What method is used in multcomp::glht to adjust the confidence intervals? I know I can specify the adjustment for p-values (Bonferroni, Holm, etc). But I can set nothing for confidence intervals. How is that handled? I would like to set the Dunnett method for the confidence intervals, but it sets only the contrast. Neither in the documentation I cannot read about except that it is "multivariate" confidence intervals. How are the confidence intervals adjusted for multiplicity?
 A: Let the ANOVA statistical model
$$
y_{ij} = \gamma + \mu_i + \sigma\epsilon_{ij}, \quad 
i = 0, 1, \ldots, m, \quad j = 1, \ldots, n_i
$$
where $\epsilon_{ij} \sim_{\textrm{iid}} \mathcal{N}(0,1)$.
Introduce the statistics 
$$
t_i = \frac{\bar{y_i} - \bar{y_0}}{\hat\sigma\sqrt{\frac{1}{n_i} + \frac{1}{n_0}}} 
$$
where 
$$
\hat\sigma^2 = \frac{\sum_{i=0}^m\sum_{j=1}^{n_i}{(y_{ij}-\bar{y}_i)^2}}{\nu}
$$ 
with $\nu = \sum_{i=0}^m n_i - (m+1)$.
Then, under the null hypothesis $H_0\colon \{\mu_0=\mu_1=\cdots = \mu_m\}$,
the random vector $\mathbf{t}=(t_1, \ldots, t_m)$ follows a multivariate Student distribution with $\nu$ degrees of freedom and scale matrix $\Sigma$ given by
$$
\Sigma_{ij} = \begin{cases}
1 & \text{if } i = j \\
\sqrt{\frac{n_i}{n_i+n_0}}\sqrt{\frac{n_j}{n_j+n_0}} & \text{if } i \neq j
\end{cases}.
$$
Under $H_0$, let $q_p$ be the equicoordinate $p$-quantile of $\mathbf{t}$:
$$
\Pr(t_1 \leq q_p, \ldots, t_m \leq q_p) = p.
$$
Then the two-sided adjusted $100p\%$-confidence interval of $\mu_i - \mu_0$ is given by
$$
(\bar{y_i} - \bar{y_0}) \pm 
q_{1-\frac{1-p}{2}}\times\frac{\hat\sigma}{\sqrt{\frac{1}{n_i}+\frac{1}{n_0}}}.
$$
A bit more details on my blog.
