Calculate Correlation Bounds Between Two Binary Outcomes

Computes the lower and upper bounds of the correlation coefficient between two binary outcomes based on their marginal probabilities, as described in Prentice (1988).

Usage

corrbound2Binary(p1, p2)

Arguments

p1

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

p2

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

Value

A named numeric vector with two elements:

L_bound

Lower bound of the correlation

U_bound

Upper bound of the correlation

Details

For two binary outcomes with marginal probabilities p1 and p2, the correlation coefficient rho is bounded by: $$\rho \in [L(p_1, p_2), U(p_1, p_2)]$$ where $$L(p_1, p_2) = \max\left\{-\sqrt{\frac{p_1 p_2}{(1-p_1)(1-p_2)}}, -\sqrt{\frac{(1-p_1)(1-p_2)}{p_1 p_2}}\right\}$$ $$U(p_1, p_2) = \min\left\{\sqrt{\frac{p_1(1-p_2)}{p_2(1-p_1)}}, \sqrt{\frac{p_2(1-p_1)}{p_1(1-p_2)}}\right\}$$

References

Prentice, R. L. (1988). Correlated binary regression with covariates specific to each binary observation. Biometrics, 44(4), 1033-1048.

Examples

# Calculate correlation bounds for two binary outcomes
corrbound2Binary(p1 = 0.3, p2 = 0.5)
#>    L_bound    U_bound 
#> -0.6546537  0.6546537 

# When probabilities are equal, upper bound is 1
corrbound2Binary(p1 = 0.4, p2 = 0.4)
#>    L_bound    U_bound 
#> -0.6666667  1.0000000 

# When p1 + p2 = 1, lower bound is -1
corrbound2Binary(p1 = 0.3, p2 = 0.7)
#>    L_bound    U_bound 
#> -1.0000000  0.4285714