The Scheffé critical value protects the family-wise error rate
for the simultaneous test of any contrast (or family of
contrasts) in a fixed-effects ANOVA, including data-driven contrasts
selected after looking at the data. It is therefore the most
conservative of the standard procedures.
The critical value, on the scale of a t-statistic, is
$$t_{\mathrm{crit}}^{\mathrm{Scheffe}} = \sqrt{(k-1)\, F_{1-\alpha,\,k-1,\,df_{\mathrm{denominator}}}},$$
so that a contrast \(\hat\psi\) with standard error \(\mathit{SE}_{\hat\psi}\) is
declared significant when
\(|\hat\psi/\mathit{SE}_{\hat\psi}| > t_{\mathrm{crit}}^{\mathrm{Scheffe}}\).
The corresponding simultaneous confidence interval is
\(\hat\psi \pm t_{\mathrm{crit}}^{\mathrm{Scheffe}} \cdot \mathit{SE}_{\hat\psi}\).
Like the Studentized range distribution underlying Tukey HSD, the
Scheffé reference is one-sided (the underlying F statistic is
non-negative), so alpha_level is not split between two tails.
The Scheffé critical value is a function of a univariate F
quantile, which base R supplies through qf, so unlike
cv_dunnett and cv_smm this function needs no
multivariate distribution machinery. Scheffé's procedure earns its
simultaneous protection over the infinite family of all possible
contrasts by projecting onto the overall F test rather than by
integrating a multivariate t density; that is why a single
univariate quantile suffices and the mvtnorm package is not
required here. Maxwell, Delaney, and Kelley (2027, Chapter 5) develop the
Scheffé method as the procedure for arbitrary contrasts within the
multiple-comparisons problem.