Cumulative Distribution Function of the Difference of Two t-Distributed Variables Using the Welch-Satterthwaite Approximation

This function calculates the cumulative distribution function (CDF) of the difference between two independent t-distributed random variables using the Welch-Satterthwaite approximation. This method provides a fast and reasonably accurate approximation, particularly useful for unequal variances and degrees of freedom. Specifically, it computes P(T1 - T2 ≤ q) or P(T1 - T2 > q) where T1 and T2 follow t-distributions with potentially different parameters.

Usage

pWS2tdiff(q, mu.t1, mu.t2, sd.t1, sd.t2, nu.t1, nu.t2, lower.tail = TRUE)

Arguments

q

A numeric value representing the quantile threshold.

mu.t1

A numeric value representing the location parameter (μ) of the first t-distribution.

mu.t2

A numeric value representing the location parameter (μ) of the second t-distribution.

sd.t1

A positive numeric value representing the scale parameter (σ) of the first t-distribution.

sd.t2

A positive numeric value representing the scale parameter (σ) of the second t-distribution.

nu.t1

A positive numeric value representing the degrees of freedom (ν) of the first t-distribution. Must be > 2 for finite variance.

nu.t2

A positive numeric value representing the degrees of freedom (ν) of the second t-distribution. Must be > 2 for finite variance.

lower.tail

A logical value; if TRUE (default), probabilities are P(T1 - T2 ≤ q), otherwise P(T1 - T2 > q).

Value

A numeric value in [0, 1] representing the cumulative probability that the difference between the two t-distributed variables is below (if lower.tail = TRUE) or exceeds (if lower.tail = FALSE) the quantile q.

Details

The Welch-Satterthwaite approximation approximates the distribution of the difference between two t-distributed variables using a single t-distribution with adjusted parameters. The method involves:

• Step 1: Calculate the pooled variance (sum of squared scale parameters): $$\sigma_{\text{pooled}}^2 = \sigma_1^2 + \sigma_2^2$$

• Step 2: Calculate the effective degrees of freedom using the Welch-Satterthwaite formula: $$\nu^* = \frac{(\sigma_1^2 + \sigma_2^2)^2}{\frac{\sigma_1^4}{\nu_1} + \frac{\sigma_2^4}{\nu_2}}$$

• Step 3: Standardize the quantile: $$q_{\text{standardized}} = \frac{q - (\mu_1 - \mu_2)}{\sqrt{\sigma_1^2 + \sigma_2^2}}$$

• Step 4: Compute the CDF using a single t-distribution with the effective degrees of freedom ν*

Advantages:

• Computationally fast and efficient

• Works well for unequal variances and degrees of freedom

• Provides good approximation in most practical scenarios

• Particularly accurate when degrees of freedom are not too small (ν > 5)

Computational considerations:

• Much faster than numerical integration methods

• Suitable for exploratory analyses and large-scale simulations

• For final analyses requiring highest accuracy, consider using numerical integration (pNIdifft)

Examples

# Calculate P(t1 - t2 > 3) for equal parameters
pWS2tdiff(q = 3, mu.t1 = 2, mu.t2 = 0, sd.t1 = 1, sd.t2 = 1,
         nu.t1 = 17, nu.t2 = 17, lower.tail = FALSE)
#> [1] 0.2421594

# Calculate P(t1 - t2 > 1) for unequal variances
pWS2tdiff(q = 1, mu.t1 = 5, mu.t2 = 3, sd.t1 = 2, sd.t2 = 1.5,
         nu.t1 = 10, nu.t2 = 15, lower.tail = FALSE)
#> [1] 0.6533185

# Calculate P(t1 - t2 > 0) for different degrees of freedom - should be ~0.5
pWS2tdiff(q = 0, mu.t1 = 1, mu.t2 = 1, sd.t1 = 1, sd.t2 = 1,
         nu.t1 = 5, nu.t2 = 20, lower.tail = FALSE)
#> [1] 0.5

# Calculate lower tail probability P(t1 - t2 ≤ 2)
pWS2tdiff(q = 2, mu.t1 = 3, mu.t2 = 0, sd.t1 = 1.5, sd.t2 = 1.2,
         nu.t1 = 12, nu.t2 = 15, lower.tail = TRUE)
#> [1] 0.3036847