The accuracy of four approximations to noncentral F

The accuracy of the two-moment, three-moment, square root, and cube root approximations to the noncentral F distribution was assessed using 7,920 entries from Tiku's (1967) power tables. Tiku's tables list exact values of beta for alpha =.005-.05, nu(1) = 1-12, nu(2)= 2-120, and phi = 0.5-3.0. Analysis of the errors showed generally satisfactory performance for all four approximations. The three-moment approximation was most accurate, registering a maximum error of only .009. The other three approximations had maximum errors of +/-.02, except for the square root approximation at nu(2) = 2, where maximum errors of .05 occur. Approximation error increased with decreases in nu(1) and, less consistently with increases in nu(2). Error was nonmonotonically related to phi. A second investigation explored the accuracy of the approximations at values of alpha ranging from .10 to .90. All four approximations degraded substantially in this situation, with maximum errors ranging from -.09 to .05. If the analysis is restricted to cases where nu(1) > 1 and nu(2) > 2, maximum errors drop to roughly +/-.03. We conclude that the approximations perform reasonably well for small a and moderately well for larger values, if certain restrictions are imposed. From a computational standpoint, however, there is little advantage to using approximate as opposed to exact methods unless exact values of F-alpha are known in advance.