Estimation and Asymptotic Properties of the Distribution of Tune-to-Tumour in Carcinogenesis Experiments

Abstract

The problem of the estimation of the cumulative hazard function for time-to-tumour is considered for tumours that are not clinically observable and irreversible. The problem is posed in a compartmental model framework like that of Dewanji and Kalbfleisch (1986). The essential data from a single animal that are considered are: its group, age at death, mode of death (natural or sacrifice), and a list of the tumours present at death. No assumption concerning the lethality of the tumour of interest is required. First an isotonic estimator is proposed for the subsurvival function for time-to-tumour, based on the estimator derived by Gómez and Van Ryzin (1987) for the subdistribution function for time-to-tumour which maintains the asymptotic behaviour, that is, it is consistent and asymptotically normal. Then, the cumulative hazard estimator is defined which is itself monotonically increasing, consistent, and asymptotically normal. The limiting variance is computed and a consistent estimator is given. The observable data are summarized in a 4-variate counting process. The intensity of this process and the corresponding martingales were derived by Gómez and Van Ryzin (1987). The martingale theory can then be applied to obtain the desired results. In particular, the limiting variance is easily achieved. Finally, the proposed methods are illustrated with Berlin, Brodsky, and Clifford data on glomerulosclerosis disease in female RF mice.