  In classical study designs, the aim is often to learn about the effects of a
treatment or intervention on a single outcome; in many modern studies, however,
data on multiple outcomes are collected and it is of interest to explore
effects on multiple outcomes simultaneously. Such designs can be particularly
useful in patient-centered research, where different outcomes might be more or
less important to different patients. In this paper we propose scaled effect
measures (via potential outcome notation) that translate effects on multiple
outcomes to a common scale, using mean-variance and median-interquartile-range
-based standardizations. We present efficient, nonparametric, doubly robust
methods for estimating these scaled effects (and weighted average summary
measures), and for testing the null hypothesis that treatment affects all
outcomes equally. We also discuss methods for exploring how treatment effects
depend on covariates (i.e., effect modification). In addition to describing
efficiency theory for our estimands and the asymptotic behavior of our
estimators, we illustrate the methods in a simulation study and a data
analysis. Importantly, and in contrast to much of the literature concerning
effects on multiple outcomes, our methods are nonparametric and can be used not
only in randomized trials to yield increased efficiency, but also in
observational studies with high-dimensional covariates to reduce confounding
bias.
