Surgical or anesthesia duration is frequently used as an independent variable in logistic regression studies of perioperative outcomes.1 References 1–5 are examples from last year. Studies including multiple specialties used category of procedure (e.g., thoracic surgery) as an explanatory variable instead of procedure itself (e.g., mediastinoscopy).1–3 Actual procedure durations were used instead of scheduled durations.1–5
In one of the studies, logistic regression was used to predict patients' risk of postoperative pulmonary complication after inpatient surgery.1 The model predicted a higher risk for thoracic surgery patients undergoing procedures with longer surgical durations.1 Yet, patients undergoing video-assisted thoracoscopic (VATS) surgery for anatomic lung resections (i.e., longer duration procedures6) have lower rates of pneumonia and shorter intensive care unit lengths of stay than those undergoing the same anatomic resections by thoracotomy.7–9 In this article, we use narrative review of the statistical literature to highlight that when there is presumptive evidence of an association between duration and outcome, the procedure itself (e.g., VATS lobectomy or thoracotomy lobectomy) needs to be evaluated for inclusion in the logistic regression, not just the category of procedure (e.g., “intrathoracic surgery” as in Reference 1). Otherwise, the odds ratios of the other variables may be biased.
Patients' risks are often modeled using actual procedure durations.1–5 An objective of the logistic regression model can be to provide an assessment of the patient's risk to guide the patient and his or her physicians when choosing among treatment options.1,3 Actual duration is known only once the period (e.g., procedure) is completed. What is known preoperatively is a forecasted estimate of duration, hereinafter referred to as the scheduled duration. Regardless of whether the actual duration better predicts outcome than does the scheduled duration, only the scheduled duration is known when a patient would be randomized in a trial and/or meets with the surgeon and anesthesiologist preoperatively for discussion of procedural benefits, risks, and options.
When studying outcome among many procedures,1–3 the use of actual duration may seem sufficiently close to scheduled duration for retrospective analysis to predict randomized trial results. For example, a mediastinoscopy taking 1 hour and VATS esophagectomy taking 8 hours would rarely have overlapping scheduled durations.6,10 Thus, it would seem that when comparing 1-hour versus 8-hour procedures, whether actual or scheduled durations are used in the logistic regression would give similar odds ratios. Yet, by reviewing the literature about logistic regression and about predicting case durations, we show that use of actual durations instead of scheduled durations can result in logistic regression results that are biased. The parameter estimates affected are not only those for the independent variable of duration but also those for the other independent variables.
MISSING CONFOUNDING VARIABLES—INFLUENCE OF PROCEDURE
In this first section, we do not differentiate between scheduled duration and actual duration. That issue is considered in the next section.
Consider 2 uncorrelated independent variables influencing a dependent variable. If the dependent variable is continuous, and the sample size is large, omitting one variable in linear regression does not influence the other variable's estimated coefficient and confidence interval for the coefficient.11 However, if the dependent variable is binary, omitting 1 of the 2 uncorrelated independent variables in the logistic regression causes change in the other variable's estimated coefficient and confidence interval.11,12 For example, consider a logistic model for intensive care unit admission that has odds ratio 2.7 for a continuous explanatory variable (e.g., each 10 mm Hg increase PaCO2) and odds ratio 7.4 for a binary variable that is uncorrelated to the first (e.g., whether or not the procedure itself is thoracotomy).12 Different data sets are generated from that logistic regression model. Omitting the binary variable causes the estimated odds ratios of the first variable to fluctuate around 2.2 rather than 2.7.12 As a second illustration, consider a different logistic model12 with odds ratio 2.7 for one continuous variable (e.g., PaCO2) and odds ratios 1.6, 2.7, or 7.4 for a second uncorrelated continuous variable (e.g., length of thoracotomy incision). Even though the 2 independent variables are uncorrelated, omitting the second variable causes the odds ratios of the first variable to fluctuate not around 2.7 but around 2.6, 2.3, or 1.8, respectively.12
Equation (7) of Mood12 shows why the results are different from those for standard linear regression (see Gail et al.11 for proofs). In linear regression, the mean of the dependent variable is a function of the independent variables. In logistic regression, both the mean and the variance of the dependent variable (i.e., the logit) are a function of the independent variables. Omission of a needed independent variable from a logistic regression model affects the variance. This leads to the downward bias seen in the estimate of the odds ratio for the other variable (or variables) in the model (e.g., from 2.7 to 2.3). Importantly, this bias occurs even when the omitted variable is uncorrelated to the other independent variables.
