Shared Frailty

Concept

Theoretically, Shared Frailty is a survival modelling approach that accounts for unobserved heterogeneity among individuals belonging to the same group or cluster. It assumes that individuals within a cluster share a common random effect, known as frailty, which influences their hazard of experiencing the event. Shared frailty models extend the Cox proportional hazards model by incorporating dependence between clustered survival times and are widely applied in health economics when analysing multicentre studies, families, hospitals or recurrent events.

Mathematically, a shared frailty model multiplies the baseline hazard by an unobserved random effect common to all individuals within a cluster. The frailty term is typically assumed to follow a Gamma or log-normal distribution with mean one and variance ?, where ? quantifies the degree of heterogeneity between clusters. Model parameters are estimated using maximum likelihood or penalised likelihood methods.

In practice, shared frailty models are estimated when survival outcomes are correlated because individuals share common but unmeasured characteristics. They are applied in clinical trials, observational studies and health economic evaluations to improve estimation of treatment effects, account for clustering and generate more reliable survival predictions.

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Purpose

Used to model clustered survival data, account for unobserved heterogeneity between groups, improve estimation of treatment effects and produce more accurate survival estimates for health economic evaluation.

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Mathematical Formulae

Primary Formula

h??(t) = z?h?(t)exp(X???)

Where:

h??(t) = hazard for individual j in cluster i

z? = shared frailty for cluster i

h?(t) = baseline hazard

X?? = covariate vector

? = regression coefficients

Supporting Formulae

Gamma frailty assumption:

z? ~ Gamma(1/?, 1/?)

Frailty variance:

Var(z?) = ?

When ? = 0, the model reduces to the standard Cox proportional hazards model.

Related Mathematical Methods

• Cox proportional hazards regression
• Gamma frailty models
• Log-normal frailty models
• Maximum likelihood estimation
• Penalised likelihood estimation
• Random effects survival models

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Example

A multicentre oncology trial enrols patients from 25 hospitals. Patients treated within the same hospital may share unmeasured characteristics such as referral patterns or clinical practice. A shared frailty model estimates a common frailty term for each hospital while simultaneously estimating treatment effects. The estimated frailty variance is ? = 0.32, indicating moderate between-hospital heterogeneity in patient survival.

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Excel Implementation

Function	Example Formula	Health Economics Application
EXP	=EXP(B2)	Calculates the exponential component of the proportional hazards model.
GAMMA.DIST	=GAMMA.DIST(A2,1/$F$1,$F$1,TRUE)	Evaluates Gamma frailty distributions for illustrative analyses.
LINEST	=LINEST(Y2:Y100,X2:X100,TRUE,TRUE)	Estimates regression coefficients in simplified proportional hazards approximations.
IF	=IF($F$1=0,"Cox Model","Shared Frailty")	Indicates whether frailty variance is present.

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VBA (Optional)

VBA can automate estimation of shared frailty models, cluster-level random effects and survival predictions for grouped patient data.

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Sources

Clayton D. A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika. 1978;65:141?151.

Duchateau L, Janssen P. The Frailty Model. Springer; 2008.

Therneau TM, Grambsch PM. Modeling Survival Data: Extending the Cox Model. Springer; 2000.

Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.

NICE. Health Technology Evaluations: The Manual. National Institute for Health and Care Excellence; 2022.