Summary

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and the resulting disease, COVID-19, have afflicted tens of millions of people globally. The urgent need for safe and effective vaccines led to a large-scale phase 3 trial to evaluate the efficacy of a candidate vaccine, BNT162b2.

In this case study, we illustrate how a Bayesian group sequential design was applied in the analysis of vaccine efficacy.

1. Trial Overview

1.1. Design and Procedure

Participants aged 16 or older were randomized 1:1 to receive two doses, 21 days apart, of either:

  • BNT162b2 (30 μg per dose), or

  • Saline placebo

Injections were administered into the deltoid muscle, and participants were monitored for 30 minutes post-vaccination.

1.2. Enrollment and Participants

  • Total screened: 44,820

  • Randomized: 43,548

  • Received treatment: 43,448 (21,720 BNT162b2, 21,728 placebo)

  • Main safety set: 37,706 participants with ≥2 months of follow-up

Key characteristics:

49% female, 83% White, 28% Hispanic/Latinx, 35% obese, 21% with coexisting conditions, median age 52.

2. Statistical Approach

Primary Efficacy Estimation

Participants contributing to efficacy evaluation received both doses and had no COVID-19 infection within 7 days after the second dose.
Vaccine Efficacy (VE) is defined as:

\(VE = 100 \times (1 – IRR)\)

where IRR is the incidence rate ratio between vaccine and placebo groups. A Bayesian beta-binomial model is used to derive:

  • A 95% credible interval (CI) for VE

  • Probability that VE exceeds 30%

3. Bayesian Group Sequential Design

Let:

  • \(\theta\): Proportion of total COVID-19 cases occurring in the vaccine group

  • \(VE = \displaystyle \frac{1 – 2\theta}{1 – \theta}\)

Assume a minimally informative prior:

\(\theta \sim \text{Beta}(0.700102, 1)\)

This prior centers around the efficacy threshold of VE = 30% (i.e., θ = 0.4118) and reflects substantial uncertainty.

Posterior

Given \(n\) total cases and \(n_v\) cases in the vaccine group, the posterior is:

\(\theta \mid \text{data} \sim \text{Beta}(0.700102 + n_v, 1 + n – n_v)\)

Probability of Efficacy

We compute:

\(P(VE \geq 30\% \mid \text{data}) = P(\theta \leq 0.4118 \mid \text{data})\)

4. Decision Rules

  • Interim efficacy is declared if posterior probability > 99.5%

  • Final efficacy is declared if posterior probability > 98.6%

At final analysis (n = 164), efficacy is declared if ≤53 cases occurred in the vaccine group.

Conclusion

This real-world example demonstrates the strength of Bayesian adaptive designs:

  • Probabilistic decision-making

  • Prior incorporation

  • Sequential monitoring

Such an approach offers transparency, flexibility, and rigor in evaluating interventions during a public health crisis.

Stay with 3 D Statistical Learning for more applied Bayesian insights.

With special thanks to Dr. Dany Djeudeu for statistical leadership and scientific rigor.