Design-based or Prediction-based Inference? Stratified Random vs Stratified Balanced Sampling

Early survey statisticians faced a puzzling choice between randomized sampling and purposive selection but, by the early 1950s, Neyman's design-based or randomization approach had become generally accepted as standard. It remained virtually unchallenged until the early 1970s, when Royall and his co-authors produced an alternative approach based on statistical modelling. This revived the old idea of purposive selection, under the new name of “balanced sampling”. Suppose that the sampling strategy to be used for a particular survey is required to involve both a stratified sampling design and the classical ratio estimator, but that, within each stratum, a choice is allowed between simple random sampling and simple balanced sampling; then which should the survey statistician choose? The balanced sampling strategy appears preferable in terms of robustness and efficiency, but the randomized design has certain countervailing advantages. These include the simplicity of the selection process and an established public acceptance that randomization is “fair”. It transpires that nearly all the advantages of both schemes can be secured if simple random samples are selected within each stratum and a generalized regression estimator is used instead of the classical ratio estimator.