Four estimators of annual infection probability were compared pertinent to Quantitative Microbial Risk Analysis (QMRA). A stochastic model, the Gold Standard, was used as the benchmark. It is a product of independent daily infection probabilities which in turn are based on daily doses. An alternative and commonly-used estimator, here referred to as the Naïve, assumes a single daily infection probability from a single value of daily dose. The typical use of this estimator in stochastic QMRA involves the generation of a distribution of annual infection probabilities, but since each of these is based on a single realisation of the dose distribution, the resultant annual infection probability distribution simply represents a set of inaccurate estimates. While the medians of both distributions were within an order of magnitude for our test scenario, the 95th percentiles, which are sometimes used in QMRA as conservative estimates of risk, differed by around one order of magnitude. The other two estimators examined, the Geometric and Arithmetic, were closely related to the Naïve and use the same equation, and both proved to be poor estimators. Lastly, this paper proposes a simple adjustment to the Gold Standard equation accommodating periodic infection probabilities when the daily infection probabilities are unknown.

Occupancy models are used in statistical ecology to estimate species dispersion. The two components of an occupancy model are the detection and occupancy probabilities, with the main interest being in the occupancy probabilities. We show that for the homogeneous occupancy model there is an orthogonal transformation of the parameters that gives a natural two-stage inference procedure based on a conditional likelihood. We then extend this to a partial likelihood that gives explicit estimators of the model parameters. By allowing the separate modelling of the detection and occupancy probabilities, the extension of the two-stage approach to more general models has the potential to simplify the computational routines used there.

Site occupancy, as estimated by the probability of presence, is used for monitoring species populations. However, the detection of species at individual sites is often subject to errors. In order to accurately estimate occupancy we must simultaneously account for imperfect detectability by estimating the probability of detection. The problem with estimating occupancy arises from not knowing whether a nondetection occurred at an occupied site due to imperfect detectability (sampling zeros), or the nondetection resulting from an unoccupied site (fixed zeros). We evaluated the performance of the basic, normal approximation, studentised and percentile methods for approximating confidence limits for occupancy and detection of species. Using coverage and average interval width, we demonstrated that the studentised estimator was generally superior to the others, except when a small sample of sites are selected. Under this circumstance and when calculating limits for detection, no estimator produced reliable results. The experimental factors we considered include: (i) number of sites; (ii) number of survey occasions; (iii) probabilities of presence (occupancy) and detection; and (iv) overdispersion in the capture matrix. Similar conclusions were reached both for the simulated studies and a case study. Overall, estimation near the boundaries of the probability of occupancy and detectability was difficult.

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