Log-logistic distribution

In probability and statistics, the log-logistic distribution is a two-parameter probability distribution sometimes used in survival analysis to model processes whose rate increases initially and decreases later. It is the probability distribution of a random variable whose logarithm has a logistic distribution. If Y is a random variable with a logistic distribution, then X = exp(Y) has a log-logistic distribution; likewise, if X has a log-logistic distribution, then log(X) has a logistic distribution. Log-logistic is also written log logistic.

The log-logistic distribution is similar in shape to the log-normal distribution, but unlike the log-normal its cumulative distribution function can be written in closed form. It can be written using a number of different parameterisations, one of which is:

$F(t)=[1+(\lambda t)^{-1/\gamma }]^{-1}\qquad t>0,\,\lambda >0,\,\gamma >0.$

In the same parameterisation, the probability density function is

$f(t)={\frac {(\lambda t)^{1/\gamma }}{\gamma t[1+(\lambda )^{1/\gamma }]^{2}}}$

The distribution may also be parameterised using κ = 1/ γ, which would result in somewhat simpler forms for the above expressions. However, γ behaves more like a conventional dispersion parameter (similar to the parameter σ in the log-normal distribution) in that the distribution has greater dispersion (is more spread out) for larger values of γ.

The parameter λ is a rate parameter, so its inverse 1/λ is a scale parameter and is in fact the median of the distribution.

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 γ < 1, the hazard function has a single maximum (when γ > 1, the hazard decreases monotonically).

The log-logistic distribution is one of a number of distributions that can be used as the basis of an accelerated failure time model, in this case by allowing λ to differ between groups, or more generally by introducing covariates that affect λ but not γ. (Collett 2003).

References

• Collett, D. (2003), Modelling Survival Data in Medical Research (2nd ed.), CRC press, ISBN 1584883251
• Cox, David Roxbee; Oakes, D. (1984), Analysis of Survival Data, CRC Press, ISBN 041224490X
• StataCorp (2005), "streg – Fit parametric survival models", Stata version 9 Reference Manual, College Station, TX: Stata Press

Further reading

• Bennett, Steve (1983), "Log-Logistic Regression Models for Survival Data", Applied Statistics, 32 (2): 165–171