Approximating Probability Levels for Testing Null Hypotheses with Noncentral F Distributions

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Abstract
Laubscher (1960) offered two approximations for normalizing noncentral F distributions and concluded that a formula based on the square root of the chi-square distribution (SRA) gave values closer to the particular exact probabilities chosen than did a formula derived from a cube root of chi-square distribution (CRA). However, the range of values selected for degrees of freedom and A, the noncentrality parameter, are not representative of those in behavioral investigations. The present study compared the two approximations at six percentile points on the noncentral F distributions for numerator degrees of freedom in the range 1(1)6, 8, 12, 24, denominator degrees of freedom 6(2)30, 40, 60, 120, 240 and for Cohen's small, medium, and large effect sizes (f = .1, .25, .4) against exact values calculated by means of Tang's (1938) recurrence formula. It was found the CRA was clearly superior to the SRA and generally provided an excellent approximation for non-central F, particularly at levels of significance traditionally employed in psychology and education. Probability values for testing directional null hypotheses specifying a size of effect were typically correct to three decimals.

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