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Cumulative Distribution Function of the Difference of Two Independent t-Distributed Variables via Numerical Integration

Source: R/ptdiff_NI.R
ptdiff_NI.Rd

Calculates the cumulative distribution function (CDF) of the difference between two independent non-standardised t-distributed random variables using numerical integration. 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_NI(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 2 for finite variance.

nu_c

A numeric scalar giving the degrees of freedom of the t-distribution for the control group. Must be greater than 2 for finite variance.

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 upper-tail probability is obtained via the convolution formula: $$P(T_t - T_c > q) = \int_{-\infty}^{\infty} F_{t(\mu_c, \sigma_c^2, \nu_c)}(x - q)\, f_{t(\mu_t, \sigma_t^2, \nu_t)}(x)\, dx$$ where \(f_{t(\mu_t, \sigma_t^2, \nu_t)}\) is the density of \(T_t\) and \(F_{t(\mu_c, \sigma_c^2, \nu_c)}\) is the CDF of \(T_c\). The integral is evaluated by adaptive Gauss-Kronrod quadrature via stats::integrate.

When the input parameters are vectors, mapply applies the scalar integration function across all parameter sets.

Advantages:

  • Exact within numerical precision.

  • Handles arbitrary parameter combinations.

Computational note: this method is substantially slower than the Moment-Matching approximation (ptdiff_MM) because it calls integrate() once per parameter set. For large-scale simulation (many parameter sets), prefer CalcMethod = 'MM' in pbayesdecisionprob1cont.

Examples

# P(T_t - T_c > 3) with equal parameters
ptdiff_NI(q = 3, mu_t = 2, mu_c = 0, sd_t = 1, sd_c = 1,
          nu_t = 17, nu_c = 17, lower.tail = FALSE)
#> [1] 0.24857

# P(T_t - T_c > 1) with unequal scales
ptdiff_NI(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] 0.6478101

# P(T_t - T_c > 0) with different degrees of freedom
ptdiff_NI(q = 0, mu_t = 1, mu_c = 1, sd_t = 1, sd_c = 1,
          nu_t = 5, nu_c = 20, lower.tail = FALSE)
#> [1] 0.4999528

# Lower tail: P(T_t - T_c <= 2)
ptdiff_NI(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] 0.3100353

# Vectorised usage
ptdiff_NI(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] 0.7453587 0.7171434 0.6815365