An algorithm for calculating probabilities for the F distribution (including the chi square, t, and normal distributions as special cases)

Abstract

A number of numerical methods exist for computing prob­ abilities associated with variables that have F distributions. Most of these procedures depend on approximations and yield rather poor accuracy when the degrees of freedom associated with either the numerator (n.) or denominator (n 2 ) are small. The approach presented here is to combine an approximation (paulson, 1942) which is sufficiently accurate for a large number of situations with a computation algorithm based on the incom­ plete Beta distribution. The incomplete Beta distribution is simply related to the F distribution and can be evaluated exactly by means of a recurrence relationship (Abramowitz & Stegun, 1964). Since the chi-square and t distributions are special cases of the F distribution, probabilities associated with these distri­ butions are easily calculated from slight modifications of the same recurrence formula. A FORTRAN function subroutine is provided to compute these statistical probabilities with a known degree of accuracy. Also included is the option of calculating probabilities for variables with standard normal distributions (mean = 0 and variance = 1.0). Method. The exact right-hand tail F distribution probabili­ ties are calculated by a recurrence relationship unless the approx­ imate formula yields a relative error rate' of less than .5% for probabilities greater than .01. The chi-square distribution is related to the F distribution (F =x 2 In has F distribution with n, =degrees of freedom and n2 =00)and it is, therefore, easy to calculate chi-square probabil­ ities from a' procedure that computes F distribution probabili­ ties. Probabilities associated with the t distribution can be similarly calculated from the F distribution (F = t 2 has the F distribution with n, = 1 and n2 = degrees of freedom). Normal distribution probabilities are necessary for the computational algorithm and are also included as a separate option. The cal­ culated normal probabilities have absolute error rates never exceeding 108