Concept Architecture
Concept
Theoretically, the Log-Normal Model is a parametric survival model that assumes survival times follow a log-normal distribution, such that the logarithm of survival time is normally distributed. It is founded on parametric survival analysis and accelerated failure time (AFT) theory, where covariates act multiplicatively on survival time rather than proportionally on the hazard. The model is appropriate when the hazard function initially increases and subsequently decreases over time.
Mathematically, the model assumes ln(T) follows a normal distribution with mean ? and standard deviation �. Survival, probability density and hazard functions are derived from the cumulative distribution and density of the normal distribution. Model parameters are estimated using maximum likelihood estimation and may include regression coefficients within the accelerated failure time framework.
In practice, the log-normal model is fitted to individual patient survival data using maximum likelihood methods. Goodness of fit is assessed using likelihood-based criteria, graphical diagnostics and residual analysis. In health economics, it is widely used for extrapolating survival beyond trial follow-up in cost-effectiveness models when empirical hazards exhibit a non-monotonic pattern.
Purpose
Used to model and extrapolate survival data when event times are positively skewed and hazards rise before declining, supporting long-term survival estimation in health economic evaluation.
Mathematical Formulae
Primary Formula
ln(T) ~ N(?, ��)
Supporting Formulae
S(t) = 1 ? �((ln(t) ? ?) / �)
f(t) = (1 / (t��(2�))) ? exp(?(ln(t) ? ?)� / (2��))
h(t) = f(t) / S(t)
ln(T) = X? + ��
Related Mathematical Methods
- Accelerated Failure Time (AFT) Model
- Maximum Likelihood Estimation
- Log-Normal Distribution
- Survival Function
- Hazard Function
- Akaike Information Criterion (AIC)
- Bayesian Information Criterion (BIC)
Example
Overall survival for patients receiving a new oncology treatment is modelled using a log-normal model. Maximum likelihood estimation produces ? = 2.80 and � = 0.75. The fitted model predicts a median survival of exp(2.80) � 16.4 months and is selected over alternative parametric models because it provides the lowest AIC and closely matches the observed Kaplan-Meier curve.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LN | =LN(A2) | Calculate log-transformed survival times. |
| NORM.DIST | =1-NORM.DIST(LN(A2),2.8,0.75,TRUE) | Estimate survival probabilities from the fitted log-normal model. |
| EXP | =EXP(B2) | Convert estimated log survival time back to the original time scale. |
| SOLVER | Maximum likelihood optimisation | Estimate model parameters ? and �. |
VBA (Optional)
Automate maximum likelihood estimation, model comparison and survival extrapolation for multiple candidate parametric survival models.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- Collett D. Modelling Survival Data in Medical Research.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
Frequently Asked Questions (7)
What is a log normal model?
A survival model based on the log-normal distribution, capable of representing a hazard that rises then falls, similar in shape to log-logistic.
Source: Collett 2015
What is a log-normal model?
A log-normal model is a survival model based on the log-normal distribution, capable of representing a hazard that rises then falls, a unimodal shape similar to the log-logistic. It suits survival data where risk increases initially, peaks, and then declines, which monotonic models cannot capture. Fitted to data, the log-normal model provides a smooth survival curve for description and extrapolation, and it is one of the candidate parametric distributions considered in survival analysis and health economic modelling.
Source: Collett 2015
What parametric survival model is a log-normal model?
A log-normal model is a parametric survival model in which the logarithm of event times is assumed to follow a normal distribution. Like the log-logistic form, it can produce a hazard that rises to a peak and then declines, so it suits data showing that pattern. It is one of the candidate distributions assessed when fitting and extrapolating survival curves for economic evaluation. Analysts compare it with alternatives using measures of fit alongside judgement about the long-term shape it implies. Its tail behaviour can strongly affect extrapolated results.
Source: Latimer 2013
What hazard pattern does the log-normal model capture?
The log-normal model captures a hazard that rises to a peak and then declines toward zero over the long term, a unimodal, non-monotonic pattern. This turning hazard, shared with the log-logistic model, distinguishes it from models with monotonic hazards and allows it to represent processes where risk builds, peaks, and then falls. Such a pattern occurs in some diseases and recovery, making the log-normal model appropriate where the hazard is expected to rise then decline.
Source: Collett 2015
When is a log-normal model used?
A log-normal model is used when survival data show a hazard that rises then falls, and where it 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 long-term hazard declines, implying long projected survival, its fit within the data and the plausibility of its extrapolation are compared with other distributions, particularly the log-logistic, before it is selected.
Source: Latimer 2013
How does the log-normal model compare with the log-logistic model?
The log-normal and log-logistic models both represent unimodal hazards that rise then fall and often fit survival data similarly, but they differ in the underlying distribution and in tail behaviour, which can lead to different extrapolations. Because they can match observed data comparably yet project differently over the long term, both are considered as candidates, and the choice is guided by fit and the plausibility of the extrapolated survival. Their similarity means they are often compared together when a turning hazard is expected.
Source: Collett 2015
What are the limitations of the log-normal model?
The log-normal model's hazard follows a specific unimodal form, so it cannot represent all patterns, such as monotonically rising hazards, and its declining long-term tail can imply implausibly long survival when extrapolated far beyond the data. Its extrapolation 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 the log-normal model 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
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- Term code
- HE-EM-SM-048
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