Introduction

Several multiple comparison procedures involve the numerical computation of multidimensional integrals, which are not easily handled. A prominent example is to find simultaneous confidence intervals for all-pairwise comparisons in general linear models. Current integration methods typically behave well for low to moderate numbers of treatment groups (say less than 10, see Bretz, Hayter and Genz, 2001). Independently of such general integration routines, research also focused on approximating the arising integrals by lower order expressions. The Tukey-Kramer procedure is a well known approximation for the all-pairwise comparisons, where the true covariance matrix of the estimates is substituted by the identity matrix. Two questions arise from this approach. Firstly, because the Tukey-Kramer procedure has not been proven to be conservative for all designs, there is still room for other procedures (where an approximation is called conservative, if its critical values are larger than the true ones). Secondly, Somerville (1993) showed that the conservatism of the Tukey-Kramer procedure can be quite substantial, so that sharper methods may exist, which are still conservative. These facts motivate us to look at and compare different approximations to multivariate normal and $t$ probabilities.

The paper is organized as follows. Section 2 introduces some basic notation. Section 3 presents the methods used for our numerical comparisons. The methods are divided into inequality based methods and methods making use of approximative correlation matrices. The results of the numerical study are given in Section 4. The final Section 5 gives some conclusions.