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.