Concise Definitive Review Series Editor, Jonathan E. Sevransky, MD, MHS

Bayesian Methodology for the Design and Interpretation of Clinical Trials in Critical Care Medicine: A Primer for Clinicians What we want (from clinical trials) is not what we are getting. —Robert Matthews (1) Andre C. Kalil, MD, MPH1; Junfeng Sun, PhD2

Objectives: To review Bayesian methodology and its utility to clinical decision making and research in the critical care field. Data Source and Study Selection: Clinical, epidemiological, and biostatistical studies on Bayesian methods in PubMed and Embase from their inception to December 2013. Data Synthesis: Bayesian methods have been extensively used by a wide range of scientific fields, including astronomy, engineering, chemistry, genetics, physics, geology, paleontology, climatology, cryptography, linguistics, ecology, and computational sciences. The application of medical knowledge in clinical research is analogous to the application of medical knowledge in clinical practice. Bedside physicians have to make most diagnostic and treatment decisions on critically ill patients every day without clear-cut evidence-based medicine (more subjective than objective evidence). Similarly, clinical researchers have to make most decisions about trial design with limited available data. Bayesian methodology allows both subjective and objective aspects of knowledge to be formally measured and transparently incorporated into the design, execution, and interpretation of clinical trials. In addition, various degrees of knowledge and several hypotheses can be tested at the same time in a single clinical trial without the risk of multiplicity. Notably, the Bayesian technology is naturally suited for the interpretation of clinical trial findings for the individualized care of critically ill patients and for the optimization of public health policies.

Division of Infectious Diseases, Department of Internal Medicine, University of Nebraska Medical Center, Omaha, NE. 2 Department of Critical Care, National Institutes of Health, Bethesda, MD. Dr. Sun has disclosed government work. Dr. Kalil has disclosed that he does not have any potential conflicts of interest. For information regarding this article, E-mail: [email protected] 1

Copyright © 2014 by the Society of Critical Care Medicine and Lippincott Williams & Wilkins DOI: 10.1097/CCM.0000000000000576

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Conclusions: We propose that the application of the versatile Bayesian methodology in conjunction with the conventional statistical methods is not only ripe for actual use in critical care clinical research but it is also a necessary step to maximize the performance of clinical trials and its translation to the practice of critical care medicine. (Crit Care Med 2014; 42:2267–2277) Key Words: Bayes; clinical trial; critical care

T

he choice for the quote above was not a random choice. The question we clinicians want to ask every time we have in our hands the publication of a new clinical trial is: What is the probability that the new treatment is better, given the trial result? It sounds very logical, doesn’t it? However, this question cannot be answered by the way most clinical trials have been performed, that is, by the conventional (i.e., frequentist) statistical methodology. Fortunately, this important question for everyone who works at the bedside of critically ill patients can be properly answered if we add the Bayesian methodology to the design and interpretation of clinical trials. Bayes’ rule combines mathematical simplicity and cleverness. It states that if we multiply the likelihood of the data with the known knowledge or beliefs (prior probability) regarding a parameter of interest, we can produce the posterior probability of this parameter; this new probability will became part of future new prior probabilities as new data arise. In other words, in science, as in real life, we should learn from our experiences and readjust our knowledge according to the accumulation of growing information on a given subject. Bayes’ theorem can be described mathematically as follows: Probability (Hypothesis |Data) =

Probability (Data |Hypothesis) × Probability (Hypothesis) Probability (Data)

Many scientific fields have fully adopted and applied Bayes’ technology to their research methodology, for example, www.ccmjournal.org

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astronomy, engineering, chemistry, genetics, physics, geology, paleontology, climatology, cryptography, linguistics, ecology, and computational sciences, just to cite a few. The uptake of the Bayesian methodology by the medical sciences has been slow but steady. In 1994, a PubMed search for “Bayesian Clinical Trials” generated a total of only 23 articles, while 20 years later, the same search generated 1,852 articles—of which 32 were associated with critical or intensive care. Furthermore, the Food and Drug Administration has released an official guidance for the use of Bayesian statistics in medical device clinical trials in 2010 and the agency has already used Bayesian trials and analyses for its advisory committee panel meetings (2). Bayesian guidelines from the European Medicines Agency have also been released (3). It is not the scope of this article to discuss this disparity between Medicine and other scientific fields, but it suffices to say that this impressive 80-fold increase in medical publications using Bayes methods in just 20 years is the tip of the iceberg. In addition, the rapid progress made in computational technology in the last few decades has allowed the application of Bayesian methodology in ways never before imagined. We strongly believe that by being more proactive in both understanding and applying this technology to critical care research, we will be closer to our ultimate goal as physicians and scientists: to improve the survival outcomes of our critically ill patients. In this article, we aim to introduce the reader of critical care medicine to Bayesian methodology and demonstrate the utility of this methodology for critical care research. We Table 1.

will neither address the historical/philosophical controversies between Bayesian and conventional statistics nor the methodology’s mathematical principles; however, we submit the readers to a set of didactic articles geared to the medical field (4–16). We provide a table highlighting the major differences between both methods regarding clinical trial research (Table 1) adapted from five studies (4–7, 14).

