The thesis of this paper is that many proposed moderators in personnel psychology are probably illusory, having been created solely by belief in the law of small numbers. Evidence is presented that race as a moderator of test validity is one such illusory moderator. In addition, a model for validity generalization is described which, in addition to eliminating the need for criterion-related validity studies under certain circumstances, strongly calls into question the idea that situations moderate test validity, i.e., the traditional doctrine of situational specificity of test validities. Calculations are presented which show that adequate statistical power in moderator research requires much larger sample sizes than have typically been employed. This requirement is illustrated empirically using validity data for the Army Classification Battery for 35 jobs and 21,000 individuals. These analyses show that (1) even when a moderator is generally assumed to be large, large samples are required to gauge its effect reliably and (2) large sample research may show that moderators that appear plausible and important a priori are nonexistent or trivial in magnitude. The practice of pooling across numerous small sample studies to obtain statistical power equivalent to that of large sample studies is recommended. In light of the evidence that many proposed moderators may not exist, the authors hypothesize that the true structure of underlying relationships in personnel psychology is considerably simpler than personnel psychologists have generally imagined it to be.