct (labor) cost reductions would be achieved from faster recovery in those ORs (because staff are paid for the entire day, regardless of their workload), intangible (indirect) value may still be achieved from more predictable recovery (e.g., fewer frustrated surgeons complaining to hospital administrators). We used the AIMS data to test for association between prolonged extubation times and longer than average times from OR exit to start of surgery of the next case.

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METHODS

The relevant Institutional Review Board approved our retrospective study of fully deidentified AIMS and OR information management system (ORIMS) data as “exempt.” The AIMS data were used preferentially, supplemented by ORIMS data in the few40,41 situations in which events needed for analysis were not recorded by the anesthesia provider. Cases studied were all those in which tracheal intubation and extubation took place in the OR and an end-tidal concentration of desflurane ≥3%, isoflurane ≥0.6%, or sevoflurane ≥1% was measured at some time during the anesthetic.

The times of extubation and end of surgery were retrieved from 36,307 cases performed over a 3-yr period (see Ref. 41 for details including explanation of outliers). Differences were calculated and the sole method of data cleaning was to trim the 5% outliers symmetrically. Values equal to the trimming thresholds of 0.1 min and 29 min were retained, resulting in 34,524 cases performed by 234 surgeons. We studied the probability distributions of the times to extubation of the patients of the 95 surgeons who each had many42 (≥59) patients meeting the inclusion criteria. The patients of the surgeons studied represented collectively 95% of the extubation times (SYSTAT 12, Systat Software, San Jose, CA). Fits were performed by maximum likelihood estimation.

Prolonged extubation times prompt some surgeons to leave the suite and not return promptly to start the next case.43 Therefore, we studied the subset of the 32,792 cases followed by another case in the same OR by the same surgeon (n = 6544 cases of 94 surgeons). We tested for association between (A) the binary variable of whether extubation time was longer than 15 min (see Results) and (B) time from out of OR to start of surgery (i.e., skin incision or the equivalent).41 The analysis needed to take 2 issues into account. First, the dependent variable (B) had many outliers (e.g., 3 h). Second, behavior varies among individuals and typical cleanup, setup, and anesthesia induction times differ among subspecialties. Therefore, we needed to compensate for outliers and to stratify by surgeons. One way that we did so was by using the Wilcoxon Mann-Whitney test to compare (A) and (B). This test relies only on the ranks of the dependent variables, not their values, and can be calculated with stratification (StatXact 8, Cytel, Cambridge, MA). However, this test does not provide an estimate of effect size. Therefore, we also applied mixed effects modeling to 5% trimmed data,* with the independent variable (A) treated as a fixed effect and the independent variable of surgeons modeled as a random effect.

A previous meta-analysis comparing desflurane to sevoflurane found that desflurane reduced the mean time to extubation by 1.3 min.34 In addition, the distribution of mean differences among studies was not associated with patient age or duration of anesthesia.34 We included the additional (secondary) end point of time to following commands, because results comparing recovery may be sensitive to the selection of the end point.24 We updated the 2005 meta-analysis by including all articles in MEDLINE on December 31, 2008, with the following terms in any field: sevoflurane and desflurane and 1) recover* (where* is the wildcard character) or 2) time* and (extub* or [follow* command*]), with limits of “humans” and “randomized controlled trial.” No restriction was placed on language or type of article. The 56 articles were read and classified independently by the first and third authors, with one study inclusion mistake by one corrected by the other. There were 29 articles that included random assignment of patients to desflurane versus sevoflurane and reported means and either standard deviations or standard errors for at least 1 of the 2 end points. Review of the articles' citations identified 2 other15,18 articles that were not found by our MEDLINE search because they lacked abstracts. The total of 31 articles made 35 comparisons between agents (Table 1).

