Modern Bayesian Statistics in Clinical Research

Abstract

Statistics is a science of quantitative reasoning. It provides a framework for assessing information contained in the data in the midst of uncertainty. Statistics is used to quantify the probability of an event so that a proper inference can be made in theerves evidence-based medicine. There are two main approaches for statistical inference: the predominant frequentist approach and the lesser known Bayesian approach. In this "Expand Your Thinking Lecture," these two approaches will be compared and contrasted, with special emphasis on the development and use of Bayesian statistics. Learning statistics can help sharpen quantitative reasoning skills. As a result, more efficient clinical trials can be designed and more accurate inference can be attained to advance knowledge in clinical research. Although the famous Bayes theorem developed by Reverend Thomas Bayes was published posthumously in 1763 - long before frequentist methods became popular - the Bayesian method has historically been underutilized and underappreciated compared to its frequentist counterpart. The Bayesian method treats an unknown parameter (e.g., the real treatment effect) as random and the known data as fixed. The Bayesian approach calculates the probability of the parameters given the data; whereas the frequentist approach computes the probability of the data given the parameters. Hence, these two methods are complementary. How does the Bayesian method work? Simply put, the Bayesian method follows these three steps: (1) obtain a prior distribution of the parameter of interest; (2) compute the data likelihood; and (3) synthesize these two pieces of information to form a posterior distribution. The posterior distribution then becomes the prior distribution for the subsequent evaluation. Some unique strengths of the Bayesian approach are as follows: (i) Compared to the frequentist approach, the Bayesian approach is more intuitive and addresses the problem at hand directly. For example, it can calculate the probability that the null hypothesis is true. On the other hand, the frequentist approach calculates the probability of the data given that the null hypothesis is true (e.g., the P value), which provides an indirect way to assess whether the null hypothesis is true. (ii) The Bayesian approach models the unknown parameter with a distribution and properly addresses various levels of uncertainty. For example, a hierarchical Bayes model can be constructed to evaluate the response rate of a drug in subgroups of patients and/or in multi-center trials. (iii) The Bayesian method accommodates more frequent monitoring and interim decision making during a trial; thus, it provides a platform for sequential learning. The prior distribution is updated by the data to form the posterior distribution. The resulting posterior distribution then becomes the prior distribution for a future evaluation. Most clinical trials are conducted over an extended period of time. Hence, it is desirable to frequently monitor the interim results so decisions can be made early when sufficient evidence has accumulated. (iv) The Bayesian method takes the "learn as you go" approach. The built-in "learning" feature makes the Bayesian approach adaptive in nature. The conduct of a clinical trial can be adapted according to the knowledge gained from the currently observed data. For example, a trial design that incorporates outcomeadaptive randomization can assign more patients to better treatments as the trial moves along. An adaptive sample size estimation procedure can adjust the size of the trial according to the observed outcome. (v) The Bayesian method formally incorporates prior information gathered before, during, and outside of the trial. Clinical trials often do not arise from an information vacuum; however, standard trial designs do not allow researchers to take advantage of all the existing knowledge about the experimental agent(s). The Bayesian method allows researchers to incorporate information gathered before the trial in the prior distribution. A Bayesian framework also can incorporate the information accumulated in the trial and that acquired outside of the trial into the model for making inference. Bayesian meta-analysis provides a natural way to synthesize information across multiple trials. (vi) The Bayesian method can incorporate a utility function for informed decision making. Taking the Bayesian decision theoretic approach, clinical trial investigators can specify the "utility" or "loss" of various events. For example, "what is the utility (or importance) to the patient of being cured of cancer, and what is the loss to the patient if a long-term toxicity occurred due to the treatment?" The Bayesian method formulates the subjective preference for outcomes explicitly and quantitatively to aid investigators in making informed decisions. The optimal decision of the trial conduct can be made by maximizing the utility function or minimizing the loss function. However, Bayesian methods require the specification of the prior distribution, and the result may be sensitive to this choice. Bayesian methods are also more computationally intensive - a challenge that has been alleviated by the development of better computing algorithms and faster computers. Additional infrastructure is required for implementing Bayesian designs in clinical trials. Specialized software programs are often required for the study design, simulation, conduct, and analysis. Web-based applications are particularly useful for timely data entry, interim analysis, and reporting. Trial success requires not only the development of proper tools, but also timely and accurate execution of outcome evaluation, adaptive randomization, data analysis, and inference making. Bayesian methods hold great promise for improving the efficiency and flexibility of clinical trial conduct, and are ideal for learning and adaptation. Bayesian methods provide excellent tools for searching for effective treatments and predictive markers in the quest for biomarker-based personalized medicine - with a goal of treating more patients with more effective therapies both inside and outside the trial. Examples such as the BATTLE trial, the BATTLE-2 trial, and the I-SPY 2 trial will be illustrated. The relative merit of the Bayesian and frequentist approaches continues to be the subject of debate in statistics. Better statistical methods can lead to more efficient clinical trials designs, lower sample sizes, more accurate conclusions, and better outcomes for patients enrolled in the trials and beyond. The Bayesian approach offers an attractive alternative for better trials. More such trials should be designed and conducted and demonstrate its real benefit