Kendall’s tau and kendall’s partial correlation: two BASIC programs for microcomputers

Abstract

Nonparametric rank-ordercorrelationprocedures, such as Spearman's rho and Kendall's tau, are often used as alternativesto Pearson's r, their parametric counterpart, when assumptions underlying that procedure cannot be met. Kendall's tau is a particularly useful alternative in that it may be generalizedto a partial correlation coefficient. This article describes an easy-to-use BASIC program for the calculationof both Kendall's tau and Kendall's partial rank correlation coefficient. Kendall's tau. The tau procedure, like mostother nonparametricprocedures, requiresat leastordinal leveldata and assumesthat these data have been drawn from distributionsthat are continuous. The 5 statistic, a measureof disarray, represents the numerator term in the tau procedure. The sampling distributions of 5 and tau are identical in a probability sense (Hays, 1972). Therefore, the significance of tau maybe determined by referring to either. Kendall has determined the exact sampling distribution of 5 for N=4 to N= 10 (Kendall, 1970). For N<?10 the distribution of 5 is closeto normal, and a normalapproximationto the exact distributionof 5 may be determined. Employing an algorithm presented by Brophy (1986), the program applies a significance test by dividing 5 by the standarddeviationof the samplingdistributionto obtain Z, the normal deviate. The approximate probability of Z is then computedby applying the formula presented by Poole and Borchers (1977). Kendall's partial correlation. A correlation between two variables, X and Y, may be a functionof each variable's association with a third variable, Z. Stated simply, partial correlation eliminatesany effects of variation by Z uponthe relationship betweenXand Y. Partial correlation, therefore, presumablyallowsone to assess the relationship between X and Y without the effects of Z. Although Kendallpresented tau;.• for the specialcase in whichno ties exist, Hawkes(1971)showedit to be appropriate as well for the case in which ties do exist. The majordrawback to thisprocedureis that the sampling distribution for tau;... unlike that for tau, has not yet been determined; there are, therefore, no tests of significance for the partial rank correlation coefficient at this time. In the present program the Kendall partial rank correlation (tau;.•) may be computed from raw score triads, three individual setsof raw scoredyads, or pairwise corre-