WHAT IS THE BAYES’ THEOREM? A clinical example is used to facilitate the understanding of Bayes’ theorem in the ICU setting. The patient is admitted with shock of unclear etiology. The knowledge of the shock status is not enough for the clinician to make inferences about the etiology, and without a clearer picture about the etiology, the treatment of shock will likely be compromised and potentially harm the patient. The physician will search for more data through both history and physical examination and, if available, through radiologic and laboratory tests. Now the physician finds out that the patient is a 20-year-old college student with recent onset of fevers and headaches and a physical examination with petechial rash; induction reasoning leads the physician to the diagnostic hypothesis of meningococcal meningitis. Thus, aggressive fluid replacement and immediate administration of antibiotics would be lifesaving at this time. However, if the history was different, that is, the patient was 60 years old and presented with shock and new-onset severe chest pain, and the physical

Clinical Trials: Comparison of Conventional and Bayesian Designs

Area of Contrast

Conventional

Bayesian

Critical question

What is the probability of the trial result (observed data), given the treatment effect (study hypothesis)?

What is the probability of the treatment effect (study hypothesis), given the trial result (observed data)?

Reasoning

Deduction

Induction

Prior information

Informally used with design but not with interpretation

Formally used with design and interpretation

α and β errors

Prespecified

Unnecessary

Test hypothesis

Prespecified

Unrestricted

Sample size

Prespecified

Unrestricted

Interim monitoring plan

Prespecified

Unnecessary

Data source

Random

Random

Variables of interest (e.g., risk ratio)

Fixed

Random

Summary measure

p value, estimates, and CIs

Posterior probability

Inference basis

Observed and unobserved trial data

Observed trial data and prior knowledge

Uncertainty

Unquantified

Quantified

Subgroup analysis

Multiplicity is present

Multiplicity is absent

Trial interpretation

Confusing

Straightforward

Clinical decision making

Difficult to translate to bedside

Easy to translate to bedside

Practice guidelines

Rigid and less intuitive

Flexible and more intuitive

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examination suggested cardiac tamponade, the use of aggressive fluids and antibiotics would not save the life of the patient; in fact, would lead to his/her demise. This type of induction reasoning we use every day in our clinical decision making, that is, we update our diagnostic hypothesis with the gathering of the new data is the core of the Bayes theorem. In other words, the probability that our hypothesis is true is appropriately influenced (i.e., conditional) by a new piece of evidence. Hence, the classical Bayes theorem formula: Probability (Meningitis |Physical) =

Probability (Physical |Meningitis) ´ Probability (Meningitis) Probability (Physical)

Translation: The probability of meningococcal meningitis (hypothesis) given the history and physical examination (data), also called posterior probability, is equal to the probability of these data given the hypothesis divided by the probability of the history and physical examination findings (new evidence) and multiplied by the probability of meningitis (also called prior probability). This is analogous to what we do daily in our clinical practice for positive predicted values, which has been well captured by a previous article by Gill et al (17). For a comparison, here is the Bayesian approach for the positive predictive value: Probability (Disease |Test Positive) =

Probability (Test Positive |Disease) ´ Probability (Disease) Probability (Test Positive)

As sensitivity and specificity alone are poorly informative (and even misleading) without knowing the prevalence (prior) of the disease to determine the positive predictive value, so are the p value and power of a clinical trial without knowing the prior probability of the treatment effect (15). As Matthews (1) described, “Bayes’ theorem is ultimately an expression of an unavoidable mathematical fact: if we want to answer the question we are interested in—what is the probability that the treatment is efficacious, given the trial result—we must include any prior insights we have.”

