samples/texts/2651514/page_3.md

with

$$h_{ijk}^{(s)}(t)=h_0^{(s)}(t)u_{ij}\exp ({\beta }^{\mathrm{\top }}{\mathbf{x}}_{ijk}^{(s)})\mathrm{.}(1)h\_\{ijk\}^\{(s)\}(t)\; =\; h\_\{0\}^\{(s)\}(t)u\_\{ij\}\; \backslash exp(\backslash beta^\{\backslash top\}\backslash mathbf\{x\}\_\{ijk\}^\{(s)\}).\; \backslash quad\; (1)$$

Here, $u_{ij} > 0$ are random effects, or frailties, shared by all individuals within the same group, and $h_0^{(s)}(t)$ is the baseline hazard function for stratum s. Clearly the survival times (either failed or censored) within the same group are correlated. The random effects are traditionally assumed not to depend on the regression parameter $\beta$. Without loss of generality, we assume that the design matrix is of full rank.

Here, we will focus on three-level hierarchical Cox models with the following nested random effects structure. Suppose the cohort is composed of $m$ independent clusters indexed by $i$. Within each cluster $i$, there are $J_i$ correlated sub-clusters indexed by $(i, j)$. Further, within each sub-cluster $(i, j)$ there are $n_{ij}$ individuals whose survival times are given by (1). One such hierarchy example was presented by Sastry (1997) where the children were clustered at both community and family levels.

We introduce a class of models with nested random effects based on the class of Tweedie exponential dispersion model distributions denoted by $\text{Tw}_r(\mu, \sigma^2)$, where $\text{Tw}_r(\mu, \sigma^2)$ includes the normal ($r=0$), Poisson ($r=1$), gamma ($r=2$), compound Poisson ($1<r<2$) and inverse Gaussian ($r=3$) distributions as special cases (Jørgensen, 1997). More specifically, we assume that the cluster level random effects $u_1, \dots, u_m$ are independently identically distributed random effects following the Tweedie distribution, with

$$U_1,\dots ,U_m\sim {\text{Tw}}_r(1,{\sigma }^2)\mathrm{.}(2)U\_1,\; \backslash dots,\; U\_m\; \backslash sim\; \backslash text\{Tw\}\_r(1,\; \backslash sigma^2).\; \backslash qquad\; (2)$$

We further assume that, given the cluster level random effects $\mathbf{U}* = \mathbf{u}* = (u_1, \dots, u_m)$, the sub-cluster level random effects $U_{11}, \dots, U_{mJ_m}$ are conditionally independent, and that the conditional distribution of $U_{ij}$, given $\mathbf{U}* = \mathbf{u}*$, depends on $u_i$ only, and is given by

$$U_{ij}\mathrm{\mid }{U}_i=u_i\sim {\text{Tw}}_q(u_i,{\omega }^2),(3)U\_\{ij\}\; |\; U\_i\; =\; u\_i\; \backslash sim\; \backslash text\{Tw\}\_q(u\_i,\; \backslash omega^2),\; \backslash qquad\; (3)$$

Assumptions (1)-(3) together provide a full specification of a nested random effects Cox model. To avoid non-positive random effects, we require $r \ge 2$ and $q \ge 2$. Here, the multiplicative sub-cluster random effect $u_{ij}$ represents the effect of the $(i,j)$th sub-cluster on the individual relative risk due to the fixed effect $\beta$. Under these assumptions, the hazard function in (1) can be rewritten as

$$h_{ijk}^{(s)}(t)=h_0^{(s)}(t)u_i{v}_{ij}\exp ({\beta }^{\mathrm{\top }}{\mathbf{x}}_{ijk}^{(s)}),(4)h\_\{ijk\}^\{(s)\}(t)\; =\; h\_\{0\}^\{(s)\}(t)u\_\{i\}v\_\{ij\}\; \backslash exp(\backslash beta^\{\backslash top\}\backslash mathbf\{x\}\_\{ijk\}^\{(s)\}),\; \backslash quad\; (4)$$

where $V_{ij} = U_{ij}/U_i$. It can be easily verified that $E(V_{ij}) = 1$ and $\text{Cov}[U_i, V_{ij}] = 0$. In the literature, $V_{ij}$ and $U_i$ are usually assumed to be independent, with $V_{ij}$ referred to as sub-cluster random effect instead of $u_{ij}$.