  Simultaneous confidence intervals (SCIs) that are compatible with a given
closed test procedure are often non-informative. More precisely, for a
one-sided null hypothesis, the bound of the SCI can stick to the border of the
null hypothesis, irrespective of how far the point estimate deviates from the
null hypothesis. This has been illustrated for the Bonferroni-Holm and
fall-back procedures, for which alternative SCIs have been suggested, that are
free of this deficiency. These informative SCIs are not fully compatible with
the initial multiple test, but are close to it and hence provide similar power
advantages. They provide a multiple hypothesis test with strong family-wise
error rate control that can be used in replacement of the initial multiple
test. The current paper extends previous work for informative SCIs to graphical
test procedures. The information gained from the newly suggested SCIs is shown
to be always increasing with increasing evidence against a null hypothesis. The
new SCIs provide a compromise between information gain and the goal to reject
as many hypotheses as possible. The SCIs are defined via a family of dual
graphs and the projection method. A simple iterative algorithm for the
computation of the intervals is provided. A simulation study illustrates the
results for a complex graphical test procedure.
