Log-Logistic Distribution

Log-Logistic Distribution: Concept Architecture

What it represents

The log-logistic distribution describes a positive time to an event, such as death, progression, relapse, or device failure, when the logarithm of event time follows a logistic distribution. It is a two-parameter, continuous survival distribution. Its hazard can rise to a peak and subsequently fall, making it a candidate for some observed time-to-event patterns and for extrapolation beyond follow-up. A declining hazard means the instantaneous event rate among those still event-free falls; the survival probability still decreases toward zero. The curve alone does not establish a biological mechanism or a cured subgroup.

Let $T>0$ denote time measured from a clearly specified origin, $t\geq 0$ a time point, $\alpha>0$ the scale (in the same time units as $T$), and $\beta>0$ the dimensionless shape. Software may call these parameters scale and shape, respectively. The density and hazard have units of inverse time; probabilities do not. The parameterization must be stated because other software uses location and scale on log time or alternative shape conventions.

Core probability and survival functions

Write $z=(t/\alpha)^\beta$. For $t>0$:

$$S(t)=P(T>t)=\frac{1}{1+z},\qquad F(t)=P(T\leq t)=\frac{z}{1+z}=1-S(t).$$

$$f(t)=\frac{\beta}{\alpha}\frac{(t/\alpha)^{\beta-1}}{[1+(t/\alpha)^\beta]^2},\qquad h(t)=\frac{f(t)}{S(t)}=\frac{\beta}{\alpha}\frac{(t/\alpha)^{\beta-1}}{1+(t/\alpha)^\beta}.$$

$$H(t)=\int_0^t h(u)\,du=-\log S(t)=\log[1+(t/\alpha)^\beta].$$

Here $f(t)$ is density, $h(t)$ is the conditional instantaneous event rate, and $H(t)$ is cumulative hazard. Check the relationships $f=hS$, $S=\exp(-H)$, $F=1-S$, and $S(0)=1$. As $t\to\infty$, $S(t)\to0$ and $H(t)\to\infty$ for every valid $\alpha,\beta$; this ordinary log-logistic model has no positive cure fraction.

The event-time quantile for $0<p<1$ is

$$Q(p)=\alpha\left(\frac{p}{1-p}\right)^{1/\beta},$$

so the median is $Q(0.5)=\alpha$. The time at which a specified survival level $s$ remains is $t_s=\alpha[(1-s)/s]^{1/\beta}$ for $0<s<1$. In particular, a scale estimate is a median time, not generally a mean time.

Shape, tail, and moments

When $\beta>1$, the peak occurs at $t_{\mathrm{peak}}=\alpha(\beta-1)^{1/\beta}$. For large $t$, $S(t)$ is asymptotically proportional to $t^{-\beta}$, while $h(t)$ approaches $\beta/t$. This heavy survival tail can make long-run outcomes sensitive to the estimated shape even when fits are similar during trial follow-up. The unconditional mean exists only when $\beta>1$:

$$E[T]=\int_0^\infty S(t)\,dt=\alpha\frac{\pi/\beta}{\sin(\pi/\beta)},\qquad \beta>1.$$

For $0<\beta\leq1$, the integral diverges: reporting a finite unrestricted mean from this model would be mathematically wrong. The variance requires $\beta>2$. A restricted mean through a finite horizon $\tau$ is well defined for every $\beta>0$:

$$\mathrm{RMST}(\tau)=E[\min(T,\tau)]=\int_0^\tau S(t)\,dt.$$

This finite-horizon quantity is often more relevant to an economic model than the unrestricted lifetime mean. Model lifetime results also depend on the chosen horizon, background mortality, and whether the extrapolated event is all-cause death or another endpoint.

Worked example: keep hazard and survival distinct

Suppose $\alpha=4$ years and $\beta=2$. These numbers are illustrative, not empirical estimates. At four years, $S(4)=1/2$, so four years is the median event time. At eight years, $S(8)=1/(1+2^2)=0.20$; a smaller fraction remains event-free. The instantaneous hazards are $h(4)=0.25$ per year and $h(8)=0.20$ per year. The hazard reaches its peak at $4(2-1)^{1/2}=4$ years, yet survival continues falling after year four.

The unrestricted mean is $E[T]=4(\pi/2)/\sin(\pi/2)=2\pi\approx6.283$ years. Through year eight, $\mathrm{RMST}(8)=\int_0^8[1+(t/4)^2]^{-1}dt=4\tan^{-1}(2)\approx4.429$ years. The two means answer different questions. The unrestricted value depends especially on the unobserved tail; the restricted value counts survival only through eight years.

