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.
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%).
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
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.
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.
†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.
‡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.
§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.
∥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.