WHERE DO THE PRIORS COME FROM? In the meningitis example above, the prior came from the epidemiological data captured by the physician’s history taking. The prevalence determines the prior for the Bayes’ rule. This is classically depicted by the diagnostic positive predictive values for a specific disease, which will increase with the increasing prevalence of the disease; this disease prevalence is exactly the definition of the prior information, which we use as part of our diagnostic tools when we see new patients. How can we use priors to update our view of the benefits of a given treatment? Similar to diagnostic decisions, if we have data derived from randomized trials, meta-analyses, observational studies, or institutional experiences with a specific treatment, then all these data can be formally entered into the Bayes’ rule to Critical Care Medicine

compose single or multiple priors (1, 5, 18–24). The availability of multiple priors is essential to understand the flexibility and advantage of Bayes for treatment approaches. Clinicians can have different interpretations of the very same data from the same trial; this may (or may not) modify their current view of a given treatment according to their prior information. This constant data gathering and accumulation (prior) gives us the de facto opportunity to update our current knowledge status with new posteriors in order to make the best treatment decisions. Of note, the definition of a prior does not require a temporal order, that is, data from previous, concurrent, or selected trials can be incorporated into the prior (7, 22, 25, 26). What about if we are going to design a phase II or III clinical trial for a new therapy, where do we get the priors? The way it has been done for decades is by finding the best “educated guess” as a single prior to design a new phase II or III trial with a single hypothesis. This may explain in part the failure of sepsis trials in the last 30 years (27). Here, all possible evidence, be it small or large, has to be evaluated and this includes animal studies, human pharmacokinetic and pharmacodynamics studies, drug biological activity, drug effect on biomarkers, activity of similar drugs or similar classes of drugs used for the same or other disease processes, and last, expert opinion. How do we get the priors in the worst case scenario, in which no external information is available? One Bayesian approach is by assuming minimal or no external information through the use of noninformative priors. This means that the posteriors are calculated with the assumption that no other related data is available to inform the value of the prior. The other approach is by methodically constructing priors through the evaluation of different degrees of belief among many experts and formally translating these various degrees into mathematical calculations to evaluate them in the context of both study design and interpretation. It has been suggested by Moyé (16) that the active incorporation of counterintuitive prior information should be performed in this worst case scenario as “the more passionate the investigator, the greater the protection the priors require from their strongly held opinion.” Hence, the reader should be aware that for small sample size scenarios for priors, the expert opinion can be overly influential. The good news is that this extreme scenario of no external information is very rare before the execution of phase III efficacy trials since they are usually preceded by preclinical studies and phase I, IIa, and IIb trials. Once the new data accumulate, the precise source and accuracy of the original prior information will naturally become less relevant because the posteriors will progressively become less influenced by the initial priors (1). In other words, the constant updating process naturally allowed by the Bayesian approach will build more robust posterior predictions (i.e., decrease uncertainty), which is exactly what we search for to best interpret and translate the trial results to our patient care. Thus, the major benefit of the Bayesian approach is that all priors are explicitly derived and reported in the clinical trial with several treatment hypotheses, whereas the conventional approach describes only implicitly the single prior for the single treatment hypothesis. www.ccmjournal.org

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BAYESIAN APPLICATION TO THE INTERPRETATION OF CLINICAL TRIALS Bayes’ rule touches one of the most important issues faced by Medicine as a scientific field and as a clinical practice in the 21st century: Why do practicing physicians commonly interpret the results of the very same trial so differently? If we ask two critical care physicians about how they would approach a patient with septic shock—considering that both trained in the same fellowship program and now practice in the same hospital and both read the last version of the Surviving Sepsis Campaign (28)—they could come up with very different treatment regimens. In reality, this can be seen with the wide range of compliance with the sepsis bundles that has been documented in many different studies from many countries (29–36). If the evidence and current information they have available is the same, why would their treatment approach be different? We conjecture that conventional statistics is poorly suited for the trial interpretation and individualization of patient care. We demonstrated in a previous comprehensive Bayesian analyses of sepsis trials that the misinterpretation of the evidence is a major barrier to improve the outcome of patients with severe sepsis (37). Table 2 shows the research hypothesis (15, 38–40) to provide a didactic example: for the same clinical trial with

type I error of 5% and 80% power, the probability of a positive finding (i.e., p < 0.05) being true is dramatically different for a different prior probability that a new vasopressor is better than the control. Would you accept to use the vasopressor results from the first interpretation and apply it to your patient care? We hope not! The accurate interpretation of this clinical trial is entirely dependent on the prior probabilities, which are not used at all with conventional methodology. Bayesian analysis can improve the trial interpretation and at the same time can allow the clinician to translate this information into bedside clinical practice, that is, shifting the focus from statistical to clinical significance. Here is a real-life example. The trial on early goal-directed therapy (EGDT) for patients with septic shock (41) showed one of the largest survival benefits in the history of critical care medicine: 16% absolute risk reduction for mortality or a 42% relative risk reduction with the 95% CI ranging from 13% to 62% and a p = 0.009. By the conventional statistics, this is a very strong result. However, several other studies (42–51) have shown inconsistent results regarding survival benefits with this approach. A straightforward Bayesian analysis which incorporate information from these additional studies is shown in Table 3: if you believe that the none of these other studies should change your

