Using the Fast Fourier Transform to Compute Some Multiple Comparisons Critical Values

Multiple comparison methods are heavily used. For balanced designs, to obtain the critical value of a multiple comparison method at a given confidence level, a double integral equation must be solved. Current computer implementations that the authors are aware of evaluate one double integral for each candidate critical value using Gaussian quadrature. For multiple comparisons with the best, subset selection, and one-sided multiple comparison with a control, the effort of repeated double Gaussian quadratures during iterative refinement of the critical value can be reduced. For these multiple comparison procedures, if one regards the inner integral as a function of the outer integration variable, then this function is a convolution which can be approximated by discrete convolution using the Fast Fourier Transform (FFT). Exploiting the fact that this function need not be re-evaluated during iterative refinement of the critical value, the FFT method can obtain critical values at least four times as accurate and two to five times as fast as the Gaussian quadrature method.