  Applied statistical problems often come with pre-specified groupings to
predictors. It is natural to test for the presence of simultaneous group-wide
signal for groups in isolation, or for multiple groups together. Classical
tests for the presence of such signals rely either on tests for the omission of
the entire block of variables (the classical F-test) or on the creation of an
unsupervised prototype for the group (either a group centroid or first
principal component) and subsequent t-tests on these prototypes.
  In this paper, we propose test statistics that aim for power improvements
over these classical approaches. In particular, we first create group
prototypes, with reference to the response, hopefully improving on the
unsupervised prototypes, and then testing with likelihood ratio statistics
incorporating only these prototypes. We propose a (potentially) novel model,
called the "prototype model", which naturally models the two-step
prototype-then-test procedure. Furthermore, we introduce an inferential schema
detailing the unique considerations for different combinations of prototype
formation and univariate/multivariate testing models. The prototype model also
suggests new applications to estimation and prediction.
  Prototype formation often relies on variable selection, which invalidates
classical Gaussian test theory. We use recent advances in selective inference
to account for selection in the prototyping step and retain test validity.
Simulation experiments suggest that our testing procedure enjoys more power
than do classical approaches.