Interpretation of a Clinical Trial With Positive Results (p < 0.05) Replicated 1,000 Times With Different Priors Table 2. Variable

New Drug Is Better Than Control

New Drug Is Not Better Than Control

Total

First interpretation  Prior probability that new vasopressor works = 5%   Positive trial result (p < 0.05)

40 (true-positive trial [power] = 80%)

48 (false-positive trial [p] = 5%)

88

  Negative trial result (p ≥ 0.05)

10 (false-negative trial = 20%)

902 (true-negative trial = 95%)

912

   Total

50 (prior = 5%)

950

1,000

Conclusion: New vasopressor is better in 46% (40/88) of all trials with p < 0.05, or 54% of all positive trials will have a significant result despite the fact the new vasopressor is not better (54% false-positive). Second interpretation  Prior probability that new vasopressor works = 15%   Positive trial result (p < 0.05)

120 (true-positive trial [power] = 80%)

43 (false-positive trial [p] = 5%)

163

  Negative trial result (p ≥ 0.05)

30 (false-negative trial = 20%)

807 (true-negative trial = 95%)

837

   Total

150 (prior = 15%)

850

1,000

Conclusion: New vasopressor is better in 74% (120/163) of all trials with p < 0.05, or 26% of all positive trials will have a significant result despite the fact the new vasopressor is not better (26% false-positive). Third interpretation  Prior probability that new vasopressor works = 60%   Positive trial result (p < 0.05)

480 (true-positive trial [power] = 80%)

20 (false-positive trial [p] = 5%)

500

  Negative trial result (p ≥ 0.05)

120 (false-negative trial = 20%)

380 (true-negative trial = 95%)

500

   Total

600 (prior = 60%)

400

1,000

Conclusion: New vasopressor is better in 96% (480/500) of all trials with p < 0.05, or 4% of all positive trials will have a significant result despite the fact the new vasopressor is not better (4% false-positive). 2270

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Posterior Probabilities With Early Goal-Directed Therapy Table 3.

Published p (Frequentist) Prior Probability (Bayesian)

0.009 Posterior Probability of New Treatment Being No Better Than Control

Enthusiastic

0.01

Mild skeptic

0.05

Mild-moderate skeptic

0.14

Moderate skeptic

0.25

Severe skeptic

0.41

Adapted with permission from Kalil and Sun (37). Adaptations are themselves works protected by copyright. So in order to publish this adaptation, authorization must be obtained both from the owner of the copyright in the original work and from the owner of copyright in the translation or adaptation.

interpretation of the study by Rivers et al (41), then you are part of the enthusiastic interpretation, that is, there is only 1% chance (posterior probability) that the EGDT would be no better than controls. On the other side, if you believe that some of the other studies should cast doubts on the very large survival benefits seen with the pivotal trial, that is, you are mild to moderate skeptical, then your interpretation translates into a 14% chance of EGDT being no better than controls. Based on the current conventional view that a 5% false-positive rate would be the most we would tolerate in order to call it a significant result, a 14% chance of not working would likely prevent you to use it in a standard ICU practice. If these posterior probabilities are helpful to you but not enough to make your final decision regarding EGDT, then Bayesian analysis can be used to ask an additional question regarding a meaningful clinical threshold based on your knowledge and clinical experience: What is the (posterior) probability of decreasing the relative risk of death of my patient with septic shock by at least 15% with EGDT? Table 4 shows that if you are enthusiastic about this therapy, your patient will have a 94% chance of having a 15% reduction in death, but if you are only mildly skeptic, then the chances of your patient reaching this clinical threshold are

