# 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 <= 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 <= 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 = π/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`