Differences in study results between sevoflurane and desflurane were tested using multiple statistical methods. First, the coefficients of variation (i.e., ratios of standard deviations to means) between agents were tested for each study using an asymptotic method44 and pooled by fixed effects meta-analysis. Second, the ratio of the standard deviations was tested using the Bayesian model described in the Appendix. Third, the means and standard deviations of the logarithms were calculated for each group by using Taylor series approximation.45,46 The difference between agents in the means of the logarithms was used to calculate the percentage difference in the geometric means. Pooling of the differences was performed by using random effects meta-analysis with variance among studies in differences estimated using restricted maximum likelihood estimation.47 The fixed and random effects meta-analyses were coded in Microsoft Excel using iterative calculations based on the worked example in Ref. 48.

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RESULTS

1. Use of AIMS Data from a Hospital to Develop Statistical Methodology for the Meta-Analysis

Among the trials in the meta-analysis, the 2 with the largest total sample sizes studied laparoscopic cholecystectomies (Table 1). Analysis41 of AIMS data showed that the times to extubation for laparoscopic cholecystectomies followed a distribution consistent with a Weibull distribution (Fig. 1). For readers not accustomed to the science of fitting probability distributions to times between events, such intervals often follow Weibull distributions. Weibull distributions are skewed, whereas normal distributions are symmetric.

Figure 1.

Figure 1.

Because most trials studied multiple surgical procedures and our exploration of AIMS data considered hundreds of procedures, we needed to expand beyond the study of individual procedures as in Figure 1. Therefore, to learn how to validly perform the meta-analysis, we studied the probability distributions of extubation times pooled by surgeon. We calculated the mean and standard deviation of the extubation times of the patients of each surgeon. The mean among surgeons of their patients' mean time to extubation was 9.2 min (Fig. 2). The mean among surgeons of their standard deviations of extubation times was 5.4 min. Figure 3 shows Weibull and normal probability plots of the times from the end of surgery to extubation of the patients of the surgeon with mean and standard deviation closest to these values. The circles are the data for the surgeon's patients. The straight line fit on the left shows a good fit to a Weibull distribution and the curved relationship on the right shows a (very) poor fit to a normal distribution. Given our relatively large sample sizes (mean n = 345) for fitting probability distributions, the nominal P value describing a reasonable fit to a set of data is P ≥ 0.05, with progressively smaller values indicating progressively poorer fits. Among all 95 surgeons, goodness of fit to normal distributions as assessed by χ2 fit tests were P > 0.05 for 8% of surgeons, P > 0.01 for 14% of surgeons, and P > 0.001 for 26% of surgeons (i.e., P < 0.001 for 74% of surgeons). In contrast, goodness of fits to Weibull distributions were P > 0.05 for 81% of surgeons, P > 0.01 for 92% of surgeons, and P > 0.001 for 99% of surgeons.

Figure 2.

Figure 2.

Figure 3.

Figure 3.

We consider the Weibull in more detail, because its parameters are not the mean and standard deviation, unlike for the more familiar normal distribution. The Weibull's 2 parameters are called “shape” and “scale.”48 We created Figure 4 to show how shape and scale can be interpreted. Figure 4 shows 2 Weibull distributions with shape parameter of 2, one with a scale parameter of 10, and the other of 12. Because the shape parameter is the same, the 2 distributions have the same coefficients of variation of 52%. In Figure 4, the Weibull distribution with a scale parameter of 12 has a mean of 10.6 min and a standard deviation of 5.5 min. An intervention (e.g., different volatile agent) that reduces the scale to 10 without changing the shape causes a 20% decrease in both the mean and standard deviation.

Figure 4.

Figure 4.

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2. Meta-Analysis of Coefficients of Variation of Time to Extubation

Table 2 and Figure 5 show that we could perform the meta-analysis while reasonably assuming that randomization to sevoflurane or desflurane does not affect the coefficient of variation. The coefficient of variation did not differ significantly (P > 0.10) for most (26 of 29) studies comparing time to extubation with sevoflurane or desflurane (Table 2). The pooled difference in the coefficient of variation was −1%. The fixed effects meta-analysis 95% confidence interval (CI) equaled −3% to 1% (P = 0.22) (Fig. 5).

