Using the Fast Fourier Transform to Compute Multiple Comparisons With the Best and Subset Selection Critical Values

Abstract

Multiple comparison methods are widely implemented in statistical packages and heavily used. To obtain the critical value of a multiple comparison method for a given confidence level, a double integral equation must be solved. Current computer implementations evaluate one double integral for each candidate critical value using Gaussian quadrature. Consequently, iterative refinement of the critical value can slow the response time enough to hamper interactive data analysis. However, for balanced designs, to obtain the critical value for multiple comparisons with the best, subset selection, and one-sided multiple comparison with a control, if one regards the inner integral as a function of the outer integration variable, then this function can be obtained 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, it is shown that the FFT method obtains critical values at least four times as accurate and two to five times as fast as the Gaussian quadrature method.

Keywords:

• multiple comparisons with the best
• subset selection
• multiple comparisons with a control
• critical values
• Fast Fourier Transform