GLL: The Generalized Log-Logistic Distribution

In hce: Design and Analysis of Hierarchical Composite Endpoints

The Generalized Log-Logistic Distribution

Description

Density, distribution function, quantile function, hazard, cumulative hazard, and random generation for the generalized log-logistic distribution.

Usage

dGLL(x, rate, shape = 1, theta = 0, log = FALSE)

pGLL(q, rate, shape = 1, theta = 0)

qGLL(p, rate, shape = 1, theta = 0)

rGLL(n, rate, shape = 1, theta = 0)

hGLL(x, rate, shape = 1, theta = 0)

HGLL(x, rate, shape = 1, theta = 0)

Arguments

x, q	vector of quantiles.
rate	positive rate parameter of the distribution.
shape	positive shape parameter of the distribution.
theta	non-negative additional parameter of the distribution, with default value 0.
log	logical; if TRUE, densities are given as logarithms.
p	vector of probabilities.
n	number of observations. Must be a single non-negative integer.

Details

The generalized log-logistic distribution with rate parameter \lambda > 0, shape parameter \alpha > 0, and parameter \theta \ge 0 has survival function

S(t \mid \lambda, \alpha, \theta) = \left(1 + \theta \lambda t^\alpha\right)^{-1 / \theta}, \qquad \theta > 0.

The corresponding cumulative distribution function is

F(t \mid \lambda, \alpha, \theta) = 1 - \left(1 + \theta \lambda t^\alpha\right)^{-1 / \theta}.

The quantile function is

Q(p \mid \lambda, \alpha, \theta) = \left( \frac{(1 - p)^{-\theta} - 1}{\theta \lambda} \right)^{1 / \alpha}, \qquad 0 \le p \le 1,\ \theta > 0.

The cumulative hazard function is

H(t \mid \lambda, \alpha, \theta) = \frac{\log\left(1 + \theta \lambda t^\alpha\right)}{\theta}.

The density function is

f(t \mid \lambda, \alpha, \theta) = \lambda \alpha t^{\alpha - 1} \left(1 + \theta \lambda t^\alpha\right)^{-1 / \theta - 1}.

For dGLL(..., log = TRUE), the logarithm of the density is returned. For \theta > 0,

\log f(t \mid \lambda, \alpha, \theta) = \log(\lambda) + \log(\alpha) + (\alpha - 1)\log(t) - \left(\frac{1}{\theta} + 1\right)\log\left(1 + \theta \lambda t^\alpha\right).

The hazard function is

h(t \mid \lambda, \alpha, \theta) = \frac{\lambda \alpha t^{\alpha - 1}} {1 + \theta \lambda t^\alpha}.

Random generation is performed by inverse transform sampling. If U \sim \mathrm{Uniform}(0,1), then

T = Q(U \mid \lambda, \alpha, \theta)

has the generalized log-logistic distribution.

As \theta \to 0, this reduces to the Weibull distribution with survival function

S(t \mid \lambda, \alpha) = \exp\left(-\lambda t^\alpha\right).

The corresponding quantile function is

Q(p \mid \lambda, \alpha) = \left( \frac{-\log(1 - p)}{\lambda} \right)^{1 / \alpha}.

In this case, dGLL(..., log = TRUE) returns the corresponding Weibull log-density. This corresponds to the Weibull distribution in stats::dweibull(), stats::pweibull(), stats::qweibull(), and stats::rweibull() with shape = shape and scale = rate^(-1 / shape).

When \theta = 1, the survival function becomes

S(t \mid \lambda, \alpha, 1) = \left(1 + \lambda t^\alpha\right)^{-1},

corresponding to the log-logistic distribution.

Value

For dGLL(), pGLL(), qGLL(), hGLL(), and HGLL(), a numeric vector of the same length as the main input (x, q, or p). For rGLL(), a numeric vector of length n.

References

Wienke A. Frailty Models in Survival Analysis. Chapman and Hall/CRC (2010).

See Also

rweibullGF() for simulation of a Weibull distribution with gamma frailty.

Examples

## Example 1: Compare the density of GLL with different values of `theta`
x <- seq(0, 10, length.out = 100)
y1 <- pGLL(x, rate = 0.5, shape = 2, theta = 1)
y0 <- pGLL(x, rate = 0.5, shape = 2, theta = 0)
plot(x, y1, type = "l", ylab = "Cumulative distribution function")
lines(x, y0, col = "red", lty = 2)

hce documentation built on May 13, 2026, 9:06 a.m.