This paper concerns the analysis of an animal survival/sacrifice experiment designed to investigate the incidence of a particular disease of interest. The disease is assumed to be irreversible, and detectable only at death, for example by a necropsy. Each observation can be of one of three types: (i) death caused by the disease, (ii) death from a competing cause such as sacrifice, with the disease present, or (iii) death with the disease absent. A two-dimensional EM algorithm is proposed for the nonparametric maximum likelihood estimation of the distributions of the time to onset and of the time to death from the disease. These can be compared with nonparametric estimators recently proposed by Kodell , Shaw and Johnson (1982, Biometrics 38, 43-58) and by Dinse and Lagakos (1982, Biometrics 38, 921-932). A slight modification of the algorithm permits the construction of likelihood-based interval estimates of quantiles of the distributions. Some extensions and generalizations are indicated.