Prior Ground: Selection of Prior Distributions When Analyzing Clinical Trial Data Using Bayesian Methods

Increasingly, investigators are choosing to use Bayesian methods for the analysis of clinical trial data. Unlike classical statistical methods that treat model parameter values (such as treatment effects) as fixed, Bayesian methods view parameters as following a probability distribution. As we have written previously,¹ by analyzing clinical trial data using Bayesian methods one can obtain quantities that may be of interest to clinicians, providers, and patients, such as the probability that a treatment effect is more or less than 0, that is, the probability that a treatment is effective.

Both classical and Bayesian methods require a model for the data. However, Bayesian inference also requires the analyst to specify an additional piece of information: the prior distribution of the parameters in the statistical model. These prior distributions (priors) are an opportunity for the analyst to incorporate any information that might be available before observing the data. Through the machinery of Bayes’ theorem, the statistical model and the prior distributions are combined to produce the posterior distribution — the distribution of the model parameters after the data have been observed — from which one can obtain quantities such as the probability a treatment is effective.

Priors tend to fall into one of two categories: informative priors and noninformative priors. Noninformative priors are chosen to “let the data speak for themselves” and to have minimal or no effect on posterior inferences such as the posterior distribution of the treatment effect. For example, a noninformative prior for a treatment effect might be a normal distribution centered at 0 (no effect) with a relatively large standard deviation. Closely related to noninformative priors are weakly informative priors that are designed to have only a modest effect on posterior inferences — just enough to keep inferences in a reasonable range when data are sparse.²

Informative priors are used when the Bayesian analyst wishes to incorporate additional information into the analysis. In the clinical trial setting, this information might come from subject matter expertise or from historical data. The latter include pilot studies, previous trials of the same treatment in the same population, or previous trials of the same treatment in a different population. Priors can also be used to formally incorporate any optimism/pessimism on the part of the investigator. Compared with noninformative priors, informative priors tend to be less diffuse and may be shifted away from a distribution centered at no treatment effect. As a result, informative priors will have a greater influence on posterior distributions and can result in more precise inferences or reduce the required sample size in a trial. This latter point is especially relevant when dealing with rare diseases or hard-to-study populations.

In this issue of NEJM Evidence, Morpeth et al.³ report the results from an open-label, pragmatic, randomized clinical trial of the antiviral nafamostat in noncritically ill hospitalized patients with Covid-19. The primary end point was death or receipt of new invasive or noninvasive ventilation or vasopressor support, within 28days. The data were analyzed using a Bayesian logistic regression model where the primary estimand of the treatment effect was the adjusted posterior log odds ratio of nafamostat versus usual care, that is, a number less than 0 indicates better outcomes with nafamostat. Effectiveness was assessed by calculating the posterior probability that the log odds ratio is less than 0, with a prespecified effectiveness threshold of 99% to conclude effectiveness of nafamostat.

The trial enrolled 160 participants beginning in May 2021. In August 2022, the trial was closed to further enrollment due to a combination of slowed recruitment, a low event rate, and funding constraints, making it unlikely that the trial would be able to meet prespecified stopping criteria. The subsequent analysis found an adjusted odds ratio of 0.40 (95% credible interval [0.12, 1.34]) and a posterior probability of effectiveness of 93%. As prespecified in the trial statistical analysis plan (SAP), a trial decision of effectiveness is to be made if the posterior probability of effectiveness is greater than 99%. Because their results did not meet this threshold, the study investigators concluded that the effectiveness of nafamostat could not be determined.

The use of weakly informative priors was prespecified in the SAP. However, it is possible that the current trial is the last clinical trial of nafamostat in patients with Covid-19. In light of the trial’s lower-than-anticipated sample size and the posterior probability of effectiveness that is close to the prespecified threshold, one wonders whether the study investigators could have constructed informative priors to incorporate additional information into their analysis.

As mentioned in the article, there were three previous randomized clinical trials of nafamostat in patients hospitalized with Covid-19,^4–6 two of which had a clinically relevant primary end point.^5,6 These were small trials with similar, but not identical, end points to those used in Morpeth et al.³ Also, using these results to elicit informative priors would not be easy. Still, neither of these trials appeared to show any benefit of nafamostat. It seems reasonable then to use an informative prior distribution for the treatment effect in the current trial that is more heavily centered around 0 (no treatment effect). Indeed, in their SAP the investigators discuss the possibility of performing sensitivity analyses that allow priors to be more informative than weakly informative priors.

An analysis using informative priors based on information from the previous trials would result in an adjusted odds ratio closer to 1.0 and a posterior probability of effectiveness less than 93%. Thus, the study conclusions would not necessarily change but the resulting inferences would now incorporate all available information on the effects of nafamostat in patients hospitalized with Covid-19 and may give decision makers greater confidence in the study’s null results.