Compute Joint Bivariate Binary Probabilities from Marginal Rates and Correlation

Converts the parameterisation \((\pi_{j1}, \pi_{j2}, \rho_j)\) into the four-cell joint probability vector \((p_{j,00}, p_{j,01}, p_{j,10}, p_{j,11})\) for a bivariate binary outcome in group \(j \in \{t, c\}\). The conversion uses the standard moment-matching identity for the Pearson correlation of two Bernoulli variables, and checks that the requested correlation is feasible for the supplied marginal rates.

Usage

getjointbin(pi1, pi2, rho, tol = 1e-10)

Arguments

pi1

A single numeric value in (0, 1) giving the marginal response probability for Endpoint 1 (\(\pi_{j1}\)) in group \(j\).

pi2

A single numeric value in (0, 1) giving the marginal response probability for Endpoint 2 (\(\pi_{j2}\)) in group \(j\).

rho

A single numeric value giving the Pearson correlation (\(\rho_j\)) between Endpoint 1 and Endpoint 2 in group \(j\). Must lie within the feasible range \([\rho_{\min}, \rho_{\max}]\) determined by pi1 and pi2 (see Details). Use rho = 0 for independence.

tol

A single positive numeric value specifying the tolerance used when checking whether rho lies within its feasible range and whether the resulting probabilities are non-negative. Default is 1e-10.

Value

A named numeric vector of length 4 with elements p00, p01, p10, p11, where p_lm = Pr(Endpoint 1 = l, Endpoint 2 = m) for \(l, m \in \{0, 1\}\). All elements are non-negative and sum to 1.

Details

For a bivariate binary outcome \((Y_{i1}, Y_{i2})\) of patient \(i\) (\(i = 1, \ldots, n_j\)) in group \(j\) with marginal success probabilities \(\pi_{j1} = \Pr(Y_{i1} = 1)\) and \(\pi_{j2} = \Pr(Y_{i2} = 1)\), the Pearson correlation is $$ \rho_j = \frac{p_{j,11} - \pi_{j1} \pi_{j2}} {\sqrt{\pi_{j1}(1 - \pi_{j1})\, \pi_{j2}(1 - \pi_{j2})}}. $$ Solving for \(p_{j,11}\) gives $$p_{j,11} = \rho_j \sqrt{\pi_{j1}(1-\pi_{j1})\,\pi_{j2}(1-\pi_{j2})} + \pi_{j1} \pi_{j2},$$ from which the remaining probabilities follow: $$p_{j,10} = \pi_{j1} - p_{j,11}, \quad p_{j,01} = \pi_{j2} - p_{j,11}, \quad p_{j,00} = 1 - p_{j,10} - p_{j,01} - p_{j,11}.$$

For all four cell probabilities to lie in [0, 1], the correlation must satisfy $$ \rho_{\min} = \frac{\max(0,\; \pi_{j1} + \pi_{j2} - 1) - \pi_{j1}\pi_{j2}} {\sqrt{\pi_{j1}(1-\pi_{j1})\,\pi_{j2}(1-\pi_{j2})}} \;\le\; \rho_j \;\le\; \frac{\min(\pi_{j1}, \pi_{j2}) - \pi_{j1}\pi_{j2}} {\sqrt{\pi_{j1}(1-\pi_{j1})\,\pi_{j2}(1-\pi_{j2})}} = \rho_{\max}. $$ The function raises an error if rho falls outside this range (subject to tol).

Examples

# Example 1: Independent endpoints (rho = 0)
getjointbin(pi1 = 0.3, pi2 = 0.4, rho = 0.0)
#>  p00  p01  p10  p11 
#> 0.42 0.28 0.18 0.12 

# Example 2: Positive correlation
getjointbin(pi1 = 0.3, pi2 = 0.4, rho = 0.3)
#>       p00       p01       p10       p11 
#> 0.4873498 0.2126502 0.1126502 0.1873498 

# Example 3: Negative correlation
getjointbin(pi1 = 0.3, pi2 = 0.4, rho = -0.2)
#>        p00        p01        p10        p11 
#> 0.37510011 0.32489989 0.22489989 0.07510011 

# Example 4: Verify cell probabilities sum to 1
p <- getjointbin(pi1 = 0.25, pi2 = 0.35, rho = 0.1)
sum(p)  # Should be 1
#> [1] 1

# Example 5: Verify marginal recovery
p <- getjointbin(pi1 = 0.25, pi2 = 0.35, rho = 0.1)
p["p10"] + p["p11"]  # Should equal pi1 = 0.25
#>  p10 
#> 0.25 
p["p01"] + p["p11"]  # Should equal pi2 = 0.35
#>  p01 
#> 0.35