  Binomial data with unknown sizes often appear in biological and medical
sciences. The previous methods either use the Poisson approximation or the
quasi-likelihood approach. A full likelihood approach is proposed by treating
unknown sizes as latent variables. This approach simplifies analysis as maximum
likelihood estimation can be applied. It also facilitates us to gain a lot more
insights into efficiency loss across models and estimation precision within
models. Simulation assesses the performance of the proposed model. An
application to the surviving jejunal crypt data is discussed. The proposed
method is not only competitive with the previous methods, but also gives an
appropriate explanation of the inflated variation of expected sizes.
