litter 32, although litter 32 was the only litter with tumours occurring in all three littermates. Examination of the data revealed that all three rats in litter 13 had exceptionally low tumour onset times (Figure 3).
Figure 1, 2 and 3 are approximately here.
6 Discussion
In this paper, we have introduced a Poisson modelling approach to random effects Cox models. We have specifically focussed on Cox models with two levels of nested random effects. We may consider models with more than two levels of random effects. For such models, our method remains valid with $(i, j, k)$ replaced by higher dimensional indices. The proposed Poisson modelling approach can also be extended to random effects Cox models with time dependent covariates in the following way. Suppose that all covariates assume constant values between two distinct failure times, as reflected by the corresponding step functions for the cumulative failure times. The incorporation of such time dependent covariates can be simply achieved by replacing $\mathbf{x}{ijk}^{(s)}$ by $\mathbf{x}{ijk}^{(s)}(t) = \mathbf{x}{ijk}(\tau{sh})$ for $Y_{ijk,h}^{(s)}$ in the model.
For the Cox model with one level of random effects ($J_i = 1, \omega^2 = 0$, with $u_{ij} = u_i$), the random effects have been previously characterized by gamma (Clayton 1991), positive stable (Hougaard 1986a, 1986b) and log-normal (McGilchrist 1993) distributions. Our framework effectively covers the gamma, log-normal and inverse Gaussian distributed random effects.
Our Poisson approach is not limited to Cox models with the nested random effects structures. Taking $u_{ij} = v_i v_j$ for balanced designs will lead to crossed random effects. For Cox models with only time dependent subject frailties $u_i(t)$ for each subject $i$, we can employ the techniques developed for Poisson models with an AR(p) structure on the latent variable $u_i(t)$. Since the distinct failure times are not equally spaced, a specific time series structure for time dependent frailties may not be appropriate. In the Cox model specified by (1)-(3), taking the second level random effects $u_{it} = u_i(t)$ as conditional on the subject random effect $u_i$, where $t$ represents the distinct failure times in the stratum of the $i$th subject, we have correlated time dependent frailties for each subject.
Acknowledgements
This research was supported by the Health Effects Institute and by a grant A8664 from the Natural Sciences and Engineering Research Council of Canada to D. Krewski.