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 Incoming links: Shared frailty model

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 32 / 6 / Solution 2026-10-07

The episodes from one patient share treatment, biological susceptibility and prior history, so they are not independent observations. Treating every row as an unrelated subject can substantially understate uncertainty. Two approaches address this within-patient recurrent-event dependence:

A patient-clustered sandwich covariance matrix retains a suitable working Cox mean/intensity model but groups score contributions by patient. If $U_{i}$ is the patient's score contribution and $A$ the observed information, its form is $A^{- 1} (\sum_{i} U_{i} U_{i}^{T}) A^{- 1}$ . Independent patients, rather than independent rows, determine the sampling units. This corrects uncertainty for within-patient dependence when the working estimating equation is appropriate; it does not repair an incorrect mean model.
A shared frailty model introduces a common latent positive multiplier $V_{i}$ for all of patient $i$ 's episode hazards, often with a specified gamma or lognormal distribution. Given frailty and the relevant history, the event mechanism is modeled through that patient's intensity. Estimate treatment and frailty parameters by integrating or profiling the latent effect. This models persistent heterogeneity directly, but relies on the frailty assumptions and generally gives a conditional treatment-effect interpretation.

A patient-level bootstrap is another way to retain dependence: resample complete histories, not individual episode rows. It is not necessary to assume that robust marginal/working and frailty-conditional effect estimates target exactly the same parameter.

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Shared frailty model 2026-10-07

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A shared frailty model assigns the same latent positive hazard multiplier

V_{i}

to correlated event histories or group members. Gamma or lognormal frailty distributions are common examples. Modeling dependence through this latent effect yields conditional covariate effects; integrating out frailty can change the form of marginal hazards. Eligibility and event history still have to be encoded in the at-risk indicator.

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Within-patient recurrent-event dependence 2026-10-07

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Repeated episodes from one person share susceptibility, treatment and history, so row-wise independence is generally inappropriate. Patient-clustered sandwich covariance matrices estimate uncertainty using whole-patient score contributions under a suitable working model. Shared frailty models instead model persistent latent heterogeneity. These methods require independent sampling units at the patient level and do not necessarily estimate the same marginal versus conditional effect.

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