Sample Size Considerations for Hierarchical Populations

In applied sciences in general, and in particular when dealing with animal health and welfare assessments, one is often confronted with the collection of correlated data (not independent data). This generic term embraces a multitude of data structures, such as multivariate observations, clustered data, repeated measurements, longitudinal data, spatially correlated data, etc. The lack of independence may be a consequence of the natural hierarchy of the population under study: for instance, animals in a flock/holding as in cluster sampling, clearly violating the assumption of independence and results in a situation where animals selected from the same cluster (e.g., flock/holding) are likely to be more similar than animals selected from different clusters. During the phases of designing a survey the population natural hierarchy prevents the use of existing methodological results to estimate the sample size needed when the outcome of interest is of the binary type. Established sample size equations either assume absence of a hierarchy, or else assume a quantitative outcome. In this report the population under consideration is exhaustively described as well as issues that it poses when estimating sample size and modelling later the data gathered. A theoretical approximation is proposed for settings that are commonly encountered in risk assessment. A simulation framework is described in order to assess how the theoretical approximations could be corrected when the population hierarchy is of a particular type and the interest is focused on the estimation of prevalence of a certain condition (conditional or marginal). The results from the simulations when compared to the theoretical derivations, show for the case of a two level hierarchical population structure the need to increase the sample size at the highest level with a percentage that depends on the variability associated to the highest level but in general an increase of 30% is advised. For the case of the three levels population structure the discrepancies observed between the theoretical derivations power and the simulated power suggest a correction of about 10%. The difference between these two corrections for two and three level structures might be due to the approximation methods used, while for the two level population structure an adaptive Gaussian Quadrature method was used for the three levels a Laplace approximation was used instead. The difference might also be due to the fact that the assumed outcome variability at the highest level was smaller for the three level structure than for the two level structure, based on analyses of pilot studies.