Log-Normal Distribution

Concept

Theoretically, the Log-Normal Distribution is a continuous probability distribution in which the logarithm of a positive random variable follows a normal distribution. It is commonly used to model positively skewed data arising from multiplicative biological or stochastic processes. In health economics, the log-normal distribution is widely applied to model survival times, healthcare costs, resource utilisation and uncertainty in probabilistic sensitivity analysis because these variables are non-negative and frequently exhibit substantial right skew.

Mathematically, the log-normal distribution is characterised by the mean and standard deviation of the logarithm of the variable rather than the variable itself. This formulation produces a positively skewed distribution whose probability density, cumulative distribution and survival functions have closed-form representations based on the normal distribution. Model parameters are typically estimated using maximum likelihood estimation.

In practice, the log-normal distribution is fitted to individual patient data using statistical software. It is routinely compared with Weibull, Gompertz, log-logistic and generalised gamma distributions when modelling survival data and is also used to represent uncertainty in positive model parameters within probabilistic sensitivity analyses. The fitted distribution is subsequently used to estimate survival probabilities, life expectancy, quality-adjusted life-years and healthcare costs.

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Purpose

Used to model positive, right-skewed continuous variables, estimate survival distributions and represent uncertainty in health economic models and probabilistic sensitivity analyses.

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Mathematical Formulae

Primary Formula

ln(T) ~ N(?, ��)

where:

• T = positive random variable
• ? = mean of ln(T)
• � = standard deviation of ln(T)

Supporting Formulae

Probability density function:

f(t) = (1 / (t��(2�))) ? exp(?(ln(t) ? ?)� / (2��))

Mean:

E(T) = exp(? + �� / 2)

Variance:

Var(T) = [exp(��) ? 1] ? exp(2? + ��)

Model parameters are estimated using:

?? = arg max L(?)

Related Mathematical Methods

• Maximum likelihood estimation
• Parametric survival modelling
• Log-normal model
• Weibull distribution
• Log-logistic distribution
• Generalized gamma distribution
• Akaike Information Criterion (AIC)
• Bayesian Information Criterion (BIC)

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Example

A clinical trial evaluating a chronic disease treatment demonstrates survival times with a long right tail. A log-normal distribution is fitted to the patient-level data because it better captures prolonged survival among a subset of patients than a Weibull distribution. The fitted distribution is then used to project lifetime quality-adjusted life-years and healthcare costs in the economic evaluation.

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Excel Implementation

Function	Example Formula	Health Economics Application
LOGNORM.DIST	=LOGNORM.DIST(A2,Mu,Sigma,FALSE)	Calculate the log-normal probability density for survival or cost data.
LOGNORM.INV	=LOGNORM.INV(RAND(),Mu,Sigma)	Generate random values for probabilistic sensitivity analysis.
LN	=LN(A2)	Transform observed values prior to parameter estimation.
AVERAGE	=AVERAGE(LNRange)	Estimate the mean of the log-transformed observations.

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VBA (Optional)

Automate fitting of log-normal distributions, compare alternative survival distributions and generate random samples for probabilistic sensitivity analysis.

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Sources

• Lawless JF. Statistical Models and Methods for Lifetime Data.
• Collett D. Modelling Survival Data in Medical Research.
• 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.