reduced to 62%, which would be too low to be introduced into standard clinical practice. Hence, instead of putting too much or too little trust in the opinions of an editorialist or an expert on the field, the ICU clinician can feel empowered to make his/ her best clinical decisions by the use of the Bayesian interpretation in conjunction with the conventional statistical reports. Because of the controversies surrounding EGDT (52), there are currently three phase III clinical trials being executed in the United States, United Kingdom, and Australia. The Bayesian interpretation provided here should not only help the ICU physician to direct his/her own clinical practice for patients with severe sepsis while this trials’ results are not available but also provide the prior to put the findings of these new clinical trials in the context of the entire available evidence. During the peer-review process of this paper, the ProCESS trial, which evaluated the efficacy of EGDT in the United States was published (53); the results showed no significant mortality differences between EGDT, protocol-based therapy, and usual care arms. If we update the original report by Rivers et al (41) with the ProCESS trial results and utilize the probabilities we provide in tables 3 and 4 of this manuscript, the Bayesian posterior probability that EGDT is no better than control is 0.25 (i.e., probability of a false-positive result with EGDT = 25%), and the posterior probability that EGDT reduces the relative risk of death by at least 15% is 6% (i.e., a 94% chance that EGDT does not reduce mortality by a clinically meaningful threshold).

BAYESIAN APPLICATION TO THE DESIGN OF CLINICAL TRIALS The implementation of Bayesian design in clinical trials has grown in the last two decades (12, 54–56). Two thirds of the articles on this subject have been published on biostatistical journals (54). Here, we provide a summary of the strengths and limitations of the Bayesian methods on phase III clinical trials, meta-analysis, and comparative-effectiveness research. Phase I and II clinical trials are described elsewhere (57–64) due to space limitation. Bayesian Phase III Trials Phase III trials are characteristically multicenter, randomized, comparative studies aimed at confirming the efficacy of a new

Posterior Probabilities of Relative Risk Reduction (RRR) With Early Goal-Directed Therapy Table 4.

Posterior Probability of Mortality Relative Risk Reduction Prior Probability (%)

RRR > 15%

RRR > 20%

RRR > 25%

Enthusiastic

94

87

78

Mild skeptic

62

41

22

Mild-moderate skeptic

27

9

2

Moderate skeptic

6

1

0

Severe skeptic

0.4

0

0

Adapted with permission from Kalil and Sun (37). Adaptations are themselves works protected by copyright. So in order to publish this adaptation, authorization must be obtained both from the owner of the copyright in the original work and from the owner of copyright in the translation or adaptation.

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treatment. One attractive feature (and point of controversy) of Bayesian designs is the choice of a prior probability distribution that summarizes the prior knowledge (or the lack of). Jennison and Turnball (23) divided prior distributions into five categories: 1) clinical priors elicited from experts; 2) skeptical priors centered at zero with a small probability of having a meaningful benefit; 3) enthusiastic priors with a small probability of no beneficial effect; 4) noninformative priors that convey as little prior information as possible; and 5) “pragmatic Bayes” proposed by Grossman et al (24) with a skeptical prior that is equivalent to an existing study with zero difference between treatments and a sample size 25%, the size of the planned study. Studies based on Markov Chain Monte Carlo simulations have shown that with five or fewer interim analyses, the type I error rate is controlled at 0.05 (24). One area of important application for Bayesian designs is the use of adult data to guide pediatric trials. Children are routinely excluded from adult trials, so when pediatric trials are proposed, there are usually substantial amount of adult data available. Therefore, it is very attractive to be able to utilize adult data for the design and analyses of pediatric trials, especially for rare diseases for which a standalone trial is unfeasible. Schoenfeld et al (65) proposed a Bayesian hierarchical modeling approach that allows pediatric trial data to be pooled with adult trial data and showed a substantial increase in power. Goodman and Sladky (66) proposed a Bayesian noninferiority trial for pediatric Guillain-Barré syndrome. They used a meta-analysis of adult data to generate a prior for the difference between the new treatment and the positive control. They generated the priors from the meta-analysis results by disregarding a small beneficial effect of the new treatment and then by reducing the information from the adult trials by half. The predominance of frequentist designs in drug trials motivated the development of “hybrid” methods (67). These methods use prior information in the design of trials, but the analysis remains frequentist. For example, frequentist power calculations assume the true variable (e.g., risk ratio) to be a fixed number, whereas Bayesians assumes it to be random with a (prior) distribution. For large phase III trials, interim analysis is important in case convincing evidence that has been accumulated before the planned termination of the trial. For example, investigators may be ethically obligated to terminate a trial early if the new treatment is shown to be worse than the control treatment so that no more participants would be harmed. On the other hand, if the new treatment is shown to be superior, the trial could also be terminated early so that the control patients would not need to continue on suboptimal therapy. Under the frequentist framework, with each interim analysis, the rate of false-positive findings increases due to the repeated evaluation of the ongoing data gathering. To allow interim analyses and control the probability of false-positive findings, group sequential (GS) methods are widely used (23) with the frequentist approach. One criticism of the GS methods is that by simply looking at the data in interim, it could affect the final finding of the whole study, especially when the final p value is near the nominal 0.05. Under the Bayesian framework, there 2272

