Authors: Giles Partington, Sarah Simpson, Lauren Rojas, Emily Foreman, Ian Wadsworth.
Bayesian statistics is one of the two main fields of thought for statistics. Whilst the more familiar frequentist statistics focuses on observing events to attempt to disprove a null hypothesis at a specific p-value and power; Bayesian statistics uses the ideas behind Bayes theorem to use observed events as a way of updating prior beliefs (Lee 2012).[1]
Prior beliefs are quantified as prior distributions which can be created through the use of historical data or expertise (or can be left vague) and are combined with observed events to create posterior distributions (which can themselves be updated by further observed events).
A good way to think about this is to take a sports event, such as Wimbledon, to calculate the odds of a specific tennis player winning Wimbledon in 2023. We cannot run this sporting event 100 times and then see how many times that player wins out of 100 to work out a percentage chance of them winning. We would need to take into account their previous performance in similar large pressure tournaments, how they play on similar surface courts, how they have been playing recently and potentially other factors such as weather/other players’ performances/etc. This is all considered “historic information” and any new information like a sudden injury or who wins the grass court tournaments that lead up to Wimbledon is equivalent to observed events and could change how we look at these factors. With Bayesian, this new information can be incorporated to update the priors, indicating the important adaptive nature of Bayesian statistics.
For clinical trials, this can be extremely useful when there is already a large amount of historic data available. For example, when there are multiple smaller trials/sources of information which can then be combined into one systematic meta-analysis, or when it is difficult or even impossible to incorporate a large number of observed events (Gelman et al. 2013).[2] When translating this to clinical trials, this is extremely useful in several different fields:
- Rare disease/small population trials
- Early phase oncology
- Simulation for sample size calculation
- Adaptive designs
For rare disease/small population trials, the use of Bayesian statistics can help overcome the strict cap on possible sample size they can achieve. Their limited size will often result in weaker powered trials and less reliable outcomes. By enhancing these trials with informed priors, a more reliable outcome can be achieved, and a Bayesian framework means that outcomes do not have to be restricted by power calculations.
In 2020 the FDA (Food and Drug Administration) started Project Optimus, a strategy to advance the way oncology trials are handled in early phases, this expected coprimary endpoints of efficacy and toxicity. Some Bayesian methodologies such as BOIN12 are used to accomplish this coprimary trial. The family of BOIN (Bayesian Optimal Interval) designs work as an adaptation of the 3+3 design, except they allow new cohorts of patients to be added to the same, a lower, or a higher dose level throughout the trial. If a cohort of patients enters the same dose level as a previous cohort, the results of the two cohorts are combined thus giving a larger enriched cohort of outcomes when considering the usefulness of the dose level.
Bayesian methods are also useful for sample size considerations, with the ability to set up priors that can be used for simulating trial outcomes. These simulations can be used to run hundreds of iterations of the trial without having to recruit any patients. By averaging the results that come out from these simulations, a sample size can be calculated that allows for reduced patient numbers and confidence that the trial will be able to produce results that are strongly representative without excessively large sample sizes.
Adaptive designs are trials in which decisions about the trial design can be changed during interim analyses. For example, a trial with 5 different potential dosages of a drug can be analysed after each dosage has had 20 patients recruited to it, and the worst performing 2 could be dropped from further recruitment. In a frequentist framework there would be issues with multiple testing, requiring increased sample sizes to avoid false positive results. However, with a Bayesian framework, this can be done without issue due to Bayesian updating.
The basic rules of clinical trials are the same for frequentist and Bayesian methods, so it is important that the type of analysis to be used is chosen in advance of observing any data. For a Bayesian trial, this would mean that decisions regarding prior distributions are decided in advance as well as the data to be collected in the trial, and the analysis method used to combine the two. This may be slightly more complex than more familiar frequentist trials, but the opportunities for efficient and effective approaches to clinical trials warrant this extra effort.
References
[1] Peter M. Lee. 2012. Bayesian Statistics: An Introduction (4th. ed.). Wiley Publishing.
[2] Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., & Rubin, D.B. 2013. Bayesian Data Analysis (3rd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/b16018
