Statistics and Its Interface

Volume 14 (2021)

Number 3

Extracting scalar measures from functional data with applications to placebo response

Pages: 255 – 265



Thaddeus Tarpey (Department of Population Health, New York University, New York, N.Y., U.SA.)

Eva Petkova (Department of Population Health, New York University, New York, N.Y., U.SA.)

Adam Ciarleglio (Department of Biostatistics and Bioinformatics, Milken Institute School of Public Health, George Washington University, Washington, D.C., U.S.A.)

Robert Todd Ogden (Department of Biostatistics, Columbia University, New York, N.Y., U.SA.)


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.


average tangent slope, longitudinal data, ordering curves, placebo effects

2010 Mathematics Subject Classification

Primary 62J99, 62P10. Secondary 62-07.

This work was supported by NIMH grants R01 MH099003 and K01 MH113850.

Received 1 January 2020

Accepted 29 August 2020

Published 9 February 2021