  Considerable interest has recently been focused on studying multiple
phenotypes simultaneously in both epidemiological and genomic studies, either
to capture the multidimensionality of complex disorders or to understand shared
etiology of related disorders. We seek to identify {\em multiple regulators} or
predictors that are associated with multiple outcomes when these outcomes may
be measured on very different scales or composed of a mixture of continuous,
binary, and not-fully-observed elements. We first propose an estimation
technique to put all effects on similar scales, and we induce sparsity on the
estimated effects. We provide standard asymptotic results for this estimator
and show that resampling can be used to quantify uncertainty in finite samples.
We finally provide a multiple testing procedure which can be geared
specifically to the types of multiple regulators of interest, and we establish
that, under standard regularity conditions, the familywise error rate will
approach 0 as sample size diverges. Simulation results indicate that our
approach can improve over unregularized methods both in reducing bias in
estimation and improving power for testing.
