Consistent estimators are derived for the age-specific and lifetime incidence rates of late-onset diseases from censored, biased samples, using a discrete time model and a traditional assumption for size-biased sampling. The estimators are alternatives to the so-called index case estimators often used by epidemiologists under similar circumstances. The asymptotic normality of the proposed estimators is established and an adaptation of Greenwood's formula is given for estimating and comparing their asymptotic variances. We show that index case estimators can be inconsistent and find a necessary and sufficient condition for their consistency. The results are applicable in the study of familial aggregation of diseases that are not manifest at birth when the sample consists of families identified through a family member who has the disease of interest. A survey of families of nursing home patients with a narrowly defined form of Alzheimer's disease, undertaken to clarify its mode of inheritance is an example of such a study. Our work shows a possible lack of consistency of the estimators obtained in that study and in other similar studies. Besides familial aggregation, the results may also be applicable to the study of slowly working environmental causes of a disease insofar as they affect individuals as members of special groups, identified through some of their diseased members; for example, factory workers exposed to certain toxic materials.