# Llogis: The log-logistic distribution In flexsurv: Flexible Parametric Survival and Multi-State Models

 Llogis R Documentation

## The log-logistic distribution

### Description

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.

### 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.

### 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.

### 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.

### Author(s)

Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>

### References

Stata Press (2007) Stata release 10 manual: Survival analysis and epidemiological tables.

dweibull