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is no need to adjust for interim analysis, since each analysis straightforwardly update the posterior distribution, hence, eliminating the risk for multiplicity. Berry (67) suggested that decision should be also made based on the benefits and costs of the different outcomes. Another way of interim monitoring is the stochastic curtailment (23). Conditioning on the data at the current evidence and fixed variable values for the distribution of the remaining data, a decision will be made regarding whether the trial will be stopped early. For example, the trial can be stopped early if the probability of failing to reject the null hypothesis is low under the alternative hypothesis since continuing the study is “futile” or if the probability of rejecting the null hypothesis is high under the null hypothesis since rejection is mostly assured. Note here the variables are assumed fixed, without accounting for the uncertainty. A hybrid approach is to use the posterior distribution of these values to calculate the probabilities of positive findings in the final frequentist analysis, which accounts for the uncertainty and incorporates all available information, also called the “Predictive Probability of Success” (69). Adaptive design has been a very intense area of research in recent years (54, 70–75). The major difference between traditional and adaptive trials is regarding the accumulating data while the trial is ongoing, which can be used to modify the trials’ course in the adaptive, but not in the traditional design. This approach has already been used in clinical research, but uncommonly through the Bayesian approach, which allows for several types of adaptation at the same time, for example, more efficient drug-dosing evaluation, the need for dropping or adding drug doses or study arms, seamless transition between drug development phases within the same trial, more efficient distribution of patients to each study arm, and modifications of accrual rate (70). The Pharmaceutical Research and Manufacturers of America working group defined adaptive design as “a clinical study design that uses accumulating data to decide how to modify aspects of the study as it continues, without undermining the validity and integrity of the trial” (74). They emphasize the “by design” nature of adaptive designs. Adaptive designs include a broad range of designs: Kairalla et al (73) categorized adaptive designs into three phases: 1) Learning phase is composed of adaptive dose response studies based on toxicity and/or efficacy. 2) Combined (seamless) phase includes seamless phase I–IIa and phase IIb–III designs. Adaptive seamless designs combine more than one phase into a single study to speed up the drug development, and all data are used in the analysis to be more efficient. 3) Confirmatory phase includes adaptive randomization, GS, sample size reestimation (SSE), and combination of GS and SSE. Adaptive randomization methods include covariate adaptive randomization, which attempts to make treatment groups more balanced with respect to key covariates, and response adaptive randomization, which tries to assign more patients to the superior treatment group. SSE allows sample size to be updated using the accumulating data during the trial. The frequentist properties and the interpretability of adaptive designs may be rigid and confusing (73). In comparison, Bayesian methods are October 2014 • Volume 42 • Number 10

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very attractive for adaptive designs due to their flexibility. It has been argued that Bayesian methods allow data to be analyzed any time during the ongoing trial without the well-known limitations of the conventional statistics (26). Importantly, there are limitations pertinent to conventional statistical methods that also apply to Bayesian methods, for example, the presence of bias and/or confounding can make the results of clinical trials less relevant or even invalidate them. These problems cannot be resolved alone by either conventional or Bayesian inferences. This is essential to recognize because no currently available statistical methodology is fool proof against bias and confounding (76). Bayesian Analysis to Evaluate the Need for Confirmatory Phase III Trials If a phase III trial shows strong and convincing efficacy results, it is unlikely that its new findings would be much influenced by skeptic priors. However, if the results are not strong, then they can be possibly changed by priors. Because the definition of strong or weak efficacy results necessarily changes according to the specific disease process, the clinical relevance of the results of a single phase III trial can be more properly evaluated by different priors to test the strength of the new evidence in the context of the clinical needs expressed by different priors. For example, if after a positive phase III trial a major efficacy difference is observed between noninformative and skeptic priors, this may suggest that a confirmatory trial is warranted (i.e., potential false-positive trial). On the other side, if after a negative phase III trial a major efficacy difference is observed between noninformative and enthusiastic priors, this may suggest that another trial is warranted (i.e., potential false-negative trial). This is an example of how even noninformative priors can be highly relevant to clarify the need for confirmatory trials. This type of sensitivity analysis is an excellent tool to assess the need or not for a confirmatory phase III trial (37, 77, 78). In addition, the needed sample size for the next (confirmatory) phase III trial can be estimated, optimized, and most importantly, designed with a substantially smaller number of patients by straightforward Bayesian tools compared with conventional methods. This was recently demonstrated in a severe sepsis trial in which 600 patients were needed for a confirmatory trial with Bayesian design, whereas 2,000 patients were needed for the same trial by conventional design (78). Bayesian Reporting for Clinical Trials There are two guidelines (79, 80) and three comprehensive analyses (37, 77, 81) regarding the items that should be reported in Bayesian studies. Table 5 summarizes the main points to be reported.