Table 2

Table 2

Figure 5.

Figure 5.

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3. Additional Use of AIMS Data for Statistical Methodology

To proceed methodologically, we combined the findings of the preceding Results sections: 1) better fits of extubation times to Weibull than to normal distributions from a studied hospital, and 2) lack of appreciable changes in the coefficients of variation between sevoflurane and desflurane among the 29 clinical trials. Because surgeons perform diverse procedures, when the mean time to extubation of the patients of each surgeon is calculated, the means differ among surgeons (e.g., small means for breast surgery and long means for pancreatic surgery) (Fig. 2). If extubation times for each surgeon or study followed a normal distribution, desflurane could reduce the mean time to extubation by 1.3 min relative to sevoflurane34 both for surgeons and studies with small means and large means. Our 2 preceding findings show that the statement “desflurane reduces the mean time to extubation by 1.3 min relative to sevoflurane”34 poorly describes the decrease, because the value would differ if, for example, the reduction were from an extubation time of 15 min or 5 min with sevoflurane. A more meaningful description would be the percentage reduction in the mean time to extubation. Similarly, different surgeons have markedly different standard deviations of extubation times (Fig. 6). The surgeons with small standard deviations are generally those with brief mean extubation times (Spearman r = 0.78, 95% CI 0.70–0.87). Thus, there is little statistical meaning to pooling clinical trial results to test for the difference in the standard deviation of extubation times between desflurane and sevoflurane. What has greater meaning is to estimate the percentage reduction in the standard deviation.

Figure 6.

Figure 6.

Table 3

Table 3

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4. Meta-Analysis of Percentage Differences in Geometric Means and Standard Deviations

Multiple methods of meta-analysis were used to analyze the previous trial results comparing sevoflurane and desflurane (Table 1). Desflurane reduced the mean by 25% (95% CI 17%–32%, P < 0.0001) and the standard deviation by 21% (95% CI 16%–26%) (Fig. 7). From the above described properties of Weibull distributions and the finding of relatively unchanging coefficients of variation (Fig. 5), the estimate for the percentage reduction in the mean was approximately the same as for the percentage reduction in the standard deviation, and vice versa. Equivalent wording would be that the estimates for the reductions in both the mean and standard deviation were 25% by random effects meta-analysis and 21% by Bayesian analysis. Figure 4 shows expected differences in the probability distributions of extubation times between the 2 volatile agents. Using fixed effects analysis of reduction in the mean as a sensitivity analysis, the estimate differed little (26%), with moderate shrinkage of the 95% CI (23%–29%). The reductions in time to following commands were similar, with mean 19% (95% CI 9%–28%, P = 0.0002) and standard deviation 22% (95% CI 17%–27%) by random effects and Bayesian analysis, respectively.

Figure 7.

Figure 7.

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5. Use of AIMS Data to Value the Reduced Variability of Time to Extubation

The principal economic value to reducing the time to extubation is from the mean reduction in time per case. The process is reviewed in the Discussion section.35–39

We used the AIMS data from a hospital to study how to value the reduction in intangible costs achieved by the 20%–25% reduced standard deviation of the time to extubation with desflurane. We considered the 15% of AIMS extubation times longer than 15 min to be prolonged. Cases with prolonged (>15 min) extubation times had 4.9 min longer times from out of the OR to the start of surgery of the surgeon's next case (95% CI 2.7–7.1 min, P < 0.0001). Surgeons whose patients' times to extubation are often prolonged are those surgeons with large means and standard deviations (Fig. 8). Each surgeon's observed incidence of patients with prolonged extubation can be estimated from the mean and standard deviation. For example, the surgeon from Figure 3 whose cases had a mean of 9.27 min and standard deviation of 5.68 min had a Z score = (15 − 9.27)/5.68 = 1.01. The corresponding prediction for the % >15 min equals 100% minus the inverse of the cumulative normal distribution for that Z score. Specifically, for Z = 1.01, the prediction is 15.7%, close to the actual percentage of 15.3%. There was very high overall agreement between observed and predicted incidences of prolonged extubation times (Spearman r = 0.97, 95% CI 0.95–0.98; mean difference from observed value = 0.7%, 95% CI 0.2%–1.1%; mean absolute error = 1.8%, 95% CI 1.6%–2.1%). Reducing all surgeons' means and standard deviations by 20% would reduce predicted incidences of prolonged extubation times by 71% (95% CI 68%–73%). A 25% reduction reduces the predicted incidences by 82% (95% CI 80%–84%).

