Identifying Feasible Sample Sizes from Incomplete Design Information

Sample size calculations return the smallest size that meets a requirement once the model and inputs are specified. We ask which sample sizes satisfy the design requirements when the planning information is compatible with several data distributions. We calculate the feasible set for each specification under prespecified error constraints. If every compatible distribution yields the same feasible set, the planning information identifies that set. Otherwise, the intersection and union give sharp inner and outer bounds. A size is feasible when a prespecified rule controls the probabilities of missing a meaningful effect and making a positive decision for a negligible effect. We distinguish the prespecified rule from the performance available in a decision class. The feasible set is therefore a partially identified target. When all proper subset margins are known, the multivariate binary identified set has at most one free coordinate, permitting exact calculation under finite polynomial risk constraints. A normal example shows that a test with null boundary zero can fail at a large sample size while a minimum effect test remains feasible. Public paired vision data illustrate sharp bounds for a test with null boundary zero when the marginal probabilities do not identify the discordance probability. Sample sizes in the inner bound satisfy the constraints for every compatible distribution. For sizes between the bounds, additional planning information is needed to determine feasibility. An empty outer bound shows that the rule or design requirements must change because no candidate size works.