Exact alpha-error determination for two-stage sampling strategies to substantiate freedom from disease

Sampling strategies to substantiate freedom from disease are important when it comes to the trade of animals and animal products. When considering imperfect tests and finite populations, sample size calculation can, however, be a challenging task. The generalized hypergeometric formula developed by Cameron and Baldock (1998a) offers a framework that can elegantly be extended to multi-stage sampling strategies, which are widely used to account for disease clustering at herd-level. The achieved alpha-error of such surveys, however, typically depends on the realization of the sample and can differ from the pre-calculated value. In this paper, we introduce a new formula to evaluate the exact alpha-error induced by a specific sample. We further give a numerically viable approximation formula and analyze its properties using a data example of Brucella melitensis in the Austrian sheep population.