Parametric Survival

Concept

Theoretically, Parametric Survival refers to survival analysis based on the assumption that survival times follow a specified probability distribution. It is founded on parametric statistical theory and survival analysis, allowing the entire survival function to be described by a finite number of parameters. The approach exists to enable efficient estimation and extrapolation of survival beyond the observed follow-up period, which is essential in health economic evaluation.

Mathematically, parametric survival analysis specifies a probability distribution such as the Exponential, Weibull, Gompertz, Log-Normal, Log-Logistic or Generalised Gamma distribution. The survival, hazard and cumulative hazard functions are derived directly from the selected distribution, and model parameters are estimated using maximum likelihood estimation. The fitted distribution determines the long-term behaviour of the survival and hazard functions.

In practice, parametric survival analysis is applied by fitting several candidate distributions to observed survival data and selecting the most appropriate model using likelihood-based criteria, graphical diagnostics and clinical plausibility. In health economics, parametric survival methods are routinely used to extrapolate survival beyond clinical trial follow-up for lifetime cost-effectiveness analyses.

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Purpose

Used to estimate and extrapolate survival by assuming a predefined probability distribution, supporting long-term prediction and economic evaluation of healthcare interventions.

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Mathematical Formulae

Primary Formula

S(t) = 1 ? F(t)

where:

S(t) = survival function

F(t) = cumulative distribution function of the selected parametric distribution

Supporting Formulae

h(t) = f(t) / S(t)

H(t) = ?ln(S(t))

L = ?? f(t? | ?)?? S(t? | ?)????

where:

h(t) = hazard function

H(t) = cumulative hazard

? = parameter vector

�? = event indicator

Related Mathematical Methods

• Maximum Likelihood Estimation
• Parametric Survival Model
• Weibull Model
• Exponential Model
• Gompertz Model
• Log-Normal Model
• Log-Logistic Model
• Generalised Gamma Model

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Example

An oncology trial follows patients for four years, but the economic evaluation requires lifetime survival estimates. Several parametric survival distributions are fitted to the observed data. The Weibull model provides the best balance of statistical fit and clinical plausibility and is therefore selected to extrapolate survival over a lifetime horizon for cost-effectiveness analysis.

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Excel Implementation

Function	Example Formula	Health Economics Application
EXP	=EXP(-((A2/$B$1)^$B$2))	Calculate Weibull survival probabilities.
LN	=-LN(C2)	Calculate cumulative hazard from survival estimates.
Solver	Maximum likelihood optimisation	Estimate distribution parameters.
IF	=IF(A2<=60,C2,D2)	Apply observed survival followed by extrapolated survival where appropriate.

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VBA (Optional)

Automate fitting and comparison of multiple parametric survival distributions and generate long-term survival extrapolations for health economic models.

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Sources

• Collett D. Modelling Survival Data in Medical Research.
• Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
• Royston P, Parmar MKB. Flexible Parametric Proportional-Hazards and Proportional-Odds Models for Censored Survival Data. Statistics in Medicine. 2002.
• NICE. Health Technology Evaluation Manual.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• ISPOR Good Practice Task Force Reports.