Figure 8.

Figure 8.

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DISCUSSION

Summary and Economic Implications

Relative to sevoflurane, desflurane decreased both the average time to extubation and variability of time to extubation by 20%–25%. These decreases can be interpreted and monitored in terms of their corresponding predicted 75% reduction in hospitals' overall incidences of prolonged extubation times. The economic value of these percentage reductions will vary markedly among facilities and anesthetics.

The principal financial benefit of reducing time to extubation is achieved through the reduction in direct OR costs. In the first section of our Results, we showed that quantitative pooling (meta-analysis) of trial results should be performed based on percentage reductions in extubation times. The steps are well understood to convert mathematically from the meta-analysis' estimated overall percentage reduction to the reduction in direct OR costs.35–39 The data used are for the specific facility and cases for which desflurane would be used.35,38 First, the percentage reduction needs to be converted to time in minutes. The reduction in direct cost will be largest for surgeons' procedures with long times to extubation (e.g., 14 min in Fig. 2). Second, the impact of each 1-min reduction in time to extubation on OR time will vary depending on surgeons' and teams' workflows.52–54 The reduction in direct cost will be largest for procedures in which time to extubation is the bottleneck to OR exit. Third, reductions in OR time reduce direct costs either when the OR has overutilized time or when there is appropriately more than 8 h of staffing planned for the OR (e.g., 10 h reduces the expected inefficiency of use of OR time compared to 8 h) and the staffing can be reduced to 8 h.35–39 The reduction in direct cost will be largest for facilities at which all ORs consistently are used for more than 8 h daily. Among those ORs, each 1-min reduction in OR time results in an overall 1.1- to 1.2-min reduction in regularly scheduled labor costs.35,39

Steps to quantify the intangible value of time saved are as follows. Calculate the hospital's incidence of times longer than 15 min. A 20%–25% reduction in the mean and standard deviation reduced that incidence by approximately 75%. Multiply that product by 4.9 min for cases with a to-follow case of the same surgeon, and 0 min otherwise. For example, the potential mean time savings at the hospital with AIMS data may be obtained as (15% extubation times >15 min) × 75% × (4.9 min extra) = 0.5 min. Multiply the 0.5 min per case by the number of “to follow” cases per OR. Add the product to the minutes per OR per day saved by the reduction in direct costs of the preceding paragraph.

Our modeling of extubation times has implications for national health care policy. Direct and intangible OR labor costs are highly sensitive to the characteristics of the cases performed by individual surgeons (Figs. 2 and 6), how staffing is planned and cases are scheduled,35–39 and how workflow is managed on the day of surgery.52–57 These variables affect how many minutes (if any) of labor costs are saved from each 1-min reduction in OR time. Thus, for drugs and devices that affect OR time, there cannot be an accurate single estimate for cost utility. Estimates do not even apply from one surgeon to another at a facility. Rather, meta-analyses can be used to summarize expected reductions in means and standard deviations. Conversion to dollars, Euros, etc. can then be done locally by each facility for each surgeon's cases.35,37–39

