Inference of COVID-19 Vaccine Efficacy & Uncertainty via Bayesian Methods

A Bayesian statistical inference project analyzing clinical trial data for major COVID-19 vaccines (Pfizer-BioNTech, Moderna, AstraZeneca, etc.), quantifying efficacy point estimates, 95% Credible Intervals, and age-stratified effectiveness using JAGS (Just Another Gibbs Sampler) and MCMC sampling.

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📌 Overview & Mathematical Formulation

Estimating vaccine efficacy ($\epsilon$) from Phase III clinical trial data requires robust uncertainty quantification, particularly across small demographic sub-samples.

1. Efficacy Parameter Definition

Vaccine efficacy $\epsilon$ is defined as the relative reduction in infection risk between the vaccinated group and placebo group:

$$\epsilon = 1 - \frac{\theta_{\text{vac}}}{\theta_{\text{placebo}}}$$

where $\theta_{\text{vac}}$ and $\theta_{\text{placebo}}$ represent the attack rates (infection probabilities per person-year) in the respective trial arms.

2. Bayesian Hierarchical Likelihood & Priors

Infection counts in each arm are modeled using Binomial or Poisson likelihoods:

$$k_{\text{vac}} \sim \text{Binomial}(N_{\text{vac}}, \theta_{\text{vac}}), \quad k_{\text{placebo}} \sim \text{Binomial}(N_{\text{placebo}}, \theta_{\text{placebo}})$$

Uninformative or informative Beta/Gamma conjugate priors are assigned to infection parameters:

$$\theta_{\text{vac}} \sim \text{Beta}(\alpha_1, \beta_1), \quad \theta_{\text{placebo}} \sim \text{Beta}(\alpha_2, \beta_2)$$

Using JAGS MCMC Gibbs sampling, posterior distributions $P(\epsilon | \text{Data})$ are generated to evaluate posterior mean efficacy and 95% Highest Posterior Density (HPD) Credible Intervals.

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🔬 Key Analytical Features

• JAGS Bayesian Model Definition: Custom JAGS scripts modeling clinical trial data across different vaccine manufacturers.
• MCMC Convergence Diagnostics: Trace plots, autocorrelation analysis, and Gelman-Rubin diagnostics ($\hat{R}$).
• Posterior Probability Density Estimation: Full posterior distribution density curves highlighting 95% Credibility Intervals.
• Age-Stratified Efficacy Analysis: Demographic breakdown exploring efficacy variations between younger ($<65$) and older ($\ge 65$) age cohorts.

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📁 Repository Structure

Advance-Statistics-in-Physics_Project/
├── Final.ipynb             # Main Jupyter Notebook containing JAGS models & MCMC analysis
├── covid19w.pdf            # Written research paper / report
├── project description.pdf # Course assignment guidelines
├── data/                   # Clinical trial datasets (Pfizer, Moderna, AstraZeneca, etc.)
├── README.md               # Project documentation
└── .gitignore              # Git ignore rules

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🛠️ Setup & Requirements

Prerequisites

• Python 3.8+ & R environment
• JAGS (Just Another Gibbs Sampler) installed on your system (brew install jags on macOS or apt install jags on Linux).

R & Python Packages

Required statistical packages:

• R: rjags, coda, ggplot2, tidyverse, bayesplot
• Python: numpy, scipy, pandas, matplotlib, seaborn, rpy2

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🚀 How to Run

1. Clone the repository:

   git clone https://github.com/mojee13/Advance-Statistics-in-Physics_Project.git
   cd Advance-Statistics-in-Physics_Project

2. Open the analysis notebook:

   jupyter notebook Final.ipynb

3. Execute cells to run JAGS Gibbs sampling chains, plot posterior density distributions, and display 95% Credibility Interval tables.

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📚 References & Paper

For full narrative analysis and tables, consult the included report: 📄 covid19w.pdf