N-mixture models provide a structure for making inference about a local population
size while accounting for imperfect detection. Using a binomial likelihood, they assume
prior distributions on the size parameters and then integrate those parameters out of the full
likelihood. For large population sizes, the established frequentist methods have exhibited
computational intractability, and the Bayesian methods have exhibited poor convergence
and mixing of chains. Additionally, estimability of parameters in these types of models
has been criticized in the literature. Although originally used for determining abundance
of rare wildlife, we explore using these models for under-diagnosed or under-ascertained
infectious diseases which have large prevalence.
We derive an asymptotic approximation of the N-mixture model that does not suffer
from computational efficiency and uses information theory to provide a method for
diagnosing estimability issues. Additionally, we extend this model to account for spatial dependency. Simulation studies show improved performance over the established methods
in numerous settings, and we successfully apply the asymptotic approximation to model
ten years of Oregon Health Authority chlamydia data.