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The small sample sizes inherent in rare and pediatric disease settings offer significant challenges for clinical trial design. In such settings, Bayesian adaptive trial methods can often pay dividends, allowing the sensible incorporation of auxiliary data and other relevant information to bolster that collected by the trial itself. Previous work has also included the use of one-arm trials augmented by the participants' own natural history data, from which the future course of the disease in the absence of intervention can be predicted. Patient response can then be defined by the degree to which post-intervention observations are inconsistent with the predicted "natural" trajectory. While such trials offer obvious advantages in efficiency and ethical hazard (since they expose no new patients to a placebo, anathema to patients or their parents and caregivers), they can offer no protection against bias arising from the presence of any "placebo effect," the tendency of patients to improve merely by being in the trial. In this paper, we investigate the impact of both static and transient placebo effects on one-arm responder studies of this type, as well as two-arm versions that incorporate a small concurrent placebo group but still borrow strength from the natural history data. We also propose more traditional Bayesian changepoint models that specify a parametric functional form for the patient's post-intervention trajectory, which in turn allow quantification of the treatment benefit in terms of the model parameters, rather than semi-parametrically in terms of a response relative to some "null" model. We compare the operating characteristics of our designs in the context of an ongoing investigation of centronuclear myopathies (CNMs), a group of congenital neuromuscular diseases whose most common and severe form is X-linked, affecting approximately 1 in 50,000 newborn boys. Our results indicate our two-arm responder and changepoint methods can offer protection against placebo effects, improving power while protecting the trial's Type I error rate. However, further research into innovative trial designs as well as ongoing dialog with regulatory authorities remain critically important in rare disease research. [ABSTRACT FROM AUTHOR]