A new formula for tail probabilities of dunnett's t with unequal sample sizes

Abstract

A single integral expression is derived from, and found computationally more efficient than the double integral given in Hochberg and Tamhane 1987, (eq. 2.7) p. 141, when the total sample size is large. The integrand of the single integral contains a series, of which the first three terms are retained here as an approximation. Both the approximate formula and the double integral are compared numerically using Gauss-Hermite quadratures. For large sample size, the new formula takes much less computing time to converage to a probability value which agrees, to at least three decimal places, with the value from the double integral. This advantage of the new formula is seen for moderate total sample size, 30, and becomes more evident as sample size increases. Simpson's rule is found much slower than Gaussian quadrature when applied to the double integral. Critical values, for P=0.05 and 0.01, computed via both single and double integrals are found very close to those by SAS, PROC GLM, which uses harmonic mean as an approximation.