Approximate Correlation Matrix Methods

A second class of approximations consists of methods that use an approximation of the form ${\bf R}={\bf D}+$   $\lambda$$\lambda$$^t$ for the correlation matrix, so the t-integrals take the form (2).

An important early approximation for all-pairwise comparisons is the Tukey-Kramer (TK) method (Tukey, 1953; Kramer, 1957). Tukey and Kramer conjectured that the use of balanced critical values instead of the true ones is always conservative (i.e., the covariance matrix ${\bf V}$ of the estimates is replaced by the identity matrix). This was proved by Hayter (1984) in the unbalanced independent one-factorial design for general $I$, where ${\bf V}=$diag$(n_i^{-1})$. A proof of this conjecture for general designs (and hence possibly non-diagonal V) for $I=3$ was given by Brown (1984). The GT2 procedure proposed by Hochberg (1974) and based on Šidák's inequality replaces ${\bf R}$ by the identity matrix. This method is known to be inferior to TK, but it is conservative for all ${\bf V}$. More recently, Iyengar (1988) and Iyengar and Tong (1989) investigated replacing $\rho_{ij}$ by their common average $\bar{\rho}= \frac{2}{I(I-1)} \sum_{i<j}\rho_{ij}$. It has been shown by Iyengar (1988), that this approach is not necessarily conservative for all choices of ${\bf R}$, but no counter example has been found yet for the types of ${\bf R}$ that arise with multiple comparison problems. Royen (1987) and Hsu (1992) independently provided different techniques to find the 'closest' $\hat{{\bf R}}$, which still possesses the product correlation structure in equation (2). Both factor-analytic and linear programming methods were investigated by Hsu (1992) and Hsu and Nelson (1998). These methods are not applicable on all-pairwise comparison problems. We refer to the original articles for more details. For comparison reasons we also included the classical yet very conservative $F$-test of Scheffé. Finally, Solow (1990) described a simple way for approximating multivariate normal probabilities from univariate and bivariate marginal probabilities. Section 4 provides results from a generalization for multivariate $t$ integrals of Solow's method. This generalized method uses a decomposition of (1) into a product of conditional probabilities and approximates each term in the product using conditional expectations.

Hochberg and Tamhane (1987, p. 145) considered several other approximations tailored to the many-to-one comparisons case. But these methods are usually inferior to the methods that we have described already, so we did not include them in our study. Another approach is to consider the spectral decomposition ${\bf R}=\sum_i e_i{\bf p}_i{\bf p}_i^t$, where $e_i$ is the $i$th eigenvalue in decreasing order and ${\bf p}_i$ is the corresponding normalized eigenvector. A possible approximation is to replace the $\rho_{ij}$ by $p_i p_{j}$ and apply (2), where $p_i$ and $p_{j}$ are the elements of ${\bf p}_1$. But this method did not perform well in our comparison study, so we omitted the results from the following section.