Calculate Correlation Bounds Between Count and Continuous Outcomes

Computes the lower and upper bounds of the correlation coefficient between an overdispersed count outcome (negative binomial) and a continuous outcome (normal), as described in Homma and Yoshida (2024).

Usage

corrbound2MixedCountContinuous(lambda, nu, mu, sd)

Arguments

lambda

Mean parameter for the negative binomial distribution (lambda > 0)

nu

Dispersion parameter for the negative binomial distribution (nu > 0)

mu

Mean for the continuous outcome

sd

Standard deviation for the continuous outcome (sd > 0)

Value

A named numeric vector with two elements:

L_bound

Lower bound of the correlation

U_bound

Upper bound of the correlation

Details

The correlation bounds are calculated using the Frechet-Hoeffding bounds for copulas, as described in Trivedi and Zimmer (2007). The negative binomial distribution has mean lambda and variance: $$Var(Y_1) = \lambda + \frac{\lambda^2}{\nu}$$

The variance of the negative binomial distribution is: Var(Y1) = lambda + lambda^2/nu

References

Homma, G., & Yoshida, T. (2024). Sample size calculation in clinical trials with two co-primary endpoints including overdispersed count and continuous outcomes. Pharmaceutical Statistics, 23(1), 46-59.

Trivedi, P. K., & Zimmer, D. M. (2007). Copula modeling: an introduction for practitioners. Foundations and Trends in Econometrics, 1(1), 1-111.

Examples

# Calculate correlation bounds for NB(1.25, 0.8) and N(0, 250)
corrbound2MixedCountContinuous(lambda = 1.25, nu = 0.8, mu = 0, sd = 250)
#>    L_bound    U_bound 
#> -0.8457747  0.8457747 

# Higher dispersion parameter
corrbound2MixedCountContinuous(lambda = 2.0, nu = 2.0, mu = 50, sd = 200)
#>    L_bound    U_bound 
#> -0.9209669  0.9209671 

# Different follow-up time
corrbound2MixedCountContinuous(lambda = 1.0 * 2, nu = 1.0, mu = 0, sd = 300)
#>    L_bound    U_bound 
#> -0.8812028  0.8812028