Concept Architecture
Concept
Theoretically, the Log-Logistic Model is a fully parametric survival model that assumes survival times follow a log-logistic distribution. It is particularly suited to time-to-event data in which the hazard initially increases and subsequently decreases, making it appropriate for diseases exhibiting non-monotonic risk over time. In health economics, the log-logistic model is widely used to estimate survival, extrapolate long-term outcomes and support cost-effectiveness analyses and health technology assessments.
Mathematically, the log-logistic model specifies the survival function using scale and shape parameters that determine the probability density, survival and hazard functions. Model parameters are estimated using maximum likelihood estimation, allowing the model to represent peaked hazard functions that cannot be captured by simpler models such as the exponential or Weibull under many circumstances. Closed-form expressions for the survival and hazard functions facilitate prediction and long-term extrapolation.
In practice, the log-logistic model is fitted to individual patient survival data using specialised statistical software. Competing survival models are compared using statistical goodness-of-fit measures, graphical assessment and clinical plausibility before selecting the most appropriate model for extrapolation. The fitted model is subsequently used to estimate life expectancy, quality-adjusted life-years, healthcare costs and incremental cost-effectiveness ratios.
Purpose
Used to model survival data with non-monotonic hazards, estimate long-term survival and generate clinically plausible projections for health economic evaluation.
Mathematical Formulae
Primary Formula
S(t) = 1 / (1 + (?t)?)
where:
- S(t) = survival probability
- ? > 0 = scale parameter
- ? > 0 = shape parameter
Supporting Formulae
Hazard function:
h(t) = (??(?t)???) / (1 + (?t)?)
Probability density function:
f(t) = (??(?t)???) / (1 + (?t)?)�
Maximum likelihood estimation:
?? = arg max L(?)
Related Mathematical Methods
- Maximum likelihood estimation
- Parametric survival modelling
- Log-logistic distribution
- Weibull model
- Log-normal model
- Generalized gamma model
- Akaike Information Criterion (AIC)
- Bayesian Information Criterion (BIC)
Example
An immuno-oncology trial demonstrates a hazard of disease progression that rises during the first year before declining among long-term survivors. A log-logistic model provides a better fit than Weibull and Gompertz models, producing more clinically plausible lifetime survival projections for estimating quality-adjusted life-years and incremental cost-effectiveness ratios.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER(Lambda*A2,Gamma) | Calculate the powered time component of the log-logistic survival function. |
| LN | =LN(A2) | Calculate log survival times for exploratory modelling. |
| EXP | =EXP(B2) | Transform predicted values during model estimation. |
| Solver | Minimise negative log-likelihood | Estimate log-logistic model parameters by maximum likelihood. |
VBA (Optional)
Automate fitting of log-logistic survival models, compare competing parametric models and generate long-term survival projections for economic evaluation.
Sources
- Collett D. Modelling Survival Data in Medical Research.
- Lawless JF. Statistical Models and Methods for Lifetime Data.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 21: Flexible methods for survival analysis — Rutherford, Lambert, Sweeting, Pennington, Crowther, Abrams & Latimer, TSD 21 ed., 2020 (NICE Decision Support Unit (University of Sheffield))
Guidance extending standard survival analysis to flexible parametric methods — spline-based models, fractional polynomials, mixture and cure models, and relative-survival approaches — for capturing complex hazard functions in economic evaluation.
Frequently Asked Questions (7)
What is a log logistic model?
A survival model based on the log-logistic distribution, representing a hazard function that rises to a peak and then declines.
Source: Collett 2015
What parametric survival model is a log-logistic model?
A log-logistic model is a parametric survival model that assumes event times follow a log-logistic distribution. A distinctive feature is that it can represent a hazard which rises and then falls, giving a non-monotonic pattern that some other standard distributions cannot. This makes it a useful option when the risk of an event peaks and then declines over time. It is one of the parametric forms routinely compared when choosing a survival model for extrapolation in economic evaluation. Fit statistics and clinical plausibility guide whether it is selected.
Source: Latimer 2013
What is a log-logistic model?
A log-logistic model is a survival model based on the log-logistic distribution, representing a hazard function that rises to a peak and then declines, a unimodal shape, or that declines monotonically. This lets it fit survival data where risk increases initially, peaks, and then falls, which monotonic models like the exponential or Weibull cannot capture. Fitted to data, the log-logistic model provides a smooth survival curve for description and extrapolation, and it is one of the candidate parametric distributions in survival analysis.
Source: Collett 2015
What hazard pattern does the log-logistic model capture?
The log-logistic model captures a hazard that rises to a peak and then declines when its shape parameter exceeds one, giving a unimodal, non-monotonic hazard, or a monotonically decreasing hazard otherwise. This turning hazard distinguishes it from models with monotonic hazards, allowing it to represent processes where risk builds, peaks, and then falls. Such a pattern occurs in some diseases and recovery processes, making the log-logistic model appropriate where the hazard is expected to rise then decline over time.
Source: Collett 2015
When is a log-logistic model used?
A log-logistic model is used when survival data show a hazard that rises then falls, or declines, and where its shape fits better than monotonic distributions. It is a candidate in survival modelling and extrapolation, chosen where its unimodal hazard suits the observed and projected pattern. Because its tail declines after the peak, implying particular long-term survival, its fit within the data and the plausibility of its extrapolation are compared with other distributions before it is selected.
Source: Latimer 2013
How does the log-logistic model behave in extrapolation?
In extrapolation, the log-logistic model projects a hazard that, after its peak, declines toward zero, implying that risk falls over the long term and survival flattens, which can produce long projected survival. This may suit conditions where late risk genuinely falls but may overstate survival if the declining hazard is projected too far. Because the extrapolated tail affects mean survival, the log-logistic model's long-term behaviour is examined for plausibility and compared with alternatives before its extrapolation is relied upon.
Source: Latimer 2013
What are the limitations of the log-logistic model?
The log-logistic model's hazard follows a specific unimodal or decreasing form, so it cannot represent all patterns, such as hazards that keep rising, and its declining tail may imply implausibly long survival when extrapolated far beyond the data. Its extrapolation, like that of any parametric model, depends on the assumed form and is uncertain. As one of several candidate distributions, it may fit worse than alternatives for particular data. These limitations mean it is used where its hazard shape fits and compared with other models.
Source: Latimer 2013
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 22 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/log-logistic-model
- Term code
- HE-EM-SM-046
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