# What are correlations among between test statistics in multiple comparison procedures

In the book Multiple Comparisons Using R Bretz et al write that their R package multcomp improves power above other multiple comparison procedures by "accounting for the structural correlations between the test statistics" (pg 41). The default option for this multiple comparison procedure implements this and is called "single-step."
My questions are 1)What do they mean by "structural correlations" (their explanation eludes me) 2)Is their a standard way to refer to or cite this approach. "Single-step" is what it is called in the call to their R function, but this is a general description of multiple comparisons. In my field people are mostly familiar with the usual Bonferroni corrections and I'd like to avoid the critical eye of reviewers and also point readers toward work that can help them understand what is going on.

In the special case of linear test statistics in normally distributed random variables (t-test), the dependency can pretty well be decribed by the correlation of the effect estimators. E.g. if you test simultaneously $\mu_1 - \mu_2$ and $\mu_1 - \mu_3$, your test statistics will base on the respective means: $\bar{x}_1 - \bar{x}_2$ and $\bar{x}_1 - \bar{x}_3$. Both differences are correlated due to the structure of the hypotheses. Their covariance is the variance of $\bar{x}_1$, and so the test statistics will likewise be (asyptotically, i.e. up to studentization) correlated. That's why Sidák cannot be used. But if you incorporate the dependency as Bretz et al. suggest, you will meet your FWER(strong) less conservative than if you would use Bonferroni.