ptdiff_MM: Cumulative Distribution Function of the Difference of Two...

In BayesianQDM: Bayesian Quantitative Decision-Making Framework for Binary and Continuous Endpoints

Cumulative Distribution Function of the Difference of Two Independent t-Distributed Variables via Moment-Matching Approximation

Description

Calculates the cumulative distribution function (CDF) of the difference between two independent non-standardised t-distributed random variables using a Moment-Matching approximation. Specifically, computes P(T_t - T_c \le q) or P(T_t - T_c > q), where T_k \sim t(\mu_k, \sigma_k^2, \nu_k) for k \in \{t, c\}.

Usage

ptdiff_MM(q, mu_t, mu_c, sd_t, sd_c, nu_t, nu_c, lower.tail = TRUE)

Arguments

q	A numeric scalar representing the quantile threshold.
mu_t	A numeric scalar or vector giving the location parameter of the t-distribution for the treatment group.
mu_c	A numeric scalar or vector giving the location parameter of the t-distribution for the control group.
sd_t	A positive numeric scalar or vector giving the scale parameter of the t-distribution for the treatment group.
sd_c	A positive numeric scalar or vector giving the scale parameter of the t-distribution for the control group.
nu_t	A numeric scalar giving the degrees of freedom of the t-distribution for the treatment group. Must be greater than 4 for finite fourth moment.
nu_c	A numeric scalar giving the degrees of freedom of the t-distribution for the control group. Must be greater than 4 for finite fourth moment.
lower.tail	A logical scalar; if TRUE (default), the function returns P(T_t - T_c \le q), otherwise P(T_t - T_c > q).

Details

The difference D = T_t - T_c is approximated by a single non-standardised t-distribution t(\mu^*, {\sigma^*}^2, \nu^*) whose parameters are determined by matching the first two even moments of D:

• \mu^* = \mu_t - \mu_c.

• {\sigma^*}^2 is obtained from the second-moment equation.

• \nu^* is obtained from the fourth-moment equation.

The approximation requires \nu_t > 4 and \nu_c > 4 for finite fourth moments. It is exact in the normal limit (\nu \to \infty) and works well in practice when \nu > 10. Because it reduces to a single call to pt(), it is orders of magnitude faster than the numerical integration method (ptdiff_NI) and is fully vectorised.

Value

A numeric scalar or vector in [0, 1]. When mu_t, mu_c, sd_t, or sd_c are vectors of length n, a vector of length n is returned.

Examples

# P(T_t - T_c > 3) with equal parameters
ptdiff_MM(q = 3, mu_t = 2, mu_c = 0, sd_t = 1, sd_c = 1,
          nu_t = 17, nu_c = 17, lower.tail = FALSE)

# P(T_t - T_c > 1) with unequal scales
ptdiff_MM(q = 1, mu_t = 5, mu_c = 3, sd_t = 2, sd_c = 1.5,
          nu_t = 10, nu_c = 15, lower.tail = FALSE)

# P(T_t - T_c > 0) with different degrees of freedom
ptdiff_MM(q = 0, mu_t = 1, mu_c = 1, sd_t = 1, sd_c = 1,
          nu_t = 5, nu_c = 20, lower.tail = FALSE)

# Lower tail: P(T_t - T_c <= 2)
ptdiff_MM(q = 2, mu_t = 3, mu_c = 0, sd_t = 1.5, sd_c = 1.2,
          nu_t = 12, nu_c = 15, lower.tail = TRUE)

# Vectorised usage
ptdiff_MM(q = 1, mu_t = c(2, 3, 4), mu_c = c(0, 1, 2),
          sd_t = c(1, 1.2, 1.5), sd_c = c(1, 1.1, 1.3),
          nu_t = 10, nu_c = 10, lower.tail = FALSE)

BayesianQDM documentation built on April 22, 2026, 1:09 a.m.