Nonparametric estimation of solid cancer size at metastasis and probability of presenting with metastasis at detection.

Two probabilistic characterizations of the relationship between the size of primary cancers and the occurrence of metastases are considered. The first is the distribution of tumor size at the point of metastatic transition, while the second is the probability that detectable metastases are present when cancer comes to medical attention. We show that the general model for this problem developed by Kimmel and Flehinger (1991, Biometrics 47, 987-1004) is unidentifiable. That is, the quantities to be estimated cannot be uniquely determined by the available information. An identifiable model that includes one limiting model of Kimmel and Flehinger (1991) as a special case is developed by using odds ratios. Under the new model, both quantities of interest are estimated by using a nonparametric maximum likelihood approach. The results are applied to two lung cancer data sets.