The log-logistic distribution

Density, distribution function, hazards, quantile function and random generation for the log-logistic distribution.

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If `length(n) > 1`, the length is taken to be the number required.
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, $P(X > x)$.

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.

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