Cumulative Distribution Function of the Difference Between Two Beta-Binomial Variables

This function calculates the cumulative distribution function (CDF) of the difference between two independent beta-binomial random variables. The computation uses exact enumeration over all possible combinations of outcomes. Specifically, it computes P((Y1/m1) - (Y2/m2) ≤ q) or P((Y1/m1) - (Y2/m2) > q) where Y1 ~ BetaBinomial(m1, alpha1, beta1) and Y2 ~ BetaBinomial(m2, alpha2, beta2).

Usage

p2betabinomdiff(q, m1, m2, alpha1, alpha2, beta1, beta2, lower.tail = TRUE)

Arguments

q

A numeric value representing the quantile threshold for the difference in proportions.

m1

A positive integer representing the number of trials for the first beta-binomial distribution.

m2

A positive integer representing the number of trials for the second beta-binomial distribution.

alpha1

A positive numeric value representing the first shape parameter of the first beta distribution.

alpha2

A positive numeric value representing the first shape parameter of the second beta distribution.

beta1

A positive numeric value representing the second shape parameter of the first beta distribution.

beta2

A positive numeric value representing the second shape parameter of the second beta distribution.

lower.tail

A logical value; if TRUE (default), probabilities are P((Y1/m1) - (Y2/m2) ≤ q), otherwise P((Y1/m1) - (Y2/m2) > q).

Value

A numeric value in [0, 1] representing the cumulative probability that the difference in proportions between the two beta-binomial variables is below (if lower.tail = TRUE) or exceeds (if lower.tail = FALSE) the quantile q.

Details

The function computes the exact probability by enumerating all possible combinations of outcomes for both beta-binomial distributions and summing the joint probabilities for which the proportion difference satisfies the specified condition.

The probability mass function for BetaBinomial(m, α, β) is: $$P(Y = k) = \binom{m}{k} \frac{B(k + \alpha, m - k + \beta)}{B(\alpha, \beta)}$$

where B(·,·) is the beta function.

The beta-binomial distribution arises when the success probability in a binomial distribution follows a beta distribution, making it useful for modeling overdispersed count data. For large values of m1 and m2, the computation may become slow as it requires evaluating (m1 + 1) × (m2 + 1) combinations.

Examples

# Calculate P((Y1/12) - (Y2/12) > 0.2) for symmetric beta priors
p2betabinomdiff(0.2, 12, 12, 0.5, 0.5, 0.5, 0.5, lower.tail = FALSE)
#> [1] 0.3372581

# Calculate P((Y1/20) - (Y2/15) > 0.1) for different sample sizes
p2betabinomdiff(0.1, 20, 15, 1, 1, 1, 1, lower.tail = FALSE)
#> [1] 0.4017857

# Calculate P((Y1/10) - (Y2/10) > 0) for informative priors
p2betabinomdiff(0, 10, 10, 2, 3, 3, 2, lower.tail = FALSE)
#> [1] 0.2383331

# Calculate lower tail probability P((Y1/15) - (Y2/15) ≤ 0.05)
p2betabinomdiff(0.05, 15, 15, 1, 1, 1, 1, lower.tail = TRUE)
#> [1] 0.53125