Regression interpretation and alternative parameterization

Because $Y=\log T$ has a logistic distribution with location $\mu=\log\alpha$ and scale $\sigma=1/\beta$, one can write $\log T=\mu+\sigma\varepsilon$, where $\varepsilon$ has the standard logistic distribution. With covariates $x$, an accelerated failure time (AFT) form is $\log\alpha(x)=x^{\mathrm{T}}\gamma$ with common $\beta$. If one binary covariate changes from zero to one and its coefficient is $\gamma_j$, the corresponding time ratio is $\exp(\gamma_j)$: a positive coefficient stretches event times. This ratio should not be labeled a hazard ratio.

Under common shape, the cumulative event odds satisfy $F(t\mid x)/S(t\mid x)=[t/\alpha(x)]^\beta$, hence $\log[F(t\mid x)/S(t\mid x)]=\beta\log t-\beta x^{\mathrm{T}}\gamma$. Thus this specification also has proportional cumulative event odds across covariate groups, with an event-odds ratio of $\exp(-\beta\gamma_j)$ for that binary contrast. Its instantaneous hazard ratio is generally time-dependent, so a constant Cox-style hazard ratio is not implied. If shape also varies across groups, neither the common-shape AFT time-ratio interpretation nor constant cumulative-odds ratio follows without further qualification. Check each software package's covariate sign, reported coefficient scale, and definition of scale before comparing results.

Estimation and validation in survival analysis

For independent people with observed follow-up time $t_i$ and event indicator $\delta_i$ ($1$ for event, $0$ for right censoring), the standard parametric likelihood contribution is $f(t_i)^{\delta_i}S(t_i)^{1-\delta_i}$. This formulation relies on suitable independent, non-informative censoring conditional on modeled variables. A person entering observation only after time $a_i$ requires conditioning on survival to entry; under a matching delayed-entry design the contribution becomes $f(t_i)^{\delta_i}S(t_i)^{1-\delta_i}/S(a_i)$, with $t_i\geq a_i$. Other truncation, competing-event, interval-censoring, or recurrent-event designs require their own likelihoods and estimands.

Before fitting, define the event, time origin, units, censoring and entry rules, covariates, and population. Compare fitted survival and hazards with observed Kaplan–Meier information where meaningful, especially near the last event and where risk sets are small. Inspect numerical fit (for example AIC), uncertainty, plausible hazard shape, and external evidence over the decision horizon. A small AIC alone cannot validate a decades-long tail. Check whether predictions imply credible long-run survival relative to general-population mortality and disease knowledge; use scenario analyses for distributions and treatment-effect extrapolation. Reassess whether a single distribution or common shape can represent treatment groups adequately.

In economic evaluation, the area under a survival curve informs time alive; combining survival with state utilities, costs, and discounting informs QALYs and costs. The log-logistic tail can materially affect extrapolated outcomes when follow-up is short. For a finite model horizon, integrate only over that horizon and state how background mortality, competing risks, and treatment waning are handled. Extrapolation is a model assumption, not observed long-term evidence.

Boundaries and common errors

• Hazard versus survival: A declining hazard after its peak does not mean survival recovers or patients become cured.
• Median versus mean: $\alpha$ is the median; the mean has the stated shape condition and is undefined as a finite quantity when $\beta\leq1$.
• Restricted versus unrestricted: A finite restricted mean does not make an infinite unrestricted mean finite.
• AFT versus proportional hazards: Common shape supports an AFT model and proportional cumulative event odds, not generally proportional instantaneous hazards.
• Parameter conventions: Do not insert a log-time scale $\sigma$ into formulas that expect shape $\beta$; here $\beta=1/\sigma$.
• Tail evidence: A fit over observed follow-up gives limited information about the far tail; compare plausible alternatives and quantify uncertainty.

Sources and further study

1. Jackson C. flexsurv: log-logistic distribution documentation. Primary package reference for density, hazard, shape behavior, moments, and parameter convention.
2. Jackson C. flexsurvreg documentation. Primary software documentation for parametric survival regression, AFT specification, likelihood-based estimation, and fitting conventions.
3. Latimer NR. NICE Decision Support Unit Technical Support Document 14: survival analysis for economic evaluations alongside clinical trials. Methods guidance on distribution choice, fit, external validity, and extrapolation uncertainty.
4. NICE. Technology appraisal and highly specialised technologies guidance manual, economic evaluation. Guidance on survival extrapolation and model validation in decision making.