Introduction to the Beta-Binomial Model

Duration: 75 minutes

Learning Objectives:

Materials:

Lesson Outline:

Introduction (5 minutes)

Theory and Concepts (15 minutes)

Practical Implementation (15 minutes)

# Sample R Code
# Load libraries
# library(rstan)
# library(ggplot2)
# library(dplyr)
# Generate a sample dataset
# set.seed(123)
# successes <- c(12, 15, 17, 19, 20)
# trials <- c(20, 20, 20, 20, 20)
# data <- data.frame(successes, trials)
# Beta-Binomial model in Stan

#data {
  #int<lower=0> N;        # Number of observations
  #int<lower=0> y[N];     # Count data
  #int<lower=0> n_trials; # Number of trials (constant for a binomial distribution)
#}

# parameters {
   #real<lower=0, upper=1> theta;  # Probability parameter
   #real<lower=0> alpha;           # Beta distribution shape parameter
   #real<lower=0> beta;            # Beta distribution shape parameter
#}

#model {
  # Prior distribution
 # theta ~ beta(alpha, beta);

  # Likelihood
 # y ~ binomial(n_trials, theta);
#}
# Install and load the rstan package
#install.packages("rstan")
#library(rstan)

# Define the data
#data_list <- list(
  #N = length(your_data),
  #y = your_data,
  #n_trials = number_of_trials
#)

# Compile the Stan model
#stan_model <- stan_model(file = "path/to/your/stan/model/file.stan")

# Fit the model
#fit <- sampling(stan_model, data = data_list, chains = 4, iter = 2000, warmup = 1000)

# Print the summary of the posterior distribution
# print(fit)

Interpretation of Results (10 minutes)

# Sample R Code:
# Plot posterior distribution of theta
# posterior_samples <- as.data.frame(fit)
# ggplot(posterior_samples, aes(x = theta)) +
# geom_histogram(binwidth = 0.01, fill = "blue", color = "black") +
# labs(title = "Posterior Distribution of Theta", x = "Theta (Probability of Success)")

Hands-on Practice (15 minutes)

Q&A and Discussion (10 minutes)

Conclusion (5 minutes)

Homework Assignment: Ask participants to analyze a real-world dataset using the Beta-Binomial model in R and share their findings in the next session.

Assessment: Evaluate participants’ understanding through their ability to implement the Beta-Binomial model, interpret results, and engage in discussions during the lesson.

Additional Notes and Resources

Explanation of the Beta Binomial Bayesian Model with examples from the Health Sciences

The Beta-Binomial Bayesian model is a statistical model used in various fields, including health sciences, to analyze and make inferences about data characterized by the number of successes in a fixed number of trials, where the probability of success can vary. It combines two key distributions: the Beta distribution as a prior for the probability of success and the Binomial distribution to model the observed data. This model is particularly useful in health sciences for a wide range of applications, such as clinical trials, epidemiology, and medical testing. Let’s explore the model with a couple of health science examples:

Some technical notes:

\[ Y|\pi \sim Bin(n, \pi) \]

\[ \pi \sim Beta(\alpha, \beta) \]

\[ \pi |(Y=y) \sim Beta (\alpha + y, \ \beta +n -y) \]

  1. Prior Model

    \[ f(\pi)=\frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}\pi^{\alpha - 1} (1-\alpha)^{\beta -1}\]

  2. Data Model

    \[ Bin(n, \pi) \]

  3. Likelihood function

    \[ L(\pi |y)=\binom {n}{y}\pi^y (1-\pi)^{n-y}\]

  4. Posterior Model

    \[ f(\pi |y) \propto f(\pi)L(\pi|y) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\pi^{\alpha-1}(1-\pi)^{\beta-1}\binom{n}{y} \propto \pi^{(y+\alpha)-1}(1-\pi)^{(\beta +n-y)-1} = Beta(\alpha +y, \ \beta+n-y) \]

Suppose a pharmaceutical company is conducting a clinical trial to evaluate the success rate of a new drug in treating a specific disease. They enroll 100 patients and record how many of them respond positively to the treatment (successes). In this case, the Beta-Binomial model can be used to estimate the probability of success and its uncertainty.

  • Beta Distribution as Prior: The company might have prior beliefs about the success rate of similar drugs and express it using a Beta distribution. For example, they believe that the success rate follows a Beta(10, 10) distribution, which represents a relatively neutral prior with a mean of 0.5.
  • Binomial Distribution for Data: If, out of 100 patients, 60 respond positively to the new drug, the observed data follows a Binomial distribution with parameters n=100 (number of trials) and p (the success rate).

In this scenario, the Bayesian analysis will update the prior Beta distribution with the observed data, yielding a posterior distribution for the success rate, which provides not only a point estimate but also a credible interval representing the uncertainty around the estimate.

Epidemiologists want to estimate the prevalence of a rare disease in a population. They decide to conduct a study in which they take a random sample of 500 individuals and test them for the presence of the disease. In this case, the Beta-Binomial model can be used to estimate the prevalence of the disease in the population.

  • Beta Distribution as Prior: Epidemiologists might use a Beta(2, 8) distribution as a prior for the prevalence. This implies they have prior knowledge suggesting that the prevalence is low (mean of 0.2) but with some uncertainty.
  • Binomial Distribution for Data: If 10 individuals in the sample test positive for the disease, the observed data follows a Binomial distribution with n=500 (number of trials) and p (prevalence).

In this scenario, the Bayesian analysis will update the prior Beta distribution with the observed data, resulting in a posterior distribution for the prevalence of the disease. This posterior distribution gives a range of credible estimates for the disease prevalence in the population.

In both examples, the Beta-Binomial model not only provides a point estimate of the parameter of interest (success rate or disease prevalence) but also quantifies the uncertainty around the estimate, which is a valuable feature in health sciences where decisions can have critical consequences. It allows researchers to make informed decisions, taking both prior beliefs and observed data into account.