A NEW APPROXIMATION FOR FISHER'S Z

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Abstract
Summary: As the sample size increases, the coefficient of skewness of the Fisher's transformationz= tanh‐1r, of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the distribution of standardized z can be approximated more accurately in terms of thetdistribution with matching kurtosis than by the unit normal distribution. Thistdistribution can, in turn be subjected to Wallace's approximation resulting in a new normal approximation for the Fisher's z transform. This approximation, which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1974).Fisher (1921) suggested approximating distribution of the variance stabilizing transform z=(1/2) log ((1 +r)/(1r)) of the correlation coefficient r by the normal distribution with mean = (1/2) log ((1 + p)/(lp)) and variance =l/(n3). This approximation is generally recognized as being remarkably accurate when||Gr|is moderate but not so accurate when||Gr|is large, even whennis not small (David (1938)). Among various alternatives to Fisher's approximation, the normalizing transformation due to Ruben (1966) and atapproximation due to Kraemer (1973), are interesting on the grounds of novelty, accuracy and/or aesthetics. If r̄= r/√ (1r2) and r̄|Gr = |Gr/√(1|Gr2), then Ruben (1966) showed that (1) gn(r,|Gr) ={(2n5)/2}1/2r̄r{(2n3)/2}1/2r̄|GR, {1 + (1/2)(r̄r2+r̄|Gr2)}1/2is approximately unit normal. Kraemer (1973) suggests approximating (2) tn(r, |Gr) = (r|GR1) √ (n2), √(11r2) √(1|Gr2) by a Student'stvariable with(n2)degrees of freedom, where after considering various valid choices for |Gr1she recommends taking |Gr1= |Gr*, the median of r given n and |Gr.