FISHTAIL: PracticalF integration on small computers in BASIC and C

Abstract

Computer algorithms can be used to calculate F sig­ nificance levels directly, thus avoiding the use of tables and resolving questions that arise when the degrees of freedom fall between the values given in an available pub­ lished table or when the exact probability level is of in­ terest. An algorithm is presented that finds F probabili­ ties and that represents a good compromise among the goals of accuracy, speed, and compactness. Unlike several others, this method has the capability of interpolation for approximations to significance levels, when noninteger values are given for the degrees of freedom. The fundamental idea underlying this algorithm was in­ troduced with the Biometrika tables ofthe incomplete beta function, from which the integrals ofF, t, and some other distributions can be obtained by the use of transforma­ tions (Kendall & Stuart, 1963, pp. 375-378). Following the suggestion of Bock (1975, p. 150), a computer al­ gorithm based on incomplete beta has been developed to bypass the tables. The method uses a finite number of terms of an infinite series that is known to converge to the desired value. The convergence is rapid enough for practical computations, under conditions that can be in­ duced. The method is based on the transformation x = V2/(V2 + vIF). In this expression, F is Fisher's F statistic with VI and V2 degrees of freedom for the numerator and the denomi­ nator, respectively. The transformed variate, x, is the var­ iate of a beta distribution with parameters a = v212 and b = vl 12 . The series expression for P, the tail area of Fisher's distribution beyond the point F (Bock, 1975, p. 150), is