Design-based or prediction-based inference? Stratified random vs stratified balanced sampling

Early survey statisticians faced a puzzling choice between randomized sampl ing and purposive selection but, by the early 1950s, Neyman's design-based or randomization approach bad become generally accepted as standard. It rem ained virtually unchallenged until the early 1970s, when Royall and his co- authors produced an alternative approach based on statistical modelling. Th is revived the old idea of purposive selection, under the new name of "bala nced sampling". Suppose that the sampling strategy to be used for a particu lar survey is required to involve both a stratified sampling design and the classical ratio estimator, but that, within each stratum, a choice is allo wed between simple random sampling and simple balanced sampling; then which should the survey statistician choose? The balanced sampling strategy appe ars preferable in terms of robustness and efficiency, but the randomized de sign has certain countervailing advantages. These include the simplicity of tbe selection process and an established public acceptance that randomizat ion is "fair". It transpires that nearly all the advantages of both schemes can be secured if simple random samples are selected within each stratum a nd a generalized regression estimator is used instead of the classical rati o estimator.