Cumulative Distribution Function of the Difference Between Two Independent Beta Variables

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 \(P(\pi_t - \pi_c > q)\).

Value

A numeric scalar in [0, 1].

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.

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)
#> [1] 0.3377403

# P(pi_t - pi_c > -0.1) with informative priors
pbetadiff(-0.1, 2, 1, 3, 4, lower.tail = FALSE)
#> [1] 0.8783

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

# Lower tail: P(pi_t - pi_c <= 0.1) with symmetric priors
pbetadiff(0.1, 2, 2, 2, 2, lower.tail = TRUE)
#> [1] 0.6181518