It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment.

It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

It is similar in shape to the log-normal distribution but has heavier tails.

Unlike the log-normal, its cumulative distribution function can be written in closed form.

The one shown here gives reasonably interpretable parameters and a simple form for the cumulative distribution function.

is a scale parameter and is also the median of the distribution.

The probability density function is An alternative parametrization is given by the pair

Expressions for the mean, variance, skewness and kurtosis can be derived from this.

for convenience, the mean is and the variance is Explicit expressions for the skewness and kurtosis are lengthy.

The log-logistic distribution provides one parametric model for survival analysis.

Unlike the more commonly used Weibull distribution, it can have a non-monotonic hazard function: when

The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring.

[9] The log-logistic distribution can be used as the basis of an accelerated failure time model by allowing

to differ between groups, or more generally by introducing covariates that affect

is the marginal distribution of the inter-times in a geometric-distributed counting process.

[11] The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation.

[4][5] Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a log-normal distribution.

[12] The log-normal distribution, however, needs a numeric approximation.

The blue picture illustrates an example of fitting the log-logistic distribution to ranked maximum one-day October rainfalls and it shows the 90% confidence belt based on the binomial distribution.

The rainfall data are represented by the plotting position r/(n+1) as part of the cumulative frequency analysis.

[14] The Gini coefficient for a continuous probability distribution takes the form: where

For the log-logistic distribution, the formula for the Gini coefficient becomes: Defining the substitution

leads to the simpler equation: And making the substitution

further simplifies the Gini coefficient formula to: The integral component is equivalent to the standard beta function

Using the properties of the gamma function, it can be shown that: From Euler's reflection formula, the expression can be simplified further: Finally, we may conclude that the Gini coefficient for the log-logistic distribution

The log-logistic has been used as a model for the period of time beginning when some data leaves a software user application in a computer and the response is received by the same application after travelling through and being processed by other computers, applications, and network segments, most or all of them without hard real-time guarantees (for example, when an application is displaying data coming from a remote sensor connected to the Internet).

It has been shown to be a more accurate probabilistic model for that than the log-normal distribution or others, as long as abrupt changes of regime in the sequences of those times are properly detected.

Another generalized log-logistic distribution is the log-transform of the metalog distribution, in which power series expansions in terms of

are substituted for logistic distribution parameters

The resulting log-metalog distribution is highly shape flexible, has simple closed form PDF and quantile function, can be fit to data with linear least squares, and subsumes the log-logistic distribution is special case.