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

Source: R/ptdiff_MM.R
ptdiff_MM.Rd

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 \(P(T_t - T_c > q)\).

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.

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.

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)
#> [1] 0.248517

# 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)
#> [1] 0.6483581

# 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)
#> [1] 0.5

# 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)
#> [1] 0.3097574

# 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)
#> [1] 0.7456908 0.7174810 0.6818673