Because procedure can be expected to impact outcome, these results relate directly to the need to explore the impact of the procedure itself on outcome and on the parameter estimates for other variables, rather than simply including category of procedures. Category of procedure is often used in lieu of procedures for statistical efficiency, because some categories will contain scores of procedures, each with far smaller sample sizes.10,13–15 The consequence of using category is that estimates of the odds ratios of other independent variables may be biased. Sometimes category of procedures can reasonably be used, in lieu of procedure itself, but that should be shown by sensitivity analyses and/or formal testing of both logistic regression models. Parameter estimates should not differ substantively among procedures, and the use of category alone should not result in substantive changes in parameter estimates for other variables. The granularity of procedure to be used needs to be at least that which influences the outcome being modeled. For anesthesia studies including more than one category of procedure, such analysis is a change from typical statistical practice.1–3 For studies of the impact of duration on outcome for one category of procedure (e.g., cardiac surgery with cardiopulmonary bypass), this generally represents no change in practice.5 Specific procedures (e.g., aortic valve replacement and/or coronary artery bypass grafting) routinely are tested for inclusion along with duration (e.g., of cardiopulmonary bypass).5
Suppose that, for a cohort of patients undergoing a common category of procedure, duration does not influence outcome. Then, omitting procedure itself as an explanatory variable can result in the spurious finding of the odds ratios for duration being significantly larger than 1.00. For example, consider a logistic regression study of a continuous variable (e.g., duration) and outcome, with 1 or 2 additional explanatory variables that are uncorrelated to each other.16 These explanatory variables can be thought of as indicating procedures that are lengthy in duration in relation to other procedures of the same category of procedures. Assume that the odds ratio between outcome and the continuous variable (duration) equals 1.00 (i.e., no relationship), and that the odds ratios of the other 2 variables equal 2.00. When the correlations between the 2 variables and duration are 0.50, excluding the variables causes the estimated16 odds ratio to fluctuate around 1.96 rather than 1.00.
Frequently in anesthesia epidemiological studies, logistic regression is used not to model outcome but the odds of a patient receiving treatment (i.e., propensity score modeling).17 Then, the clinical outcomes are compared between treatments, with the treatment groups balanced on all variables influencing the odds of receiving the treatment.17 For such studies, the following result18 shows that procedure itself should be used, not19 only category of procedure and duration. Suppose that there are 2 statistically significant independent variables in a propensity score analysis modeling a patient's odds of being in one treatment group or another, one variable being duration and the other variable being binary, specifying one procedure or another.18 Suppose also that the treatment being studied protects (odds ratio 0.75) from the adverse outcome. Then, omission of the binary procedure variable from the model for receiving treatment results in the average estimated odds ratio for treatment effect to range from 0.76 to 1.00.18 The largest residual confounding of 1.00, versus the correct 0.75, would occur when the odds ratio is relatively large (3.00), both between the omitted procedure variable and treatment group and between procedure and outcome.18
LOGISTIC REGRESSION WITH MEASUREMENT ERROR—INFLUENCE OF DURATION
Three facts are known when considering the use of scheduled versus actual duration in logistic regression. (1) The scheduled duration of a scheduled procedure is known prospectively, not the actual1–3 duration. (2) In the absence of interaction with other explanatory variables in the logistic regression model (e.g., category of procedure), the scheduled duration can only change the odds of adverse outcome through its correlation with the actual duration. This is because scheduled duration is a managerial construct, whereas the actual duration represents true time affecting the patient biologically. (3) For each specified procedure, and usually category of procedure, actual duration is generally a linear function of scheduled duration.10,20–23 Applying these facts, suppose that logistic regression is performed using retrospective (observational) data with the hope that results would apply prospectively.1 Then, if actual duration is entered into the logistic regression as an independent variable, in fact what is being used is the scheduled duration plus measurement error. This is important because not only may the scheduled duration be correlated with the probability of adverse outcome (i.e., the relationship of interest), so may be the measurement error.