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Limitations

Although the principal financial benefit of reducing time to extubation is through the reduction in mean times to extubation, we doubt that the reduction could be measured other than in prospective trials and recommend instead the monitoring of prolonged extubation times. To explain why, we consider laparoscopic cholecystectomy with a mean time to extubation of 8.4 min (Fig. 1). A 20%–25% reduction in the time to extubation is 2 min. From features of clock displays on AIMS systems,41 2 min is within the routine measurement error of the data. Even if a hospital or system wanted to do an internal study of the impact of using desflurane or sevoflurane, and even if it could and would use propensity score methods to control suitably for the nonrandom use of the 2 agents, unrealistically large data collection periods would likely be required to measure accurately differences in the mean and standard deviation of extubation times. In contrast, the organization could measure differences in the incidences of prolonged extubation times, because the tracked values are just the numbers of prolonged extubation times and numbers of cases with extubation. We showed that change in the incidence of prolonged times is an omnibus indicator of change in the average time to extubation and in the variability of time to extubation.

Our statistical modeling of times to extubation was, by design, limited to the interpretation of the results of prospective clinical trials. Presence of negative times from end of surgery to extubation (i.e., patient was extubated before the end of surgery or the provider made a documentation error) would have reduced the accuracy of the 2-parameter Weibull distribution, which gives zero probability for negative values (Figs. 1, 3, and 4). We needed a fully systematic (unbiased) way to clean the many outliers in the data caused41 by user entry and clock errors. Our trimming of 5% values symmetrically eliminated extubation times <0.1 min and >29 min. The trimming of times <0.1 min was fortuitous for our purpose, because we needed to study times to extubation as reflecting those in prospective trials comparing sevoflurane and desflurane. In such studies, first surgery ends and then the volatile agents are discontinued.

Because our methodological development relied on time to extubation from AIMS data (Figs. 1–3, 6, and 8),41 our primary end point had to be time to extubation. However, context sensitive decrement times may differ more between volatile anesthetics for later end points such as time to recovery of pharyngeal function.24

At the hospital with AIMS data, prolonged (>15 min) extubation times had indirect (behavioral) effects of prolonging the nonoperative times of surgeons' to-follow cases. The effect was very strong, detectable using multiple statistical methods while controlling for multiple potential confounders.§ However, these results were from only one hospital. Therefore, we do not know how they were influenced by health system decisions. We also do not know if the cause of longer nonoperative times was surgeons' leaving the immediate area, because we lack the necessary secondary data.

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APPENDIX: TEST FOR PROPORTIONAL TREATMENT EFFECT ON STANDARD DEVIATION

Our prior knowledge is from the AIMS data. Suppose that we were designing a clinical trial that was comparing sevoflurane to desflurane and we wanted to know the probability distribution for the mean time to extubation. If we did not yet know the characteristics of the procedures to be studied, our prior knowledge would suggest the use of Figure 2, because the mean extubation times among surgeons follows a normal distribution. In fact, we are studying the standard deviations of time to extubation. Thus, our prior knowledge is represented by Figure 6. The inverses of the sample variances (precisions) among surgeons follow gamma distributions.

The WinBUGS58 code is given in Table 3, with the explanations here referred to by row.

At A, the “i” refers to the ith of r studies. For analysis of time to extubation, there are r = 29 studies.

At B, we rely on a gamma distribution and its 2 parameters to model the precisions. Because there is only 1 observed standard deviation per treatment group for each trial, there is enough information in the data to estimate only 1 of the 2 parameters of the gamma distributions. We keep the first of the 2 parameters fixed at the value estimated from the surgeons (Fig. 6) and let the second parameter “b” drive both the mean and the variance. The “prec” refers to “precision,” the inverse of variance. The “sev” represents sevoflurane and the “des” represents desflurane.

At C, we parameterize the “b” in terms of the quantities we want to estimate (random trial effect for the sevoflurane group α and fixed incremental effect of desflurane β). The use of the logarithm assures that the second parameter at Step B is always positive, as should be for the gamma distribution.

At D, the random effect among studies is considered normally distributed, as for the traditional random effects meta-analysis used for the ratio of means.47

At E, the weight attributed to each study is proportional to the average sample size in each group. This is the same as in the traditional random effects meta-analysis that we performed with a common pooled sample variance.47

At F, treatment specific precision values are obtained from the observed standard deviations “sd.”

