# 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?

## 1 Answer

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.

• This is excellent answer. Thank you. This should be really part of the documentation, like in SAS pages. Just one more question. So here I can see this gives me simultaneous confidence intervals. The same method seems to be used for Dunnett and Tukey. Am I right, that only the contrast is different (pairwise vs all-to-control), and the method of adjustment is the same? Is this comparable to Bonferroni? I am asking as my reviewer asked me to adjust p-values for each comparison using Bonferroni and Holm. He also asked me to adjust the CIs. I assume I could do that by correcting the alpha Feb 15 '20 at 12:41
• and display just the univariate Cis, calculated using the corrected alpha, according to the method I want, yes? Feb 15 '20 at 12:49