From the developed science of predicting case durations, we know that there are 3 reasons for the measurement error. First, there is bias, which is like aiming for a target and consistently hitting to the left.24 Bias (e.g., actual durations consistently longer than scheduled) usually is negligible, but when present, can be corrected using linear regression (see #3, above), as done de rigueur in scientific studies of case duration prediction.20–23 Second, even after bias adjustment, there is uncertainty about the true probability distribution of durations for each procedure and how the probability distribution of durations is centered about the scheduled duration.10,13,20,22 This so-called “parameter uncertainty” results from there being many procedures that are rare.10,13–15 For example, there can be variations on a basic procedure with different durations: unilateral right VATS thymectomy, bilateral VATS thymectomy, transcervical thymectomy, partial sternotomy thymectomy, combined transcervical and partial sternotomy (T incision) thymectomy, transsternal thymectomy, or robotic assisted thymectomy.6 Although parameter uncertainty causes most of the uncertainty in management decision making related to case durations at tertiary surgical suites, it can be compensated for statistically by using Bayesian methods suitable for log-normally distributed durations.10,13,20,22 Furthermore, practically, cohort outcome studies would exclude procedures not performed at least several times.25 Third, there is process variability (e.g., the SD of durations of 100 consecutive VATS lobectomies by a group of surgeons). This process variability is intrinsic to each procedure and can influence the logistic regression results. At a tertiary suite, process variability accounted for 28% of the variance in duration in the logarithmic scale, the rest being due to procedure (e.g., mediastinoscopy brief versus pneumonectomy long).26 For an ambulatory surgery center with fewer very long procedures, process variability accounted for 36% of the variance in log-duration.26 Combining the information about the 3 sources of measurement error, actual duration tends to be a linear function of scheduled duration for each procedure, with zero intercept when there is no bias, but regardless with substantial predictive error due to nonnormally distributed process variability10,13,20,22 that differs among procedures.10,13
Consider logistic regression with an independent variable such as duration following a log-normal distribution,27 an appropriate model.28,29 The odds ratio for the independent variable influencing outcome is 2.6, and process variability accounts for 20% of the variance in the independent variable. Then, without statistical compensation for the measurement error, estimated odds ratios do not fluctuate around 2.6, but around 2.1 or 2.2, depending on the correlation between the independent variable with measurement error and the other independent variables.27 The important issue is not whether the estimate would be 2.1 or 2.2, but that the estimate is biased (i.e., not 2.6). If the process variability were larger (35%) (e.g., as might be observed if data for categories of procedures were included in the logistic regression rather than procedure itself), then the error from using actual duration in lieu of scheduled duration would result in the odds ratios being shrunk even lower to 1.4 or 1.5.27 What these results show is that when an increase in duration causes an increase in the odds of a worse outcome, odds ratios estimated using actual durations generally overestimate those that will be achieved in prospective studies that use scheduled durations. Results are unchanged when the error term is skewed,27 as would happen depending on how duration were entered into the model.10,22,30,31 Similar results are also reported by Cheng.32
This result does not imply that scheduled duration needs to always be used in lieu of actual duration. Suppose that stepwise logistic regression were used and there is no significant effect of duration on outcome. This condition can apply when duration is correlated with other independent variables, those other variables influence outcome, but duration itself does not directly cause a change in outcome. Then, the use of actual duration seems reasonable provided duration is not included in the final logistic regression model.
Consider that instead of modeling the relationship between outcome and variables such as duration, the objective of the logistic regression is to evaluate the relationship between outcome and exposure to a treatment (e.g., preoperatively administered drug4) or a patient characteristic (e.g., race2), while controlling for variables such as duration. If one is interested in the conditional effect of treatment on outcome, and duration influences outcome, then scheduled duration should be used in the logistic regression model, because conditional models are useful prospectively, and prospectively the scheduled duration is known, not the actual duration. If the marginal effect of treatment on outcome were desired (i.e., average across all other variables in accordance with their incidence of occurrence),12 then actual duration may seem suitable, because actual duration is inherently a more accurate predictor of outcome than is scheduled duration. However, logistic regression gives biased estimates for marginal effects, which is why propensity score models are used to estimate the marginal effect of a treatment on outcome.17 Yet, the propensity score model for the assignment of a patient to a treatment group would use information that clinicians know at the time of assignment, which would be the scheduled duration, not the actual duration, as considered at the end of the preceding section.
