Bootstrapping multiple systems estimates to account for model selection

Multiple systems estimation using a Poisson loglinear model is a standard approach to quantifying hidden populations where data sources are based on lists of known cases. Information criteria are often used for selecting between the large number of possible models. Confidence intervals are often reported conditional on the model selected, providing an over-optimistic impression of estimation accuracy. A bootstrap approach is a natural way to account for the model selection. However, because the model selection step has to be carried out for every bootstrap replication, there may be a high or even prohibitive computational burden. We explore the merit of modifying the model selection procedure in the bootstrap to look only among a subset of models, chosen on the basis of their information criterion score on the original data. This provides large computational gains with little apparent effect on inference. We also incorporate rigorous and economical ways of approaching issues of the existence of estimators when applying the method to sparse data tables.