Density, distribution function, hazards, quantile function and random generation for the log-logistic distribution.
Usage
dllogis(x, shape = 1, scale = 1, log = FALSE)
pllogis(q, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)
qllogis(p, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)
rllogis(n, shape = 1, scale = 1)
hllogis(x, shape = 1, scale = 1, log = FALSE)
Hllogis(x, shape = 1, scale = 1, log = FALSE)Arguments
- x, q
vector of quantiles.
- shape, scale
vector of shape and scale parameters.
- log, log.p
logical; if TRUE, probabilities p are given as log(p).
- lower.tail
logical; if TRUE (default), probabilities are \(P(X \le x)\), otherwise, \(P(X > x)\).
- p
vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.
Value
dllogis gives the density, pllogis gives the
distribution function, qllogis gives the quantile function,
hllogis gives the hazard function, Hllogis gives the
cumulative hazard function, and rllogis generates random
deviates.
Details
The log-logistic distribution with shape parameter
\(a>0\) and scale parameter \(b>0\) has probability
density function
$$f(x | a, b) = (a/b) (x/b)^{a-1} / (1 + (x/b)^a)^2$$
and hazard
$$h(x | a, b) = (a/b) (x/b)^{a-1} / (1 + (x/b)^a)$$
for \(x>0\). The hazard is decreasing for shape \(a\leq 1\), and unimodal for \(a > 1\).
The probability distribution function is $$F(x | a, b) = 1 - 1 / (1 + (x/b)^a)$$
If \(a > 1\), the mean is \(b c / sin(c)\), and if \(a > 2\) the variance is \(b^2 * (2*c/sin(2*c) - c^2/sin(c)^2)\), where \(c = \pi/a\), otherwise these are undefined.
Note
Various different parameterisations of this distribution are
used. In the one used here, the interpretation of the parameters
is the same as in the standard Weibull distribution
(dweibull). Like the Weibull, the survivor function
is a transformation of \((x/b)^a\) from the non-negative real line to [0,1],
but with a different link function. Covariates on \(b\)
represent time acceleration factors, or ratios of expected
survival.
The same parameterisation is also used in
eha::dllogis in the eha package.
References
Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.