Motivation

Consider the $k-$variate $t$ integral

$$T_k({\bf a}, {\bf b}; {\bf R}, \nu) = \frac{\Gamma(\frac{\nu+k}{2... ...frac{ {\bf x}^t {\bf R}^{-1} {\bf x}}{\nu} \right)^{-\frac{\nu+k}{2}} d{\bf x},$$ (1)

with $\nu$ degrees of freedom and correlation matrix ${\bf R}=\{\rho_{ij}\}$. The integration region ${\bf A}=\{{\bf x}\in \mathbb{R}^k: {\bf a}\leq {\bf C}{\bf x}\leq {\bf b}\}$ is a convex polyhedron for a given $q \times k$ constraint matrix ${\bf C}$ and ${\bf a}, {\bf b}\in \bar{\mathbb{R}}^q$. The main interest lies in determining a critical value ${\bf t}_{1-\alpha}=(t_{1-\alpha}, \ldots, t_{1-\alpha})^t$, such that $T_k(-{\bf t}_{1-\alpha}, {\bf t}_{1-\alpha}; {\bf R}, \nu) = 1- \alpha$ for a given probability $\alpha \in (0,1)$. Since this latter problem can be traced back to repeated evaluations of $T_k({\bf a}, {\bf b}; {\bf R}, \nu)$ (Genz and Bretz, 2000), we focus on the numerical evaluation of integral (1).

It is well known that particular structures of ${\bf R}$ allow a substantial dimension reduction of the initial integration problem (Curnow and Dunnett, 1962). Consider as an example the two-way no-interaction model $Y_{ijl}=\mu+\alpha_i+\beta_j+\epsilon_{ijl}$, $i=1, \ldots, I, j=1, \ldots, J$ and $l=1, \ldots, n_{ij}$. Suppose we are interested in the many-to-one comparisons $\alpha_2-\alpha_1, \ldots, \alpha_I-\alpha_1$. Then, ${\bf R}={\bf D}+$$\lambda$$\lambda$$^t$, where $D=$diag$(1-\lambda_i^2)$ and $\lambda_i=(1+w_1/w_{i+1})^{-1/2}, i=1, \ldots, I-1$ as long as the cell sizes satisfy $n_{ij}=w_i n_{.j}$ for all $i$ and $j$, $n_{.j}=\sum_i n_{ij}$ (Hsu, 1992). Thus, ${\bf R}$ is of one-factorial structure and the integral (1) is reduced to a two-dimensional integral, regardless of $I$,

$$\int\limits_0^\infty\int\limits_{-\infty}^{\infty} \prod\limits_{... ...lambda_i y}{\sqrt{1-\lambda_i^2}} \right) \right] \varphi(y) dy \chi_\nu(s) ds,$$ (2)

where $\varphi$ is the standard normal pdf and $\chi_\nu$ is the $\sqrt{\chi_\nu^2}$ pdf and $\Phi'=\varphi$. But if above proportionality rule is violated for $I>4$, the integral (1) is not reducible in general. A similar problem arises for all-pairwise comparisons in one-way layouts. As long as the group sample sizes $n_i$ are equal to a common $n$, the integral (1) can be reduced to the double integral

$$I\int\limits_0^\infty\int\limits_{-\infty}^{\infty} \left[ \Phi(y) - \Phi ( y - \vert b\vert s) \right]^{I-1} \varphi(y) dy \chi_\nu(s) ds,$$ (3)

which can be efficiently computed (for the purpose of computing critical values, $-{\bf a}={\bf b}=(b, \ldots, b)^t$). But this reduction fails if $n_i \neq n$ for some $i$.