BAYESIAN APPLICATION TO META-ANALYSIS Meta-analytic methodology has expanded substantially in the last decade and the limitations of the conventional methods are increasingly being resolved by the application of Bayesian technology (82, 83). The advantages of including the Bayesian approach into the performance of meta-analyses include the following (79, Critical Care Medicine

Table 5. Optimal Reporting of Bayesian Clinical Trials Introduction  Background  Intervention  Objectives Methods  Design of study  Statistical model    Prior distribution     Rationale for choice     Sensitivity analysis   Likelihood of observed data    Posterior distribution     Variable estimates     Posterior densities    Joint probabilities if multiple comparisons     Bayes’ factors    Computation    Checks for convergence if Markov Chain Monte Carlo used    Methods for generating posteriors    Software used, how validated if not publically available Results  Description of all posterior distributions  Description of all sensitivity analyses if alternative priors used Limitations  Description of all design and methodological limitations

82–85): 1) the full spectrum of uncertainty is accounted for and measured in all analyses based on the entire state of knowledge; 2) it enables evidence from different sources to be formally incorporated in the probability calculations; 3) studies with different designs can be evaluated with appropriate weights to combine comparable studies through Bayesian hierarchical modeling, also called cross-design synthesis; 4) the borrowing strength concept can be applied through Bayesian hierarchical models; this concept allows a fully Bayesian implementation of random effect models, which accounts for the uncertainty in the estimated random effect distribution; 5) no asymptotic normal approximation is necessary, thus making the occurrence of zero events in an arm of a study mathematically treatable; 6) multiple testing issues, a frequent problem with conventional statistics, is eliminated with Bayesian analysis; 7) the final estimates of the Bayesian meta-analysis are reported as posterior probabilities, which are easily translated to individual-patient application; 8) the results from the Bayesian meta-analysis may be used for the design of future studies; and 9) the Bayesian framework naturally allows www.ccmjournal.org

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same parameter estimates as the three previously published meta-analyses (i.e., risk ratio or Study name Events / Total Risk ratio and 95% CI odds ratio) (Fig. 1A), and then Risk Lower Upper we applied a Bayesian analytic ratio limit limit Steroids Control approach. Of note, we started Bollaert 1998 0.50 0.25 1.02 7 / 22 12 / 19 the analysis without any skeptiChawla 1999 0.55 0.24 1.25 6 / 23 10 / 21 cism by using a noninformative Briegel 1999 0.75 0.19 2.93 3 / 20 4 / 20 Yildiz 2002 0.67 0.35 1.27 8 / 20 12 / 20 prior probability. Our findAnnane 2002 0.89 0.73 1.08 82 / 151 91 / 149 ings of the three meta-analyses Confalonieri 2005 0.08 0.00 1.29 0 / 23 6 / 23 consistently showed that in the Tandan 2005 0.85 0.62 1.15 11 / 14 13 / 14 best case scenario, low-dose Oppert 2005 0.99 0.52 1.88 10 / 23 11 / 25 steroids had a 57% probability Rinaldi 2006 0.86 0.33 2.21 6 / 26 7 / 26 Cicarelli 2006 0.63 0.35 1.12 7 / 14 12 / 15 of reducing the relative risk of Meduri 2007 0.57 0.27 1.20 10 / 42 8 / 19 death by 15% in patients with Sprung 2008 1.09 0.85 1.40 86 / 251 78 / 248 septic shock. Figure 1B shows 0.84 0.72 0.97 236 / 629 264 / 599 in a didactic form the Bayesian 0.1 0.2 0.5 1 2 5 10 results parallel to the forest Favors Steroids Favors Control plot previously published by Annane et al (87). The Bayesian graph is highly transparent by B showing the changes in survival probability after the addition of each trial until the report of the most updated posterior probability (57%) after the last performed trial by Sprung et al (90). This calculated probability (57%) of benefits with steroids was not far from flipping a coin, and this could never be appreciated if we only used the conventional methodology. The estimated probability would be even lower if we had used any skeptic prior probabilities. In addition, we found that the probability of increasing the risk of superinfections, hyperglycemia, and gastroinFigure 1. A, Conventional analysis of low-dose steroids mortality relative risk reduction. B, Bayesian analysis testinal bleeding were all eleof low-dose steroids. RRR = relative risk reduction. Adapted with permission from Kalil and Sun (86). Adaptavated with low-dose steroids, tions are themselves works protected by copyright. So in order to publish this adaptation, authorization must 81%, 99%, and 73%, respecbe obtained both from the owner of the copyright in the original work and from the owner of copyright in the translation or adaptation. tively. Based on the disagreements the design differences for accounting cost and utilities, which is ideal for guidelines and and potential reasons for the negative results found in the last public health policy decisions. phase III trial by Sprung et al (89), we redid the entire Bayesian Here is a practical example of the utility of the Bayesian meta-analyses with and without the trial by Sprung et al (89): meta-analytic approach in critical care medicine. The use of the overall results remained similar. In addition, two real-life low-dose steroids in septic shock remains controversial due large ICU cohorts by Ferrer et al (91) (n = 2,786) and Casserly et to contradictory findings from different randomized trials al (91) (n = 8,992) did not find significant survival benefits with (86). Our research group hypothesized that the re-evaluation low-dose steroids in patients with septic shock, which further of three meta-analysis (87–89) done with the conventional confirmed our Bayesian results. Taken together, the efficacy and methods through the Bayesian lens would provide further safety findings from our combined Bayesian and conventional insight into this controversy (86). We replicated the methods analyses brought scientific clarity to the field and were highly (i.e., fixed-effect or random-effects modeling) and used the applicable to the bedside practice of critical care. Notably, the

