Estimates from non-replicated population surveys rely on critical assumptions

N-mixture and occupancy models are often used to account for non-detections in population surveys. The consensus has been that the methods require data that are replicated in space, as well as within a short period of time while the population at each site remains closed, in order for parameters such as detection probabilities and expected abundances to be identifiable. The requirement of replication prohibits the use of N-mixture and occupancy models for many surveys in practice. Recently, some studies have argued that N-mixture and occupancy models for surveys with only one visit at each site are identifiable when covariates for both detection probabilities and expected abundances, with at least one distinct covariate for each, are available (Journal of Plant Ecology, 5, 2012, 22; Environmetrics, 23, 2012, 197). We investigate the reasons for why detection probabilities have traditionally been considered unestimable from non-replicated counts and how the new methods sidestep these issues. We further use simulations to investigate properties of the new estimators. We show that detection probabilities of the single-visit models with covariates are non-identifiable and that absolute abundances cannot be estimated when particular link functions are employed (log links for both expected abundance and detection probability). Further, assumptions about the range within which detection probabilities vary are necessary to render estimability. The possibility of estimating abundance from single-visit surveys therefore implicitly hinges on knowledge about the link functions. Simulations show that estimates of abundance can be highly variable and sensitive to the choice of link function. We further show how a reduced parameterization of an N-mixture model for surveys repeated over time, without replication under closure but where detection probabilities are constant over time, corresponds to a Poisson model. Non-robust estimation can result in misleading conclusions about population abundance. When estimating abundance from count data that are not replicated, it is therefore important to be aware of how imprecise estimators may be and how sensitive they are to model assumptions.