Hierarchical likelihood estimator for a stratified sample design with observation error

The ultimate goal of a scientific study is to gather evidence for a given hypothesis or parameter value, however, traditional frequentist and Bayesian statistical methods do not provide objective measures of evidence from data.  The likelihood function, a fundamental component of both statistical methods, has been proposed as a means for representing and interpreting data as evidence.  The likelihood function gives the relative plausibility of parameter values given observed data and a statistical model.  We present a likelihood based method to construct interval estimates for bull trout, Salvelinus confluentus, redd (spawning nest) abundance from a stratified sample design when redd abundances are counted with error consisting of both omission of true redds and false identification of stream features as redds.  The interval estimates account for uncertainty due to counting only a portion of the streams (i.e. sampling error) and counting errors in selected streams (observation error), and can incorporate data on redd detection probabilities and false count rates.  Unlike traditional frequentist or Bayesian confidence intervals, the likelihood intervals directly represent redd abundances supported by data.  Additionally, there is no penalty for gathering additional samples after estimates are calculated (so-called ‘peeking’ at the data) or for constructing multiple intervals from the same data (e.g. for each individual stratum).