Frailty models introduce latent positive heterogeneity into an event intensity. They can represent shared cluster dependence or unmeasured subject susceptibility, but marginalizing the latent variable changes the hazard structure. Conditional and population-level interpretations must therefore be separated.
Multiplicative frailty
Definition. A multiplicative frailty model is \[ \lambda(t\mid Z,U) =U\lambda_0(t)e^{\beta^\top Z}, \qquad U>0, \] where \(U\) is independent of \(Z\), commonly normalized by \(EU=1\). Write \[ \mathcal L_U(s)=E(e^{-sU}) \] for its Laplace transform.
Theorem (marginal survival and hazard). Conditional and marginal survival are \[ S(t\mid Z,U) =e^{-Ue^{\beta^\top Z}\Lambda_0(t)}, \qquad S_m(t\mid Z) =\mathcal L_U\{e^{\beta^\top Z}\Lambda_0(t)\}. \] If \(\mathcal L_U\) is differentiable, the marginal hazard is \[ \lambda_m(t\mid Z) =\lambda_0(t)e^{\beta^\top Z} \frac{-\mathcal L_U'\{e^{\beta^\top Z}\Lambda_0(t)\}} {\mathcal L_U\{e^{\beta^\top Z}\Lambda_0(t)\}}. \]
Proof. Integrating the conditional hazard gives the first survival formula. Taking expectation over \(U\) gives the Laplace transform. Differentiate \(-\log S_m(t\mid Z)\) with respect to \(t\), using \[ \frac{\mathrm{d}}{\mathrm{d}t} \{e^{\beta^\top Z}\Lambda_0(t)\} =e^{\beta^\top Z}\lambda_0(t). \]
Gamma frailty
Example. Let \[ U\sim\operatorname{Gamma}(1/\theta,\text{ rate }1/\theta), \qquad EU=1,\quad\operatorname{Var}(U)=\theta. \] Then \[ \mathcal L_U(s)=(1+\theta s)^{-1/\theta}, \] so \[ S_m(t\mid Z) =\{1+\theta e^{\beta^\top Z}\Lambda_0(t)\}^{-1/\theta}, \] \[ \lambda_m(t\mid Z) =\frac{\lambda_0(t)e^{\beta^\top Z}} {1+\theta e^{\beta^\top Z}\Lambda_0(t)}. \] For \(z_1,z_0\), the marginal hazard ratio is \[ e^{\beta^\top(z_1-z_0)} \frac{1+\theta e^{\beta^\top z_0}\Lambda_0(t)} {1+\theta e^{\beta^\top z_1}\Lambda_0(t)}, \] which varies with time unless \(\theta=0\) or the covariate contrast is null.
Derivation. The gamma Laplace transform follows by integrating \[ \frac{(1/\theta)^{1/\theta}}{\Gamma(1/\theta)} u^{1/\theta-1}e^{-u/\theta}e^{-su} \] over \(u>0\). Substitute it into the general theorem and differentiate. The time variation in the marginal hazard ratio is a selection effect: highly frail individuals fail earlier, so later risk sets are enriched for lower frailty.
Posterior frailty among survivors
Proposition. Under gamma frailty, conditional on survival beyond \(t\), \[ U\mid(T>t,Z) \sim\operatorname{Gamma}\left( \frac1\theta,\quad \text{rate }\frac1\theta+e^{\beta^\top Z}\Lambda_0(t) \right), \] and therefore \[ E\{U\mid T>t,Z\} =\frac{1} {1+\theta e^{\beta^\top Z}\Lambda_0(t)}. \]
Proof. Bayes' formula multiplies the gamma density by the survival likelihood \(e^{-u e^{\beta^\top Z}\Lambda_0(t)}\). This preserves the gamma shape and adds \(e^{\beta^\top Z}\Lambda_0(t)\) to the rate. The displayed mean is shape divided by rate.
Shared frailty and cluster dependence
Definition. For members \(j=1,\ldots,m_k\) of cluster \(k\), \[ \lambda_{kj}(t\mid U_k,Z_{kj}) =U_kY_{kj}(t)\lambda_0(t)e^{\beta^\top Z_{kj}}. \] Conditional on \(U_k\), event times are independent; marginally they are positively associated.
Proposition. For two members with cumulative conditional hazards \(H_j(t_j)=e^{\beta^\top Z_j}\Lambda_0(t_j)\), \[ P(T_1>t_1,T_2>t_2\mid Z_1,Z_2) =\mathcal L_U\{H_1(t_1)+H_2(t_2)\}. \] Under gamma frailty, the induced survival copula is Clayton and Kendall's concordance is \[ \tau_K=\frac{\theta}{\theta+2}. \]
Proof. Conditional independence gives \[ P(T_1>t_1,T_2>t_2\mid U) =e^{-U(H_1+H_2)}. \] Expectation over \(U\) gives the joint survival. Substituting the gamma Laplace transform yields the Clayton survival copula. Its Kendall's tau follows from the Archimedean-copula identity \(\tau_K=1+4\int_0^1\varphi(u)/\varphi'(u)\,\mathrm{d}u\) with gamma-frailty generator.
Likelihood and estimation
Proposition (cluster likelihood). Let cluster \(k\) have total event count \(D_k\) and total frailty-free cumulative exposure \[ H_k(\beta,\Lambda_0) =\sum_j\int Y_{kj}(t)e^{\beta^\top Z_{kj}(t)} \,\mathrm{d}\Lambda_0(t). \] After integrating out \(U_k\), its likelihood contribution contains \[ (-1)^{D_k}\mathcal L_U^{(D_k)}\{H_k(\beta,\Lambda_0)\} \prod_j\prod_{t:\Delta N_{kj}(t)=1} e^{\beta^\top Z_{kj}(t)}\,\mathrm{d}\Lambda_0(t). \]
Proof. Conditional on \(U_k=u\), the cluster likelihood contains \[ u^{D_k}e^{-uH_k} \prod_{\text{events}} e^{\beta^\top Z_{kj}(t)}\,\mathrm{d}\Lambda_0(t). \] Since \[ E(U^{D_k}e^{-sU}) =(-1)^{D_k}\mathcal L_U^{(D_k)}(s), \] integrating over \(u\) gives the formula.
Estimation may use penalized partial likelihood, EM or Laplace approximation, Gaussian quadrature, or full Bayesian integration. Each method makes a distributional choice for \(U\); a cluster-robust sandwich estimator without frailty instead targets an average relative-risk parameter and does not estimate latent heterogeneity.
Identifiability and interpretation
Proposition (scale normalization). Without fixing the scale of \(U\), frailty and baseline hazard are not separately identifiable: for every \(c>0\), \[ U\lambda_0(t) =\left(\frac{U}{c}\right)\{c\lambda_0(t)\}. \]
Proof. The transformed pair gives the same conditional intensity for every subject and therefore the same observation law. A normalization such as \(EU=1\) is necessary.
Even after normalization, one event per independent subject provides little information about the shape of a frailty distribution without parametric restrictions: baseline heterogeneity and frailty mixing can mimic one another. Repeated events or clustered members provide within-cluster replication and stronger identification.
Scientific distinction. A frailty variance is not automatically “unmeasured confounding.” Frailty is a model for latent outcome dependence. Confounding concerns common causes of treatment and outcome and requires an assignment model or causal assumptions. Adding a random effect can improve dependence modeling while leaving a causal bias untouched.