Two-stage sampling in surveys to substantiate freedom from disease

Disease in livestock populations tends to cluster at the herd level. In order to account for this--and to overcome the problems of simple random sampling from a very large population--large-scale livestock surveys usually involve two-stage sampling. However, the use of two-stage sampling presents particular problems for sample-size calculation and analysis. We developed a probability formula for two-stage sampling, initially based on the assumption of a perfect test. We used this formula to demonstrate how combinations of first-stage (number of herds) and second-stage (number of animals in selected herds) sample sizes can be altered to achieve a least-cost survey, and used simulation to validate the formula. To overcome the unrealistic assumption of a perfect test, we then applied an exact-probability formula (which takes imperfect tests and finite population sizes into account) to the two-stage sampling design. An example is given which shows how implementing the formula with the FreeCalc computer program allows least-cost first and second-stage sample sizes to be calculated.