Dunnett's procedure controls the family-wise error rate for the
special case of comparing each of \(k - 1\) treatments to a single
control (the many-to-one comparisons setting), using a multivariate
t reference with constant pairwise correlation \(1/2\). That
correlation is exact for a balanced design: each comparison is
\((\bar Y_i - \bar Y_0)\), all sharing the one control mean \(\bar Y_0\)
of variance \(\sigma^2/n\), while each comparison has variance
\(2\sigma^2/n\), so any two comparisons correlate
\((\sigma^2/n)/(2\sigma^2/n) = 1/2\). Maxwell, Delaney, and Kelley (2027,
Chapter 5) develop the many-to-one comparisons problem and tabulate these
critical values.
How it is computed. The critical value is a quantile of a
\((k-1)\)-dimensional multivariate t distribution whose
correlation matrix has a unit diagonal and off-diagonal entries of 1/2.
Because that correlation is a single common value, the comparisons have the
one-factor representation \(Z_i = \sqrt{1/2}\,W + \sqrt{1/2}\,U_i\) with a
shared factor \(W\) and independent \(U_i\), all standard normal.
Conditioning on \(W\) and on the common scale estimate
\(S = \sqrt{\chi^2_{df}/df}\) makes the comparisons independent, so the
\((k-1)\)-dimensional integral collapses to two nested one-dimensional
integrals:
$$P\!\left(\max_i T_i \le d\right) = \int_0^\infty\!\!\int_{-\infty}^{\infty}
\Bigl[\Phi\bigl((d\,s - \sqrt{1/2}\,w)/\sqrt{1/2}\bigr)\Bigr]^{k-1}
\phi(w)\,dw\; f_S(s)\,ds$$
for the one-sided value (the two-sided value replaces the bracket with the
probability that \(|T_i| \le d\)). This package evaluates those integrals
with integrate and inverts with
uniroot, so the returned value is deterministic and
accurate to the solver tolerance, with no Monte Carlo simulation and no
random seed. When \(k - 1 = 1\) the value is the ordinary one- or
two-sided t critical value.