This article derives the power curves for a Wald test that can be applied to randomized response models when small prevalence rates must be assessed (e.g., detecting doping behavior among elite athletes). These curves enable the assessment of the statistical power that is associated with each model (e.g., Warner's model, crosswise model, unrelated question model, forced-choice models, item count model, cheater detection model). This power analysis can help in choosing the optimal model and sample size and in setting model parameters in survey studies. The general framework can be applied to all existing randomized response model versions. The Appendix of this article contains worked-out numerical examples to demonstrate the power analysis for each specific model. (Contains 3 tables, 7 footnotes and 13 figures.)