Probability Mass Function of Bivariate Binomial Distribution

Calculates the probability mass function of the bivariate binomial distribution for given parameters, as described in Homma and Yoshida (2025).

Usage

dbibinom(N, y1, y2, p1, p2, rho)

Arguments

N

Sample size (number of trials)

y1

Observed value(s) of the first random variable (0 to N)

y2

Observed value(s) of the second random variable (0 to N)

p1

True probability of responders for the first outcome (0 < p1 < 1)

p2

True probability of responders for the second outcome (0 < p2 < 1)

rho

Correlation coefficient between the two binary outcomes

Value

Probability mass function value(s) for the bivariate binomial distribution. If y1 and y2 are vectors, returns a vector of probabilities.

Details

The bivariate binomial distribution BiBin(N, p1, p2, gamma) has probability mass function given by equation (3) in Homma and Yoshida (2025): $$P(Y_1 = y_1, Y_2 = y_2) = f(y_1|N, p_1) \times g(y_2|y_1, N, p_1, p_2, \gamma)$$ where $$g(y_2|y_1, N, p_1, p_2, \gamma) = \frac{1}{(1+\gamma)^N} \sum_{m \in \mathcal{M}} \binom{y_1}{m} \binom{N-y_1}{y_2-m} (\xi+\gamma)^m (1-\xi)^{y_1-m} \xi^{y_2-m} (1-\xi+\gamma)^{N-y_1-(y_2-m)}$$ with \(\xi = p_2 + \gamma(p_2 - p_1)\) and \(\mathcal{M} = \{m : m = \max(0, y_2-(N-y_1)), \ldots, \min(y_1, y_2)\}\).

References

Homma, G., & Yoshida, T. (2025). Exact power and sample size in clinical trials with two co-primary binary endpoints. Statistical Methods in Medical Research, 34(1), 1-19.

Examples

# Calculate single probability mass
dbibinom(N = 100, y1 = 30, y2 = 50, p1 = 0.3, p2 = 0.5, rho = 0.5)
#> [1] 0.007981836

# Verify that probabilities sum to 1
N <- 20
p1 <- 0.3
p2 <- 0.5
rho <- 0.5
sum(outer(0:N, 0:N, function(x, y) dbibinom(N, x, y, p1, p2, rho)))
#> [1] 1