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Bayesian methods discussed above are also an excellent fit for the performance of individual-patient meta-analyses (52), as well as for network meta-analyses (93, 94).

BAYESIAN APPLICATION TO CLINICAL PRACTICE GUIDELINES AND COMPARATIVEEFFECTIVENESS RESEARCH There is a growing body of literature that is already applying Bayesian methodology to guideline development (95–102). In addition, Diamond and Kaul (103) recently proposed a new methodology to report clinical practice guidelines based on Bayesian principles; they suggest that the use of this Bayesian approach could potentially reduce practice variation and increase the adherence of practice guidelines, which in turn would culminate in better clinical outcomes and lower healthcare cost. The growing application of Bayes in both metaanalyses and practice guidelines should propel this method to be used in comparative-effectiveness research. Berry et al (104) recently provided the basis and rationale for why Bayes technology is perfectly suited for comparative-effectiveness research. As a real-world example on comparative-effectiveness research, a recently published study by the American Heart Association Task Force on Practice Guidelines (99) demonstrated that percutaneous coronary intervention, similar to coronary artery bypass surgery, improved survival for patients with unprotected left main coronary artery disease compared with medical therapy. The application of Bayesian decision making to evidence-based medicine and public health has been comprehensively reviewed by Ashby and Smith (105) and Lilford and Braunholtz (106).

CONCLUSIONS The addition of the Bayesian approach to clinical trials in critical care medicine can realistically and timely lead to the better performance of trials of all phases, including the concrete benefits of faster trial completion with the reduction of unnecessary placebo exposure (positive trial) or treatment exposure (negative trial), fewer false-positive trials, and fewer false-positive findings in subgroup analyses. In addition, the Bayesian application can lead to more rapid pace from drug development to drug approval and improved translation of trial results into public health policies. Further, the bedside utilization of Bayesians posteriors can provide the ICU clinician with a more individualized application of trial results to determine the best course of action to treat critically ill patients. Both conventional and Bayesian approaches have their own strengths and weaknesses. Both approaches carry a substantial degree of subjectivity, implicitly with the conventional approach, and explicitly with the Bayesian approach. If either approach were fully objective as one would dream of, all medical statistics would be done by automated processes without the need for statisticians. This is not different from clinical medicine, which also cannot be fully objective; otherwise, we would not need physicians anymore, just computers to make perfect diagnoses and deliver the ideal treatments. Critical Care Medicine

What we propose in this article is that today we already have the clinical, mathematical, and computational skills to apply the Bayesian technology in conjunction with the conventional methodology to maximize the design, execution, and interpretation of clinical trials in the field of critical care. The failure to recognize the fact that we could be doing better clinical research in the critical care field is the same as the physician who provides a less than optimal treatment to a critically ill patient because he/she does not know enough about a known optimal treatment. This is inadmissible in today’s clinical medicine, so the same high standards should hold for clinical research. We hope that our comprehensive review and suggestions to improve clinical research in our field of critical care will spark the interest and excitement of clinical researchers with this versatile Bayesian world.

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