At G, the second (scale) parameter in Figure 6 is 0.0016734. WinBUGS uses the inverse of this number. The prior mean of the trial specific alpha equals ln(1/0.0016734) = 6.4. The second parameter sets the standard deviation equal to the mean (i.e., 0.024414 = 1/[6.4]2) as a weak informative prior distribution.

At H, the same standard deviation is used as in G.

At I, “prec.alpha” represents the variability among studies in the logarithm of the scale parameter. The variance of the natural logarithms of the estimated Weibull scale parameters from the n = 95 surgeons was 0.029825. Multiplying by the mean sample size per group among studies from Step E gives the value 0.98064. The mean prec.alpha is the inverse of that value.

WinBUGS uses Markov chain Monte Carlo methods to draw samples from the posterior distribution of all model parameters. Three independent Markov chains were run from overdispersed initial values. The Brooks-Gelman-Rubin diagnostic was used to assess convergence. The first 9000 iterations were discarded and inference reported based on 30,000 additional iterations. The mean and median of simulation results were the same to 3 significant digits.

Two sensitivity analyses were our 2 random effects analyses of ratio of means, as described in the Results section. Both gave 25% (95% CI 17%–32%) as compared with the Bayesian analysis of ratios giving 21% (95% credible interval CI 16%–26%).

We repeated the Bayesian analysis with uninformative priors. The precisions in Steps G and H were set equal to 0.001 and both parameters in Step I were set equal to 0.001. The absolute differences in results were <0.2%.

We increased and reduced the precisions in Steps G, H, and I 10-fold. The absolute differences in results were <0.2%.

When we increased the first parameter in Step B 10-fold (and accordingly increased the mean of 6.4 in Step G), the point estimate was unchanged to within 0.2%, but the 95% CI was substantially narrower: 20%–22%. Likewise, when we decreased the first parameter 10-fold, the 95% CI was wider: 6.0%–35%.

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*After trimming, the resulting 6226 cases by 93 surgeons had out of OR to start of next surgery mean 75 min, standard deviation 33 min, range 31–233 min, and 90th percentile 113 min. For the 5298 cases following an extubation time ≤15 min, the 93 surgeons had mean 74 min, standard deviation 33 min, range 31–233 min, and 90th percentile 112 min. The other 928 cases by 82 surgeons had mean 79 min, standard deviation 35 min, range 31–233 min, and 90th percentile 124 min.
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†For 4 of 95 mean and standard deviation pairs, the estimated standard deviations in the log domain were >0.7. As described in Ref. 46, these 4 had large (≥5%) bias in the estimated raw mean when calculated using the estimates of the mean and standard deviation in the log domain. For those 4, we used the moment-based approach. We also repeated the random effects meta-analysis calculating the variances directly as described in Ref. 47, and results were identical to 3 significant digits.
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‡Although other positively valued skewed distributions could have been used (e.g., gamma or log normal distributions), neither had fits as good as the Weibull distribution. Using gamma, P > 0.05 for 62% of surgeons, P > 0.01 for 74% of surgeons, and P > 0.001 for 80% of surgeons. Using log normal, P > 0.05 for 10%, P > 0.01 for 20%, and P > 0.001 for 28% of surgeons.
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§P < 0.00001 by the Wilcoxon Mann-Whitney test stratifying by surgeon. The mixed effects modeling was repeated with control for another variable entered as a fixed effect: 4-wk period, age, time from into OR to end of surgery, case sequence in OR, weekday, specialty, ASA physical status, sex, and body mass index. Estimates of mean difference >4.0 min and all P < 0.0014.
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∥This result is consistent with the preceding findings comparing fits to normal and Weibull distributions. The surgeon of Figure 3 has χ2 goodness of fit to normal P < 0.00002 versus Weibull P = 0.58. The explanation is that the extubation times exceeding the mean of 9.27 min follow a folded normal distribution (P = 0.45). Thus, the normal distribution is a good fit on the right side of the distribution, as pointed out in Figure 1.
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