Combining planned and discovered comparisons in observational studies.

In observational studies of treatment effects, it is common to have several outcomes, perhaps of uncertain quality and relevance, each purporting to measure the effect of the treatment. A single planned combination of several outcomes may increase both power and insensitivity to unmeasured bias when the plan is wisely chosen, but it may miss opportunities in other cases. A method is proposed that uses one planned combination with only a mild correction for multiple testing and exhaustive consideration of all possible combinations fully correcting for multiple testing. The method works with the joint distribution of $\kappa^{T}\left( \mathbf{T}-\boldsymbol{\mu}\right) /\sqrt {\boldsymbol{\kappa}^{T}\boldsymbol{\Sigma\boldsymbol{\kappa}}}$ and $max_{\boldsymbol{\lambda}\neq\mathbf{0}}$$\,\lambda^{T}\left( \mathbf{T} -\boldsymbol{\mu}\right) /$$\sqrt{\boldsymbol{\lambda}^{T}\boldsymbol{\Sigma \lambda}}$ where $\kappa$ is chosen a priori and the test statistic $\mathbf{T}$ is asymptotically $N_{L}\left( \boldsymbol{\mu},\boldsymbol{\Sigma}\right) $. The correction for multiple testing has a smaller effect on the power of $\kappa^{T}\left( \mathbf{T}-\boldsymbol{\mu }\right) /\sqrt{\boldsymbol{\kappa}^{T}\boldsymbol{\Sigma\boldsymbol{\kappa} }}$ than does switching to a two-tailed test, even though the opposite tail does receive consideration when $\lambda=-\kappa$. In the application, there are three measures of cognitive decline, and the a priori comparison $\kappa$ is their first principal component, computed without reference to treatment assignments. The method is implemented in an R package sensitivitymult.