Extracting scalar measures from functional data with applications to placebo response

In controlled and observational studies, outcome measures are often observed longitudinally. Such data are difficult to compare among units directly because there is no natural ordering of curves. This is relevant not only in clinical trials, where typically the goal is to evaluate the relative efficacy of treatments on average, but also in the growing and increasingly important area of personalized medicine, where treatment decisions are optimized with respect to a relevant patient outcome. In personalized medicine, there are no methods for optimizing treatment decision rules using longitudinal outcomes, e.g., symptom trajectories, because of the lack of a natural ordering of curves. A typical practice is to summarize the longitudinal response by a scalar outcome that can then be compared across patients, treatments, etc. We describe some of the summaries that are in common use, especially in clinical trials. We consider a general summary measure (weighted average tangent slope) with weights that can be chosen to optimize specific inference depending on the application. We illustrate the methodology on a study of depression treatment, in which it is difficult to separate placebo effects from the specific effects of the antidepressant. We argue that this approach provides a better summary for estimating the benefits of an active treatment than traditional non-weighted averages.