ported by Mantel and Ciminera (1979). This experiment involved 50 sets of three female weanling rats selected from within the same litter, with one animal assigned to a treatment group exposed to a putative carcinogen, and the remaining two serving as litter-matched controls. The time to tumour occurrence or censoring was recorded to the nearest week for each of the 150 animals employed in this study. This experiment thus involved a single binary covariate with values of 0 and 1 indicating assignment to the control or treated group, respectively.
Because of the possibility of intra-litter correlation (Gart et al. 1986), we included a random effect for each litter. The corresponding Cox regression model assumes that, given the random effects, the hazard functions for individuals are conditionally independent, with the hazard function for individual j from litter i given by
where $x_{ij}$ is the indicator variable, reflecting exposure to the test agent. The litter random effect $u_i$ are assumed to follow independent and identical Tweedie distributions with unity mean and dispersion parameter $\sigma^2$ described in (2).
Parameter estimates for both the standard and random effects Cox models are shown in Table 1 where the Peto-Breslow approximation (Cox and Oakes 1984) for tied failure times was used in both analyses. The estimates of the regression parameter $\beta$ associated with the treatment effect are comparable under both models, as are the standard errors of these estimates. Based on the ratio of these estimates to their respective standard errors, the treatment effect is significant under both models.
Table 1: Parameter estimates for the animal carcinogenesis data.
| Parameter Estimates | ||
|---|---|---|
| Cox Model | β ± SE | σ² |
| Standard | 0.898 ± 0.317 | - |
| Random effects | 0.902 ± 0.312 | 0.293 |
Scatter plot of the litter random effects is shown in Figures 1. These 50 litters were labelled as 1, 3, ..., 99 by Mantel and Ciminera (1979) and are re-numbered as 1, 2, ..., 50 here for convenience. Litters 3, 21, 22, 25 and 37 demonstrated the lowest litter-specific relative risks, whereas litter 13 had the highest (Figure 1). Figure 2 shows that the litter random effects match the number of tumour occurrences in the corresponding litter; the higher the litter-specific relative risk, the higher the litter tumour occurrence. The one exception is litter 13, which had a higher litter-specific relative risk than