APL approximations for common statistical critical values

Abstract

Evans and Gilfillan (1986) presented APL functions ZTAB, FTAB, CHlSQ, and BINOM, which produce quantile values from z scores, F values, x2 values, and frequencies, respectively. Quantiles of t scores may be obtained from function FTAB. Statistical analyses frequently requirethatcriticalvalues be obtained fromprobabilityvaluesunder the z, t, X , andF distributions when the requiredprobability valuesare not tabled. Oftensuch requirements arise under Bonferroni multiple comparison procedures, where in order to maintaina given level of familywise Type I error over many comparisons, the researcher mayneeda t or F criticalvaluecorresponding to a nontabled probability value. Other uses stem from the analysisof simpleeffects in factorial analysisof variance. This paper presents four APL functions that compute Z, t, X , and F critical values from upper-tailprobability values. The functions are ideal for constructing tables of critical values of the z, t, X , F, and incomplete beta distributions for given probability levels. The function that computes the F values (PTOF) depends on functions for the natural logarithmof the complete beta function (LNBETA), the incomplete beta function (BETAIN), and the inverse incomplete beta function (XINBTA). These three functions may be used for other applications. The functions are presented in Listing 1. Each of the seven functions has been written and tested in APL* PLUS/PCfromSTSC, Inc. mM APLdoesnotacceptthe diamond separator; enter the expressions separated by the diamonds in order from left to right. The execution times were obtained on an mM PC-AT compatible with an 80287 internal arithmetic coprocessor. Inverse Normal Function. Monadic function PTOZ acceptsa list of one or more upper-tailprobabilityvalues in its right argument and returns a vector of the corresponding approximate signed normal deviates. Thus, the result for rightargument [0.950.05 0.025] is [-1.645 1.645 1.960]. PTOZ uses the approximation given by Odeh and Evans (1974). Execution time for a random sampleof 100upper-tailprobability values was0.94 sec. Inverse t Function. Dyadic function PTOT accepts a list of one or more degrees-of-freedom values in its left argument and a list of associated upper-tail probability values in its right argument. PTOT returns a vector of the corresponding approximateStudent's t values. Thus,