Information loss for 2 × 2 tables with missing cell counts: binomial case

摘要

We formulate likelihood-based ecological inference for 2 × 2 tables with missing cell counts as an incomplete data problem and study Fisher information loss by comparing estimation from complete and incomplete data. In so doing, we consider maximum-likelihood (ML) estimators of probabilities governed by two independent binomial distributions and obtain simplified expressions for their covariance. These expressions reflect well the additional uncertainty arising from the unobserved data compared to complete data tables. We also discuss an approximation to the expected conditional variance of the unobserved entries and ML parameter bias correction. An empirical example is used to demonstrate the results.