pbetadiff: Cumulative Distribution Function of the Difference Between...

In BayesianQDM: Bayesian Quantitative Decision-Making Framework for Binary and Continuous Endpoints

Cumulative Distribution Function of the Difference Between Two Independent Beta Variables

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent Beta-distributed random variables. Specifically, computes P(\pi_t - \pi_c \le q) or P(\pi_t - \pi_c > q), where \pi_j \sim \mathrm{Beta}(\alpha_j, \beta_j) for j \in \{t, c\}.

Usage

pbetadiff(q, alpha_t, alpha_c, beta_t, beta_c, lower.tail = TRUE)

Arguments

q	A numeric scalar in [-1, 1] representing the quantile threshold for the difference in proportions.
alpha_t	A positive numeric scalar giving the first shape parameter of the Beta distribution for the treatment group.
alpha_c	A positive numeric scalar giving the first shape parameter of the Beta distribution for the control group.
beta_t	A positive numeric scalar giving the second shape parameter of the Beta distribution for the treatment group.
beta_c	A positive numeric scalar giving the second shape parameter of the Beta distribution for the control group.
lower.tail	A logical scalar; if TRUE (default), the function returns P(\pi_t - \pi_c \le q), otherwise P(\pi_t - \pi_c > q).

Details

The upper-tail probability is obtained via the convolution formula:

P(\pi_t - \pi_c > q) = \int_0^1 F_{\mathrm{Beta}(\alpha_c, \beta_c)}(x - q)\, f_{\mathrm{Beta}(\alpha_t, \beta_t)}(x)\, dx

where f_{\mathrm{Beta}(\alpha_t, \beta_t)} is the density of \pi_t and F_{\mathrm{Beta}(\alpha_c, \beta_c)} is the CDF of \pi_c. Boundary cases are handled automatically by pbeta (0 for x - q \le 0, 1 for x - q \ge 1), so integrating over [0, 1] is safe.

This single-integral convolution replaces an equivalent double-integral formulation based on Appell's F1 hypergeometric function, yielding the same result with lower computational cost because both pbeta and dbeta are implemented in compiled C code.

Value

A numeric scalar in [0, 1].

Examples

# P(pi_t - pi_c > 0.2) with symmetric Beta(0.5, 0.5) priors
pbetadiff(0.2, 0.5, 0.5, 0.5, 0.5, lower.tail = FALSE)

# P(pi_t - pi_c > -0.1) with informative priors
pbetadiff(-0.1, 2, 1, 3, 4, lower.tail = FALSE)

# P(pi_t - pi_c > 0) with equal priors -- should be approximately 0.5
pbetadiff(0, 1, 1, 1, 1, lower.tail = FALSE)

# Lower tail: P(pi_t - pi_c <= 0.1) with symmetric priors
pbetadiff(0.1, 2, 2, 2, 2, lower.tail = TRUE)

BayesianQDM documentation built on April 22, 2026, 1:09 a.m.