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powerprior: Conjugate Power Priors for Bayesian Analysis of Normal Data

Overview

The powerprior package implements conjugate representations of power priors for efficient Bayesian analysis of normal data. This package provides closed-form computation of power priors and posterior distributions, eliminating the need for MCMC sampling.

Key Features: - Univariate normal data using Normal-Inverse-Chi-squared (NIX) distributions - Multivariate normal data using Normal-Inverse-Wishart (NIW) distributions - Closed-form posterior updating (no MCMC required) - Support for both informative and vague (non-informative) initial priors - Efficient posterior sampling algorithms - Ideal for simulation studies and clinical trial design

Theoretical Background

Power priors provide a flexible framework for incorporating historical data with discounting:

\[P(\theta|x, a_0) = L(\theta|x)^{a_0}P(\theta)\]

where: - \(x\) is historical data - \(\theta\) are parameters of interest - \(a_0 \in [0,1]\) is the discounting parameter - \(a_0 = 0\): no borrowing from historical data - \(a_0 = 1\): full borrowing (treat as equivalent to current data)

Dependencies

The package requires: - stats (base R) - MASS - LaplacesDemon

Install dependencies:

install.packages(c("MASS", "LaplacesDemon"))

Quick Start

Univariate Example

library(powerprior)

# Generate historical and current data
set.seed(123)
historical <- rnorm(50, mean = 10, sd = 2)
current <- rnorm(30, mean = 10.5, sd = 2)

# Compute power prior with moderate discounting
pp <- powerprior_univariate(historical, a0 = 0.5)
print(pp)

# Update with current data to get posterior
posterior <- posterior_univariate(pp, current)
print(posterior)

# Sample from posterior
samples <- sample_posterior_univariate(posterior, n_samples = 1000)

# Summarize
cat("Posterior mean of mu:", mean(samples[, "mu"]), "\n")
cat("95% CI for mu:", quantile(samples[, "mu"], c(0.025, 0.975)), "\n")

Multivariate Example

library(powerprior)
library(MASS)

# Generate bivariate data
Sigma_true <- matrix(c(4, 1.5, 1.5, 3), nrow = 2)
historical <- mvrnorm(60, mu = c(10, 5), Sigma = Sigma_true)
current <- mvrnorm(40, mu = c(10.3, 5.2), Sigma = Sigma_true)

# Compute power prior
pp <- powerprior_multivariate(historical, a0 = 0.6)

# Update with current data
posterior <- posterior_multivariate(pp, current)

# Sample from marginal distribution of mu
samples <- sample_posterior_multivariate(posterior, n_samples = 1000, marginal = TRUE)

cat("Posterior mean:", colMeans(samples), "\n")

Main Functions

Univariate Analysis

Multivariate Analysis

Detailed Examples

Example 1: Sensitivity Analysis

Explore how different discounting parameters affect posterior inference:

a0_values <- seq(0, 1, by = 0.1)
results <- data.frame(a0 = a0_values, mean = NA, sd = NA)

for (i in seq_along(a0_values)) {
  pp <- powerprior_univariate(historical, a0 = a0_values[i])
  post <- posterior_univariate(pp, current)
  samples <- sample_posterior_univariate(post, n_samples = 2000, marginal = TRUE)
  
  results$mean[i] <- mean(samples)
  results$sd[i] <- sd(samples)
}

print(results)

Example 2: Informative Prior

Use an informative NIX initial prior:

pp_inform <- powerprior_univariate(
  historical_data = historical,
  a0 = 0.5,
  mu0 = 9.5,      # Prior mean
  kappa0 = 2,     # Prior precision
  nu0 = 5,        # Prior degrees of freedom
  sigma2_0 = 4    # Prior scale parameter
)

posterior_inform <- posterior_univariate(pp_inform, current)

Example 3: Clinical Trial Simulation

Assess operating characteristics through simulation:

n_simulations <- 1000
coverage_count <- 0
true_mean <- 10.5

for (sim in 1:n_simulations) {
  hist <- rnorm(50, mean = 10, sd = 2)
  curr <- rnorm(30, mean = true_mean, sd = 2)
  
  pp <- powerprior_univariate(hist, a0 = 0.5)
  post <- posterior_univariate(pp, curr)
  samples <- sample_posterior_univariate(post, n_samples = 1000, marginal = TRUE)
  
  ci <- quantile(samples, c(0.025, 0.975))
  if (true_mean >= ci[1] && true_mean <= ci[2]) {
    coverage_count <- coverage_count + 1
  }
}

cat("Coverage probability:", coverage_count / n_simulations, "\n")

Computational Advantages

The conjugate representation provides significant computational benefits:

• No MCMC required: Closed-form posterior computation
• Fast sampling: Direct sampling from known distributions
• Simulation studies: Ideal for repeated posterior sampling
• Clinical trial design: Efficient operating characteristic evaluation

Traditional power prior analysis with MCMC might take hours for 1000 simulations; this package completes them in seconds.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.