The opportunity to study the effect of anesthesia duration, surgical times, etc., on outcome is exciting, especially because anesthesia durations are available from billing data33 and differ among facilities.34 However, because procedure, not category of procedure, determines duration and because logistic regression is nonlinear, procedure needs to be used when duration is studied. For some specialties and outcomes, category of procedure may be sufficient to predict outcome in lieu of procedure, but that should be tested rather than assumed for statistical simplicity. In addition, when duration influences outcome, results should not be interpreted as being applicable to prospective studies unless scheduled duration is shown to work as well as actual duration. The latter condition may not be satisfied. Generally, it is for the relatively long cases (e.g., >3.5 hours) that duration influences outcome.35 Also, generally, these long cases have the largest absolute predictive errors in case duration, because predictive variability in case duration is inherently proportional (e.g., 22% of scheduled).23
Throughout our paper, we relied on examples from general thoracic surgery. In part, this was because thoracic surgery has served as a model system for multiple studies of case duration prediction.6,23,36 Many of its procedures have relatively large predictive (absolute) errors in case duration,10,23,36 and future preoperative interventions are unlikely to reduce the error substantively.36–38 However, the scientific results that have been reviewed have nothing to do with any one specialty. The results rely on previously performed statistical studies with known (true) odds ratios.11,12,16,18,27,32 When logistic regression is used, and a covariate is both correlated to outcome and excluded, the other covariates' odds ratios will be biased.11,12,16,18 Likewise, when logistic regression is used, and a covariate is measured with error, the other covariates' odds ratios will be biased.27,32 For any given application (specialty), the magnitude of bias from not including procedure per se and using actual duration instead of scheduled duration will depend on multiple factors. This is why we summarize statistical results as being “biased,” in comparison with having small or large bias.
Our review suggests the need for research in 3 areas. One is learning how to model the relationship mathematically between the logit of adverse events and duration. Typically, procedure would be treated as a random effect,20,25,26 because there are nearly an unlimited number of procedures.10,14,15,23 From studies of logistic regression with biomarkers, it is known that because of logistic regression's nonlinearity, odds ratios are highly sensitive to the statistical model used to represent the relationship between the independent variable and the logit function for adverse events.39 We do not think that it is clear whether duration, logarithm of duration, or ordered categories of duration would best be used in the model.39 Second, some investigators using large (e.g., national) datasets with administrative data necessarily have only actual durations, not scheduled ones.33,34 For such multicenter observational data, forecasted duration may not practically be obtained. Yet, as reviewed in this paper, statistical methods relating actual and scheduled durations are well developed.10,13,20,22,24 Furthermore, statistical methods have been developed to correct logistic regression models for measurement error.32,40–42 Future work can evaluate whether these 2 statistical methods can be combined. We doubt that such modeling would be necessary, though, for outcomes shown to be unaffected by actual duration, because then scheduled duration can have no direct effect (i.e., actual duration is a suitable independent variable for screening). Third, suppose that the random effects of procedure, surgeon, and center were replaced by the single variable of scheduled (or actual) duration. Then, superficially, this would seem to reduce the degrees of freedom (i.e., parameters to be estimated) and provide a more explanatory model. However, this appearance of statistical simplification is generally false, because the scheduled duration is chosen (one hopes) preoperatively on the basis of the procedure and surgeon/center combinations.6,10,13,20–22, 26,43 The equivalence between the categorical data and scheduled duration is clearer when the models are presented as separate (simultaneous) statistical equations. Understanding how best for the analyst of outcome to model these relationships formally is currently unknown.
Name: Franklin Dexter, MD, PhD.
Contribution: This author helped design the study, conduct the study, and write the manuscript.
Attestation: Franklin Dexter approved the final manuscript.
Name: Elisabeth U. Dexter, MD, FACS.
Contribution: This author helped conduct the study and write the manuscript.
Attestation: Elisabeth U. Dexter approved the final manuscript.
Name: Johannes Ledolter, PhD.
Contribution: This author helped design the study, conduct the study, and write the manuscript.
Attestation: Johannes Ledolter approved the final manuscript.
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