Bayesian Bite: How could a Beta-Binomial model streamline analyses in your next clinical study?

If your trial uses a binary endpoint such as responder vs. non-responder or you are interested in safety monitoring via AE rates — a Bayesian Beta-Binomial model could be the option for you!

What is it?  

A beta-binomial model combines: 

• Binomial likelihood (e.g. the data you observe in the trial such as the number of responders out of n participants in each arm), and 

• Beta prior for the believed response/event rate p (can be vague ‘flat’ prior) 

to give a Beta-Binomial predictive posterior distribution. This leverages the conjugacy of the beta prior distribution with the binomial likelihood to maintain simplicity for the posterior distribution. 

Intuitive probabilistic statements can be made from your posterior distribution enabling communication of results (e.g. what’s the probability there are more AEs in the active group than placebo?)  

Another key benefit is the mechanistic approach to update your beliefs about responder/event rates as data accumulates throughout the study. This makes it an excellent candidate approach for AE rate monitoring or decision making at intermediate timepoints e.g. interim analysis DMCs or for incorporating knowledge from previous phases to increase certainty of your results. 

📊 Efficacy 

Suppose for each independent group, active and placebo, you start with a Beta(α, β) prior distribution for the response rate. To make this a vague prior α=1, β=1 could be used. 

After observing: 

• x_T responders out of n_T in the active arm 
• x_P responders out of n_P in the placebo arm

The posteriors become: 

• p_T ∼ Beta(α + x_T,  β + n_T − x_T )
• p_P ∼ Beta(α + x_P,  β + n_P − x_P )  

Compute the posterior probability of active treatment superiority 

= P(p_T − p_P > c  | data)

That is, by randomly drawing values from the independent distributions of p_T and p_P, what proportion of the random draws met the condition p_T− p_P > c, where c is a clinically meaningful treatment difference. Typically, c=0 to demonstrate superiority in response rate of active treatment vs. placebo. We set a predefined threshold for success (e.g., 0.95), and if the probability is greater than the threshold then we may consider the active treatment effective. Ensure a large enough number of samples is used to ensure consistent results up to an acceptable level of precision between random draws with different seeds. 

Beta-binomial models can be particularly useful in: 

• adaptive or seamless trial designs 

• small or rare disease populations (sparse data) 

• Early phase “go/no-go” decision making (perhaps in dose-finding studies) 

 ⚠️ Safety Monitoring: Comparing AE Rates 

In almost the same way as for efficacy above, the same beta–binomial framework applies to estimating and comparing AE rates between treatment groups. 

For an AE of interest: 

• x_T participants with AEs out of n_T participants in the active arm 

• x_P participants with AEs out of n_P participants in the placebo arm 

P(p_T > p_P | data) gives the posterior probability that the active AE rate is greater than placebo AE rate.

A threshold of 80-95% may be set, as appropriate to the study, whereby the safety data will be independently reviewed with consideration to stop the study if P(p_T > p_P | data) > threshold. 

Here’s where it gets really useful! 

If you are revisiting the analysis repeatedly, e.g. in the case of AE rate monitoring at DMCs, the predictive posterior distribution can be obtained simply by updating the Beta distribution shape and scale parameters according to the number of participants with events and participants without events that have been observed since the last analysis (simple addition of new observations to the pre-existing predictive posterior). 

Key insight: as more information is observed (from 10 participants in each arm to 20) in the plots below the density close to the mean increases, i.e. we observe greater certainty that response rate in the active arm is greater than in the placebo arm despite the presence of a placebo effect. Therefore as your study progresses the updated posteriors reflect the newly observed information.