  We describe a general framework for weighted parametric multiple test
procedures based on the closure principle. We utilize general weighting
strategies that can reflect complex study objectives and include many
procedures in the literature as special cases. The proposed weighted parametric
tests bridge the gap between rejection rules using either adjusted significance
levels or adjusted $p$-values. This connection is possible by allowing
intersection hypotheses to be tested at level smaller than $\alpha$, which may
be needed for certain study considerations. For such cases we introduce a
subclass of exact $\alpha$-level parametric tests which satisfy the consonance
property. When only subsets of test statistics are correlated, a new procedure
is proposed to fully utilize the parametric assumptions within each subset. We
illustrate the proposed weighted parametric tests using a clinical trial
example.
